Brain Functional Assessment With fMRI · Abstract Functional MRI (Magnetic Resonance Imaging),...

Brain Functional Assessment With fMRI

Joint Bayesian Activity Detection and Hemodynamic Response

Estimation in Brain function

DAVID MIGUEL PÊGAS AFONSO

Thesis for the Master degree in

BIOMEDICAL ENGINEERING

Jury

President: Prof. Teresa Peña

Tutors: Prof. João Sanches

Prof. Martin Lauterbach

External: Prof. Silva Carvalho

September 2007

Acknowledgments

I would like to thank my IST tutor, Prof. João Sanches, for the overall commitment and workhe putted in the collaboration effort that resulted in this thesis. To my FML tutor, Dr. MartinLauterbach, for all the interest, motivation, availability and support. To the Philips physics techni-cian Nuno Loução for the occasionally needed enlightenments. To Daniel Handwerker for providingessential data to this work. A note of appreciation to the Institute for Systems and Robotics whosefinancial support was essential in publishing part of this thesis scientific work. A special thanks tomy friend Henrik Halvorsen for (even in the works of becoming a father) reading, trying to compre-hend, and correcting my immature English on all of the scientific papers which content ended upin this document. Also to Daniel Lopes and to my "other-half" Cátia Fonseca for the occasionalgrammatical corrections.

3

Abstract

Functional MRI (Magnetic Resonance Imaging), fMRI, is a new imaging tool to study and evaluatethe brain neural processes. The Blood-Oxygenation-Level-Dependent (BOLD) signal is currentlyused to detect the activated brain regions due to a stimulus application, e.g., visual or auditive. Ina block design approach, the stimuli (called paradigm in the fMRI scope) are designed to enhancethis detection with maximized certainty. However, corrupting noise in MRI volumes acquisition,subject motion, image "ghostings" and the normal brain activity interference makes this detection adifficult task. The standard activation detection fMRI data analysis, here called SPM-GLM [1], usesa linear regression methodology with a conventional statistical inference based on the t-statistics,and assumes a rather rigid shape on the BOLD Hemodynamic Response Function (HRF).

With the purpose of extending the state-of-the-art in fMRI brain activity detection analysis, adifferent approach is presented in this thesis; a new Bayesian method, here called SPM-MAP, forthe joint detection of brain activated regions and estimation of the underlying HRF. This approachpresents two main advantages: 1) The activity detection benefits from the adaptative nature of theHRF shape estimation; 2) It provides local, space variant, HRF estimation which could be valuablefor possible behavioral, neural and/or vascular local considerations.

Furthermore, the HRF model incorporated in the proposed algorithm is a novel approach wherea simple linear function is developed from reasonable physiological assumptions about oxygen con-sumption and vasodilatation processes. The derived transfer function is estimated on real single-events fMRI data, showing small values for the fitting error, resulting, at average, in better fittingresults than those achievable with the standard basis gamma functions [2,3].

The SPM-MAP method’s brain activity detection error probability is compared against thestandard SPM-GLM on synthetic Monte Carlo tests, showing, at average, superior performance.

Also, synthetic Monte Carlo test on the SPM-MAP joint activity detection and HRF estimation,demonstrates the algorithm’s robustness to noise and flexibility over the underlying HRF, whileretaining a considerably low error probability. Finally, the behavior of the algorithm is analysed onexperimental fMRI data and compared against some results from the SPM-GLM.

Keywords: fMRI, Bayesian, Neurology, Hemodynamic Response, Activity Detection.

5

Resumo

A ressonância magnética funcional (fMRI), é uma nova ferramenta imagiológica que permite estudare avaliar processos neurais. O sinal BOLD (dependente do nível de oxigenação do sangue) é actual-mente usado para detectar regiões de actividade cerebral causadas pela aplicação de estimulo(s),e.g. visuais ou auditivos. O delineamento do paradigma de estimulação é normalmente efectuado deforma a maximizar a certeza dessa detecção. No entanto esta é uma tarefa difícil devido ao ruído,aos "fantasmas" que corrompem os volumes de imagens, movimento do participante e à actividadecerebral normal não correlacionada com o estímulo. O método standard de detecção de actividadeem dados de fMRI, aqui intitulado SPM-GLM, faz uso de uma metodologia de regressão linear aosdados com inferência estatística clássica baseada em testes t, assumindo uma resposta hemodinâmica(HRF) com fortes restrições de forma.

Com o objectivo de estender o estado da arte na detecção de actividade neural com fMRI, nestatese é apresentada uma abordagem diferente; uma nova metodologia Bayesiana, intitulada SPM-MAP, que estima conjuntamente a HRF subjacente aos dados e as áreas de actividade cerebral.Esta abordagem apresenta duas principais vantagens: 1) A detecção de actividade beneficia danatureza adaptativa do método em relação à forma da HRF; 2) fornece uma estimativa local daHRF que poderá ser útil para considerações fisiológicas vasculares e neuronais.

O modelo da HRF incorporado no algoritmo proposto é uma nova abordagem, onde uma funçãolinear simples é desenvolvida a partir de assunções fisiológicas razoáveis sobre o consumo de oxigénioe o processo de vasodilatação. A função de transferência deduzida é estimada em dados reais single-event, demonstrando baixos valores para o erro de ajuste, resultando, em média, num melhor ajustedo que o possível com a base de funções gamma standard [2,3].

A probabilidade de erro do algoritmo SPM-MAP na detecção de actividade cerebral é comparadacom a do algoritmo SPM-GLM em testes de Monte Carlo com dados sintéticos, demonstrando, emmédia, melhor performance.

Testes de Monte Carlo sintéticos na estimação conjunta da HRF e da actividade cerebral eviden-ciam a robustez do algoritmo ao ruído e a sua flexibilidade para com a HRF subjacente aos dados,mantendo uma probabilidade de erro razoavelmente baixa. Por fim, o comportamento do algoritmoé analisado em dados experimentais e comparado com alguns resultados obtidos com o SPM-GLM.

Keywords: fMRI, Neurologia, Bayesiano, Resposta Hemodinâmica, Detecção de Actividade.

7

Contents

Acknowledgments 3

Abstract 5

Resumo 7

List of Abbreviations 11

1 Introduction 151.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Hemodynamic Response Function Model 212.1 PBH Model Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Joint Activity and HRF Estimation Method 273.1 Problem and Method Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.1 Step One: b estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Step Two: h estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.3 Step Three: ProjFIR [ProjIIR(g)] . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Results 354.1 PBH Model Estimation on Real Data . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 SPM-MAP Activity Detection Part v.s. SPM-GLM on Synthetic Data . . . . . . . . 404.3 SPM-MAP Results on Synthetic Data . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3.1 Activity Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3.2 HRF Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4 SPM-MAP Results on Real Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

9

10 CONTENTS

5 Conclusions and Future Work 53

Bibliography 55

6 Appendices 596.1 EMBC2007 Published Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.2 ICASSP2008 Submitted Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

List of Abbreviations

AWGN Additive White Gaussian NoiseBOLD Blood-Oxygenation-Level-Dependent

CAT Computed Axial TomographyCBF Cerebral Blood FlowCBV Cerebral Blood Volume

CMRO2 Cerebral Metabolic Rate of OxygenEPI Echo-Planar ImagesFIR Finite Impulse Response

fMRI Functional Magnetic Resonance ImagingHRF Hemodynamic Response FunctionICA Independent Component Analysisi.i.d Independent and Identically DistributedIIR Infinite Impulse ResponseLTI Linear Time InvariantMR Magnetic Resonance

MAP Maximum A PosterioriMRI Magnetic Resonance ImagingMRF Markov Random Field (MRF)MSE Minimum Square ErrorPBH Physiologically Based HemodynamicPCA Principal Component AnalysisROI Region of InterestSNR Signal-to-Noise RationSPM Statistical Parametric Mapping

SPM-GLM SPM done in the GLM frameworkSPM-MAP SPM done in the MAP bayesian framework

TR Time of RepetitionVoxel Volume Element

11

12 CONTENTS

List of Figures

1.1 Schematic of the fMRI data structure and its acquisition process. . . . . . . . . . . . 17

2.1 Schematic representation of the common features of the fMRI BOLD response to ashort period of neural stimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Block diagram of the proposed physiologically based hemodynamic (PBH) modelbehind the HRF on BOLD fMRI data . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 Example of activated and non activated regions in an fMRI image, overlaid on a higherresolution structural MRI image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 BOLD signal generation model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Paradigm example with three block-designed stimulus. . . . . . . . . . . . . . . . . . 283.4 FIR → IIR → FIR projections scheme. . . . . . . . . . . . . . . . . . . . . . . . . . 293.5 Fluxogram of the proposed SPM-MAP algorithm. . . . . . . . . . . . . . . . . . . . 31

4.1 Results from the PBH model estimation on 4 experimental data with initial dip . . . 364.2 Results from the PBH model estimation results on 4 experimental data without un-

dershoot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Results from the PBH model estimation results on 4 experimental data with undershoot 384.4 Example of synthetic data used for SPM-MAP and SPM-GLM Pe Monte Carlo tests 424.5 Error probability differences, ∆Pe, between the SPM-MAP and the SPM-GLM meth-

ods for two different graphic views. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.6 Synthetic image, used on Monte Carlo tests, representing a single BOLD data slice . 454.7 SPM-MAP activation detection results for σy = 0.5 and σy = 1 . . . . . . . . . . . . 464.8 Mean HRF estimation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.9 SPM-MAP activity detection results on real data compared against data processed

by a neurologist on SPM-GLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.10 Example of two voxel time-courses, from two different brain areas, detected as acti-

vated by the SPM-MAP that were not detected by the SPM-GLM. . . . . . . . . . . 51

13

14 LIST OF FIGURES

Chapter 1

Introduction

Medical imaging is a fundamental area in modern medicine. The gross part of the developed countriespopulation has used, or will use, at least one medical imaging modality in the course of their lives.Of the several available modalities (such as projection radiography, computed tomography, nuclearmedicine, ultrasound imaging an magnetic resonance imaging) that allow to "look" in different waysat the inside of the human body, Magnetic Resonance Imaging (MRI) stands out as a being a non-invasive technique and risk-free (so far as we know), with high spatial resolution (only surpassed byComputed Axial Tomography (CAT)) and with flexibility to "look at" different body characteristics.

Of the several MRI modalities, functional MRI (fMRI), is a new and exciting method that, amongother purposes, allows the determination of which parts of the brain are involved in a particulartask. This is done by observing the activation of brain structures in response to almost any kind ofbrief stimulation, ranging from sounds, to visual images, to gentle touching of the skin. This recentand growing imaging technique has already been able to established itself as the most prominentmethod used for functional brain imaging, and will certainly have a large impact in the future ofNeurology. As it is, fMRI is being used worldwide as a powerful neuroscientific research tool tostudy how the brain works, although some medical applications are being discovered as well. Thisknowledge is a cradle for an infinite virtual number of future applications, such as neurosurgicalplanning and orientation, pain management, understanding the physiological basis for neurologicaldisorders, and understanding the physiological basis for cognitive and perceptual events [4].

Although MRI have been around since the 1980s for anatomical clinical purposes (e.g., for braintumor stroke and multiple sclerosis analysis) it was only in 1990 that Ogawa [5] showed that the MRIwater signals can be sensitized to cerebral oxygenation, using deoxyhemoglobin as an endogenoussusceptibility contrast agent. Using gradient-echo imaging, a form of MRI image encoding sensitiveto local inhomogeneity of the static magnetic field, Ogawa demonstrated (in an animal model) thatthe appearance of the brains blood vessels changed with blood oxygenation. Within two years, hisand two other groups had published papers using this Blood-Oxygenation-Level-Dependent (BOLD)contrast MRI to detect brain activation in humans [6–8], and, today, an explosion of studies use this

15

16 CHAPTER 1. INTRODUCTION

so-called fMRI technique to map human brain function.

Neural activity can be broadly or particularly defined. It could be set to include all of theelectrical, magnetical, chemical and biological events associated with it, which is, at the currentscientific state-of-the-art, impossible to measure in an unified way. In the case of fMRI, it is veryparticularly defined, because BOLD signal does not measure brain function directly. Rather, BOLDfMRI brain activation studies use brain perfusion as a proxy for brain function. Without makingdirect measurements of brain function (i.e., without measuring computations performed in neuronalcell bodies, action potentials traveling along axons, or neurotransmitter trafficking at synaptic junc-tions), these approaches take advantage of neuro-vascular physiological events (e.g, perfusion, oxygenmetabolism) that locally change the diamagnetic oxygen presence near neural activity. Specifically,BOLD fMRI sensitizes MRI acquisitions to the local decreases in deoxyhemoglobin due to reactivehyperemia [9,10] accompanying neuronal activation. This excess of that is needed to support theincreased oxygen consumption due to neuronal activation, results in a local decrease in the concen-tration of deoxyhemoglobin, which yields an increase in the gradient-echo MRI signal. Of courseother physiological events influence the BOLD signal dynamics, some of which are presented anddiscussed in chapter 2.

For BOLD fMRI, image data are typically acquired slicewise using single-shot echo planar imag-ing (EPI). A frequency-selective RF (Radio Frequency) pulse is applied in the presence of a staticmagnetic field gradient to selectively excite nuclear spins in a virtual slice; the slice-select gradientis then turned off, and the signals from these spins are encoded along the dimensions of the sliceusing rapidly switched magnetic field gradients. Within approximately 50 ms, a dataset is acquired,which, when Fourier transformed, will yield an image of the slice in question. This is rapidly re-peated for all the slices (n in Fig. 1.1, typically around 30 to 40) in the brain, such that a completemultislice volume (represented as cubes in Fig. 1.1) is built up within a Time of Repetition (TR),after which the process is repeated. For a typical TR of 3 s, if 200 volumes are acquired, thenwe have a volume movie consisting of 200 volumes of the brain (each consisting of 30 to 40 slices)acquired over 600 s. During these 600 s, a neurobehavioral paradigm is played out in which theresearch participant is exposed to sensory stimuli or asked to perform some set of mental and motortasks or some combination of them. So we have a situation where 600 s of temporally structuredbrain activity (e.g., watching flashing lights every other 30 s, tapping one’s fingers every other 20s, or reading words) are accompanied by the acquisition of a brain volume movie with 3 s temporalresolution, which core unit is the brain movie of each Volume Element (voxel) intensity time-course.In Fig. 1.1, drawn in green, is a schematic representation of a time-course signal obtained from anactivated voxed.

This 3D brain movie has the interesting property that any single image from it contains no infor-mation about brain function. Instead, information about brain function is encoded in the varianceof image intensity over time. Unfortunately retrieval of this information is a challenging problembecause BOLD signal changes are very small when compared with the images dynamic scale (givingpotential false negatives), and the number of voxels simultaneously interrogated across the image

17

Figure 1.1: Schematic of the fMRI data structure and its acquisition process. A structured brainstimulus is applied to the participant and on each time-point a volume (cubes) of n (in this example,axial) images are taken and a 3D movie of the brain with sample time of TR is obtained. Theobtained intensity 1D movie of each voxel (example in green) and the applied paradigm are thenused for univariate data analysis.

volume is very large (giving potential false positives). Furthermore, fMRI data is corrupted withseveral artifacts that lower the Signal-to-Noise Ratio (SNR), affecting the brain activity detectionsensitivity. Of these artifacts, head and brain motion, random noise, magnetic field inhomogeneities,and echo planar imaging "ghosting", are the most notable. Hence the computational fMRI data anal-ysis is not trivial and is a subject open to much developments. The conventional approach consistsof three steps [11]: preprocessing, statistical analysis, and inference.

• In preprocessing, the raw images are subjected to spatial registration to correct for small headmotions, temporally interpolated to compensate for the fact that different slices are acquiredat different times, spatially smoothed to enhance SNR, and sometimes spatially normalized ortransformed into a common space to facilitate group analyses and neuroanatomical labeling.

• In statistical analysis, usually a linear regression is done where the paradigm timing is con-volved with an estimate of the Hemodynamic impulse Response Function (HRF). The resultanthemodynamically lagged and blurred version of the paradigm timing is used as a regressor (orExplanatory Variable (EV)) of interest, forming a General Linear Model (GLM), which is thenfit to the data, allocating temporal variance in the (preprocessed) data among such regressors.

• In inference, the spatial map of statistical analysis coefficients resulting from the regressionstep is thresholded for significance (usually with a p-value over a T or F -statistics (see section4.2)), labelling the suprathreshold regions as activated brain areas, and often overlaying themwith color on a (higher resolution) anatomical MR image (see example in Fig. 3.1).


This standard inferential univariate fMRI data analysis approach, which is implemented in sev-eral fMRI data analysis packages (e.g. Statistical Parametric Mapping [12], FMRIB Software Library[13], BrainVoyager [14]), can be seen as a good machine for testing prior temporal hypotheses. How-ever, it cannot discover unanticipated structure in the data, i.e., it cannot detect brain activationswith timings that were unanticipated by the investigator due to the fixed nature of the explanatoryvariables used in the GLM. To cope with this, in the last decade, investigators have developed avariety of data-driven or exploratory techniques that can discover brain activity not anticipated inadvance, such as clustering methods, Principal Component Analysis (PCA) [15] and IndependentComponent Analysis (ICA) [16]. Unfortunately these methods are not as efficient when trying toidentify voxels that show signal changes varying with the changing brain states of interest across theserially acquired images.

1.1 Objectives

In this thesis we pursued the standard functional neurology imaging goal of determining whichbrain areas were activated in a particular task, i.e, determine which brain areas display, over time, adeterministic positive influence from the paradigm stimulus, and which do not. The objective was todevelop a robust method with greater, or at least similar, activity detection sensitivity as comparedto the standard GLM procedure (presented above and described in section 4.2), but that would behighly flexible in terms of the underlying BOLD dynamics. Undertaking this, and focusing on thestatistical analysis and inference steps referred above, a new perspective robust Bayesian algorithm(here called SPM-MAP), that jointly estimates the local HRF and provides neural activity detection,was intended. Besides the mentioned higher flexibility for deviant HRF shapes (which can be veryuseful in pathological cases, for instance), and the local HRF estimation (which can be useful inneuro-vascular physiological studies), this method is parameter-free presented. This means thatits results could provide an interesting comparison with parameter-dependent methods (like the p-value dependent standard GLM procedure, here called SPM-GLM ) yielding a valuable hint into thecommon practical question: "what is the best parameter value to be used for this data?"

1.2 Original Contributions

Most of the work presented in this document is an original contribution to knowledge and has beenpublished or submitted.

• The PBH HRF model and its estimation results on real data (see chapter 2 and results section4.1) was published at the international IEEE EMBS 2007 conference in Lyon, France ([17],section 6.1) and an extended abstract and poster were presented at the RecPad 2007 nationalconference.

• The SPM-MAP activity detection part and comparison to SPM-GLM on synthetic data (see

1.3. THESIS ORGANIZATION 19

chapter 3 and results section 4.2) was submitted to the IEEE ICASSP 2008 internationalconference in Las Vegas, USA ([18], section 6.2) and an extended abstract and poster werepresented at the RecPad 2007 national conference.

• The complete SPM-MAP algorithm and its joint activity detection and HRF estimation resultson synthetic data (see chapter 3 and results section 4.3) was submitted to the IEEE ICASSP2008 international conference in Las Vegas, USA ([19], section 6.2).

• The SPM-MAP algorithm performance in synthetic and real fMRI data was submitted to theIEEE Transactions on Medical Imaging international magazine [20].

• Further scientific articles with parts and extensions of this work are also programmed to besubmitted to the IEEE ISBI 2008 international conference in Paris, France and to the IEEEICIP 2008 international conference in San Diego, USA.

1.3 Thesis Organization

The rest of the thesis is organized as follows. Chapter 2 starts with a short introduction to thehemodynamics characteristics behind the BOLD fMRI. A presentation of the newly developed modelfor the HRF follows (section 2.1), along with an explanation of the physiological considerations andevents incorporated in it. The mathematical construction of the assumed HRF model and how itcan be data-estimated is then presented in section 2.2.

Chapter 3 starts with a formulation of the Activity Detection problem and the introductionand mathematical construction of the assumed model behind the BOLD data generation, and itsintervening elements such as the stimulus paradigms, the HRF model and the noise model used(section 3.1). Section 3.2 closes this chapter and formulates the core of the work, which is theproposed Bayesian mathematical algorithm to, adaptively and simultaneously, detect brain activityand estimate the HRF.

Chapter 4 groups the several results obtained in testing several aspects of the proposed work,and is divided in four sections: Section 4.1 presents the estimation results of the PBH model onreal single-event data and how the proposed model closely mimics the several data; Section 4.2shows how the activity detection portion (i.e. without HRF estimation) of the SPM-MAP proposedalgorithm compares against the standard SPM-GLM method in terms of their error probability onMonte Carlo testing; Section 4.3 display the results of the SPM-MAP joint brain activity detection(subsection 4.3.1) and HRF estimation (subsection 4.3.2) on synthetic Monte Carlo testing; Section4.4 ends the results chapter with the behavior of the proposed method on real datasets, against dataprocessed with SPM-GLM by an expert neurologist.

Finally, chapter 5 concludes this thesis and discusses some limitations and possible future exten-sions of this work.

Chapter 2

Hemodynamic Response FunctionModel

During functional magnetic resonance imaging, a brief focal neural activation evokes what is calleda Hemodynamic time-course Response Function (HRF) that mainly depends on tissue physiology[11]. But before describing the putative biological and physical origins of the various signal featuresit is useful to schematically divide the typical HRF into three epochs, as shown in figure 2.1. First,immediately after electrical activity commences there may be, depending on the magnetic fieldstrength used, a brief period of approximately 0.5−1s during which the MRI signal decreases slightlybelow baseline (≈ 0.5%), usually intituled initial dip. Subsequently, the BOLD response increases,yielding a robust positive BOLD response which peaks 5 − 8s after the stimulus commences. It isthis positive BOLD response that is used by most fMRI researchers. Finally, upon cessation of thestimulus, there is a return of the BOLD response to baseline, often accompanied by a post-stimulusundershoot, during which the response passes through the baseline and remains negative for severaltens of seconds, eventually returning to baseline.

Although variability exist on who, where, when and how the data was acquired and processed[21,22], standard HRF models are often an essential basis of fMRI data analysis. Several assumptionsare usually made, namely that all neural impulse events produce the same or similar HRF (assumingminimal variability across subject, brain region and acquisition system) and that a BOLD time seriesdata is modeled as the convolution between this invariant HRF and an impulse train of neural events[23]. Even though some evidences have shown these assumptions to be erroneous [21,22], its impacton many fMRI statistical analysis studies has not been considered relevant enough to generallyabandon the simplification advantages.

In the literature there are two different approaches in modeling this HRF. The most commonapproach is purely heuristic, using known functions (e.g. gamma functions [2,3], or, to a less extent,Poisson and Gaussian distributions [24], that satisfactorily modulate the rather invariant shape

21

22 CHAPTER 2. HEMODYNAMIC RESPONSE FUNCTION MODEL

Figure 2.1: Schematic representation of the common features of the fMRI BOLD response to a shortperiod of neural stimulation. During the first epoch a small negative initial dip may be observed.Subsequently, a more robust positive BOLD response is observed. Following cessation of the stimulusa return to baseline accompanied by a post-stimulus undershoot is often seen.

of the BOLD time-signal to short stimuli. These analytical models are most commonly used asregressors, after convolving with the paradigm stimulus, in the GLM as mentioned in chapter 1.The second approach is physiological, modeling and providing information on the events that causethe BOLD signal, e.g. the Balloon Model [25] which is often used and augmented [26]. But,unfortunately, due to much higher conceptual and computational complexity and higher number ofstate variables and parameters to be estimated, these models have been usually remitted to studiesin which the knowledge of the physiological events are important or essential.

In this work we propose a new HRF model [17], with a new perspective, to combine the bestof both typical approaches referred above. A simple and light model that would not compromiseits use on standard, straight ahead activity detection studies (which usually happens to complexphysiological models), but that, at the same time, would also provide some information on significantunderlying physiological events of the vascular and neural tissues.

Unfortunately, the exact coupling between brain activity, vascular response (affecting blood per-fusion) and Cerebral Metabolic Rate of Oxygen (CMRO2) that leads to the BOLD response are notwell understood. But there are several experimental evidences that lead to conclusions and assump-tions that provides a sketchily, but important, view of the physiological features that influences theBOLD signal.

Following an increase in neuronal activity, local Cerebral Blood Flow (CBF) and Cerebral BloodVolume (CBV) increases. The consequent increase in perfusion, in excess of that needed to sup-port the increased oxygen consumption due to neuronal activation, results in a local decrease in theconcentration of deoxyhemoglobin in the vessels, capillaries and surrounding tissues. As deoxyhe-moglobin is paramagnetic, a reduction in its concentration results in an increase in the homogeneityof the static magnetic field which yields an increase in the gradient-echo MRI signal, resulting in the

2.1. PBH MODEL PRESENTATION 23

positive BOLD response (Fig. 2.1). However, the BOLD time course signal has transient features(see Fig. 2.1) at the onset and end of the stimulus: an initial dip and a post-stimulus undershoot ;that are not explained by the coupling of flow and metabolism referred above.

It has been shown [27] that an increase in cerebral neuronal activity generally leads to an im-mediate co-localized surge in tissue deoxyhaemoglobin levels, which leads to the initial dip. Thisstimulus-driven increase has been putatively accounted for energy consumption by neural and glialbrain cells, hence the assumed relation between the BOLD signal and neural activity. So the initialdip, corresponding to an increase in local deoxyhaemoglobin, has been interpreted as evidence for aninitial increase in oxygen extraction before flow increase. Furthermore, the post-stimulus undershootis accounted for an elevated oxygen extraction after the flow has returned to baseline. These modelsfor the transient features based on uncoupling, coupling and reuncoupling of vascular response andCMRO2 provide a, possibly rough but valuable, key element in generating a physiological model thatevokes an output similar to the HRF. In fact these interpretations have been recently reinforced bymultimodal fMRI studies [28].

In the following section we propose [17] a simple linear model of the hemodynamic responsefunction based on a modulation of basic physiological processes behind the BOLD signal, with themain objective of being used as basis for fRMI activation mapping statistical analysis. This hasbeen done considering uncoupling, coupling and reuncoupling processes of the vascular response andCMRO2 variables and accounting for neural demand and systemic myogenic feedback control on thevascular response. The model was tested on real data fMRI BOLD signal time-courses (see section4.1). A final Z-transform function is presented, which has obvious frequency analysis advantagesproviding computational processing efficiency.

2.1 PBH Model Presentation

Brain activation is accompanied by a series of physiologic alterations, including focal changes inthe vascular response (CBF and CBV), blood oxygenation and cerebral oxygen consumption. Wehave tried to encapsulate all of these physiological variables in a simple based linear discrete modelthat would translate their effects on a HRF estimation (see Fig.2.2). The Physiologically BasedHemodynamic (PBH) model input is the neural activation, r(n), and the output is the BOLD contrastsignal, y(n). The reference value Ref is the baseline of the vascular properties. Since this work isfocused on incremental variations of all its variables, this parameter is constantly null. All modelblocks are zero and first order linear functions, which provide empirically reasonable approximations.Three main groups can be distinguished in the PBH model: a brain group which modulates the neuraland glial cells oxygen CMRO2 and vascular response demands; a vessel group which modulates thesummed effect of CBV and CBF vascular changes on the rate of deoxyhaemoglobin concentration inand around blood vessels; a control group for the systemic myogenic negative feedback control overvasodilatation.


Figure 2.2: Block diagram of the proposed physiologically based hemodynamic (PBH) model behindthe HRF on BOLD fMRI data. It incorporates vascular response demand and oxygen metabolismconsumption by brain tissue, vascular response producing changes in both CBV and CBF andsystemic negative myogenic feedback control system of the vascular response. Baseline vascularproperties and neural activation stimulus are considered in the Ref and r(n) inputs respectively.

The PBH model was developed upon the fundamental consideration of separate dynamics be-tween CMRO2 and vascular response features (CBV and CBF). This means that the transient initialdip and poststimulus undershoot of the HRF are then modeled as an uncoupling of these features,where the negative influence of CMRO2 to the BOLD signal is not counterbalanced by the briefervascular response. This has been an old assumption [11] that has recently gained strength throughmultimodal fMRI studies [28]. The uncoupling considered indicates that the vascular response de-mand is probably not due to the oxygen metabolism pathway, but due to the neurotransmitterspathway, hence the separation of both blocks of the PBH models brain group (CMRO2 and vasculardemand blocks). Still, they are both a consequence of the brain activity, which is their input signal.

The dynamics of the actual vascular response are accounted in the vessel group block function.This deals with the reasonable assumption that, upon tissue demand for more blood delivery, thevascular response to this request is delayed and constrained. Besides these aspects, the gain in thisblock is also responsible for the relation between the vascular response features and the amplitudeeffect they cause on the BOLD signal. On the other hand, vascular response demand by brain tissuesis most likely bigger than their needs, and alongside the vascular answering dynamic referred, thereis a systemic myogenic vascular response control modeled in the control group that reduces theamplitude of the vascular response. Notice again that the gain of this block reflect the amplitudeimpact on the BOLD signal of the vasodilatation control.

2.2. MODEL ESTIMATION 25

2.2 Model Estimation

The transfer function of the discrete time Infinite Impulse Response (IIR) PBH model, displayed inFig.2.2, is

H(z) =(AV − B) + (a + v − bAV )z−1 − (avB)z−2

(1 − az−1)(1 − bz−1)(1 + C(s)V − vz−1)(2.1)

which can be rewritten as follows

H(z) =Y (z)R(z)

=b0 + b1z

−1 + b2z−2

1 + a1z−1 + a2z−2 + a3z−3(2.2)

where

b0 =AV − B

1 + V C(s)

b1 =a + v − bAV

1 + V C(s)

b2 = − avB

1 + V C(s)

a1 = −(a + b +v

1 + V C(s))

a2 = ab + (a + b)v

1 + V C(s)

a3 = −abv

1 + V C(s)

(2.3)

Assuming the simpler controller (more complex controllers will be considered in the future),C(z) = K, the correspondent difference equation is the following,

y(n) =2∑

k=0

bkr(n − k) −3∑

l=1

aly(n − l). (2.4)

The estimation of the parameters, bk and ak is performed with the minimum square error (MSE)method,

p = arg minp

E(y, r,p) (2.5)

where y = {y(1), y(2), ..., y(N)} is the vector with the N experimental points, r = {r(1), r(2), ..., r(N)}is the stimulus signal that is unknown and must also be estimated, p = [b0, b1, b2, a1, a2, a3] is thevector of parameter to be estimated and E(y, r,p) is the function to be minimized,

E(y, r,p) =N∑

n=0

[y(n) −

2∑k=0

bkr(n − k) +3∑

l=1

alr(n − l)

](2.6)


which can be written as follows using matrix notation,

E(y, r,p) = (Y − Φp)T (Y − Φp) (2.7)

where Φ = [Φr

... − Φy] with

Φr =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

r(1) 0 0r(2) r(1) 0r(3) r(2) r(1)r(4) r(3) r(2)... ... ...

r(N − 1) r(N − 2) r(N − 3)r(N) r(N − 1) r(N − 2)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2.8)

Φy =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0y(1) 0 0y(2) y(1) 0y(3) y(2) y(1)... ... ...

y(N − 2) y(N − 3) y(N − 4)y(N − 1) y(N − 2) y(N − 3)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (2.9)

The minimization of E(y, r,p) is performed by finding its stationary point by computing

∇pE(y, r,p) = 0 (2.10)

, were ∇ where (∇) is the gradient operator. This as the following solution,

p =[ΦT Φ

]−1Φ︸︷︷︸

Ψ

y (2.11)

where Ψ is called pseudoinverse of Φ. The neural activity signal r(n) is not completely known andtherefore must be estimated. It is assumed that

r(n) =

⎧⎨⎩1 0 ≤ n ≤ n0

0 otherwise(2.12)

where n0 is unknown (notice that the data used in this work [22] was previously aligned and shifted).

For each data set five values of n0 were tested, n0 = [M, ...,M −4] in which M is the time instantwhere the maximum of the experimental data occurs, that is, the position of the larger maximum.The final solution is obtained by choosing the set of parameters that lead to the minimum error,

[p, n0] = arg minp,n0

E(y, r(n0),p) (2.13)

Chapter 3

Joint Activity and HRF EstimationMethod

The present thesis focuses on a new analysis perspective that combines the activity estimationproblem, which leads to a functional brain map, and the local HRF estimation problem. Thecombination of these problems in one, provides an activity detection method that is more sensible tounderlying HRF shapes that deviate from the "standard shape" generally assumed. Simultaneouslythe HRF it provides might be useful for studying the neuro-vascular events and physiological eventsbehind local brain activity. In order to achieve the binary (activated or not) information on eachvoxel (volume element), we present a Bayesian method that forces this binary solution and doesa semi-model-free estimation of the HRF. This is done by estimating a model-free Finite ImpulseResponse (FIR) and then projecting it into the PBH model IIR space, here the shape is somehowconstrained and some physiological considerations might be done. Several Monte Carlo tests onsynthetic data for this method are presented in chapter 4.

3.1 Problem and Method Formulation

Let us consider the voxels displayed in Fig. 3.1. Each voxel, after the application of a givenparadigm, may be activated by one or more applied stimulus (∃k : βk = 1) or may not be activatedat all (∀k : βk = 0).

In this theis we consider the BOLD signal associated to a single voxel at a time - time course -with the following data observation model, displayed in Fig. 3.2,

y(n) = h(n) ∗N∑

k=1

βkpk(n) + η(n) (3.1)

where η(n) is modeled as Additive White Gaussian Noise (AWGN), h(n) is the HRF of the brain

27

28 CHAPTER 3. JOINT ACTIVITY AND HRF ESTIMATION METHOD

Figure 3.1: Example of activated and non activated regions in an fMRI image, overlaid on a higherresolution structural MRI image.

Figure 3.2: BOLD signal generation model.

tissues, pk(n) are the stimulus signals along time (see Fig. 3.3) and βk are unknown binary variablesto model the activation of the voxel by the kth stimulus. For instance, Fig. 3.1 shows the result ofapplication of a two stimulus paradigm where three voxels are referenced: i) a voxels was activatedby the first stimulus, β1 = 1 and β2 = 0, ii) a voxel was not activated, β1 = 0 and β2 = 0, and iii)a voxel was only activated by the second stimulus, β1 = 0 and β2 = 1.

Figure 3.3: Paradigm example with three block-designed stimulus.

In this thesis we describe a Bayesian Statistical Parametric Mapping algorithm (SPM) basedon the Maximum a Posteriori (MAP) [29] criterion called SPM-MAP1. The proposed algorithmjointly estimates the vector b = {β1, β2, ..., βN}T , associated with each voxel and the correspondinghemodynamic response, h(n), which can be denoted in vectorial form, h = {h(1), h(2), ..., h(N)}T .

The hemodynamic signal is assumed to exist in the class of functions provided by the PBH model[17], presented in section 2.1, i.e. the response of a third order IIR Linear Time Invariant (LTI)system (eq. 2.2).

1Statistical parametric mapping is generally used to identify functionally specialized brain responses[1]

3.1. PROBLEM AND METHOD FORMULATION 29

The estimation process is performed by minimizing an energy function depending on the binaryunknowns βk, on the hemodynamic response h(n), and on the observations y(n) (see Fig. 3.2).The direct estimation of the H(z) coefficients is a difficult task because it is not easy to definesimple priors for these coefficients based on the desired time response h(n) function. Therefore, toovercome this difficulty, instead of estimating the ak and bk IIR coefficients, a FIR is estimated,g = {g(1), g(2), ..., g(F )}T , with length F minor or equal to the observations length, F ≤ L. In eachiteration this estimated response is projected into the H(z) space, i.e., a set of coefficients ak and bk

are estimated in order to minimize ‖g(n) − h(n)‖ by estimating the coefficients ak and bk. The PBHak and bk estimated coefficients are then used to re-project H(z) into the FIR space by computinga new finite response h, which is used to obtain a new estimate of the binary unknowns βk. Thisprocess is schematically represented in Fig. 3.4, where each circle represents the set of admissibleresponses for each one of the IIR and FIR systems.

Figure 3.4: FIR → IIR → FIR projections scheme.

Let x = {x(1), x(2), x(3), ..., x(L)}T (see Fig. 3.2), which may be expressed as follows

x = θb (3.2)

where

θ =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

p1(1) p2(1) p3(1) ... pN (1)p1(2) p2(2) p3(2) ... pN (2)p1(3) p2(3) p3(3) ... pN (3)

......

... ......

p1(L) p2(L) p3(l) ... pN (L)

⎞⎟⎟⎟⎟⎟⎟⎟⎠

(3.3)

The output vector of h(n) displayed in Fig. 3.2, z = {z(1), z(2), z(3), ...z(L)}T , is obtained byz(n) = h(n) ∗ x(n), where h(n) is a F length FIR and ∗ denotes the convolution operation. Theoutput signal may be expressed in the two following ways

z = Hx (3.4)


z = Φh (3.5)

where H and Φ are the following L × L and L × p Toeplitz matrices respectively

H =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

h(1) 0 0 0 0 0h(2) h(1) 0 0 0 0h(3) h(2) h(1) 0 0 0

......

......

......

0 ... h(p) h(p − 1) ... h(1)

⎞⎟⎟⎟⎟⎟⎟⎟⎠

(3.6)

Φ =

⎛⎜⎜⎜⎜⎝

x(1) 0 0 0 0x(2) x(1) 0 0 0

......

...... 0

x(L) x(L − 1) ... ... x(L − P + 1)

⎞⎟⎟⎟⎟⎠ (3.7)

The observed BOLD signal y(n), y = {y(1), y(2), ..., y(L)}T , can therefore be obtained with thefollowing two ways

y = Ψb + n (3.8)

y = Φh + n (3.9)

where Ψ = Hθ and n = {η(1), η(2), ..., η(L)}T is a vector of Independent and Identically Distributed(i.i.d) zero mean random variables normally distributed, that is, p(η(k)) = N(0, σ2

y). This AWGNis usually used to model the corruption process in functional MRI [30], although other models mayalso be used, e.g., Rice [31] and Rayleigh [32]. In this work we suppose that the motion correctionpreprocessing step was efficient enough to remove most of the temporal correlation between voxels,and so its influence is included and corrected along with the corruption noise.

3.2 Estimation

The MAP estimation is obtained by minimizing the following energy function

E(y,x(b),h) = Ey(y,x(b),h) + Eb(b) + Eh(x(b)) (3.10)

3.2. ESTIMATION 31

where the data fidelity term is

Ey(y,x(b),h) = − log(p(y|x(b))) (3.11)

and the prior terms associated to the unknowns to be estimated, b = {β1, ...βN} and h = {h(0), ..., h(p−1)} are

Eb(b) = − log(p(b)) (3.12)

Eh(x(b)) = − log(p(h)). (3.13)

These priors incorporate the a priori knowledge about the unknowns to be estimated: i) βk arebinary and ii) h(n) is smooth.

Figure 3.5: Fluxogram of the proposed SPM-MAP algorithm.

The estimation process is performed in the following three steps, as shown in Fig. 3.5,

bt = arg minβ

E(y,x(bt−1),ht−1) (3.14)

g = arg minh

E(y,x(bt),ht−1) (3.15)

ht = ProjFIR [ProjIIR(g)] (3.16)

where ()t means estimation at tth iteration and Proj stands for the projection operation by using


the Minimum Square Error (MSE) criterion. The ProjIIR problem is not trivial [33,34]. In this workthe approximation algorithm proposed by Shanks [35,36] is used. Other approximations proposedby Prony [36,37] and Padé [36] may also be used, but lead to worse results in the MSE criterion.

The independence assumption on the time-course observations may not be realistic. However,it is a reasonable assumption and a convenient mathematical simplification, because it separatesthe observations dependence and the estimated data dependence, simplifying a considerable num-ber of expressions. Furthermore, the inclusion of the observations statistical dependence in themathematical formulation may not lead to relevant improvement on the final solution, as noted in[38].

The observations independence means that p(y|x(b),h) =∏L

i=1 p(y(i)|(x ∗ h)(i)). The adoption

of AWGN model leads to, p(y(i)|(x ∗ h)(i)) = 1√2πσ2

y

e− (y(i)−(x∗h)(i))2

2σ2y where σ2

y is the observations

noise variance. The parameters βk to be estimated are also assumed independent, that is,

p(b) =N∏

i=1

p(βk) (3.17)

where p(βk) is a bi-modal distribution defined as a sum of two Gaussian distributions centered atzero and one, with σ2

β variance,

p(βk) =12[N(0, σ2

β) + N(1, σ2β)]

(3.18)

because βk are binary variables, βk ∈ {0, 1}. In order to better approximate the binary answer, theσβ parameter should be as small as possible but numerical stability reasons prevent the adoption oftoo small values. The prior term Eb(b) may therefore be written as

Eb(b) =N∑

k=1

[2β2

k − 2βk + 14σ2

β

− log

(cosh

[2βk − 1

4σ2β

])]. (3.19)

To impose smoothness [39] on the estimated HRF, h(n) is assumed to be a Markov Random Field(MRF), which means, by the Hammersley-Clifford theorem [40], that p(h) is a Gibbs distribution:

p(h) =1

Zhe−α

∑Nn=2 (h(n)−h(n−1))2 (3.20)

which leads to

Eh(x(b)) = − log(p(h)) = α(∆h)T (∆h) + C (3.21)

where α is a parameter that tunes the smoothing degree for h(n), C is a constant, Zh is a partitionfunction and ∆ is the following difference operator

3.2. ESTIMATION 33

∆ =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 ... 0 −1−1 1 0 ... 0 00 −1 1 ... 0 0...

...... ... 0 0

0 0 0 ... −1 1

⎞⎟⎟⎟⎟⎟⎟⎟⎠

(3.22)

The energy function (3.10) to be minimized, E(y,x(b),h), has the following formats in step one(see section 3.2.1) and step two (see section 3.2.2) respectively

E1(y,x(b),h) =1

2σ2y

(Ψb − y)T (Ψb − y) + Eb(b) + C1 (3.23)

E2(y,x(b),h) =1

2σ2y

(Φh − y)T (Φh − y) + αhT (∆T ∆)h + C2. (3.24)

The MAP estimate is obtained by finding the E(y,x(b),h) stationary point,

∇E(y,x(b),h) = 0. (3.25)

3.2.1 Step One: b estimation

In the first step the following equation must be solved

∇bE1 = ΨT (Ψb − y) +σ2

y

σ2β

[b − 1

2R(b)

]= 0 (3.26)

where ∇b is the gradient operator with respect to b and R(b) is a column vector with N elementsrk,

rk = 1 + tanh

[2βk − 1

4σ2β

]. (3.27)

The solution of (3.26) may be obtained by using the fixed point method which leads to thefollowing recursion

bt = (ΨT Ψ + λI)−1(ΨT y +λ

2R(bt−1)) (3.28)

where λ = σ2y/2σ2

β is a parameter, I is a N dimensional identity matrix and bt is the b estimate attth iteration.


3.2.2 Step Two: h estimation

In the second step, where a new estimate of h(n) is obtained, the following equation is solved

∇hE2 = ΦT (Φh − y) + 2λσ2yLh = 0 (3.29)

where ∇h is the gradient operator with respect to h and L = ∆T ∆ (3.22).The solution of (3.29) is

g =[ΦT Φ + 2λσ2

yL]−1

ΦT y (3.30)

where Φ(x(b)) is computed (3.7) with the b estimate, bt, obtained in step one (3.28).

3.2.3 Step Three: ProjFIR [ProjIIR(g)]

A HRF smoothness constrain is much more easy to define on a FIR hemodynamic response, thanon the IIR space. However, this FIR response does not necessarily belong to the responses space ofthe adopted IIR PBH model, presented in section 2.1, which imposes harder restrictions on h(n).To cope with this, restriction to its shape is adopted, in each iteration, by projecting the finitelength hemodynamic response, g, estimated in step two (3.30) into the IIR space of H(z) and thenre-projecting it into the FIR space to obtain a new estimate of h, ht, as schematized in Fig. 3.4.

For the g → IIR projection, we use the Shanks’s method [35,36] which provides the least,although far from ideal, MSE when compared with Padé’s [36] and Prony’s [36,37] method.

The IIR → FIR projection is achieved by simply sampling the estimated continuous IIR functioninto a discrete signal of length F , which is the best possible estimate in a MSE sense.

These three steps (3.2.1, 3.2.2 and 3.2.3) are repeated until convergence, when very low variabilityof b, from the previous iterations to the current one, is achieved (stop criteria refered in Fig. 3.5).Hence, the estimated elements of b, βk, are not binary but, close to 0 or 1, real numbers. Toaccomplish the desired binary nature of b the following threshold is applied to βk

bk =

⎧⎨⎩0 βk < 0.5

1 otherwise.(3.31)

and this is the final activation estimation that provides information on whether the brain arearepresented in the corresponding voxel was activated by each of the paradigm stimulus or not. Ifthe answer is positive, then the estimated HRF provides a possibly valuable insight into the BOLDlocal dynamics.

Chapter 4

Results

4.1 PBH Model Estimation on Real Data

To test the PBH model we used the same normalized data, as used by the authors of [22]. Assuch, it might also provide a means of comparison to their results and remarks. Note that the datawas acquired in four different brain areas of twenty-seven male subjects with no history of neuronalor psychiatric diseases. T2*-weighted Echo-Planar Images (EPI) were acquired at 4 Teslas, withvariations in the TR and time resolution. For a more complete description on materials and methodsused please see [22].

The selection of representative experimental results are organized in three sets: i) data thatdisplays the initial dip (Fig.4.1), emphii) data without poststimulus undershoot (Fig.4.2) and iii)data with poststimulus undershoot (Fig.4.3).

For each experimental curve a set of modeled parameters was estimated as well as the optimalstimulus duration, which is unknown. Unfortunately, we did not had access to the paradigm infor-mation. But even if we did, we do not have direct access to the real duration time of the neuralactivation in each data sample.

From the displayed results it is visible that the PBH model manages to accurately explain, ina MSE basis, the experimental curves. Upholding to this argument are the low values of the meansquare error (0.018) and the mean error variance (0.011) for the overall 80 datasets. This providesa considerable confidence for the base assumptions upon which our PBH model is built. Note thatmany of these experimental data shapes rather deviate from the range of shapes that the HRFgamma functions are able to produce. Conversely our PBH model was successful in providing suchform variability.

Although these results encourage the use of the PBH model in brain activity detection and HRFshape estimation studies, some apprehension should be taken when analysing the physiological con-siderations provided by the model estimation. In fact, after a PBH model system (2.3) inversion(which could be analytically or numerically obtained) the amplitude and rate of response for each

35

36 CHAPTER 4. RESULTS

Figure 4.1: Results from the PBH model estimation on 4 experimental data with initial dip. Red -real data; Blue - model; Black - stimulus.

4.1. PBH MODEL ESTIMATION ON REAL DATA 37

Figure 4.2: Results from the PBH model estimation results on 4 experimental data without under-shoot. Red - real data; Blue - model; Black - stimulus.


Figure 4.3: Results from the PBH model estimation results on 4 experimental data with undershoot.Red - real data; Blue - model; Black - stimulus.

4.1. PBH MODEL ESTIMATION ON REAL DATA 39

of the physiological events considered (except for the myogenic controller which only provides am-plitude) in the PBH model blocks (see Fig. 2.2) might provide important physiological feedback.Unfortunately the parameterization presented above suffers from the general problem in parameter-based models: parameter variability (see the poles/zeros maps in each image in Fig. 4.1, 4.2 and4.3). In order to obtain greater confidence in the physiological variables obtainable, it is importantto compare these with in-vivo measurements, and possibly further restrict the parameter estimationto some magnitude ranges to reduce variability.


4.2 SPM-MAP Activity Detection Part v.s. SPM-GLM on

Synthetic Data

The proposed SPM-MAP method formalizes the neural activation detection problem in step one(section 3.2.1), and the HRF estimation problem in the following steps (sections 3.2.2 and 3.2.3).The results of each step has a strong influence on the others. So, in order to compare our methodto the standard SPM-GLM [1,11], only the activation detection procedure is considered. Two maindifferences must be stressed between both methods:

1. In the SPM-GLM method the whole N period signal is sometimes broken into N piecescorresponding to each paradigm period and the resulting observation pieces are averaged toreduce the noise corrupting the observations Y . The matrix θ, defined in (3.3), is built byusing only a single paradigm period. In the proposed method, instead of braking the signal,it is dealt with as a whole signal. And the same goes for the paradigm signal. The noisereduction is performed in a Bayesian framework where a realistic observation model is used tocope with it. In the case of AWGN both methods are very similar, but if other noise models(e.g. multiplicative) are used this would no longer be true. This is because the averagingprocedure is only adequate for certain noise models.

2. In the SPM-GLM method the estimation of each βk is based on the well known classical t-test[41] applied to the estimated coefficients βk. This statistical inference technique is based on thenull hypothesis test, Ho, where the activation probability of a given voxel is computed with acertain confidence degree. This test is performed over the estimated coefficients, βk, obtainedwith the GLM. These coefficients used to linearly combine the EVs (usually a convolutionbetween the paradigm stimulus and the HRF model (s)) are estimated by using the MSEcriterion. These real coefficients reflect the estimated "presence" amplitude of each EV in theobserved data. In the SPM-MAP method the coefficients set are assumed to be binary andare estimated in a Bayesian framework where a prior distribution forces its values to be closeof {0, 1}. Once again, our concern is to follow a realistic model where it is assumed that agiven voxel was activated or not by a given stimulus. Partial voxel activation is not acceptablein this scope: it is totally activated or it is not activated at all, by a given stimulus, pk(n).

To better understand the difference between both methods, a short description of the SPM-GLMmethod is presented where the t-test is used to determine if a given voxel is or is not activated by asingle EV , p(n) ∗ h(n), which means that b is a scalar, b = [β].

The observed BOLD signal is assumed to be obtained from the following model

y = β (θ ∗ h(n))︸︷︷︸EV

+e (4.1)

4.2. SPM-MAP ACTIVITY DETECTION PART V.S. SPM-GLM ON SYNTHETIC DATA 41

where θ is defined in (3.3) but with only one stimulus, θ = {p(1), p(2), ..., p(L)}T , h(n) is thecanonical gamma function [1,11,42] and e is the residual error vector not explained by the model.The β estimation given by the GLM method, in this very simple case, is computed as follows

β = χ+Y (4.2)

where χ = θ ∗ h(n) is the EV and χ+ = (χT χ)−1χT is the so called pseudoinverse of χ [39].

The SPM-GLM method core, tags each voxel as activated, b = 1, or as inactivated, b = 0, bycomputing the probability of a random generation of the analysed y(n) time-course with a confidencelevel α, that is,

b =

⎧⎨⎩1 (Active) if p < α; (reject H0)

0 (No Active) Otherwise; (acept H0)(4.3)

where H0 is the null hypothesis which assumes no activation with a confidence level α.

The p-value is obtained as follows

p = P (t ≥ T ) = 1 − I LL+T2

(L/2, 0.5) (4.4)

where Ix(a, b) is the incomplete Beta-function [41] defined as

Ix(a, b) =Γ(a + b)x

Γ(a)Γ(b)

∫ x

0

τa−1(a − τ)b−1dτ (4.5)

and

T = β/σβ (4.6)

is the T estimator associated t-statistics, where β is the estimation value of β, and σβ the standarddeviation. T is large if the estimated value is much larger than the estimator variance and T is smallif the estimated value is comparable with the corresponding estimator variance.

The estimator variance, σ2β , may be numerically estimated using the following expression

σ2β = σ2

y

L∑n=1

p2(n) (4.7)

where p(n) is the nth θ element and σ2y is the estimated noise energy

σ2y =

1L − 1

L∑n=1

[y(n) − β.p(n)

]2. (4.8)


To access the effectiveness of the proposed SPM-MAP method against the presented SPM-GLM,synthetic 1D-block-designed single stimulus data (see example in Fig. 4.4), with several AWGNnoise levels (σy) and several stimulus epochs (periods in a block design paradigm approach), N , areused in Monte Carlo simulations.

Figure 4.4: Example of synthetic data used for SPM-MAP and SPM-GLM Pe Monte Carlo testswith noise level of σy = 0.5 and 5 epochs, N = 5. Paradigm in blue, noisy BOLD data in red andBOLD data without noise in green.

On this data, the error probability (Pe), was obtained for each method as follows

Pe(σ,N) =1R

R∑i=1

|bi − bi| (4.9)

where R = 250 is the number of data repetitions used in the Monte Carlo tests. The HRF functionis assumed to be known and was selected from the PBH model estimation on real data [17].

The resulting Pe differences between SPM-MAP and SPM-GLM, i.e.: ∆Pe = Pe(SPM−MAP ) −Pe(SPM−GLM), was computed for each experiment, and the average results for the different noiselevels and epochs are displayed in Fig.4.5 and in table 4.1. Notice that in table 4.1, σy values thatresulted in all null Pe are not displayed.

Although the performance of both algorithms decreases, as expected, with the amount of noiseand with the decrease in epochs number, the ∆Pe obtained is negative for most of the (N,σ)data pairs tested, which means that the SPM-MAP outperforms the SPM-GLM method for al-most every configuration tested. This is confirmed by Table 4.1, where the summation of ∆Pe,∑

i,j ∆Pe(Ni, σj) = −0.46, is negative. The number of negative values of ∆Pe, #(∆Pe < 0) = 23and the number of positive values of ∆Pe, #(∆Pe > 0) = 6, which confirms that the SPM-MAPsurpasses the traditional SPM-GLM method. Still it is obvious that both methods present high

4.2. SPM-MAP ACTIVITY DETECTION PART V.S. SPM-GLM ON SYNTHETIC DATA 43

accuracy in these tests due to the exact knowledge of both the HRF and noise distribution.It is important to notice that in the SPM-GLM, approach the HRF is indirectly estimated and

used for every time-course activation analysis. On the contrary, in the SPM-MAP method, this HRFis jointly estimated with the activation variables and is space variant. The SPM-MAP is thereforeadaptative, which allows the proposition that in real conditions the performance gap should increase.

Figure 4.5: Error probability differences, ∆Pe, between the SPM-MAP and the SPM-GLM methodsfor two different graphic views. N is the number of paradigm epochs and σ is AWGN standarddeviation level.


N\σy 1 2 4

2 -0.012 0 -0.1683 -0.012 0.004 -0.1084 -0.008 -0.012 -0.0445 -0.004 -0.012 -0.0326 0 -0.008 -0.0167 0 -0.008 -0.0088 0 -0.004 09 0 0 -0.00410 0 0 0.00411 0 -0.004 0.01212 0 0 0.00413 0 -0.004 014 0 -0.004 015 0 -0.004 0.00416 0 0 -0.00417 0 0 018 0 0 0.00419 0 0 -0.00420 0 0 -0.008

Table 4.1: ∆Pe = Pe(SPM−MAP ) − Pe(SPM−GLM) for 2 ≤ N ≤ 20 and σy = {1, 2, 4}. For all othertested values of σy = {0.01, 0.02, 0.05, 0.1, 0.2, 0.5}, Pe = 0

4.3 SPM-MAP Results on Synthetic Data

4.3.1 Activity Detection

In this section Monte Carlo tests of the complete proposed method are presented in order to evaluatethe performance of the algorithm. Two synthetic binary images of 128x128 pixels where generated,which represent a single BOLD slice signal, as can be seen overlapped in Fig. 4.6. In it, coloredvoxels (red, yellow and white) where activated by, at least, a stimulus paradigm and the blackpixels where not activated at all. So according to the mathematical notation presented above, red:b = {1, 0}T ; yellow: b = {0, 1}T ; white: b = {1, 1}T and black: b = {0, 0}T , which is the activationground truth to be estimated for each voxel.

The BOLD signal, y(n), is generated by using the model presented in Fig. 3.2. A reasonabletwo stimuli block-design paradigm, p1(n) and p2(n), of 10 seconds task duration followed by a 30second rest period each in 5 epochs, were used in order to obtain a non superposition of p1(n) andp2(n) while allowing for the BOLD signal to decay to rest. The true impulse HRF signal, h(n), wasgenerated from a representative IIR, selected from the PBH estimation on real single-event data[17], and the following noise energies were used: σy = {0.2, 0.5, 0.7, 0.8, 1}. These noise energiesare better evaluated when compared against the BOLD signal energy level, which is done with thesignal-to-noise ratio (SNR = 10 log

∑N1

[(β1×p1(n)+β2×p2(n))∗h(n)]2

σ2y

) for the two data cases in which

4.3. SPM-MAP RESULTS ON SYNTHETIC DATA 45

Figure 4.6: Synthetic image, used on Monte Carlo tests, representing a single BOLD data slicewith brain areas activated by two paradigms (red and yellow), with a functional overlapping region(white) and non activated areas (black).

the BOLD signal is present:∑2

k=1 b(k) ≥ 1 (see Table 4.2 ).This generated synthetic data is equivalent to 2 × 128 × 128 = 32768 independent y(n) time-

courses, containing all possible combinations for the b vector. These are used on Monte Carlo teststo compute the Pe (4.9). The results obtained are graphically presented in Fig. 4.7 and in Table4.2). These values were computed as the ratio of the total number of wrong estimations over thetotal number of Monte Carlo tests (32768).

σy 0.2 0.5 0.7 0.8 1SNR(dB) 7.3;11 -0.63;2.6 -3.6;-0.37 -4.7;-1.5 -6.7;-3.5

Pe(%) 0.0427 0.0916 0.168 0.260 4.27

Pe(%) 0 0 0.0244 0.0245 1.51Pe(%) 0 0 0.0073 0.0061 0.513

Table 4.2: Monte Carlo Pe (4.9) of SPM-MAP for several values of AWGN σy and correspondentSNR values for the

∑2k=1 b(k) = 1 and

∑2k=1 b(k) = 2 two different signal energy situations.

Spatial correlation correction is exemplified in Pe and Pe where one and two isolated pixels weredismissed, respectively.

Table 4.2 shows that even for the high noise levels of σy = 0.8 the proposed SMP-MAP methodpresents a Pe < 0.3%. It is important to point-out that although the SNR in MRI depends on alarge number of variables, it is usually more than 1dB [11,43]. So the most realistic σy values wouldbe situated between 0.2 and 0.5, for the data used. In this range the method achieves values ofPe < 0.1%. Furthermore, for the very high noise amount of σy = 1 (SNR = [−6.7;−3.5]) the Pe

stays below 5%, resulting in the right image in Fig. 4.7. Notice that when looking at Fig. 4.7, theintuitive notion on the error probability might seem higher than the refered 0.1% and 0.5% values,due to the fact that the images are actualy a overlap of two images.

It is intuitive when looking at the results in figure 4.7, that the accuracy of the method can be


Figure 4.7: SPM-MAP activation detection results for σy = 0.5 (left) and σy = 1 (right). Samecolor code as in Fig. 4.6

improved if spacial correlation information is included, removing several of those isolated, spatiallyuncorrelated, voxels. For illustration purposes, the Pe is recalculated after removing areas of one(Pe) and two (Pe) isolated voxels in an 8 voxels neighborhood. The resultant error probabilities(see Table 4.2) decreases for all the noise amounts, yielding null for the 0.2 and 0.5 σy values. Thegain in performance with this simple morphological correction applied, is unquestionable in thisexample because there are no small regions. If real regions of 1 or 2 voxels existed in the data, thentheir elimination would provide false-negative errors. Therefore, the inclusion of spatial correlationbetween voxels should follow a more robust methodology.

The Pe results cannot be directly compared with the results in section 4.2, because there, the HRFis exactly known and uses only one stimulus in the paradigm. Here, the HRF is jointly estimatedwith the brain activation indicators, βk, and the paradigm is comprised of two stimulus. Theseaspects greatly influence the Pe due to an explosion in the possible number of solutions, particularlyif we consider the great degree of freedom in the HRF estimation (as mentioned in section 3.2.3). Inboth tests, the exact noise distribution (AWGN) knowledge is an advantage that does not exist inreal data. Adding to this, the existence of other factors like field inhomogeneity and motion errors,the Pe on real data should be higher. Unfortunately, ground truth in real fMRI is difficult to obtain.Only rough expectations based on the prior knowledge of which brain areas should be activated bythe applied stimulus, may be used to validate the results.

4.3.2 HRF Estimation

The HRF estimation results are harder to analyze. For each one of the 32768 voxels a h(n) HRFis estimated, but only the ones corresponding to activated brain areas βk = 1 are relevant. So, inthe false-negative case, that information is discarded. On the other hand, the false-positive case isnot discarded and the estimated h(n) tends to follow the random AWGN form, around zero. Thesereduce the amplitude of the HRF estimated mean, computed over all positive estimated voxels βk,

4.4. SPM-MAP RESULTS ON REAL DATA 47

including false-positives. This effect can be seen in Fig. 4.8, where the false-positives effect isremoved. Considering that in real data we do not have information on false-positives, a correctioncould be done dismissing estimated h(n) functions with AWGN distribution, but since the primarygoal of this work is the activation detection, this is left for further works. Furthermore there is aglobal decrease in amplitude of every HRF estimation in the FIR → IIR (section 3.2.3) projectionoperation, which can be seen on figure 4.8 right column. In fact, when there is not an IIR filter thatperfectly describes the estimated g(n) FIR, the IIR computed by the Shanks [35,36] algorithm isalways of lower amplitude. In spite of all this, the HRF estimation provided reasonable results (Fig.4.8) and proved robust even in high noise levels.

Figure 4.8: Mean HRF estimation results considering all βk �= 0 estimations (left) and for only thebk �= 0 correct ones (right), for σy = {0.05, 0.5, 1} from top to down. The Real HRF used for datageneration is in green, the estimated FIR average in red, and the estimated IIR average in blue.

4.4 SPM-MAP Results on Real Data

Subjects and Data Collection

Three volunteers with no history of neurological or psychiatric diseases participated on stimulatedverbal, motor and trajectory activity during fMRI data acquisition on a Philips Intera AchievaQuasar Dual 3T whole-body system with a 8 channel head-coil. T2*-weighted echo-planar images(EPI) 23cm square field of view with a 128 × 128 matrix size resulting in an in-plane resolutionof 1.8 × 1.8mm for each 4mm slice, echo time = 33ms, flip angle = 20o were acquired with aTR = 3000ms.

Three motor paradigms, one verb generation paradigm and one trajectory generation paradigm,summing up to the total of five paradigms, described in the following table, were applied to thesubjects.


(a) Verbs Stimulus: Seeing nouns and thinking of related verbs.Baseline: Seeing the "#####" string.

(b) Trajectories Stimulus: Reminding routes through familiar places.Baseline: Silently counting numbers.

(c) Right Foot Stimulus: Moving right foot fingers.Baseline: Complete rest with closed eyes.

(d) Right Hand Stimulus: Closing/opening right hand.Baseline: Complete rest with closed eyes.

(e) Hands Stimulus: Closing/opening both hands.Baseline: Complete rest with closed eyes.

Table 4.3: Description of the paradigms conditions applied to the participant subjects.

These paradigms were all structured on the same block-design, with 20 samples per epoch (mean-ing 10 samples of stimulus following 10 samples of baseline, summing up to 60s time per epoch) and4 total epochs.

Data Analysis and Results

The fMRI data was preprocessed with the standard procedures implemented in the BrainVoyagersoftware [14], namely decrease of data distortions due to motion or other phase changes over time(registration) and spacial smoothing. This data was then statistically processed by the BrainVoyagerSPM-GLM and SPM-MAP algorithms, and the resolving results are plotted in Fig. 4.9 for a selectionof slices that have considerable activity areas in them. Since the obtainable brain maps by SPM-GLM highly depends on the selected p-values (see section 4.2), a neurologist provided the results,for each data set, which he considered more correct (reference result), based on its experience. Sincethis result is subjective, he also provided two other results which we considered loose and restricted.

Visual inspection of the results in Fig. 4.9 show some expected resemblance between the rea-sonable SPM-GLM brain maps, selected by the neurologist, and the brain maps obtained by theproposed SPM-MAP algorithm. Still the compared algorithms are methodologically different. First,SPM-MAP does not rely on the null hypothesis for activity detection as SPM-GLM, but rather con-siders two hypothesis (the null and the alternative) and so is able to answer different classes ofquestions not handled by the standard fMRI analysis. Second, SPM-MAP jointly estimates theHRF shape and the associated activity brain map, and therefore, should theoretically be more pre-cise in its activity results. These main differences should account for most of the dissimilaritiesobserved in each image set (identified as a, b, c, d and e on Fig. 4.9) between the algorithms results.

In several of the brain map image sets obtained, there are brain regions detected as activated bySPM-MAP that were not detected by SPM-GLM. These are unlikely false positives. Consideringthat the error probability has been shown considerably low (see section 4.3.1), the probability ofseveral false positives occurring grouped in a small image area, instead of randomly dispersed in theimage, is infinitesimally small. Therefore, new brain areas detected by the proposed method should


be considered reliable in most cases, depending on its size (number of voxel in that group). Thedetection of these "new" areas happens when a voxel time-course does fluctuates in correlation withthe stimulus paradigm, but with an HRF of deviant shape from the EVs considered in the GLM (seeexamples in Fig. 4.10). Still, there are possibly some false positive regions (e.g. activated groups ofonly one or two voxels) in the SPM-MAP results that have been taken care of by means of clusteringin the SPM-GLM results by the BrainVoyager software. Further developments of SPM-MAP areplanned to include spatial correlation priors to correct this handicap and improve results.

To enhance confidence in the the "new" activated brain regions detected by the proposed algo-rithm, a closer neurological functional analysis should be done. This would involve a neurologistcareful evaluation, on each case, following a spatial transformation of the SPM-MAP brain mapsinto a standard mapped space like the Talairach stereotaxic coordinates [11]. Although this wouldinvolve another study, it is noticeable that many of these "new" areas are located in the frontalcortex, which is the focus of the conscious activity. We suspect that these may well be involved inperforming the respective paradigm tasks, since all of the paradigm stimulus applied to the subjectshave a strong conscious influence (see Tab. 4.3).


(a) Verbs generation paradigm results

(b) Trajectory generation paradigm results

(c) Right foot paradigm results

(d) Right hand paradigm results

(e) Bilateral hands paradigm results

Figure 4.9: SPM-MAP activity detection results on real data compared against data processed bya neurologist on SPM-GLM. The first image on each image-set ((a), (b), (c), (d) and (e)) is theSPM-MAP result. The following three images are the SPM-GLM results with left to right increasein restriction on the p-value, where the middle image was intuitively set as the reference result.The background fMRI image in the SPM-MAP results is the smoothed data used for the statisticalanalysis, and in SPM-GLM is the unsmoothed data.


Figure 4.10: Example of two voxel time-courses, from two different brain areas, detected as activatedby the SPM-MAP that were not detected by the SPM-GLM. Binary paradigm is in red and thevoxel time-course in blue.

Chapter 5

Conclusions and Future Work

The characterization of brain regions from a functional point of view can be performed by usingBOLD contrast fMRI technology. In order to do so, statistical data analysis is needed to extractreliable information from the noisy and low amplitude BOLD signal. In this thesis a new Bayesianmethod is presented, where the neural activity detection is jointly obtained along with the HRFestimation. This approach presents two main advantages: 1) The activity detection benefits from theadaptative nature of the HRF shape estimation; 2) It provides local, space variant, HRF estimation.This might provide a useful tool to evaluate and compare estimated brain activity between regionsor subjects and for possible behavioral [44], neural and vascular local consideration.

In this thesis we propose a linear model, called PBH model, that describes the BOLD signalchange after the application of an external stimulus, based on the a priori knowledge and reasonableassumptions of the physiologic behavior of the vascular and neural tissues. The neuron oxygenconsumption, the vascular response demand induced by brain activity and the vascular responseprocess itself are modeled by first order linear systems. The systemic vasodilatation control of thevessels is modeled by a simple negative-feedback proportional controller.

A representative selection of twelve experimental curves, out of eighty, are presented, upon whichthe corresponding model is estimated. The model parameters are obtained using the minimum squareerror (MSE) method, where the square error between the model response and the observations isminimized, resulting in good shape modeling. Additionally, the true stimulus, which is also unknown,is also estimated and displayed. Physiological considerations on the tissues were the model is adjustedmight be done, but in-vivo studies must be performed to further validate this information. Also,building up the PBH model by adding blocks for more physiological events might be considered but,the consequential increase in complexity and number of parameters, could deviate the PBH modelfrom its original purpose (see chapter 2), into the complex physiological approach (e.g. the BalloonModel [25]).

In the joint Bayesian method presented, noisy observations are modeled by the AWGN modeland the stimulus activation indicators are modeled by binary variables that are estimated. The prior

53

54 CHAPTER 5. CONCLUSIONS AND FUTURE WORK

associated with the binary indicators is a bimodal Gaussian distribution around the 0 and 1 valuesto cope with the uncertainty related to data noise. Notice that the neuron activity is all-or-nothing,but within a voxel some neurons might be activated and others non-activated. Hence, this binaryactivity is a reasonable but limited constrain. The HRF is adaptively estimated on the restrictedspace responses of the adopted IIR PBH model [17].

The neural activity detection section of the proposed method is compared with the standardmethod proposed in [1] that uses a classical statistical inference methodology based on the t-testmethod. Monte Carlo tests on synthetic-1D-block-designed data with both methods have shown thatthe proposed Bayesian method, called SPM-MAP, outperforms the classical one based on the generallinear model, here called SPM-GLM. The performance evaluation was based on the computation ofthe error probability, Pe(N,σy), for each method which proved to be smaller for the proposed SPM-MAP method than the corresponding ones obtained with the SPM-GLM method, for almost everytested conditions: different noise levels, σy and number of paradigm epochs, Ni, in a block designframework.

On Monte Carlo tests of the whole method with, again, synthetic data, the Pe obtained waslower than 0.1% for noise levels close to expected real ones. But even for extreme noise levels themethod showed itself considerably reliable. This Pe should increase on real data, but consideringits low values, a prominent increase is not expected. Preliminary results after a very simple spatialcorrelation on the activity detection results are presented and foretells some achievable accuracyimprovement. In fact, future developments of the SPM-MAP are planned to include priors on voxellocation within the brain. This is, the consideration that a voxel has higher activation probabilityif his neighbors are activated and has lower activation probability if he is surrounded by non-activated neighbors. Regarding the HRF estimation, the average results showed a robust-to-noiseclose similarity between the real synthetic data HRF and the estimated HRF, although there is ajustified small bias towards an amplitude reduction.

The real fMRI data tests showed that the SPM-MAP was able to detect most of the activitydetected by SPM-GLM and also detect other activated brain regions. Further analysis of these newbrain areas will be done in future works.

In the end we presented and a Bayesian method for statistical analysis and inference of fMRIdata with a new perspective and reliable results validated on synthetic and real data. This methodis parameter-free presented (if the noise distribution is know) while many methods are parameter-dependent, like the p-value dependant SPM-GLM standard algorithm. This may be very useful forstandalone analysis or for comparison against results from other methods.

Bibliography

[1] K. J. Friston, “Analyzing brain images: Principles and overview,” in Human Brain Function,R.S.J. Frackowiak and K.J. Friston and C. Frith and R. Dolan and J.C. Mazziotta, Ed. Aca-demic Press USA, 1997, pp. 25–41.

[2] S. L. Z. Nicholas Lange, “Non-linear fourier time series analysis for human brain mapping byfunctional magnetic resonance imaging.” Applied Statistics, vol. 46, no. 1, pp. 1–29, 1997.

[3] K. J. Friston, P. Fletcher, O. Josephs, A. Holmes, M. D. Rugg, and R. Turner, “Event-relatedfMRI: characterizing differential responses,” Neuroimage, vol. 7, no. 1, pp. 30–40, Jan 1998.

[4] “Functional mri research center, columbia university.” [Online]. Available: http://www.fmri.org/

[5] S. Ogawa, T. M. Lee, A. R. Kay, and D. W. Tank, “Brain magnetic resonance imaging withcontrast dependent on blood oxygenation,” Proc Natl Acad Sci U S A, vol. 87, no. 24, pp.9868–9872, Dec 1990.

[6] S. Ogawa, D. W. Tank, R. Menon, J. M. Ellermann, S. G. Kim, H. Merkle, and K. Ugurbil,“Intrinsic signal changes accompanying sensory stimulation: functional brain mapping withmagnetic resonance imaging,” Proc Natl Acad Sci U S A, vol. 89, no. 13, pp. 5951–5955, Jul1992.

[7] P. A. Bandettini, E. C. Wong, R. S. Hinks, R. S. Tikofsky, and J. S. Hyde, “Time course EPIof human brain function during task activation,” Magn Reson Med, vol. 25, no. 2, pp. 390–397,Jun 1992.

[8] S. Ogawa, R. S. Menon, D. W. Tank, S. G. Kim, H. Merkle, J. M. Ellermann, and K. Ugurbil,“Functional brain mapping by blood oxygenation level-dependent contrast magnetic resonanceimaging. A comparison of signal characteristics with a biophysical model,” Biophys J, vol. 64,no. 3, pp. 803–812, Mar 1993, comparative Study.

[9] R. Zimmer, R. Lang, and G. Oberdorster, “Post-ischemic reactive hyperemia of the isolatedperfused brain of the dog,” Pflugers Arch, vol. 328, no. 4, pp. 332–343, 1971.

55

56 BIBLIOGRAPHY

[10] J. K. Gourley and D. D. Heistad, “Characteristics of reactive hyperemia in the cerebral circu-lation,” Am J Physiol, vol. 246, no. 1 Pt 2, pp. 52–58, Jan 1984.

[11] P. Jezzard, P. M. Matthews, and S. M. Smith, Functional magnetic resonance imaging: Anintroduction to methods. Oxford Medical Publications, 2006.

[12] “Statistical parametric mapping software.” [Online]. Available:http://www.fil.ion.ucl.ac.uk/spm/

[13] “Fmrib software library.” [Online]. Available: http://www.fmrib.ox.ac.uk/fsl/

[14] “Brainvoyager software.” [Online]. Available: http://www.brainvoyager.com/

[15] R. Baumgartner, L. Ryner, W. Richter, R. Summers, M. Jarmasz, and R. Somorjai, “Compari-son of two exploratory data analysis methods for fMRI: fuzzy clustering vs. principal componentanalysis,” Magn Reson Imaging, vol. 18, no. 1, pp. 89–94, Jan 2000, comparative Study.

[16] F. Årup Nielsen, “Bibliography on independent component analysis in functional neuroimag-ing,” 2007. [Online]. Available: http://www2.imm.dtu.dk/ ˜ fn/bib/Nielsen2001BibICA/Nielsen2001BibICA.html

[17] D. M. Afonso, J. Sanches, and M. H. Lauterbach, “Neural physiological modeling towards ahemodynamic response function for fMRI,” Conf Proc IEEE Eng Med Biol Soc, vol. 1, pp.1615–1618, 2007, jOURNAL ARTICLE.

[18] J. Sanches, D. Afonso, K. Bartnykas, and M. Lauterbach, “Robust bayesian brain activitydetection in fmri,” Apr 2008.

[19] D. Afonso, J. Sanches, and M. Lauterbach, “Joint bayesian detection of brain activated regionsand local hrf estimation in functional mri,” Apr 2008.

[20] J. Sanches, D. Afonso, K. Bartnykas, and M. Lauterbach, “Joint bayesian detection of brainactivated regions and hemodynamic response estimation in functional mri,” Nov 2008.

[21] G. K. Aguirre, E. Zarahn, and M. D’esposito, “The variability of human, BOLD hemodynamicresponses,” Neuroimage, vol. 8, no. 4, pp. 360–369, Nov 1998, clinical Trial.

[22] D. A. Handwerker, J. M. Ollinger, and M. D’Esposito, “Variation of BOLD hemodynamicresponses across subjects and brain regions and their effects on statistical analyses,” Neuroimage,vol. 21, no. 4, pp. 1639–1651, Apr 2004.

[23] G. M. Boynton, S. A. Engel, G. H. Glover, and D. J. Heeger, “Linear systems analysis offunctional magnetic resonance imaging in human V1,” J Neurosci, vol. 16, no. 13, pp. 4207–4221, Jul 1996.

BIBLIOGRAPHY 57

[24] J. C. Rajapakse, F. Kruggel, J. M. Maisog, and D. Y. von Cramon, “Modeling hemodynamicresponse for analysis of functional MRI time-series,” Hum Brain Mapp, vol. 6, no. 4, pp. 283–300, 1998.

[25] R. B. Buxton, E. C. Wong, and L. R. Frank, “Dynamics of blood flow and oxygenation changesduring brain activation: the balloon model,” Magn Reson Med, vol. 39, no. 6, pp. 855–864, Jun1998.

[26] K. J. Friston, A. Mechelli, R. Turner, and C. J. Price, “Nonlinear responses in fMRI: TheBalloon model, Volterra kernels and other hemodynamics,” NeuroImage, vol. 12, pp. 466–477,2000.

[27] D. Malonek and A. Grinvald, “Interactions between electrical activity and cortical microcircu-lation revealed by imaging spectroscopy: implications for functional brain mapping,” Science,vol. 272, no. 5261, pp. 551–554, Apr 1996.

[28] H. Lu, X. Golay, J. J. Pekar, and P. C. M. van Zijl, “Sustained poststimulus elevation incerebral oxygen utilization after vascular recovery,” J Cereb Blood Flow Metab, vol. 24, no. 7,pp. 764–770, Jul 2004.

[29] K. M. Hanson, “Introduction to Bayesian image analysis,” Medical imaging VII: image process-ing (ed. M. H. Loew), Proc. SPIE, vol. 1898, no. 8, pp. 716–731, 1993.

[30] A. M. Wink and J. B. Roerdink, “Denoising functional MR images: a comparison of waveletdenoising and Gaussian smoothing,” IEEE Trans Med Imaging, vol. 23, no. 3, pp. 374–387, Mar2004, comparative Study.

[31] H. Gudbjartsson and H. S. Patz, “The Rician distribution of noisy MRI data,” Magn ResonMed, vol. 34, no. 6, pp. 910–914, Dec 1995.

[32] Z. Q. Wu, A. Ware, and J. Jiang, “Wavelet-based rayleigh background removal in mri,” pp.603– 604, 2003.

[33] N. Vucic and H. Boche, “Equalization for MIMO ISI Systems using Channel Inversion underCausality, Stability and Robustness Constraints,” in Proc. IEEE International Conference onAcoustics, Speech, and Signal Processing (ICASSP 2006), Toulouse, France, May 2006.

[34] S. S. Rao and A. Ramasubrahmanyan, “Evolving iir approximants for fir digital filters,” inASILOMAR ’95: Proceedings of the 29th Asilomar Conference on Signals, Systems and Com-puters (2-Volume Set). Washington, DC, USA: IEEE Computer Society, 1995, p. 976.

[35] John L. Shanks, “Recursion filters for digital processing,” Geophysics, vol. 32, pp. 33–51, 1967.

[36] M. H. Hayes, Statistical Digital Signal Processing and Modeling. Wiley, March 1996.

58 BIBLIOGRAPHY

[37] G. Baron de Prony, “Essai expérimental et analytique sur les lois de la Dilatabilité des uidesélas-tique et sur celles de la Force expansive de la vapeur de l’eau et de la vapeur de l’alkool,ádiérentes températures,” Jour. de L’Ecole Polytechnique, vol. 1, no. 1, pp. 24–76, 1795.

[38] E. Rignot and R. Chelappa, “Segmentation of polarimetric sunthetic aperture radar data,” IEEETrans. Image Processing, vol. 1, no. 1, pp. 281–300, 1992.

[39] T. K. Moon and W. C. Stirling, Mathematical methods and algorithms for signal processing,2000.

[40] S. Geman and D. Geman, “Stochastic relaxation, gibbs distributions, and the bayesian restora-tion of images,” pp. 564–584, 1987.

[41] N. Kutner and N. Wasserman and C. J. Nachtsheim and W. Wasserman, Applied Linear Sta-tistical Models. The McGraw-Hill Companies, Inc., 1996.

[42] G. H. Glover, “Deconvolution of impulse response in event-related BOLD fMRI,” Neuroimage,vol. 9, no. 4, pp. 416–429, Apr 1999, clinical Trial.

[43] ——, “On signal to noise ratio tradeoffs in fmri,” Apr 1999.

[44] M. Fukunaga, S. G. Horovitz, P. van Gelderen, J. A. de Zwart, J. M. Jansma, V. N. Ikonomidou,R. Chu, R. H. R. Deckers, D. A. Leopold, and J. H. Duyn, “Large-amplitude, spatially correlatedfluctuations in BOLD fMRI signals during extended rest and early sleep stages,” Magn ResonImaging, vol. 24, no. 8, pp. 979–992, Oct 2006.

Neural physiological modeling towards a hemodynamic response functionfor fMRI

David M. Afonso Joao M. Sanches Martin H. Lauterbach (MD)[email protected] [email protected] [email protected]

Instituto de Sistemas e Robotica - Instituto Superior Tecnico

Abstract— The BOLD signal provided by the functionalMRI medical modality measures the ratio of oxy- to deoxy-haemoglobin at each location inside the brain. The detection ofactivated regions upon the application of an external stimulus,e.g., visual or auditive, is based on the comparison of thementioned ratios of a rest condition (pre-stimulus) and of astimulated condition (post-stimulus). Therefore, an accurateknowledge of the impulse response of the BOLD signal to neuralstimulus in a given region is needed to design robust detectorsthat discriminate, with a high level of confidence activatedfrom non activated regions. Usually, in the literature, thehemodynamic response has been modeled by known functions,e.g., gamma functions, fitting them, or not, to the experimentaldata. In this paper we present a different approach based onthe physiologic behavior of the vascular and neural tissues.

Here, a linear model based on reasonable physiologicalassumptions about oxygen consumption and vasodilatationprocesses are used to design a linear model from which atransfer function is derived. The estimation of the modelparameters is performed by using the minimum square error(MSE) by forcing the adjustment of the stimulus response tothe observations.

Experimental results using real data have shown that theproposed model successfully explains the observations allowingto achieve small values for the fitting error.

I. INTRODUCTION

During functional magnetic resonance imaging (fMRI), a brieffocal neural activation evokes what is called a hemodynamic time-course response function (HRF) that mainly depends on tissuephysiology [1]. Although variability exist on who, where, whenand how the data was acquired and processed [2,3], standard HRFestimates are often an essential basis of fMRI analysis. Severalassumptions are usually made, namely that all neural impulseevents produce the same HRF (assuming minimal variability acrosssubject, brain region and acquisition system) and that a time seriesdata is modeled as an impulse train of neural events convolvedwith this invariant HRF [4]. Even though some evidences haveshown these assumptions to be erroneous, its impact on manyfMRI statistical analysis studies has not been considered relevantenough to generally abandon the simplification advantages. In factthe most commonly used HRF are parametric analytical functions,namely gamma functions [5]–[7], and to a less extent Poisson orGaussian distributions [8], that satisfactorily modulate the ratherinvariant form of the blood-oxygen-level dependent (BOLD) time-signal to short stimuli. Still, the accuracy and relative superiority ofthese HRF models cannot be entirely questioned because of theiranalytical nature, without incorporation of the slightest knowledge

This work was supported by Fundacao para a Ciencia e a Tecnologia(ISR/IST plurianual funding) through the POS Conhecimento Programwhich includes FEDER funds. This work was done in partial collaborationwith the Hospital da Cruz Vermelha de Lisboa and the fMRI group of theHospital de Santa Maria.

of the physiological processes behind the BOLD signal itself. Andthough the exact coupling between brain activity, vascular responseand cerebral metabolic oxygen rate that leads to the BOLD responseare not well understood, there are several experimental evidencesthat lead to conclusions and assumptions that give us a sketchily,but important, view of the BOLD physiological features.

It has been shown that an increase in cerebral neuronal activitygenerally leads to co-localized increases in cerebral metabolic rateof oxygen (CMRO2), followed by a much larger co-localizedincrease in local cerebral blood flow (CBF) and volume (CBV).These effects are consequence of energy consumption by neuraland glial brain cells, leading to an increased ratio of oxy- to de-oxyhaemoglobin (for which blood-oxygenation-level is the obviouscomplement under normal conditions) in the vessels, capillariesand surrounding tissues. Particularly, this energy consumption isputatively accounted for neuron synapse activity, hence the assumedrelation between the BOLD signal and neural activity. However,the BOLD time course signal has several transient features atthe onset and end of the stimulus: an initial dip and a post-stimulus undershoot; that are not explained by the coupling offlow and metabolism referred above. The initial dip, correspondingto an increase in local deoxyhaemoglobin, has been interpretedas evidence for an initial increase in oxygen extraction beforeflow increase; and the post-stimulus undershoot as an elevatedoxygen extraction after the flow has returned to baseline. Thesemodels for the transient features based on uncoupling, coupling andreuncoupling of vascular response and CMRO2 provide a, possiblyrough but valuable, key element in generating a physiological modelthat evokes an output similar to the HRF. In fact these interpretationshave been recently reinforced by multimodal fMRI studies [9].Buxton et.al. [10] have done such a task, resulting in one of themost interesting physiological models in the area, although thereis some strong skepticism on some of its base assumptions [9].Still, this model is currently being extended and perfected, but hasnot had much practical application contrary to the proposed gammafunction HRF’s. This is most probably due to two important featuresof the latter: simplicity and computational efficiency.

In this paper we propose a simple linear model of the hemody-namic response function based on a modulation of basic physio-logical processes behind the BOLD signal, with the main objectiveof being used as basis for fRMI activation mapping statisticalanalysis. This has been done considering uncoupling, couplingand reuncoupling processes of the vascular response and CMRO2

variables and accounting for neural demand and systemic feedbackcontrol of vascular response. The final results were tested on realdata fMRI BOLD signal time-courses. A final Z-transform functionis presented, which has obvious frequency analysis advantagesproviding computational processing efficiency.

II. MODEL PRESENTATION

Brain activation is accompanied by a series of physiologic alter-ations, including focal changes in the vascular response (cerebralbrain flow and blood volume), blood oxygenation and cerebral

Proceedings of the 29th Annual InternationalConference of the IEEE EMBSCité Internationale, Lyon, FranceAugust 23-26, 2007.

FrA03.6

1-4244-0788-5/07/$20.00 ©2007 IEEE 1615

oxygen consumption. We have tried to encapsulate all of thesephysiological variables in a simple based linear discrete modelthat would translate their effects on a HRF estimation (see Fig.1).The physiologically based hemodynamic (PBH) model input isthe neural activation, r(n) and the output is the BOLD contrastsignal, y(n). The reference value Ref is the baseline of the vascularproperties. Since this work is focused on incremental variations ofall its variables, this parameter is constantly null. All model blocksare zero and first order linear functions, which provide empiricallyreasonable approximations. Three main groups can be distinguishedin the PBH model: a brain group which modulates the neural andglial cells oxygen consumption (CMRO2) and vascular responsedemands; a vessel group which modulates the summed effect ofCBV and CBF vascular changes on the rate of deoxyhaemoglobinconcentration in and around blood vessels; a control group for thesystemic negative feedback control over vasodilatation.

Fig. 1. Block diagram of the proposed physiologically based hemodynamic(PBH) model behind the HRF on BOLD fMRI data. It incorporates vascularresponse demand and oxygen metabolism consumption by brain tissue,vascular response producing changes in both CBV and CBF and systemicnegative feedback control system of the vascular response. Baseline vascularproperties and neural activation stimulus are considered in the Ref and r(n)inputs respectively.

The PBH model was developed upon the fundamental consider-ation of separate dynamics between CMRO2 and vascular responsefeatures (CBV and CBF). This means that the transient initialdip and poststimulus undershoot of the HRF are then modeled asan uncoupling of these features, where the negative influence ofCMRO2 to the BOLD signal is not counterbalanced by the briefervascular response. This has been an old assumption [1] that hasrecently gained strength through multimodal fMRI studies [9]. Theuncoupling considered indicates that the vascular response demandis probably not due to the oxygen metabolism pathway, hence theseparation of both blocks of the PBH models brain group. Still,they are both a consequence of the brain activity.

The dynamics of the actual vascular response are accounted inthe vessel group block function. This deals with the reasonableassumption that, upon tissue demand for more blood delivery, thevascular response to this request is delayed and constrained. Besidesthese aspects, the gain in this block is also responsible for therelation between the vascular response features and the amplitudeeffect they cause on the BOLD signal. On the other hand, vascularresponse demand by brain tissues is most likely bigger than theirneeds, and alongside the vascular answering dynamic referred, thereis a systemic vascular response control modeled in the control groupthat reduces the amplitude of the vascular response. Notice againthat the gain of this block reflect the amplitude impact on the BOLD

signal of the vasodilatation control.

III. MODEL ESTIMATION

The transfer function of the discrete time PBH model displayedin Fig.1 is

H(z) =(AV − B) + (a + v − bAV )z−1 − (avB)z−2

(1 − az−1)(1 − bz−1)(1 + C(s)V − vz−1)(1)

which can re rewritten as follows

H(z) =Y (z)

R(z)=

b0 + b1z−1 + b2z−2

1 + a1z−1 + a2z−2 + a3z−3(2)

where

b0 =AV − B

1 + V C(s)(3)

b1 =a + v − bAV

1 + V C(s)(4)

b2 = − avB

1 + V C(s)(5)

a1 = −(a + b +v

1 + V C(s)) (6)

a2 = ab + (a + b)v

1 + V C(s)(7)

a3 = −abv

1 + V C(s)(8)

Assuming the simpler controller (more complex controllers willbe considered in the future), C(z) = K, the correspondentdifference equation is the following,

y(n) =2X

k=0

bkr(n − k) −3X

l=1

aly(n − l) (9)

The estimation of the parameters, bk e ak is performed with theminimum square error (MSE) method,

p = arg minp

E(Y,R, p) (10)

where Y = y(0), y(1), ..., y(N − 1) is the vector with the Nexperimental points, R = r(0), r(1), ..., r(N − 1) is the stim-ulus signal that is unknown and must also be estimated, p =[b0, b1, b2, a1, a2, a3] is the vector of parameter to be estimatedand E(Y, R, p) is the function to be minimized,

E(Y, R, p) =NX

n=0

"y(n) −

2Xk=0

bkr(n − k) +3X

l=1

akr(n − k)

#

which can be written as follows using matrix notation,

E(Y,R, p) = (Y − Φp)T (Y − Φp) (11)

where Φ = [ΦR

... − ΦY ] with

ΦR =

0BBBBBBBB@

r(0) 0 0r(1) r(0) 0r(2) r(1) r(0)r(3) r(2) r(1)r(4) r(3) r(2)... ... ...

r(N − 1) r(N − 2) r(N − 3)r(N) r(N − 1) r(N − 2)

1CCCCCCCCA

.

ΦY =

0BBBBBBBB@

0 0 0y(0) 0 0y(1) y(0) 0y(2) y(1) y(0)y(3) y(2) y(1)... ... ...

y(N − 2) y(N − 3) y(N − 4)y(N − 1) y(N − 2) y(N − 3)

1CCCCCCCCA

.

1616

The minimization of E(Y,R, p) is performed by finding itsstationary point, ∇pE(Y, R, p) = 0, with the following solution,

p =hΦT Φ

i−1

Φ| {z }Ψ

Y (12)

where Ψ is called pseudoinverse of Φ. The stimulus r(n) is notcompletely known and therefore must be estimated. It is assumedthat

r(n) =

(1 0 ≤ n ≤ n0

0 otherwise(13)

where n0 is unknown (notice that the data used in this paper [3]was previously aligned and shifted).

For each data set five values of n0 were tested, n0 = [M, ..., M−4] in which M is the time instant where the maximum of the ex-perimental data occurs, that is, the position of the larger maximum.The solution is obtained by choosing the set of parameters that leadto the minimum error,

[p, n0] = arg minp,n0

E(Y, R(n0), p) (14)

IV. EXPERIMENTAL RESULTS

To test the PBH model we used the same normalized data, asused by the authors of [3]. As such it might also provide a meansof comparison to their results and remarks. Note that the data wasacquired in four different brain areas of twenty-seven male subjectswith no history of neuronal or psychiatric diseases. T2*-weightedecho-planar images (EPI) were acquired at 4 Teslas, with variationsin the TR and time resolution. For a more complete description onmaterials and methods used please see [3].

The experimental results are organized in three sets: i) datawith poststimulus undershot (Fig.4), ii) data without poststimulusundershot (Fig.3) and iii) data that displays the initial dip (Fig.2).

For each experimental curve a set of modeled parameters wasestimated as well as the optimal stimulus duration, which isunknown. Unfortunately, we did not had access to the paradigminformation. But even if we did, we do not have direct access tothe real duration time of the neural activation in each data sample.

From the displayed results it is concluded that the PBH modelmanages to explain well, in a MSE basis, the experimental curves.This provides a considerable confidence for the base assumptionsupon which our PBH model is built. Note that many of theseexperimental data shapes rather deviate from the range of shapesthat the HRF gamma functions are able to produce. Conversely ourPBH model was successful in providing such form variability.

V. CONCLUSIONS

The characterization of brain regions from a functional pointof view can be performed by using BOLD contrast fMRI. Thistechnique uses the ratio of oxy- to deoxyhaemoglobin before andafter the application of an external stimulus, e.g. visual or auditive,to identify the activated regions. The design of robust and reliabledetectors that discriminate activated from non-activated regions,need accurate models for the HRF of the neural tissues. In theliterature, usually, this hemodynamic response is obtained by fittingthe experimental observations with known functions, e.g. gammafunctions.

In this paper we propose a linear model, called PBH model,that describes the BOLD signal change after the application of anexternal stimulus, based on the a priori knowledge and reasonableassumptions of the physiologic behavior of the vascular and neuraltissues. The neuron oxygen consumption, the vascular responsedemand induced by brain tissues and the vascular response processitself are modeled by first order linear systems. The systemic vasodi-latation control of the vessels is modeled by a simple proportionalcontroller.

Fig. 2. Results from the PBH model estimation with 4 experimental datawith initial dip. Red - real data; Blue - model; Black - stimulus.

Twelve experimental curves are presented and the correspondingmodel estimated. The model parameters are estimated using theminimum square error (MSE) method where the square errorbetween the model response and the observations is minimized.Additionally, the true stimulus, which is also unknown, is alsoestimated and displayed.

The PBH model successfully explains the observations and willbe incorporated in the activated region detector in the future.

VI. ACKNOWLEDGMENTS

We are very grateful to D.A. Handwerker et. al. for providingthe fMRI BOLD signal time course data they used on [3], whichwas essential to validate our model and enrich the quality of thepaper.

1617

Fig. 3. Results from the PBH model estimation results with 4 experimentaldata without undershoot. Red - real data; Blue - model; Black - stimulus.

REFERENCES

[1] P. Jezzard, P. M. Matthews, and S. M. Smith, Functional magneticresonance imaging: An introduction to methods. Oxford MedicalPublications, 2006.

[2] G. K. Aguirre, E. Zarahn, and M. D’esposito, “The variability ofhuman, BOLD hemodynamic responses,” Neuroimage, vol. 8, no. 4,pp. 360–369, Nov 1998, clinical Trial.

[3] D. A. Handwerker, J. M. Ollinger, and M. D’Esposito, “Variation ofBOLD hemodynamic responses across subjects and brain regions andtheir effects on statistical analyses,” Neuroimage, vol. 21, no. 4, pp.1639–1651, Apr 2004.

[4] G. M. Boynton, S. A. Engel, G. H. Glover, and D. J. Heeger, “Linearsystems analysis of functional magnetic resonance imaging in humanV1,” J Neurosci, vol. 16, no. 13, pp. 4207–4221, Jul 1996.

[5] S. L. Z. Nicholas Lange, “Non-linear fourier time series analysis forhuman brain mapping by functional magnetic resonance imaging.”Applied Statistics, vol. 46, no. 1, pp. 1–29, 1997.

Fig. 4. Results from the PBH model estimation results with 4 experimentaldata with undershoot. Red - real data; Blue - model; Black - stimulus.

[6] K. J. Friston, P. Fletcher, O. Josephs, A. Holmes, M. D. Rugg, andR. Turner, “Event-related fMRI: characterizing differential responses,”Neuroimage, vol. 7, no. 1, pp. 30–40, Jan 1998.

[7] M. A. Burock, R. L. Buckner, M. G. Woldorff, B. R. Rosen, andA. M. Dale, “Randomized event-related experimental designs allow forextremely rapid presentation rates using functional MRI,” Neuroreport,vol. 9, no. 16, pp. 3735–3739, Nov 1998.

[8] J. C. Rajapakse, F. Kruggel, J. M. Maisog, and D. Y. von Cramon,“Modeling hemodynamic response for analysis of functional MRItime-series,” Hum Brain Mapp, vol. 6, no. 4, pp. 283–300, 1998.

[9] H. Lu, X. Golay, J. J. Pekar, and P. C. M. Van Zijl, “Sustainedpoststimulus elevation in cerebral oxygen utilization after vascularrecovery,” J Cereb Blood Flow Metab, vol. 24, no. 7, pp. 764–770,Jul 2004.

[10] R. B. Buxton, E. C. Wong, and L. R. Frank, “Dynamics of blood flowand oxygenation changes during brain activation: the balloon model,”Magn Reson Med, vol. 39, no. 6, pp. 855–864, Jun 1998.

1618

ROBUST BAYESIAN BRAIN ACTIVITY DETECTION IN FMRI

João Sanches David Afonso Kestutis Bartnykas Martin H. Lauterbach (MD)

Instituto Superior Técnico / Instituto de Sistemas e Robótica, Lisbon, PortugalVilnius Gediminas Technical University, Vilnius, Lithuania

Faculty of Medicine, University of Lisbon, Portugal

ABSTRACTThe functional MRI (Magnetic Resonance Imaging), fMRI, istoday a widespread tool to study and evaluate the brain from afunctional point of view. The blood-oxygenation-level-dependent(BOLD) signal is currently used to detect the activation of brainregions with a stimulus application, e.g., visual or auditive. In ablock design approach the stimuli (called paradigm in the fMRIscope) are designed to detect activated and non activated brainregions with maximized certainty. However, corrupting noise inMRI volumes acquisition, patient motion and the normal brainactivity interference makes this detection a difficult task. The mostused activation detection fMRI algorithm, here called SPM-GLM[1] uses a conventional statistical inference methodology based onthe t-statistics

In this paper we propose a new Bayesian approach, by mod-eling the data acquisition noise as additive white Gaussian noise(AWGN) and the activation indicators as binary unknowns that mustbe estimated. Monte Carlo tests using both methods have shownthat the Bayesian method, here called SPM-MAP, outperforms thetraditional one, here called SPM-GLM, for almost all conditions ofnoise and number of paradigm epochs tested.

Index Terms— Functional MRI, Activity Detection, Bayesian

I. INTRODUCTION

The functional Magnetic Resonance imaging (fMRI) is currentlythe most prominent method used for functional brain imaging, andit is a big step forward in the process of answering the mainquestion asked to all the functional imaging methods: What arethe brain regions involved in mediating a specific brain function?And thought the fMRI’s obvious qualities have allowed for its fastacceptance and development, its limitations are far from letting thisquestion to become a "closed problem".

The fMRI experiments usually look for the change in bloodoxygenation and blood volume resulting from altered neural activ-ity. This signal, called blood-oxygenation-level-dependent (BOLD),results from the endogenous paramagnetic contrast property of thedeoxygenated hemoglobin. Hence, increased blood volume reducesthe local concentration of deoxygenated hemoglobin causing anincrease in the MR signal on a T2*-weighted image [2]. It iscommonly accepted, and has been empirically proved, that thereis a strong correlation between neural activity and the vascular

Correspondent author: J. Sanches ([email protected]). This work wassupported by Fundação para a Ciência e a Tecnologia (ISR/IST plurianualfunding) through the POS Conhecimento Program which includes FEDERfunds. This work was done in partial collaboration with the Hospital daCruz Vermelha de Lisboa. An acknowledgment to Henrik Halvorsen forrevision

response that leads to the consequent increase of this BOLD signal[3]. This relation, know as the hemodynamic response function(HRF), is at the core of fMRI data analysis. Inferences about whichbrain regions are involved in the particular stimulated cognitive andsensorimotor functions are based on how well the BOLD signalcorrelates with this stimulation, mediated by the HRF.

Contrary to the impression one might get in a brief review of theliterature, there are not many ways to analyze fMRI time-series witha diversity of statistical and conceptual approaches. In fact, withvery few exceptions, every analysis is a variant of the general linearmodel (GLM), that expresses the observed response variable interms of a linear combination of explanatory variables [4]. Based onthis model, data analysis is usually processed in a number of stepsinvolving image processing and statistical evaluation that, in theend, produce a functional brain map. Often methods used involveseveral modules for image preprocessing, spacial transformation,statistical tests and the final inferences procedures. This workfocuses on the last two steps and how our Bayesian method com-pares with the most commonly used method, Statistical ParametricMapping (SPM) [1] when applied to single voxel time-course data.This last method makes use of univariate statistical tests (T of Ftests) at each brain voxel and subsequent statistical inferences aboutthe observed responses using a user defined p-value threshold, forthe 1D data case. This classical approach maybe a simple onewith reasonable results, but it has several disadvantages that canbe tackled. On this work we propose a Bayesian approach for thebinary (activated or not) analysis of simulated 1D block-designeddata and present the Monte Carlo results from the comparisonbetween the presented method and the standard SPM procedure.

II. PROBLEM FORMULATION

Let us consider the voxels (volume elements) displayed in Fig.1. Each voxel, after the application of a given paradigm, may beactivated by one or more applied stimulus (∃k : βk = 1) or maynot be activated at all (∀k : βk = 0).

Fig. 1. Activated and non activated regions in fMRI.

In this paper we consider the signal BOLD associated to a singlevoxel at a time - time course - with the following data observationmodel, displayed in Fig. 3,

y(n) = h(n) ∗N∑

k=1

βkpk(n) + η(n) (1)

where η(n) is modeled as additive white Gaussian noise (AWGN),h(n) is the hemodynamic response function of the brain tissues,pk(n) are the stimulus signals along time (see Fig. 2) and βk areunknown binary variables to model the activation of the voxel bythe kth stimulus. For instance, Fig. 1 shows the result of applicationof a two stimulus paradigm where three voxels are referenced: i)a voxels was activated by the first stimulus, β1 = 1 and β2 = 0,ii) a voxel was not activated, β1 = 0 and β2 = 0, and iii) a voxelwas only activated by the second stimulus, β1 = 0 and β2 = 1.

Fig. 2. Paradigm with three block-designed stimulus.

In this paper we describe a Bayesian Statistical Parametric Map-ping algorithm (SPM) based on the maximum a posteriori (MAP)criterion to estimate the vector b = {β1, β2, ..., βN}, associatedwith each voxel, called SPM-MAP1. We use the observation modeldisplayed in Fig. 3 and described by the equation (1). Monte Carlotests with synthetic data are used to evaluate the performance ofthe algorithm and compare it with the SPM based on the generallinear model (GLM), here called SPM-GLM [1], which is one ofthe most commonly used methods to detect activated voxels in thefunctional MRI scope.

Fig. 3. BOLD signal generation model.

Let x = {x(1), x(2), x(3), ..., x(L)}T (see Fig. 3) where L isthe time-course observations length. x may be expressed as x = θbwhere

θ =

⎛⎜⎜⎜⎜⎜⎝

p1(1) p2(1) p3(1) ... pN(1)p1(2) p2(2) p3(2) ... pN(2)p1(3) p2(3) p3(3) ... pN(3)

......

... ......

p1(L) p2(L) p3(L) ... pN(L)

⎞⎟⎟⎟⎟⎟⎠ (2)

1Statistical parametric mapping is generally used to identify functionallyspecialized brain responses[1]

The output vector of h(n) displayed in Fig. 3, z ={z(1), z(2), z(3), ...z(L)}T , is obtained by z(n) = h(n) ∗ x(n).Here, for sake of simplicity, h(n) is assumed to be a F lengthfinite impulse response (FIR), although infinite responses may alsobe considered [5]. Therefore the output signal may be expressed asz = Hx where H is the following Toeplitz matrix

H =

⎛⎜⎜⎜⎜⎝

h(1) 0 0 0 0 0h(2) h(1) 0 0 0 0h(3) h(2) h(1) 0 0 0... ... ... ... ... ...0 ... h(p) h(p − 1) ... h(1)

⎞⎟⎟⎟⎟⎠ (3)

The observed BOLD signal y(n), y = {y(1), y(2), ..., y(L)}T

is therefore obtained as follows

y = Ψβ + n (4)

where Ψ = Hθ, n = {η(1), η(2), ..., η(L)}T is a vector ofindependent and identically distributed (i.i.d) zero mean randomvariables normally distributed, that is, p(η(k)) = N(0, σ2). Theadditive white Gaussian noise (AWGN) is usually used to modelthe corruption process in functional MRI although other modelsmay also be used, e.g., Rice and Rayleigh.

III. ESTIMATION

The MAP estimate of b can be obtained by solving the followingequation

b = arg minb

E(y,b) (5)

where E(y,b) is an energy function defined as follows

E(y,b) = − log(p(y|x(b)))︸︷︷︸Data fidelity term

− log(p(b))︸︷︷︸Prior term

(6)

where Ey(y,b) = − log(p(y|x(b))) is the likelihood term andEb(b) = − log(p(b)) incorporates the a priori knowledge aboutthe unknowns to be estimated; in this case, that βk are binary.

Statistical independence of the observations means thatp(y|x(b)) =

∏Li=1 p(y(i)|x(b)). Since the noise is assumed

to be additive white and Gaussian (AWGN), p(y(i)|x(b)) =

1√2πσ2 e

− (y(i)−x(i))2

2σ2y . The unknowns to be estimated, βk , are also

assumed to be independent, that is,

p(b) =N∏

i=1

p(βk) (7)

where p(βk) is a bi-modal distribution defined as a sum of twoGaussian distributions centered at zero and one, of σ2

β variance

p(βk) =1

2

[N(0, σ2

β) + N(1, σ2β)]

(8)

because βk are binary variable, βk ∈ {0, 1}. In order to betterapproximate the binary answer σβ should be as small as possiblebut numerical stability reasons prevent the adoption of too smallvalues.

The MAP estimate is therefore the minimizer of the followingenergy function

E(y,b) = (Ψb− y)T (Ψb− y) + Eb(b) + C (9)

where C is a constant and

Eb(b) =N∑

k=1

[2β2

k − 2βk + 1

4σ2β

− log

(cosh

[2βk − 1

4σ2β

])]. (10)

The MAP estimate is obtained by finding the stationary point ofE(y,b),

∇bE(y,b) = 0 (11)

where ∇b is the gradient operator w.r.t. b, which may be writtenas follows

∇bE = ΨT (Ψb − y) +σ2y

σ2b

[b − 1

2R(b)

]= 0 (12)

where R(b) is a column vector with N elements rk

rk = 1 + tanh

[2βk − 1

4σ2β

](13)

The solution of (12) may be obtained by using the fixed pointmethod which leads to the following recursion

bt+1 = (ΨT Ψ + αI)−1(ΨT y + αR(bt)) (14)

where α = σ2y/2σ2

β is a parameter, I is a N × N identity matrixand bt is the b estimate at tth iteration.

The estimated elements of b, βk, are real numbers and not binary.Therefore, the binary estimate of βk, bk, is obtained as follows

bk =

{0 βk < 0.5

1 otherwise(15)

Two main differences must be stressed between the proposedSPM-MAP and the standard SPM-GLM method:

1) In the SPM-GLM method the whole N period signal is some-times broken into N pieces corresponding to each paradigmperiod and the resulting observation pieces are averaged toreduce the noise corrupting the observations Y . The matrixθ, defined in (2), is built by using only a single paradigmperiod. In the proposed method, instead of braking the signal,it is dealt with as a whole signal. And the same goes forthe paradigm signal. The noise reduction is performed in aBayesian framework where a realistic observation model isused to cope with it. In the case of AWGN both methods arevery similar, but if other noise models (e.g. multiplicative)are used this would no longer be true. This is becausethe averaging procedure is only adequate for certain noisemodels.

2) In the SPM-GLM method the estimation of each βk isbased on the well known classical t-test [6] applied to theestimated coefficients βk. This statistical inference techniqueis based on the null hypothesis test, Ho, where the activationprobability of a given voxel is computed with a certainconfidence degree. This test is performed over the estimatedcoefficients, βk, obtained with the GLM. These coefficientsused to linearly combine the EVs (usually a convolutionbetween the paradigm stimulus and the HRF model (s)) areestimated by using the MSE criterion. These real coefficientsreflect the estimated "presence" amplitude of each EV in theobserved data. In the SPM-MAP method the coefficients setare assumed to be binary and are estimated in a Bayesianframework where a prior distribution forces its values to

be close of {0, 1}. Once again, our concern is to follow arealistic model where it is assumed that a given voxel wasactivated or not by a given stimulus. Partial voxel activationis not acceptable in this scope: it is totally activated or it isnot activated at all, by a given stimulus, pk(n).

To better understand the difference between both methods, ashort description of the SPM-GLM method is presented where thet-test is used to determine if a given voxel is or is not activated bya single EV , p(n)∗h(n), which means that b is a scalar, b = [β].

The observed BOLD signal is assumed to be obtained from thefollowing model

y = β(p(h) ∗ h(n)θ) + e (16)

where θ is defined in (2) but with only one stimulus, θ ={p(1), p(2), ..., p(L)}T , h(n) is the canonical gamma function[1,4,7] and e is the residual error vector not explained by the model.The β estimation given by the GLM method, in this very simplecase, is computed as follows

β = θ+Y (17)

where θ+ = (θT θ)−1θT is the so called pseudoinverse of θ.The SPM-GLM method core, tags each voxel as activated, b = 1,

or as inactivated, b = 0, by computing the probability of beingactivated by the stimulus with a confidence level α, that is,

b =

{1 (Active) ifp < α; (reject H0)

0 (No Active) Otherwise; (acept H0)(18)

where H0 is the null hypothesis which assumes no activation witha confidence level α.


p = P (t ≥ T ) = 1 − I LL+T2

(L/2, 0.5) (19)



Γ(a)Γ(b)

∫ x

0

τa−1(a − τ )b−1dτ (20)

andT = β/σβ (21)

is the T estimator associated t-statistics, where β is the estimationvalue of β, and σβ the standard deviation. t is large if the estimatedvalue is much larger than the estimator variance and t is small ifthe estimated value is comparable with the corresponding estimatorvariance.

The estimator variance, σ2β , may be numerically estimated using

the following expression

σ2β = σ2

y

L∑n=1

p2(n) (22)

where p(n) is the nth θ element and σ2y is the estimated noise

energy

σ2y =

1

L − 1

L∑n=1

[y(n) − βp(n)

]2

. (23)

In the next section Monte Carlo simulations are presented com-paring the probability of error (Pe), obtained with both methods,SPM-MAP and the SPM-GLM described above.


To access the effectiveness of the proposed SPM-MAP methodagainst the presented SPM-GML, synthetic 1D-block-designed sin-gle stimulus data, with several AWGN noise levels (σ) and severalstimulus epochs (periods in a block design paradigm approach), N ,are used in Monte Carlo simulations. In these, the error probability(Pe), was obtained for each method as follows

Pe(σ, N) =1

R

R∑i=1

|bi − bi| (24)

where R = 250 is the number of data repetitions used in the MonteCarlo tests. The HRF function is assumed to be known and wasselected from the PBH model estimation on real data [5].

The resulting Pe differences between SPM-MAP and SPM-GLM,i.e: ∆Pe = Pe(SPM−MAP ) − Pe(SPM−GLM), was computed foreach experiment, and the average results for the different noiselevels and epochs are displayed in Fig.4 and in table I. Notice thatin table I, σ values that resulted in all null Pe are not displayed.

Although the performance of both algorithms decreases, asexpected, with the amount of noise and with the decrease inepochs number, the ∆Pe obtained is negative for most of the(N, σ) data pairs tested, which means that the SPM-MAP out-performs the SPM-GLM method for almost every configurationtested. This is confirmed by Table I, where the summation of ∆Pe,∑

i,j ∆Pe(Ni, σj) = −0.46, is negative. The number of negativevalues of ∆Pe, #(∆Pe < 0) = 23 and the number of positivevalues of ∆Pe, #(∆Pe > 0) = 6, which confirms that the SPM-MAP surpasses the traditional SPM-GLM method. Still it is obviousthat both methods present high accuracy in these tests due to theexact knowledge of both the HRF and noise distribution.

Fig. 4. Difference of probability errors, ∆Pe = Pe(SPM−MAP ) −Pe(SPM−GLM), between the SPM-MAP and the SPM-GLM meth-ods for two different views. N is the number of epochs and σ isnoise standard deviation level.

V. CONCLUSIONS

In this paper a new Bayesian method for detection of brainactivated regions in the scope of functional MRI is proposed wherethe noisy observations are modeled with additive white Gaussiannoise (AWGN) and the activation indicators are modeled by binaryvariables that are estimated. The prior associated with the binaryindicators is a bimodal Gaussian distribution around the 0 and 1values to cope with the uncertainty related with the noise.

The proposed method is compared with the one proposed in[1] that uses a classical statistical inference methodology based

N\σ 1 2 4

2 -0.012 0 -0.1683 -0.012 0.004 -0.1084 -0.008 -0.012 -0.0445 -0.004 -0.012 -0.0326 0 -0.008 -0.0167 0 -0.008 -0.0088 0 -0.004 09 0 0 -0.00410 0 0 0.00411 0 -0.004 0.01212 0 0 0.00413 0 -0.004 014 0 -0.004 015 0 -0.004 0.00416 0 0 -0.00417 0 0 018 0 0 0.00419 0 0 -0.00420 0 0 -0.008

Table I. ∆Pe = Pe(SPM−MAP ) − Pe(SPM−GLM) for 2 ≤N ≤ 20 and σ = {1, 2, 4}. For all other tested values ofσ = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5

, ∆Pe = 0

on the t-test method. Monte Carlo tests on synthetic-1D-block-designed data with both methods have shown that the proposedBayesian method, called SPM-MAP, outperforms the classical onebased on the general linear model (GLM), here called SPM-GLM.The performance evaluation was based on the computation of theerror probability, Pe(N, σ), for each method which proved to besmaller for the proposed SPM-MAP method than the correspondingones obtained with the SPM-GLM method, for almost every testedconditions: different noise levels, σj and number of paradigmepochs, Ni, in a block design framework.

VI. REFERENCES

[1] K. J. Friston, “Analyzing brain images: Principles and overview,” inHuman Brain Function, R.S.J. Frackowiak and K.J. Friston and C.Frith and R. Dolan and J.C. Mazziotta, Ed., pp. 25–41. AcademicPress USA, 1997.

[2] S. Ogawa, R. S. Menon, D. W. Tank, S. G. Kim, H. Merkle, J. M.Ellermann, and K. Ugurbil, “Functional brain mapping by bloodoxygenation level-dependent contrast magnetic resonance imaging.A comparison of signal characteristics with a biophysical model,”Biophys J, vol. 64, no. 3, pp. 803–812, Mar 1993, ComparativeStudy.

[3] D. Malonek and A. Grinvald, “Interactions between electrical activ-ity and cortical microcirculation revealed by imaging spectroscopy:implications for functional brain mapping,” Science, vol. 272, no.5261, pp. 551–554, Apr 1996.

[4] P. Jezzard, P. M. Matthews, and S. M. Smith, Functional magneticresonance imaging: An introduction to methods., Oxford MedicalPublications, 2006.

[5] David Miguel Afonso, João Sanches, and Martin Hagen Lauter-bach, “Neural physiological modeling towards a hemodynamicresponse function for fMRI,” Aug 2007, pp. 1615–1618.

[6] N. Kutner and N. Wasserman and C. J. Nachtsheim and W.Wasserman, Applied Linear Statistical Models, The McGraw-HillCompanies, Inc., 1996.

[7] G. H. Glover, “Deconvolution of impulse response in event-relatedBOLD fMRI,” Neuroimage, vol. 9, no. 4, pp. 416–429, Apr 1999,Clinical Trial.

ROBUST BAYESIAN BRAIN ACTIVITY DETECTION IN FMRI

João Sanches David Afonso Kestutis Bartnykas Martin H. Lauterbach (MD)

Instituto Superior Técnico / Instituto de Sistemas e Robótica, Lisbon, PortugalVilnius Gediminas Technical University, Vilnius, Lithuania

Faculty of Medicine, University of Lisbon, Portugal

ABSTRACTThe functional MRI (Magnetic Resonance Imaging), fMRI, istoday a widespread tool to study and evaluate the brain from afunctional point of view. The blood-oxygenation-level-dependent(BOLD) signal is currently used to detect the activation of brainregions with a stimulus application, e.g., visual or auditive. In ablock design approach the stimuli (called paradigm in the fMRIscope) are designed to detect activated and non activated brainregions with maximized certainty. However, corrupting noise inMRI volumes acquisition, patient motion and the normal brainactivity interference makes this detection a difficult task. The mostused activation detection fMRI algorithm, here called SPM-GLM[1] uses a conventional statistical inference methodology based onthe t-statistics

In this paper we propose a new Bayesian approach, by mod-eling the data acquisition noise as additive white Gaussian noise(AWGN) and the activation indicators as binary unknowns that mustbe estimated. Monte Carlo tests using both methods have shownthat the Bayesian method, here called SPM-MAP, outperforms thetraditional one, here called SPM-GLM, for almost all conditions ofnoise and number of paradigm epochs tested.

Index Terms— Functional MRI, Activity Detection, Bayesian

I. INTRODUCTION

The functional Magnetic Resonance imaging (fMRI) is currentlythe most prominent method used for functional brain imaging, andit is a big step forward in the process of answering the mainquestion asked to all the functional imaging methods: What arethe brain regions involved in mediating a specific brain function?And thought the fMRI’s obvious qualities have allowed for its fastacceptance and development, its limitations are far from letting thisquestion to become a "closed problem".

The fMRI experiments usually look for the change in bloodoxygenation and blood volume resulting from altered neural activ-ity. This signal, called blood-oxygenation-level-dependent (BOLD),results from the endogenous paramagnetic contrast property of thedeoxygenated hemoglobin. Hence, increased blood volume reducesthe local concentration of deoxygenated hemoglobin causing anincrease in the MR signal on a T2*-weighted image [2]. It iscommonly accepted, and has been empirically proved, that thereis a strong correlation between neural activity and the vascular

Correspondent author: J. Sanches ([email protected]). This work wassupported by Fundação para a Ciência e a Tecnologia (ISR/IST plurianualfunding) through the POS Conhecimento Program which includes FEDERfunds. This work was done in partial collaboration with the Hospital daCruz Vermelha de Lisboa. An acknowledgment to Henrik Halvorsen forrevision

response that leads to the consequent increase of this BOLD signal[3]. This relation, know as the hemodynamic response function(HRF), is at the core of fMRI data analysis. Inferences about whichbrain regions are involved in the particular stimulated cognitive andsensorimotor functions are based on how well the BOLD signalcorrelates with this stimulation, mediated by the HRF.

Contrary to the impression one might get in a brief review of theliterature, there are not many ways to analyze fMRI time-series witha diversity of statistical and conceptual approaches. In fact, withvery few exceptions, every analysis is a variant of the general linearmodel (GLM), that expresses the observed response variable interms of a linear combination of explanatory variables [4]. Based onthis model, data analysis is usually processed in a number of stepsinvolving image processing and statistical evaluation that, in theend, produce a functional brain map. Often methods used involveseveral modules for image preprocessing, spacial transformation,statistical tests and the final inferences procedures. This workfocuses on the last two steps and how our Bayesian method com-pares with the most commonly used method, Statistical ParametricMapping (SPM) [1] when applied to single voxel time-course data.This last method makes use of univariate statistical tests (T of Ftests) at each brain voxel and subsequent statistical inferences aboutthe observed responses using a user defined p-value threshold, forthe 1D data case. This classical approach maybe a simple onewith reasonable results, but it has several disadvantages that canbe tackled. On this work we propose a Bayesian approach for thebinary (activated or not) analysis of simulated 1D block-designeddata and present the Monte Carlo results from the comparisonbetween the presented method and the standard SPM procedure.

II. PROBLEM FORMULATION

Let us consider the voxels (volume elements) displayed in Fig.1. Each voxel, after the application of a given paradigm, may beactivated by one or more applied stimulus (∃k : βk = 1) or maynot be activated at all (∀k : βk = 0).

Fig. 1. Activated and non activated regions in fMRI.

In this paper we consider the signal BOLD associated to a singlevoxel at a time - time course - with the following data observationmodel, displayed in Fig. 3,

y(n) = h(n) ∗N∑

k=1

βkpk(n) + η(n) (1)

where η(n) is modeled as additive white Gaussian noise (AWGN),h(n) is the hemodynamic response function of the brain tissues,pk(n) are the stimulus signals along time (see Fig. 2) and βk areunknown binary variables to model the activation of the voxel bythe kth stimulus. For instance, Fig. 1 shows the result of applicationof a two stimulus paradigm where three voxels are referenced: i)a voxels was activated by the first stimulus, β1 = 1 and β2 = 0,ii) a voxel was not activated, β1 = 0 and β2 = 0, and iii) a voxelwas only activated by the second stimulus, β1 = 0 and β2 = 1.

Fig. 2. Paradigm with three block-designed stimulus.

In this paper we describe a Bayesian Statistical Parametric Map-ping algorithm (SPM) based on the maximum a posteriori (MAP)criterion to estimate the vector b = {β1, β2, ..., βN}, associatedwith each voxel, called SPM-MAP1. We use the observation modeldisplayed in Fig. 3 and described by the equation (1). Monte Carlotests with synthetic data are used to evaluate the performance ofthe algorithm and compare it with the SPM based on the generallinear model (GLM), here called SPM-GLM [1], which is one ofthe most commonly used methods to detect activated voxels in thefunctional MRI scope.

Fig. 3. BOLD signal generation model.

Let x = {x(1), x(2), x(3), ..., x(L)}T (see Fig. 3) where L isthe time-course observations length. x may be expressed as x = θbwhere

θ =

⎛⎜⎜⎜⎜⎜⎝

p1(1) p2(1) p3(1) ... pN(1)p1(2) p2(2) p3(2) ... pN(2)p1(3) p2(3) p3(3) ... pN(3)

......

... ......

p1(L) p2(L) p3(L) ... pN(L)

⎞⎟⎟⎟⎟⎟⎠ (2)

1Statistical parametric mapping is generally used to identify functionallyspecialized brain responses[1]

The output vector of h(n) displayed in Fig. 3, z ={z(1), z(2), z(3), ...z(L)}T , is obtained by z(n) = h(n) ∗ x(n).Here, for sake of simplicity, h(n) is assumed to be a F lengthfinite impulse response (FIR), although infinite responses may alsobe considered [5]. Therefore the output signal may be expressed asz = Hx where H is the following Toeplitz matrix

H =

⎛⎜⎜⎜⎜⎝

h(1) 0 0 0 0 0h(2) h(1) 0 0 0 0h(3) h(2) h(1) 0 0 0... ... ... ... ... ...0 ... h(p) h(p − 1) ... h(1)

⎞⎟⎟⎟⎟⎠ (3)

The observed BOLD signal y(n), y = {y(1), y(2), ..., y(L)}T

is therefore obtained as follows

y = Ψβ + n (4)

where Ψ = Hθ, n = {η(1), η(2), ..., η(L)}T is a vector ofindependent and identically distributed (i.i.d) zero mean randomvariables normally distributed, that is, p(η(k)) = N(0, σ2). Theadditive white Gaussian noise (AWGN) is usually used to modelthe corruption process in functional MRI although other modelsmay also be used, e.g., Rice and Rayleigh.

III. ESTIMATION

The MAP estimate of b can be obtained by solving the followingequation

b = arg minb

E(y,b) (5)

where E(y,b) is an energy function defined as follows

E(y,b) = − log(p(y|x(b)))︸︷︷︸Data fidelity term

− log(p(b))︸︷︷︸Prior term

(6)

where Ey(y,b) = − log(p(y|x(b))) is the likelihood term andEb(b) = − log(p(b)) incorporates the a priori knowledge aboutthe unknowns to be estimated; in this case, that βk are binary.

Statistical independence of the observations means thatp(y|x(b)) =

∏Li=1 p(y(i)|x(b)). Since the noise is assumed

to be additive white and Gaussian (AWGN), p(y(i)|x(b)) =

1√2πσ2 e

− (y(i)−x(i))2

2σ2y . The unknowns to be estimated, βk , are also

assumed to be independent, that is,

p(b) =N∏

i=1

p(βk) (7)

where p(βk) is a bi-modal distribution defined as a sum of twoGaussian distributions centered at zero and one, of σ2

β variance

p(βk) =1

2

[N(0, σ2

β) + N(1, σ2β)]

(8)

because βk are binary variable, βk ∈ {0, 1}. In order to betterapproximate the binary answer σβ should be as small as possiblebut numerical stability reasons prevent the adoption of too smallvalues.

The MAP estimate is therefore the minimizer of the followingenergy function

E(y,b) = (Ψb− y)T (Ψb− y) + Eb(b) + C (9)

where C is a constant and

Eb(b) =N∑

k=1

[2β2

k − 2βk + 1

4σ2β

− log

(cosh

[2βk − 1

4σ2β

])]. (10)

The MAP estimate is obtained by finding the stationary point ofE(y,b),

∇bE(y,b) = 0 (11)

where ∇b is the gradient operator w.r.t. b, which may be writtenas follows

∇bE = ΨT (Ψb − y) +σ2y

σ2b

[b − 1

2R(b)

]= 0 (12)

where R(b) is a column vector with N elements rk

rk = 1 + tanh

[2βk − 1

4σ2β

](13)

The solution of (12) may be obtained by using the fixed pointmethod which leads to the following recursion

bt+1 = (ΨT Ψ + αI)−1(ΨT y + αR(bt)) (14)

where α = σ2y/2σ2

β is a parameter, I is a N × N identity matrixand bt is the b estimate at tth iteration.

The estimated elements of b, βk, are real numbers and not binary.Therefore, the binary estimate of βk, bk, is obtained as follows

bk =

{0 βk < 0.5

1 otherwise(15)

Two main differences must be stressed between the proposedSPM-MAP and the standard SPM-GLM method:

1) In the SPM-GLM method the whole N period signal is some-times broken into N pieces corresponding to each paradigmperiod and the resulting observation pieces are averaged toreduce the noise corrupting the observations Y . The matrixθ, defined in (2), is built by using only a single paradigmperiod. In the proposed method, instead of braking the signal,it is dealt with as a whole signal. And the same goes forthe paradigm signal. The noise reduction is performed in aBayesian framework where a realistic observation model isused to cope with it. In the case of AWGN both methods arevery similar, but if other noise models (e.g. multiplicative)are used this would no longer be true. This is becausethe averaging procedure is only adequate for certain noisemodels.

2) In the SPM-GLM method the estimation of each βk isbased on the well known classical t-test [6] applied to theestimated coefficients βk. This statistical inference techniqueis based on the null hypothesis test, Ho, where the activationprobability of a given voxel is computed with a certainconfidence degree. This test is performed over the estimatedcoefficients, βk, obtained with the GLM. These coefficientsused to linearly combine the EVs (usually a convolutionbetween the paradigm stimulus and the HRF model (s)) areestimated by using the MSE criterion. These real coefficientsreflect the estimated "presence" amplitude of each EV in theobserved data. In the SPM-MAP method the coefficients setare assumed to be binary and are estimated in a Bayesianframework where a prior distribution forces its values to

be close of {0, 1}. Once again, our concern is to follow arealistic model where it is assumed that a given voxel wasactivated or not by a given stimulus. Partial voxel activationis not acceptable in this scope: it is totally activated or it isnot activated at all, by a given stimulus, pk(n).

To better understand the difference between both methods, ashort description of the SPM-GLM method is presented where thet-test is used to determine if a given voxel is or is not activated bya single EV , p(n)∗h(n), which means that b is a scalar, b = [β].

The observed BOLD signal is assumed to be obtained from thefollowing model

y = β(p(h) ∗ h(n)θ) + e (16)

where θ is defined in (2) but with only one stimulus, θ ={p(1), p(2), ..., p(L)}T , h(n) is the canonical gamma function[1,4,7] and e is the residual error vector not explained by the model.The β estimation given by the GLM method, in this very simplecase, is computed as follows

β = θ+Y (17)

where θ+ = (θT θ)−1θT is the so called pseudoinverse of θ.The SPM-GLM method core, tags each voxel as activated, b = 1,

or as inactivated, b = 0, by computing the probability of beingactivated by the stimulus with a confidence level α, that is,

b =

{1 (Active) ifp < α; (reject H0)

0 (No Active) Otherwise; (acept H0)(18)

where H0 is the null hypothesis which assumes no activation witha confidence level α.


p = P (t ≥ T ) = 1 − I LL+T2

(L/2, 0.5) (19)



Γ(a)Γ(b)

∫ x

0

τa−1(a − τ )b−1dτ (20)

andT = β/σβ (21)

is the T estimator associated t-statistics, where β is the estimationvalue of β, and σβ the standard deviation. t is large if the estimatedvalue is much larger than the estimator variance and t is small ifthe estimated value is comparable with the corresponding estimatorvariance.

The estimator variance, σ2β , may be numerically estimated using

the following expression

σ2β = σ2

y

L∑n=1

p2(n) (22)

where p(n) is the nth θ element and σ2y is the estimated noise

energy

σ2y =

1

L − 1

L∑n=1

[y(n) − βp(n)

]2

. (23)

In the next section Monte Carlo simulations are presented com-paring the probability of error (Pe), obtained with both methods,SPM-MAP and the SPM-GLM described above.


To access the effectiveness of the proposed SPM-MAP methodagainst the presented SPM-GML, synthetic 1D-block-designed sin-gle stimulus data, with several AWGN noise levels (σ) and severalstimulus epochs (periods in a block design paradigm approach), N ,are used in Monte Carlo simulations. In these, the error probability(Pe), was obtained for each method as follows

Pe(σ, N) =1

R

R∑i=1

|bi − bi| (24)

where R = 250 is the number of data repetitions used in the MonteCarlo tests. The HRF function is assumed to be known and wasselected from the PBH model estimation on real data [5].

The resulting Pe differences between SPM-MAP and SPM-GLM,i.e: ∆Pe = Pe(SPM−MAP ) − Pe(SPM−GLM), was computed foreach experiment, and the average results for the different noiselevels and epochs are displayed in Fig.4 and in table I. Notice thatin table I, σ values that resulted in all null Pe are not displayed.

Although the performance of both algorithms decreases, asexpected, with the amount of noise and with the decrease inepochs number, the ∆Pe obtained is negative for most of the(N, σ) data pairs tested, which means that the SPM-MAP out-performs the SPM-GLM method for almost every configurationtested. This is confirmed by Table I, where the summation of ∆Pe,∑

i,j ∆Pe(Ni, σj) = −0.46, is negative. The number of negativevalues of ∆Pe, #(∆Pe < 0) = 23 and the number of positivevalues of ∆Pe, #(∆Pe > 0) = 6, which confirms that the SPM-MAP surpasses the traditional SPM-GLM method. Still it is obviousthat both methods present high accuracy in these tests due to theexact knowledge of both the HRF and noise distribution.

Fig. 4. Difference of probability errors, ∆Pe = Pe(SPM−MAP ) −Pe(SPM−GLM), between the SPM-MAP and the SPM-GLM meth-ods for two different views. N is the number of epochs and σ isnoise standard deviation level.

V. CONCLUSIONS

In this paper a new Bayesian method for detection of brainactivated regions in the scope of functional MRI is proposed wherethe noisy observations are modeled with additive white Gaussiannoise (AWGN) and the activation indicators are modeled by binaryvariables that are estimated. The prior associated with the binaryindicators is a bimodal Gaussian distribution around the 0 and 1values to cope with the uncertainty related with the noise.

The proposed method is compared with the one proposed in[1] that uses a classical statistical inference methodology based

N\σ 1 2 4

2 -0.012 0 -0.1683 -0.012 0.004 -0.1084 -0.008 -0.012 -0.0445 -0.004 -0.012 -0.0326 0 -0.008 -0.0167 0 -0.008 -0.0088 0 -0.004 09 0 0 -0.00410 0 0 0.00411 0 -0.004 0.01212 0 0 0.00413 0 -0.004 014 0 -0.004 015 0 -0.004 0.00416 0 0 -0.00417 0 0 018 0 0 0.00419 0 0 -0.00420 0 0 -0.008

Table I. ∆Pe = Pe(SPM−MAP ) − Pe(SPM−GLM) for 2 ≤N ≤ 20 and σ = {1, 2, 4}. For all other tested values ofσ = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5

, ∆Pe = 0

on the t-test method. Monte Carlo tests on synthetic-1D-block-designed data with both methods have shown that the proposedBayesian method, called SPM-MAP, outperforms the classical onebased on the general linear model (GLM), here called SPM-GLM.The performance evaluation was based on the computation of theerror probability, Pe(N, σ), for each method which proved to besmaller for the proposed SPM-MAP method than the correspondingones obtained with the SPM-GLM method, for almost every testedconditions: different noise levels, σj and number of paradigmepochs, Ni, in a block design framework.

VI. REFERENCES

[1] K. J. Friston, “Analyzing brain images: Principles and overview,” inHuman Brain Function, R.S.J. Frackowiak and K.J. Friston and C.Frith and R. Dolan and J.C. Mazziotta, Ed., pp. 25–41. AcademicPress USA, 1997.

[2] S. Ogawa, R. S. Menon, D. W. Tank, S. G. Kim, H. Merkle, J. M.Ellermann, and K. Ugurbil, “Functional brain mapping by bloodoxygenation level-dependent contrast magnetic resonance imaging.A comparison of signal characteristics with a biophysical model,”Biophys J, vol. 64, no. 3, pp. 803–812, Mar 1993, ComparativeStudy.

[3] D. Malonek and A. Grinvald, “Interactions between electrical activ-ity and cortical microcirculation revealed by imaging spectroscopy:implications for functional brain mapping,” Science, vol. 272, no.5261, pp. 551–554, Apr 1996.

[4] P. Jezzard, P. M. Matthews, and S. M. Smith, Functional magneticresonance imaging: An introduction to methods., Oxford MedicalPublications, 2006.

[5] David Miguel Afonso, João Sanches, and Martin Hagen Lauter-bach, “Neural physiological modeling towards a hemodynamicresponse function for fMRI,” Aug 2007, pp. 1615–1618.

[6] N. Kutner and N. Wasserman and C. J. Nachtsheim and W.Wasserman, Applied Linear Statistical Models, The McGraw-HillCompanies, Inc., 1996.

[7] G. H. Glover, “Deconvolution of impulse response in event-relatedBOLD fMRI,” Neuroimage, vol. 9, no. 4, pp. 416–429, Apr 1999,Clinical Trial.

Brain Functional Assessment With fMRI · Abstract Functional MRI (Magnetic Resonance Imaging),...

Documents

Transcript of Brain Functional Assessment With fMRI · Abstract Functional MRI (Magnetic Resonance Imaging),...