A computational pipeline for quantification of mouse myocardialstiffness parameters

Øyvind Nordbø a, Pablo Lamata b, Sander Land b, Steven Niederer b, Jan M. Aronsen c,d,William E. Louch c,e, Ivar Sjaastad c, Harald Martens f, Arne B. Gjuvsland g, Kristin Tøndel b,Hans Torp h, Maelene Lohezic i, Jurgen E. Schneider h, Espen W. Remme e,j, Nicolas Smith b,Stig W. Omholt k, Jon Olav Vik g,n

a Department of Mathematical Sciences and Technology, Centre for Integrative Genetics, Norwegian University of Life Sciences, P.O. Box 5003,N-1432 Ås, Norwayb Department of Biomedical Engineering, King’s College London, St. Thomas' Hospital, Westminster Bridge Road, London SE17EH, UKc Institute for Experimental Medical Research, Oslo University Hospital Ullevål and University of Oslo, Kirkeveien 166, 4th Floor Building 7, 0407 Oslo, Norwayd Bjørknes College, Oslo, Norwaye KG Jebsen Cardiac Research Center and Center for Heart Failure Research, University of Oslo, 0407 Oslo, Norwayf Department of Engineering Cybernetics, Faculty of Information Technology, Mathematics and Electrical Engineering, Norwegian University of Science andTechnology, Trondheim, Norwayg Department of Animal and Aquacultural Sciences, Centre for Integrative Genetics, Norwegian University of Life Sciences, P.O. Box 5003, N-1432 Ås, Norwayh Department of Circulation and Medical Imaging, Faculty of Medicine, Norwegian University of Science and Technology, Postboks 8905, Medisinsk tekniskforskningssenter, NO-7491 Trondheim, Norwayi Radcliffe Department of Medicine, Division of Cardiovascular Medicine, University of Oxford, Welcome Trust Centre for Human Genetics, Roosevelt Drive,Oxford OX3 7BN, UKj Institute for Surgical Research, Oslo University Hospital, Rikshospitalet, Oslo, Norwayk Faculty of Medicine, Norwegian University of Science and Technology, P.O. Box 8905, N-7491 Trondheim, Norway

a r t i c l e i n f o

Article history:Received 12 March 2014Accepted 20 July 2014

Keywords:Myocardial stiffnessParameter estimationPassive inflationTransversal isotropySpeckle tracking

a b s t r a c t

The mouse is an important model for theoretical–experimental cardiac research, and biophysically basedwhole organ models of the mouse heart are now within reach. However, the passive material propertiesof mouse myocardium have not been much studied.

We present an experimental setup and associated computational pipeline to quantify these stiffnessproperties. A mouse heart was excised and the left ventricle experimentally inflated from 0 to 1.44 kPa ineleven steps, and the resulting deformation was estimated by echocardiography and speckle tracking. Anin silico counterpart to this experiment was built using finite element methods and data on ventriculartissue microstructure from diffusion tensor MRI. This model assumed a hyperelastic, transverselyisotropic material law to describe the force–deformation relationship, and was simulated for manyparameter scenarios, covering the relevant range of parameter space. To identify well-fitting parameterscenarios, we compared experimental and simulated outcomes across the whole range of pressures,based partly on gross phenotypes (volume, elastic energy, and short- and long-axis diameter), and partlyon node positions in the geometrical mesh. This identified a narrow region of experimentally compatiblevalues of the material parameters. Estimation turned out to be more precise when based on changes ingross phenotypes, compared to the prevailing practice of using displacements of the material points. Weconclude that the presented experimental setup and computational pipeline is a viable method thatdeserves wider application.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The passive mechanical properties of the heart wall play animportant role during the filling of the ventricle; for instance,stiffening of the myocardium can lead to diastolic dysfunction andheart failure [1]. Incorporating patient-specific geometries and

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Computers in Biology and Medicine

http://dx.doi.org/10.1016/j.compbiomed.2014.07.0130010-4825/& 2014 Elsevier Ltd. All rights reserved.

Abbreviations: LVP, Left Ventricular Pressure; MVO, Mitral Valve Opening; ED,End Diastole; ES, End Systole; PV, Pressure Volume

n Corresponding author. Tel.: þ47 45882998; fax: þ47 64965101.E-mail address: jonovik@gmail.com (J.O. Vik).

Computers in Biology and Medicine 53 (2014) 65–75

material properties into biomechanical models holds promise forthe improved diagnosis and treatment of heart failure [2,3]. Inbiomechanical models, the stress–strain relationship for myocar-dial tissue is described by constitutive laws whose parametersdescribe the material stiffness along the microstructural axes, e.g.fibre and sheet directions. Several studies have quantified passivemyocardial stiffness in humans [3–5] and model organisms, suchas the pig [6,7], dog [8,9], and rat [10] (see Table 1, withreferences). Less well studied, however, is the mouse, which isan important model organism in cardiac research. Transgenicmouse models have been extensively used in cardiac researchover the last decade, where manipulation of the mouse genomehas provided several novel insights in the development of cardiacdisease [11]. Alterations of the myocardial properties of the heart,both in the extracellular matrix and in the cardiomyocytes, aresuggested to play a major role in development of diastolicdysfunction and heart failure. Several mutations leading to heri-table heart disease occur in both mouse and human [11], and thehearts have similar composition and function with regards tomechanical properties [12].

Here we propose a workflow for estimation of passive materialproperties of mouse myocardium by comparing computer simula-tions of soft tissue mechanics to observed deformation underpassive inflation of an excised mouse heart. Firstly, we estimatetrajectories of material points by speckle tracking of echo datathrough the deformation from zero to end diastolic pressure levels.This deformation is used, together with fibre structure obtainedfrom diffusion tensor MRI, to build meshes for computer simula-tions. Secondly, we run model simulations and use full factorialexperimental designs to cover the biological relevant region of theparameter space for the transversely isotropic Guccione law [13].The results show that the parameters of the constitutive law arenot separately identifiable from these passive inflation measure-ments alone, in accordance with previous studies [14–16]. For thefour-dimensional parameter space of the material law, we identifyan elongated region of parameter combinations with nearlyequivalent good fit to deformation data from passive inflation.

We describe the structure of this parameter redundancy, andcompare our findings with published estimates. Following adiscussion of its advantages and limitations, we conclude thatthe presented pipeline merits wider application.

2. Methods

The quantification of the passive mechanical properties of themyocardium relies on mechanistic models whose parameters aretuned to reproduce the observation of a corresponding experiment[9,14,15]. Below we describe the passive inflation experiment withdeformation measurement, MR imaging for geometric meshing,image processing and speckle tracking to estimate deformation,mechanics simulations, and comparison of experiment andsimulation.

2.1. Passive inflation experiment

A single mouse was anesthetized with 5% isoflurane andsacrificed by cervical dislocation. The heart was then rapidlyexcised and cannulated via the aorta and mounted on a Langen-dorff setup (Fig. 1). The cannula was firstly placed over the level ofthe aortic valves, to perfuse the myocardium with cardioplegicsolution of pH 7.4, 118.3 mM NaCl, 3.0 mM KCl, 4 mM MgSO4,0.2 mM CaCl2, 2.4 mM KH2PO4, 24.9 mM NaHCO3, 10 mM glucose,2.2 mM mannitol. Some calcium was included in the superfusatesince calcium-free conditions promote breakdown of gap junc-tions and dissociation of cardiomyocytes [17], which could causeunderestimates of myocardial stiffness. The cannula was thenintroduced into the left ventricle, and thereafter retracted intothe aorta. This technique allowed the cannula both to manipulateleft ventricular pressure and to perfuse the coronary arteries; thelatter was found necessary to prevent ischemia and stiffening inpreliminary experiments. Pressure was increased from 0 kPa toapproximately 1.44 kPa over 10 s by adjusting the height of a fluidcolumn connected to the cannula. Deformation of the left ventricle

Table 1Studies estimating the parameters in the Guccione and Costa material laws. For the current study, parentheses show the range of each parameter among scenarios that fitwithin a 25% increase from the minimum value of the objective function (i.e. least lack-of-fit). The Costa law is a generalization of the Guccione law, with separate coefficientsfor each term in the parentheses in Eq. (5). See Supplementary Fig. S1 for a visualization of the stress–strain relationships corresponding to these parameter scenarios.ED¼End Diastole, ES¼End Systole, PV¼Pressure Volume, MVO¼Mitral Valve Opening

Study Experiment Referenceconfiguration

Recording Time-pointsused

Species a (kPa) b1 b2 b3

Guccione law[10]a Passive inflation Cylinder PV-curves, implanted

markers5 Rat 2.2 9.2 2.0 3.7

Dog 2.4 26.7 2.0 14.7[8] Epicardial

suctionZero pressure Tagged MRI 3 Dog 0.10–1.0 39.5–93.0 6.1–61.6 3.1–73.1

[14] Passive inflation Zero pressure Tagged MRI 5 Pig 3.0 11.1 1.8 10.0[41] In vivo MVO Tagged MRI 2 (ED, ES) Sheep 0.12–0.35 9.2–67.1 5.0–26.6 9.3–21.6[6] Passive inflation Low pressure Tagged MRI 5 Pig 0.07–0.79 8.0–83.4 6.1–36.4 8.2–62.4[44] In Vivo MVO Tagged MRI 2 (ED, ES) Sheep 0.95 49.3 19.2 17.4[9]a In Vivo MVO Tagged MRI, pressure wire 2 (MVO, ED) Dog 1.7 14.3 4.5 0.76[24] In Vivo Zero pressure Tagged MRI, pressure wire 2 Human 0.3 41.7 9.1 51.5[45] Passive inflation Mid Diastole PV-curves 3 Mouse 1.1 8.0 2.0 3.7[15]a In vivo Calculated Tagged MRI, pressure wire 4–6 Human 2 19.3 10.7 12.8Costa lawb

[46]a In vivo Ellipsoid Implanted markers 2 (ED, ES) Dog 1.8 6.0 3.0–12.0 3.0–7.0[47] Shear tests Unstressed Tissue block Several Pig 0.22 42.5 7.8–18.6 10.9–11.0[7] Shear tests Unstressed Tissue block Several Pig 0.26 37.2 9.1–18.9 12.0[48] Passive inflation Calculated PV-curves Several Human 0.3 39.0 4.2–7.6 12.8–17.2Currentstudy

Passive inflation Zero pressure Echocardiography 8 Mouse 3.1 (2.0–5.6)

5.0 (2.7–7.0)

3.7 (2.1–5.2)

2.6 (1.5–6.0)

a These studies used a definition of a that was twice the one used in the other studies. Estimates in this table have been halved to make them directly comparable withthe others.

b The orthotropic Costa law has three parameters in place of b2 and two in place of b3. The values in the b2 and b3 columns for Costa-law studies show the variabilitybetween these non-isotropic coefficients.

Ø. Nordbø et al. / Computers in Biology and Medicine 53 (2014) 65–7566

was recorded by B-mode echocardiography (Vevo2100, VisualSo-nics) using a 35 MHz scan head. Three short-axis (probe mountedat the side of the ventricle) and four long-axis views (probemounted at apex) were recorded for each pressure level (seeFig. 1 for details). Due to capillary forces in the cannula, the leftventricle pressure was not equivalent to the height of the fluidcolumn. In a subsequent experiment, we therefore calibrated leftventricular pressure vs. fluid column height by a Samba pressurecatheter (Samba 3000; Samba Sensors AB), inserted via the apex.The Samba pressure catheter was autocalibrated using the Sambamemory control unit at room temperature. Proper calibration wasevaluated by pressure at 0 mmHg at no height of the water columnand 10 mmHg at the corresponding height of the water column.

2.2. Mesh of mouse ventricle with embedded microstructure

MR imaging: Diffusion tensor MRI (DTI) measurements of anisolated mouse heart (a different individual than the one used in the

inflation experiment) embedded in 1% agarose were performed ona 9.4 T horizontal bore MR system (Agilent Technologies, Santa Clara,CA) equipped with a shielded gradient system (Gmax¼1 T/m, risetime¼130 ms, inner diameter¼60mm), using a 13 mm-diametervolume transmit/receive coil. A 3D fast spin-echo sequence was usedas reported previously [18] with the following imaging parameters:TR¼1 s, number of echoes¼8, TEeff¼44.25ms, matrix size¼256�256�192, number of averages¼4, field of view 13�13�13 mm3 (yielding a resolution of 51�51�68 mm3), maximumb-value¼688 s/mm2 (including imaging gradients and cross-termsbetween imaging and diffusion gradients). Eight non-collineargradients directions were used, arranged according to an optimizedscheme based on the electrostatic repulsion principle [19].

Post-processing: The fibre field was extracted from these 3DDTI data as described previously [20]. A smooth cubic Hermitemesh [21] of the left ventricle was built, and the fibre field wasembedded into this mesh (see Fig. 3). The estimated transmuralgradient in fibre angle was similar to previous findings [22].

Fig. 1. (a) Sketch of experimental setup for echo measurements; (b) Mouse heart with cannula and echo recorder; (c) placement of long-axis views of the heart (451 rotationbetween views, probe placed apically to the heart); (d) placement of short-axis views (2 mm vertical spacing, probe placed at the side of the heart)

Ø. Nordbø et al. / Computers in Biology and Medicine 53 (2014) 65–75 67

2.3. Speckle tracking and estimation of 3D deformation

In ultrasound imaging, speckle pattern arises because sub-resolution scatterers interfere with the incoming ultrasound pulses.Although the reflected signal contains no information about theunderlying microstructure of the tissue [23], individual pieces oftissue respond consistently to the ultrasound, and can thus be tracedthrough a sequence of images. However, there is some uncertaintydue to decorrelation of the speckle pattern between successiveframes, as may be caused by out-of-plane motion, rotation, deforma-tion, and additive thermal noise. Apart from the ultrasound gel andprobe, no other external forces where applied to the ventricle.

Because the excised heart was still beating slightly, we removedimages that were from the systolic phase, as identified by a highmean absolute signal difference between subsequent images. Weused block-matching [23] and followed endo- and epicardial pointsbetween subsequent images through the inflation, as detailed inAppendix A. The particle paths estimated from the different sliceswere aligned into 3D deformed configurations for the variouspressures (methods are explained in Appendix B), and a set ofsmooth cubic Hermite interpolated meshes (CH meshes) were thenconstructed combining the methods described in [21,24] (see Fig. 3).

2.4. Mechanical simulations

The myocardium was modelled as a hyper-elastic incompres-sible material [25] with homogenous, transversal isotropic

properties. The framework for mechanics is described in [26].Adopting the approach of Ref. [14], we used the cubic Hermitemesh corresponding to the zero pressure configuration (CH1, seeFig. 3) of the left ventricle (Fig. 4) as the reference configuration[27]. According to [13], the estimated stiffness parameters werenot particularly sensitive to the magnitude of the residual stress[28], which supports our approach.

We formulate the stress equilibrium equations by using theprinciple of virtual work on a general deformable body with initialvolume V and surface area AZVS : δEdV�

ZApappF

�TNUδudA¼ 0: ð1Þ

Here S is the second Piola–Kirchhoff stress tensor and δE is thevirtual Lagrangian strain tensor. papp is the applied ventricularpressure acting on the endocardial surface, F �T is the inverse ofthe transposed deformation gradient tensor, N is the normal vectorfor the undeformed material surface while δu is an arbitrary virtualdisplacement. For hyper-elastic materials, the components of thesecond Piola–Kirchhoff stress tensor are calculated by differentiatingthe strain energy function by the corresponding components of theLagrangian strain tensor. For incompressible materials, the strains areseparated into distortional and dilatational components, and wecalculate the second Piola–Kirchhoff stress tensor from

S¼ ∂W∂E

� JpðFTFÞ�1; ð2Þ

Fig. 2. Echo recordings of a left ventricle excised from a mouse, at zero pressure (left) and passively inflated to 1.44 kPa (right). Particle paths during inflation, estimated byspeckle tracking, are shown as red lines (the same paths are shown in both left and right images for reference). Red circles mark segmentation points of the inner and outerwalls at the respective pressures, so that the left and right images have the starting and ending positions circled, respectively. Image dimensions are 12 mm�11 mm (longaxis) and 12 mm�11 mm (short axis). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).

Ø. Nordbø et al. / Computers in Biology and Medicine 53 (2014) 65–7568

where W is the strain energy function and F is the deformationgradient tensor. The scalar P serves as a Lagrangian multiplierinterpreted as the hydrostatic pressure in the material/tissue. Thework done by the hydrostatic forces is zero due to the

incompressibility constraint, expressed on variational form as

ZVðJ�1ÞpdV ¼ 0; ð3Þ

Fig. 3. Workflow for generating cubic Hermite (CH) meshes. Material points were followed by speckle tracking from different views and aligned into quadratic interpolatedmeshes (Q-mesh). The zero pressure CH mesh (CH1) was built by fitting the fibre field (yellow lines) to the binary mask of the Q-mesh corresponding to zero pressure (Q1)[21]. Then we mapped all Gauss point positions back to Q1 and calculated their displacements corresponding to the various pressures. These displacements were usedtogether with the CH1 mesh to build cubic Hermite meshes for the various pressures [24]. The CH1 mesh was used as reference configuration for the simulation, while theother CH meshes constituted the experimental measured configurations for the various pressures. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.).

Fig. 4. Cubic Hermite interpolated meshes of the zero pressure (left) and maximal pressure (1.44 kPa) configuration (right) based on the deformation field captured byexperiments and speckle tracking. The nodes in the basal plane were constrained such that no material points contained on the basal surface were allowed to move in the zdirection. The endocardial basal node at the centre of the free wall (red dot) was not allowed to move at all, while one at the middle of the septum (green dot) was onlyallowed to move in the radial direction. Yellow lines indicate fibre angles fitted from DTI data. (For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.).

Ø. Nordbø et al. / Computers in Biology and Medicine 53 (2014) 65–75 69

where J is the determinant of the deformation gradient tensor F. Thestress–strain relation in hyper-elastic models follows from ourassumption of a strain energy function, W . The strain energy densityfunction, W , was described by the Guccione law [13]

W ¼ 12aðeQ �1Þ ð4Þ

where

Q ¼ b1E211þb2ðE222þ2E23E32þE233Þþb3ð2E12E21þ2E13E31Þ: ð5Þ

Here, a, b1, b2 and b3 are material specific parameters, while Exyare the various components of the Lagrangian strain tensor alignedto the underlying tissue microstructure. The subscripts indicatethe microstructural directions (1 is along the fibres while 2 and3 are orthogonal directions, normal to the fibres).

Table 1 summarizes published parameter estimates for theGuccione [13] and Costa [29] laws, showing that stiffness para-meters vary across two orders of magnitude. To comprehensivelycover the parameter space, simulations were performed withparameters incremented based on a log scale from the minimumto the maximum value reported in Table 1.

For the 4 Guccione law parameters, we used a full factorialdesign with eight levels, resulting initially in 84 ¼ 4096 parametersets. For each parameter set, the simulation model was run for thesame pressure steps as in the experiment (eleven levels rangingfrom 0 to 1.44 kPa).

We applied boundary conditions constraining the basal planeto not move in the z-direction, which should correspond well tothe experimental setup where the heart was attached to thecannula through the aortic valve. Additional Dirichlet boundaryconditions to avoid whole-body rotation/translation are shown inFig. 4.

Inside the cavity, we assumed a homogeneous internal pres-sure, which should be a reasonable assumption due to the lowvelocity of the flow. In the model we neglected the forces betweenthe ultrasound gel and the heart, which is the simplest possibleassumption until such forces are measured. As in previous pub-lished studies of similar setups [14], we also assumed that theinfluence from the right ventricle was negligible.

2.5. Comparison of measured and simulated deformations

In both the simulations and the measurements, the shape ofthe heart ventricle at a given pressure is represented by nodepositions and an associated cubic Hermite interpolation of thesurfaces between nodes. We tried two different objective functionsfor comparison of experimental and simulated deformation,restricting the computation to the most biologically relevantpressure range [30], 40.2 kPa.

1. Comparison of positions for all endocardial and epicardialnodes in the CH mesh. The criterion for good fit was the L2norm between experimental and simulated node position,averaged over the relevant pressures M and nodes N in themesh

Ω1 ¼1

MN∑M

m ¼ 1∑N

n ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxsim�xexpÞ2þðysim�yexpÞ2þðzsim�zexpÞ2

q� �mn

ð6Þ

2. Comparison of K¼4 aggregated phenotypes (change inVolume (ΔV), short and long axis diameter (ΔSa and ΔLa),and elastic energy (

RPdV , computed as the integral of the

pressure–volume curve). For each phenotypic variable, thedifference between simulation and experiment is nondimen-sionalized as a proportion of the respective experimental

value. Then, each variable is weighted equally in averagingover the K phenotypes and M relevant pressures

Ω2 ¼1

MK∑M

m ¼ 1∑K

k ¼ 1abs

Φsim�Φexp

Φexp

� �mk

ð7Þ

These objective functions were computed for all simulated para-meter scenarios, and we identified the parameter regions thatcorresponded to low values of each objective function. Theidentified region of the parameter space was further analyzed bymanually selecting new parameter limits and running new simu-lations for a full factorial design over the new parameter regions.We did this in an iterative way (3 times), and got an overview ofthe landscape of the parameter space and were able to see thecorrelation between estimated parameters.

We evaluated the ability of each objective function to preciselyestimate parameters (i.e. identify a small set of well-fitting para-meter scenarios) by asking “how many parameter scenarios liewithin a given % increase from the optimal value (least lack-of-fit)of the objective function?” This is valid because both objectivefunctions have a meaningful absolute zero point.

3. Results

3.1. Experimental deformation

The deformation of the experimentally manipulated heart wasestimated by speckle tracking of four long-axis and three short-axisimage sequences. The free wall generated larger longitudinal strainsthan the septum, causing the apex to tilt towards the septum duringthe inflation (Fig. 2). The pressure–volume relationship was quitelinear within the interval of applied pressures (Fig. 7a), in agree-ment with previous in vivo work [31,32]. The mesh correspondingto zero and maximum pressures are shown in Fig. 4.

3.2. Comparison of simulated vs. experimental deformation

The objective function based on aggregated phenotypes(Eq. (7)) was more selective than the one based on node positions(Eq. (6)), in terms of identifying some parameter scenarios asbetter fitting than others (Fig. 5). For example, the number ofparameter scenarios within a 50% increase from the minimum ofthe objective functions (6) and (7) was 5729 and 65, respectively.Likewise, the contrast between Figs. 6 and 7a shows how thescenarios selected by the latter criterion give a much tighter matchto the observed inflation trajectory. Furthermore, the objectivefunction (6) tended to select scenarios with much lower compli-ance than observed in the experiment (Fig. 6) and worse fits forwhole organ phenotypes (Fig. 7). Subsequent analyses were there-fore based on the objective function in Eq. (7). Especially for thevolume, short axis and elastic energy, the model was able tocapture the essential dynamics.

3.3. Correlation structure of well-fitting parameter combinations

Well-fitting parameter combinations (i.e. those with a lowvalue of the objective function) showed considerable variationalong all individual parameter axes, though there were clearpatterns of covariation among parameters (Fig. 8). Similar to thefinding in [24], the well-fitting parameter scenarios tended to lienearly along a straight line in the space of (log a, log(b1þb2þb3)).

The estimated overall stiffness parameter a was clearlybounded from below, while the values of the b were boundedfrom above. Conditional on a value for a, the b1 and b2 parameterswere also bounded from below, but the optimal estimate of b3approached parameter regions with bad convergence of the

Ø. Nordbø et al. / Computers in Biology and Medicine 53 (2014) 65–7570

mechanics solver. The tissue was stiffer along the fibres than in thedirections orthogonal to the fibres (b1¼5.0, b2¼3.7, b3¼2.6), withb2 more tightly constrained than b1 and b3 (Table 1, Fig. 8).Although the point estimate appears comfortably inside theexplored parameter region as viewed in the projections in Fig. 8,there were convergence problems for some of the parameterscenarios adjoining the best-fitting one as viewed in more thantwo parameter dimensions. Therefore, we cannot concludewhether the material parameters would be uniquely identifiablewith a more robust numerical integration solver.

4. Discussion

There are several advantages to the experimental and compu-tational pipeline presented here. Echocardiography is simple touse during pressure manipulation of excised hearts and is fasterthan tagged MRI (as used in e.g. [14]), thus reducing problemswith rigor mortis. Furthermore, it can follow internal materialpoints over the entire geometry [9], unlike the method of videorecordings of epicardial markers (used by [33]). Passive inflationexperiments avoid the sectioning of tissue required for uni-axialand biaxial material testing [34], which would disrupt the micro-structure and influence the material properties. However, thislimits the possibility of experimentally inducing shear forces asdescribed by [35]. [8] argue that passive inflation producesinsufficient transverse shear strain to quantify the material proper-ties in those directions, and therefore developed an epicardialsuction setup for isolated arrested hearts. However, our resultsindicate that the relevant parameter (b3) is constrained fromabove (Fig. 8), but shows larger relative uncertainty than the otherb parameters in the Guccione law Eqs. (4), (5).

Our numerical experiments enable parameter estimation andan overview of the structure of parameter uncertainty across abounding box of all previously published parameter estimates(Table 1; see Supplementary Fig. S1 for a visual comparison of thecorresponding stress–strain relationships). The simultaneous esti-mation of parameters is an improvement over one-at-a-timeoptimization, and highlights the possibility that many empiricaldata sets may not be able to support the complexity of commonlyused models and material laws [24,36]. The robustness of anexhaustive search comes with the associated computationalcost of a wide sweep of parameters, as compared with otherapproaches [4,24,37,38].

The definition of an adequate objective function to comparedata and model is critical for a data assimilation method. Both dataand model are limited by acquisition issues and assumptions,respectively, and a suitable choice of the space to compare them isrequired. Parameters were easier to identify using aggregatemeasures of cardiac deformation, rather than the more detailednode positions, for the goodness-of-fit criterion (objective func-tion). This may reflect an inability of our model to mimic certaindeformations that occurred in the experimental data, particularlya bending due to larger longitudinal strains in the free ventricularwall than the septum during inflation (Fig. 3, bottom), or circum-ferential variation in material stiffness, violating our assumption ofhomogeneity. Other possible causes originate in the experimentaldata, a 3D reconstruction of 2D observations that is subject tomisregistration and out-of-plane errors. Regardless of the cause,our results suggest that a gross description of mechanical defor-mation introduces the benefit of a more relevant and robustcomparison between data and model.

As opposed to the traditional and more intuitive direct use ofdisplacement fields for the objective function [9,24], recent workshave explored the use of integral deformation metrics [37], oradvanced mathematical frameworks (using the concept of cur-rents) [39]. Our proposed aggregated phenotypic metric is asimple and effective solution, and offers an intuitive interpretationof where the agreement and disagreement occurs. Preliminaryresults with our echo data, not reported here, showed that theinclusion of twist (defined as the difference in rotation betweenbase and apex as done clinically [40]) in the objective function wasdetrimental, probably due to a combination of imaging acquisitionand analysis limitations. Further research is needed to optimizethe definition of the objective functional, and to analyse thecomparative performance of the different alternatives.

Because the excised heart kept beating a little throughout theexperiment (see “Speckle tracking” in Methods), despite perfusion

0% 25% 50% 75%0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Eq. (6) (displacements)

Eq. (7) (aggregated)

Objective function value (percent increase from minimum)

EC

DF

(pro

porti

on o

f si

mul

ated

par

amet

er s

cena

rios)

Fig. 5. The objective function based on aggregated geometry measures (Eq. (7))was more precise in identifying a narrow set of well-fitting parameter scenarios,compared to the objective function based on node displacements (Eq. (6)). Linesshow empirical cumulative distribution functions, i.e. the proportion of simulatedparameter scenarios for which the objective function was less than the value givenby the horizontal axis. Colour shading for the objective function value is used forcomparison with Figs. 6, 7 and 8. (For interpretation of the references to colour inthis figure legend, the reader is referred to the web version of this article.).

0 0.5 10

10

20

30

40

50

60

Δ V

(μ

l)

LVP (kPa)

Fig. 6. Node positions made a poor goodness-of-fit criterion (Eq. (6)) for matchingsimulations (thin lines coloured with the same colour scale used in Fig. 5) toobservation (thick red line). Compliance is unreasonably low for many of theparameter scenarios that give low values of this objective function. Contrast withthe corresponding plot (Fig. 7a) for the other objective function (Eq. (7)), where thebest-fitting scenarios cluster more tightly around the experimental results. Whitedots indicate unphysiological pressure levels that were omitted in computing theobjective function. Note that this plot shows the increase in volume as a result ofimposed pressure, and so is transposed relative to the traditional P–V loop plot.(For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.).

Ø. Nordbø et al. / Computers in Biology and Medicine 53 (2014) 65–75 71

with cardioplegic solution, and we were unable to measure thecalcium level inside the heart, we cannot be sure about ourassumption that there were no active forces. This assumptiongreatly simplifies the decoupling of active and passive relaxationprocesses [15], but may overestimate the stiffness of the tissue. Onthe positive side, the residual heartbeat confirmed that the excised

heart was still capable of contraction, and so its material proper-ties remained similar to in vivo.

Our experiment showed that the fibre stress-strain relationshipfor the mouse left ventricle was quite linear over a biologicallyrelevant range of pressures (Fig. S1). This brings challenges inidentifying, separately from the overall stiffness a, the b parameters

0 0.5 10

20

40

60

Δ V

(μ l)

0 0.5 10

0.5

1

ΔΔ

Sa (m

m)

0 0.5 10

0.5

1

Δ L a (m

m)

0 0.5 10

10

20

30

PdV

(μ J

)LVP (kPa)

Fig. 7. Comparison of whole-organ geometrical characteristics (see text) as a function of left ventricular pressure (LVP), between experimental data (red) and simulated data,using a goodness-of-fit measure based on whole-organ phenotypes (Eq. (7)). Colour indicates the value of the objective function (Eq. (7)) relative to the minimum value(colour scale as in Figs. 5 and 6). Geometric measurements are shown as differences from the zero-pressure values of volume¼V0¼58.1 μL, long-axis diame-ter¼LA0¼6.88 mm, short-axis diameter¼SA0¼3.39 mm. Panel (a) contrasts with Fig. 6. White dots indicate unphysiological pressure levels that were omitted in computingthe objective function. The observed experimental relationship between pressure and volume was near-linear, in agreement with in vivo PV loop data (see references intext). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).

0 2 4 60

5

10

15

a (kPa)

b1

0 2 4 60

5

10

15

a (kPa)

b2

0 2 4 60

5

10

15

a (kPa)

b3

0 5 10 150

5

10

15

b1

b2

0 5 10 150

5

10

15

b1

b3

0 5 10 150

5

10

15

b2

b3

Fig. 8. Projections of the objective function onto two-dimensional slices of the stiffness-parameter space. Each panel shows, conditional on the two parameters named onthe axes, the best fit (as defined in Eq. (7); red dot) among all combinations of the remaining parameters. Black dots indicate simulated parameter sets; red dot shows thebest-fitting parameter combination among the simulations. Diagonal line shows 1:1 ratio of parameters for reference. Parameter scenarios outside the displayed axis limitswere also run, spanning the full parameter ranges reported in Table 1, but these did not fit well and are omitted here for clarity. (For interpretation of the references to colourin this figure legend, the reader is referred to the web version of this article.).

Ø. Nordbø et al. / Computers in Biology and Medicine 53 (2014) 65–7572

that pertain to curvature in the material law (Fig. 8, [24]). However,in contrast to [24], we find that a is well bounded from below whilethe b's are bounded from above: No parameter scenarios with low aor very high b's could be made to match the data well (Fig. 8).

The mouse myocardium was found to be stiffer along the fibresthan normal to the fibres, which is in agreement with the findingsin other model organisms [6]–[10]. The value of b1 was consider-ably higher than b2 and b3, in agreement with findings in in vivosheep and human hearts [15,41] and in tissue blocks of pig heart[7]. The stiffness parameters estimated in this work describe arelatively soft material (low stress–strain relationships) comparedwith most of the literature, and is most similar to the materialproperties found in a similar experiment on rat hearts [10].

Complementary experiments are needed to better constrainthe material parameters of this law. We inflated the ventricle from0 to 1.44 kPa, i.e. within the normal diastolic pressure range. In thisrange the stress (or pressure) vs. deformation relationship wasrelatively linear, while the nonlinear behavior occurs at higherpressures. The estimated parameters capture the passive materialproperties for normal diastolic pressures in a healthy specimen,but extrapolation outside this range, for example for the study ofincreased filling pressures, will require further characterization.

Several future refinements are possible in this parameterestimation workflow. Better speckle tracking would improve theability to find regional differences in material passive properties byfollowing material points in a Lagrangian, moving frame ofreference, as suggested for tagged MRI by [16]. Other challengesinclude the aligning of 2D data into 3D geometries. Misalignmentmay cause spurious bulging in the estimated deformation of theapex region (Fig. 4), and is probably one of the main reasons forthe lack of fit between the simulated and experimental deforma-tion data on a nodal basis (Eq. (6) and Fig. 5). Manual interactionscould be reduced, facilitating e.g. the study of individual variation,by automating the selection of valid end diastolic frames from theechocardiographic recordings. Also, mesh construction mightbenefit from being constrained by incompressibility [42]; ourcurrent meshes differed in volume by 15% from the start to theend of filling. Complementary experiments are needed to uniquelyconstrain the material parameters of this law. In particular, theapplication of higher pressures, e.g. of relevance to hypertension,might bring out nonlinearities in the force-deformation relation-ship that would constrain the possible parameter combinations.

We have demonstrated the feasibility of quantifying mousemyocardial stiffness using the experimental setup and associatedcomputational pipeline described above, and found that it is aviable method that deserves wider application. A better knowl-edge of passive properties would shed light on diastolic dysfunc-tion and is an important step towards understanding diastolicheart failure in mouse and human.

Conflict of interest statement

The authors declare no competing interests.

Acknowledgments

This work was funded by the Research Council of Norwayunder the eVITA programme, project number 178901/V30 and bythe Virtual Physiological Rat project funded through NIH grantP50-GM094503. NOTUR, the Norwegian Metacenter for Computa-tional Science, provided computing resources under projectnn4653k. JES is a BHF Senior Basic Science Research Fellow, andacknowledges funding from the Biotechnology & BiologicalSciences Research Council (BB/I012117/1).

Appendix A. Speckle tracking

Two-dimensional particle paths can be estimated from a seriesof images by block-matching [23]. The algorithm compares aregion of the current image with nearby regions in the nextimage, finding the best-matching displacement by optimizingsome matching criterion. This is repeated for each region ofinterest in the current image, for example centred on manuallychosen landmarks to be tracked.

Each series consisted of approximately 300 images (long-axis:512�384 pixels; short-axis: 512�472 pixels; pixel size0.0234 mm). However, the heart was still beating a little duringthe experiment, whereas our focus was on passive mechanics only.We therefore removed systolic images before speckle tracking,identifying these images by their large mean absolute signalintensity difference between subsequent images. Because thesampling rate was high relative to the rate of displacement, wesubsampled the data to every third image to avoid round offproblems with zero displacements. The speckle tracking algorithmwas implemented in Matlab ([43]) as follows:

� We chose limits for the search algorithm, consisting of the sizeof the matching kernel window and the maximum alloweddisplacements. Kernel windows were 17�31 and 9�17 pixelsfor long- and short-axis images, respectively. The maximumallowed displacements between two subsequent images were2 pixels (in x and y directions) for long-axis and 3 pixels forshort-axis images.

� The ventricular wall was more visible and easier to segmentfrom the high pressure images. We therefore selected materialpoints along the inner and outer surface of the ventricle for theimage corresponding to maximal pressure, and worked back-wards from high to low pressure images. The selected materialpoints were used as central points for block matching betweensubsequent images in the following automatic procedure.� The ventricle in the current image was masked based on the

position of the material points in the preceding image (echointensity was set to NaN outside the ventricle).

� The displacement for each material point in the currentimage was estimated by matching the neighbourhood of thematerial point against candidate regions in the next image,minimizing the mean absolute difference between the sub-images.

� Due to considerable noise, especially in the basal region inthe long axis images, we imposed additional constraints onthe deformation:� The material points closest to the base were allowed to

move only in horizontal direction and the outward radialsearch region was reduced.

� The local wall thickness was measured. If the wallthickness was shrinking compared with the max pres-sure image, the material points on the inner wall weremoved inwards, normal to the surface of the outer wall.

� The displacements were smoothed using a cubic spline.(Displacements from short axis images were converted topolar coordinates before smoothing).

� The coordinate position of each material point was updated,based on its estimated displacement, before the next imagewas analysed by block-matching.

Appendix B. From 2D particle paths to 3D deformation

The individual 2D particle paths were smoothed over time andinterpolated so we could estimate the position of each 2D material

Ø. Nordbø et al. / Computers in Biology and Medicine 53 (2014) 65–75 73

point for any specified LVP. We used the long-axis images to finddisplacement in longitudinal and radial direction, while circum-ferential displacements were obtained from the short-axis images.The short-axis images also contain information about the radialmovement, but this was not utilized because the out-of-planemovement is larger for these images than for long-axis images.The long-axis images were centred, based on the centre of mass ofthe basal points in the zero pressure images, and rotated about thez-axis (01, 451, 901 and 1351) to correspond to the experimentalviews (see Fig. 1a). The apical points were defined as a weightedaverage of the most apical points from the different long-axisslices. The short-axis images were aligned to match the long-axisimages with respect to the free wall and the septum. The mostapical short-axis image was aligned with the long-axis images inthe z-direction, so that the inner diameters of the long- and short-axis images were identical in the maximum pressure configura-tion. The circumferential displacements of all nodes were theninterpolated in the z-direction between the short-axis slices byusing a cubic spline (we required zero circumferential displace-ment in the base, in accordance to the experimental setup). Thenodes were then used as nodes in a triquadratic interpolatedmesh, and estimated 3D-displacements were used to calculate theconfiguration for each pressure level (see Fig. 3).

Appendix C. Supplementary material

Supplementary data associated with this article can be found inthe online version at http://dx.doi.org/10.1016/j.compbiomed.2014.07.013.

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