MODELING CARDIOVASCULAR AND RESPIRATORY …...reduction in cardiac output, elevation of left atrial...

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MODELING CARDIOVASCULAR AND RESPIRATORY DYNAMICS IN CONGESTIVE HEART FAILURE LAURA M. ELLWEIN 1 , SCOTT R. POPE 2 , ALIANG XIE 3 , JERRY J. BATZEL 4 , C.T. KELLEY 5 , METTE S. OLUFSEN 5 1) Olin Engineering Center, Marquette University, 1515 West Wisconsin Av, Milwaukee, WI 2) SAS, 100 SAS Campus Drive, Cary, NC 3) Pulmonary Physiology Laboratory, William S. Middleton Veterans Hospital, 2500 Overlook Tarace, Madison, WI 4) Inst. for Math. and Scientific Computing, Univ. of Graz, Heinrichstrasse 36, Graz, Austria 5) Dept. of Mathematics, North Carolina State University, Campus Box 8205, Raleigh, NC Abstract. This study develops a coupled cardiovascular-respiratory model that predicts cerebral blood flow velocity (CBFV), arterial blood pressure, end-tidal CO 2 , and ejection fraction for a patient with congestive heart failure. The model is a lumped parameter model giving rise to a system of ordinary differential equations. We use sensitivity analysis and subset selection to identify a set of model parameters that can be estimated given the patient data. Gradient based nonlinear optimization is used to estimate the subset of parameters. Optimization was caried out first for the cardiovascular submodel and subsequently for the respiratory model. Once a set of optimal parameters were found, the coupled model was computed to confirm that the model is still able to predict the observed data. Results showed that with the approach and methods presented in this paper it is possible to examine and quantify identifiability of model parameters. Using this approach we identified 5 key cardiovascular parameters and 4 key respiratory parameters. Nonlinear optimization techniques was used to estimate these parameters and we tested that values for all parameters were physiologically reasonable for a patient with congestive heart failure. 1. Introduction. The strong pathophysiological links between sleep-related breathing disorders and cardiovascular diseases have recently received raised attention among clinicians and researchers. On one hand, sleep-related respiratory disorders, especially obstructive apneas, have been identified as an independent risk factor for hypertension, myocardial infarction and stroke due to the associ- ated recurrent upper airway vibration, intermittent hypoxia, surges of sympathetic neural activities, and repetitive arousals and resulting sleep fragmention a (see for example [23, 16]). Central sleep apnea has been considered a marker for congestive heart failure (CHF) and also been associated with the deterioration of heart function in CHF [21]. On other hand, epidemiological data reveal that about one-half of patients with CHF develop periodic breathing [17, 53]. CHF is characterized by a reduction in cardiac output, elevation of left atrial pressure and pulmonary capillary wedge pressure, accumulation of fluid in the lung, reduced functional residual capacity (FRC), and prolongation of circulation time. All of these factors may destabilize breathing. In fact, a greater central and pe- ripheral chemosensitivity to CO 2 (i.e., controller gain) has been observed in CHF patients [54, 9] and CHF animal models [57]. Respiratory changes duiring sleep combined with these cardiovascular and control sensitivity changes can contribute to the occurrence of sleep apnea ([12, 37, 4]. Furthermore, the low cardiac output also reduces brain perfusion [9, 36] and attenuates cerebrovascular chemosen- sitivity [22, 63]. The combination of hypoperfusion and low cerebrovascular reactivity impairs the CBF protective mechanism for breathing stability [64]. Finally, CHF and its concomitant cardiovas- cular insufficiency and potential for reduced cerebral perfusion has been associated with impaired cognitive function [5], while syncope is an issue in patients with CHF [26]. In general, although the Key words and phrases. Cardiovascular modeling, Respiratory modeling, Parameter estimation, Model validation, Sensitivity analysis, Subset selection. 1

Transcript of MODELING CARDIOVASCULAR AND RESPIRATORY …...reduction in cardiac output, elevation of left atrial...

Page 1: MODELING CARDIOVASCULAR AND RESPIRATORY …...reduction in cardiac output, elevation of left atrial pressure and pulmonary capillary wedge pressure, accumulation of fluid in the lung,

MODELING CARDIOVASCULAR AND RESPIRATORY DYNAMICS IN

CONGESTIVE HEART FAILURE

LAURA M. ELLWEIN1, SCOTT R. POPE2, ALIANG XIE3, JERRY J. BATZEL4, C.T. KELLEY5, METTES. OLUFSEN5

1) Olin Engineering Center, Marquette University, 1515 West Wisconsin Av, Milwaukee, WI2) SAS, 100 SAS Campus Drive, Cary, NC3) Pulmonary Physiology Laboratory, William S. Middleton Veterans Hospital,

2500 Overlook Tarace, Madison, WI4) Inst. for Math. and Scientific Computing, Univ. of Graz, Heinrichstrasse 36, Graz, Austria5) Dept. of Mathematics, North Carolina State University, Campus Box 8205, Raleigh, NC

Abstract. This study develops a coupled cardiovascular-respiratory model that predicts cerebralblood flow velocity (CBFV), arterial blood pressure, end-tidal CO2, and ejection fraction for apatient with congestive heart failure. The model is a lumped parameter model giving rise toa system of ordinary differential equations. We use sensitivity analysis and subset selection toidentify a set of model parameters that can be estimated given the patient data. Gradient basednonlinear optimization is used to estimate the subset of parameters. Optimization was caried outfirst for the cardiovascular submodel and subsequently for the respiratory model. Once a set ofoptimal parameters were found, the coupled model was computed to confirm that the model is stillable to predict the observed data. Results showed that with the approach and methods presentedin this paper it is possible to examine and quantify identifiability of model parameters. Usingthis approach we identified 5 key cardiovascular parameters and 4 key respiratory parameters.Nonlinear optimization techniques was used to estimate these parameters and we tested that valuesfor all parameters were physiologically reasonable for a patient with congestive heart failure.

1. Introduction. The strong pathophysiological links between sleep-related breathing disordersand cardiovascular diseases have recently received raised attention among clinicians and researchers.On one hand, sleep-related respiratory disorders, especially obstructive apneas, have been identifiedas an independent risk factor for hypertension, myocardial infarction and stroke due to the associ-ated recurrent upper airway vibration, intermittent hypoxia, surges of sympathetic neural activities,and repetitive arousals and resulting sleep fragmention a (see for example [23, 16]). Central sleepapnea has been considered a marker for congestive heart failure (CHF) and also been associated withthe deterioration of heart function in CHF [21]. On other hand, epidemiological data reveal thatabout one-half of patients with CHF develop periodic breathing [17, 53]. CHF is characterized by areduction in cardiac output, elevation of left atrial pressure and pulmonary capillary wedge pressure,accumulation of fluid in the lung, reduced functional residual capacity (FRC), and prolongation ofcirculation time. All of these factors may destabilize breathing. In fact, a greater central and pe-ripheral chemosensitivity to CO2 (i.e., controller gain) has been observed in CHF patients [54, 9] andCHF animal models [57]. Respiratory changes duiring sleep combined with these cardiovascular andcontrol sensitivity changes can contribute to the occurrence of sleep apnea ([12, 37, 4]. Furthermore,the low cardiac output also reduces brain perfusion [9, 36] and attenuates cerebrovascular chemosen-sitivity [22, 63]. The combination of hypoperfusion and low cerebrovascular reactivity impairs theCBF protective mechanism for breathing stability [64]. Finally, CHF and its concomitant cardiovas-cular insufficiency and potential for reduced cerebral perfusion has been associated with impairedcognitive function [5], while syncope is an issue in patients with CHF [26]. In general, although the

Key words and phrases. Cardiovascular modeling, Respiratory modeling, Parameter estimation, Model validation,Sensitivity analysis, Subset selection.

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2 L.M. ELLWEIN, S.R. POPE, A. XIE, J. J. BATZEL, C.T. KELLEY, M.S. OLUFSEN

underlying mechanisms for the interaction between the respiratory disorders and cardiaovasculardisorders have not fully understood, the two disorders eventually exacerbate each other, creating avicious cycle.

Modeling and especially models that can be applied in the clinical setting can contribute to thedevelopment of methods for assessment of respiratory function such as estimating central and periph-eral control gain and CO2 reactivity. Models can also provide insight into the control mechanismsthat govern respiratory function, respiratory interaction with cardiovascular control, and mutualregulation between ventilation and cerebral blood flow (CBF) through partial pressure of arterialCO2.

Recent models that address aspects of the above topics include work by Dong and Langford [13],which developed a model to study factors affecting stable behavior of the cardiovascular-respiratorysystem in heart failure and Bidini et al. [38, 39] who applied a combined cardiovascular-respiratorymodel to study the Valsalva maneuver and factors affecting cerebral blood flow. Recent experimentalstudies include work investigating the differential role of partial pressures of CO2 and sympatheticresponse on vasoconstriction [1], and the role of cerebral vascular function in setting ventilatoryresponse via influence on CO2 [63].

Thus, mathematical models that integrate ventilatory and cardiovascular system dynamics andcontrol functions have the potential to be used as a diagnostic tool as well as to obtain moredetailed knowledge of the dynamics of the underlying complex systems including aspects related tocardiovascular diseases. Furthermore, models that can be used to investigate the cardiovascular-respiratory coupling both in health and disease has potential to help understanding and diagnosingclinical conditions. To understand this complex dynamics models should be applicable for analyzingdynamics during typical orthostatic and ventilatory tests such postural change, step changes ininspiratory CO2 concentrations, or rhythmic breathing at various rates. These requirements indicatethat models should include a fair degree of complexity. On the other hand, for a given modelto have an impact in clinical settings, it is essential that the model can predict patient specificoutcomes such as systemic and cerebrovascular resistances. The latter is often studied by solvingan inverse problem, using a model combined with experimental data to predict patient specificparameters that minimize the difference between observed and computed quantities such as arterialblood pressure, cerebral blood flow velocity, or end-tidal CO2. However, estimating parametersfrom complex nonlinear models is difficult, in particular since parameters may be correlated or notsensitive to given observations. Developing better techniques for analyzing such complex data is themain focus in this manuscript.

To achieve an effective compromise between model complexity and clinical application requiresa modular approach to model analysis. This entails several considerations. First, model reductionshould be applied to reflect only the key functional elements under investigation. This simplifiesthe parameter estimation problem but care needs to be taken to avoid loss of the models’ predictivepower. Second, a given model must be analyzed with regard to available data so that key model pa-rameters sensitive to the observed data can be effectively and reliably estimated within a reasonabletime frame. Other less sensitive parameters should be estimated using information from literature,statistics, or allometric scaling. Conversely, sensitivity analysis can provide suggestions for exper-imental design. However, sensitivity analysis alone does not fully enable estimation of all modelparameters. Therefore, this analysis must be coupled with a more advanced analysis investigatingcorrelations between model parameters, e.g., two correlated sensitive parameters should not both beestimated.

This study develops a coupled pulsatile cardiovascular-respiratory model applicable to clinicalproblems related to CHF. We use sensitivity analysis to rank model parameters from the most tothe least sensitive with respect to the available data. Furthermore, we study correlation of modelparameters using subset selection.

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CARDIOVASCULAR & RESPIRATORY DYNAMICS IN CONGESTIVE HEART FAILURE 3

RightVentricleLeft

Ventricle

Lungs

Systemic Tissue

Cerebral Tissue

Body

Head

Ccv

pa,g

pD1,g

pD2,g

pD3,g

VD

pI,g

pexp,CO2

qpRp

CpaVpa

ppa

Vrv

Cpv

ppv

qpv

Vpv

RmvRpv

ca,g

Rc qc

Rs

MS,g

cS,g

cB,g

qs

ElvVlvErv

Rtv

qpa

Vsv Csv

qcv

psv

pcv pca

Rcv Rca

qca

Rav

plvprv

psa

VsaCsa

VcaCca

qsa

Thorax

Vcv

MB,g

qsv

csv,g

Figure 1. Compartmental model of systemic and pulmonary circulations. Systemic (subscript s),cerebral (subscript c), and pulmonary (subscript p) arteries (subscript a) and veins (subscript v);Vessels carrying oxygenated blood (systemic arteries and pulmonary veins) are red, while vesselscarrying deoxygenated blood (systemic veins and pulmonary arteries) are blue. All vascular com-partments represent a group of vessels with similar pressure p [mmHg]. Each vascular compartmentis characterized by its volume V [ml] and compliance C [ml/mmHg]. The left (dark red, subscript lv)and right (dark blue, subscript rv) ventricles generate pulsatile pressure plv, prv [mmHg]. These twocompartments are defined using time-varying elastance E [mmHg/ml]. Flow q [ml/sec] between com-partments are opposed by constant resistances R [mmHg sec/ml]. Tissue compartments (magenta)account for exchange of gases (O2 and CO2). Each vascular bed is characterized by a metabolicrate M [mlSTPD/sec], and the gas concentrations in the tissues are denoted by c [mlSTPD/ml]. Thelungs are represented by three dead-space compartments each predicting the partial pressure of thegases pDi,g [mmHg]. The partial pressure of the gases in the inspired air pI,g, and the end tidalpartial pressure of expiratory CO2 pexp,CO2

are marked separately.

2. Methods. The cardiorespiratory model depicted in Figure 1 is a lumped parameter compart-mental model designed to predict systemic arterial blood pressure psa, cerebral blood flow velocityvc, and end tidal partial pressure of expiratory CO2 pexp,CO2

for a CHF patient resting in supineposition.

For the purposes of this paper, we are considering dynamic oscillatory variations in cardiovascularand respiratory system quantities around a resting steady state. Hence, control mechanisms are notresponding to system stress and control responses are not directly invoked. For this reason we donot consider control submodels in this study. Note also that H and VIE are inputs to the model sothat variations in the controls of these quantities are implicitly included in the model.

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4 L.M. ELLWEIN, S.R. POPE, A. XIE, J. J. BATZEL, C.T. KELLEY, M.S. OLUFSEN

2.1. Cardiovascular system. The cardiovascular system (see Figure 1) is represented by a closedcircuit with 3 arterial compartments, 3 venous compartments, and 2 ventricular compartments.Vascular compartments represent systemic arteries and veins in the body and the brain, as well aspulmonary arteries and veins. Each compartment lumps vessels with similar transmural pressurep(t) [mmHg], volume V (t) [ml], and compliance C [ml/mmHg] (constant). For each compartment i,the stressed volume Vi,str(t) is given by

Vi,str(t) = Vi(t) − Vi,unstr = Cipi(t), (1)

where Vi(t) [ml] is the time-dependent total volume and Vi,unstr [ml] (constant) is the unstressedvolume, at zero transmural pressure.

Flow between adjacent compartments are characterized by a constant resistance R [mmHg sec/ml]and volumetric flow rate q(t) [ml/sec]. Incorporating Ohm’s law, the net change in volume for eachcompartment is given by

dVi(t)

dt= qin(t) − qout(t), qin(t) =

pi−1(t) − pi(t)

Rin

, qout(t) =pi(t) − pi+1(t)

Rout

. (2)

The subscripts i − 1 and i + 1 refer to upstream and downstream compartments in relation tocompartment i, respectively. A system of differential equations is obtained by differentiating (1)and equating with (2),

dpi(t)

dt=

qin(t) − qout(t)

Ci

. (3)

An equation of this form is associated with each vascular compartment.The heart is represented by the left and right ventricles, modeled to generate the driving pressures

for the systemic and pulmonary systems. This is realized by imposing a time-varying pressurepredicted as a function of ventricular volume. Thus, for the ventricular compartments we obtain adifferential equation by imposing volume conservation, i.e.,

dViv(t)

dt= qin(t) − qout(t), i = l, r (4)

where the flows q are determined similarly to (2). We use a time-varying elastance model to predictventricular pressures piv(t) [48, 51, 55, 60],

piv(t) = Eiv(t)[Viv(t) − Vid], i = l, r, (5)

where Vd [ml] denotes the volume at zero end-systolic pressure [48, 52] and Eiv(t) [mmHg/ml] denotesthe time-varying elastance. In the above equations subscript i denote the left and right ventricles,respectively. For each ventricle, time-varying elastance E(t) is modeled using a piecewise sinusoidalfunction of the form

E(t) =

(EM − Em)

[

1 − cos

(

πt

TM

)]

/2 + Em 0 ≤ t ≤ TM

(EM − Em)

[

cos

(

π(t − TM )

TR

)

+ 1

]

/2 + Em TM ≤ t ≤ TM + TR

Em TM + TR ≤ t ≤ T,

(6)

where TM and TR denote the time for maximum (systolic) elastance (TM ) and the remaining time torelaxation (TR). To account for varying heart rate, we express these times as fractions of the heartperiod T , i.e., TM,frac = TM/T and TR,frac = TR/T . We assume that peak elastance occurs at thesame time for both left and right ventricles; thus we assume that the values of the two parametersTM and TR remain the same between the two sides of the heart. Em and EM denote that minimumand maximum elastance of each ventricle. The maximum and minimum elastance differ significantlybetween the right and the left ventricle, in particular for the CHF patient studied in this manuscript.

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CARDIOVASCULAR & RESPIRATORY DYNAMICS IN CONGESTIVE HEART FAILURE 5

Thus we include four elastance parameters Eml, Emr and EMl, EMr and two timing parameters TM

and TR.Similar to previous studies [45, 14, 50], the ventricular valves are modeled using time-varying

resistances {Rmv(t), Rav(t), Rtv(t), Rpv(t)} defined as a function of the pressure drop across thevalves. A small baseline resistance is used to define an “open” valve (subscript “o”) and a resistancethat is several orders of magnitude larger is used to define the “closed” valve (subscript “c”). Anexponential function is applied to describe the degree of openness as a function of the pressuregradient. The effective resistance of a valve is then defined by

Rvalve = min[

Rvalve,o + e−k(pin−pout), Rvalve,c

]

. (7)

The parameter Rvalve,o is the small resistance allowing flow out of the ventricle, k describes thespeed of the transition from open to closed, and Rvalve,c is a value large enough to effectively shutoff the flow through the valve. Since this function is non-smooth at the junctions of the exponentialand Rvalve,c, a smoothing function [11]

minε

(x) = −ε log

(

i

exp(xi/ε)

)

,

where 0 < ε < 1 denotes the degree of smoothness (in this study we used ε = 0.5) and x denote thevector to be minimized.

All equations and initial conditions for the cardiovascular model are given in the Appendix.

2.2. Respiratory system. The respiratory system is modeled using components allowing for gasexchange, transport, and metabolism. We model the dynamics of the two metabolite gases, O2 andCO2, while we do not account for inert gases such as N2. As shown in Figure 1, a tissue compartmentconnects each peripheral arterial compartment with each peripheral venous compartment. Dynamicgas concentrations in each tissue compartment reflect O2 consumption and CO2 production duringmetabolism as well as convection via blood flow into and out of each compartment. The modelalso includes two pulmonary components. The alveolar space has a dynamic volume, in which O2

and CO2 are exchanged between the lungs and the pulmonary vasculature. The lungs have beenmodeled using rigid dead space compartments connecting the alveolar space with the atmosphere.The rigid-walled dead space is divided into three compartments of the same size to account for someof the effects of the pulmonary branching. Gas concentrations and partial pressures are predicted inall compartments. The inputs to the model include HR and volumetric airflow VIE , which providesinformation about tidal volume VT and ventilation frequency fR.

Standard material balance equations describe respiration in the tissue compartments. The fol-lowing symbol convention will be used: c represents concentration, T represents a generic tissuecompartment which can be chosen to be systemic tissue S or brain tissue B. Generic gas quantityis denoted by g and gas fractional amount by F . Venous outflow is denoted by v and arterial inflowby a. Thus we denote the total amount of a gas in a tissue compartment by AT,g [ml] which is givenby

AT,g = VT,gcT,g.

This equation describes AT,g by the product of the effective tissue volume V [ml] (constant) availablefor the gas and the concentration c [mlgas/mlblood] of the gas in the volume. The change in amount ofgas in a compartment is equal to the sum of the amount of gas produced or consumed by metabolismM [ml/sec] and the amount added or removed by the bloodstream qT [ml/sec].

dAT,g

dt=

dVT,g

dtcT,g + VT,g

dcT,g

dt= MT,g + qT (ca,g − cT,g).

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6 L.M. ELLWEIN, S.R. POPE, A. XIE, J. J. BATZEL, C.T. KELLEY, M.S. OLUFSEN

Note, the concentration in the systemic arteries (i.e., in the body and the brain) are the same, sinceno metabolism has been accounted for on this side. Hence we denote the arterial concentration forca. Thus, Assuming dVT,g/dt = 0, this equation reduces to

VT,g

dcT,g

dt= MT,g + qT (ca,g − cT,g). (8)

The tissue compartments are considered well-mixed and equilibrated with exiting venous concentra-tion. Note that the gas concentration csv,g is calculated as a mixture of the concentrations from theincoming systemic venous stream qs and cerebral stream qcv so that

csv,g =cS,gqs + cB,gqcv

qs + qcv

,

where cS,g and cB,g are the systemic and cerebral concentrations of each gas, respectively.Let A denote the alveolar compartment; then the quantity of alveolar gas is given by

VA,g = VAFA,g,

which is the product of alveolar volume VA and gas fraction FA,g. The change in quantity of alveolargas is represented by a mass balance relation parallel to the one developed for the tissue compartment,taking into account this time a time-varying alveolar volume, and with gas transport via fluid flowingto and from the lungs via pulmonary capillaries, and gas exchange with environment via inspirationand expiration. Let subscript p represent the pulmonary compartment. Thus we have,

dVA,g

dt= FA,g

dVA

dt+ VA

FA,g

dt=

dVA

dtFi,g + qp(csv,g − ca,g).

In this representation Fi,g denotes the fraction of gas in the air that is either being inspired or expiredinto the alveolar compartment. Hence, i = D3 during inspiration since inspired air is coming fromthe adjacent dead space region and i = A during expiration since air leaving the alveoli is alveolarair equilibrated with the pulmonary capillaries. Rearranging to express the rate of change of alveolargas fraction gives

VA

dFA,g

dt=

dVA

dt(FD3,g − FA,g) + qp(csv,g − ca,g), inspiration

qp(csv,g − ca,g), expiration.

Gas fractions are converted to partial pressures via the relationship

FA,g = pA,g/(pamb − pwater) = pA,g/713,

where pamb is the ambient air pressure of 760 mmHg and water vapor partial pressure pwater equals47 mmHg at body temperature of 37◦, i.e.,

VA

713

dpA,g

dt=

dVA

dt

(pD3,g − pA,g)

713+ qp(csv,g − ca,g), inspiration

qp(csv,g − ca,g), expiration.

Because blood gas concentrations are reported in [mlSTPD/ml] (STPD is the Standard Temperature(0◦ C), barometric Pressure at sea level and Dry gas), but alveolar volumes are in BTPS (standard-ized to Body Temperature, barometric Pressure at sea level, Saturated with water vapor: bodytemperature and pressure, saturated), we convert tissue concentrations to BTPS. Incoming air isimmediately humidified once it enters the nasal passages [7, 8] and expired air is a composition ofalveolar air and dead space air at BTPS, therefore terms with those quantities are not converted.The final equations are obtained by using the conversions and multiplying through by 713, i.e.,

VA

dpA,g

dt=

dVA

dt(pD3,g − pA,g) + 863 · qp(csv,g − ca,g), inspiration

863 · qp(csv,g − ca,g), expiration.(9)

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CARDIOVASCULAR & RESPIRATORY DYNAMICS IN CONGESTIVE HEART FAILURE 7

Additional modeling considerations concern the relationship between the pulmonary capillaries andthe systemic arteries. First, an anatomical shunt is present in the pulmonary circulation. This con-sists of O2-deficient blood that bypasses the alveoli for physiological or pathological reasons. Typicalshunt values range from 2-5%, with a larger shunt percentage possibly indicating a pathological con-dition [8, 7]. The subject studied has no known pulmonary health issues so we assume a 2% shunt.Therefore only 98% of the cardiac output becomes oxygenated, altering the alveolar equations toread as

VA

dpA,g

dt=

dVA

dt(pD3,g − pA,g) + 0.98 · 863qp(cv,g − ca,g), inspiration

0.98 · 863qp(cv,g − ca,g), expiration.(10)

Second, we note that the thin alveolar wall allows for almost immediate equilibration of gases betweenthe alveoli denoted by A and the pulmonary capillaries; thus we assume that the concentrations ofblood gases is the same in the pulmonary capillaries and in the systemic arteries. This concentrationis denoted by psa,g, i.e., we assume pA,g = psa,g as an auxiliary equation.

Connected to the alveolar compartment are three compartments of equal volume with a total vol-ume representing anatomical dead space. Each dead space is considered a well-mixed compartmentwith units VBTPS , in ml. Material balance equations for the dead space compartments reflect changein gas levels due to airflow, with opposite directions of flow for inspiration versus expiration. Therelation FA,g = pA,g/713 holds, and equation units are all in BTPS. Thus following the derivationof (10),

Inspiration: VD1

dpD1,g

dt=

dVA

dt(pI,g − pD1,g), (11)

VDi

dpDi,g

dt=

dVA

dt(pDi−1,g − pDi,g), i=2,3.

Expiration: VDi

dpDi,g

dt=

dVA

dt(pDi,g − pDi+1,g), i=1,2, (12)

VD3

dpD3,g

dt=

dVA

dt(pD3,g − psa,g).

Pressure pI,g is the partial pressure of the gas in the inspired air.Note that dVA/dt is positive during inspiration and negative during expiration. The rate of change

of alveolar volume dVA/dt is equivalent to the ventilation airflow VIE . Thus, the alveolar volumecan be predicted as

VA =

VIE dt. (13)

To give a correct value for the alveolar volume, additional information is needed to offset the inte-gration constant. We have chosen to adjust this assuming that the minimum alveolar volume shouldmatch the FRC [ml], which we determine as a function of height H [cm] and weight W [kg], i.e., weassume

min(VA) = FRC, FRC = (3.8 · H/100 − 3.41 · W/H − 2.74) · 1000

The above estimation for FRC is based on work by Stocks and Quanjer [56]. Furthermore, it should

be noted that the use of the ”dot” notation in VIE is a standard way to indicate a volumetric flowrateas opposed to a rate of change of volume. The airflow VIE is measured during experimentation andis used as an input to the model to drive the gas concentration dynamics.

O2 and CO2 have different affinities for hemoglobin, therefore behave differently in the gas ver-sus liquid phases. Gas dissociation laws are used to convert alveolar gas pressures to blood gasconcentrations.

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8 L.M. ELLWEIN, S.R. POPE, A. XIE, J. J. BATZEL, C.T. KELLEY, M.S. OLUFSEN

In this study we use equations also used by Batzel et al. [4],

cT,CO2= KCO2

pT,CO2+ kCO2

,

cT,O2= K1(1 − e−K2pT,O2 )2.

Note that the law for CO2 is linear while the law for O2 is exponential. This reflects general behavior,but does not account for the the dependencies of each gas on the other (Bohr and Haldane effects)as well as factors such as pH and temperature.

All respiratory equations and initial values are listed in the Appendix.

3. Experimental Methods. The data analyzed in this study include continuous (sampled at128 Hz) measurements of systemic arterial blood pressure (psa), cerebral blood flow velocity vc

measured from the middle cerebral artery (MCA), and end tidal partial pressure of CO2 in expiratoryair. In addition we have a measure for left ventricular ejection fraction, as well as anthropometricmeasurements of height, weight, and gender. These data are obtained from Dr. Skatrud’s group,Department of Medicine, University of Wisconsin.

The subject is a male age 55, height 178 cm, weight 82.3 kg with CHF but no known sleep apneasyndrome. Data used for this study were recorded while the subject was in semirecumbent positionduring normal breathing (for 10 min). Experiments were started between 8.00 and 9.00 am, tostandardize the effect of diurnal variability of cerebral vasomotor reactivity [2].

A 2-MHz pulsed Doppler ultrasound system (Neurovision 500 M; Multigon Industries, Yonkers,NY) was used to continuously measure cerebral blood flow velocity in the proximal segment of themiddle cerebral artery (MCA). The MCA was insonated through the right temporal bone windowusing search techniques described in Otis and Ringelstein [46]. After detection and optimization ofthe Doppler signal, the probe was mechanically secured using a headband device and probe holderto provide a fixed angle of insonation for the duration of the experiment. The subject, in thesemirecumbent position, was asked to keep his head still and eyes open throughout the experiment.Heart rate was obtained from the electrocardiogram, and arterial pressure (psa) was measured beatby breat in the middle finger of the left hand by photoelectric plethysmography (Finapres, Ohmeda,Louisville CO). Tidal volume (VT ) and breathing frequency were measured with a pneumotachographModel 3700, Hans Rudolph, Kansas City, MO) that was attached to a leak-free nasal mask. Endtidal PETCO2 tensions was sampled from the mask and measured by gas analyzers (#S-3A/I &CD-3; Ametek, Pittsburgh PA). Detailed descriptions of the experimental protocol can be foundin [63].

4. Parameterization.

4.1. Cardiovascular parameters. Nominal parameter values and initial conditions for the cardio-vascular model were estimated from the subject’s anthropometric measurements. A complete list ofall initial parameter values can be found in Table 1. We start by setting up predictions for nominalparameter values used to predict time-varying elastance. Parameters needed include maximum andminimum elastance of the left ventricle (EMi, Emi, i = l, r), as well as fractions predicting the timingof the cardiac cycle (TM,frac, TR,frac).

Nominal parameter values for end diastolic elastance are obtained from estimated diastolic ven-tricular pressure pdia, end diastolic volume EDV, and zero pressure volume Vd as

Emi =piv,dia

EDViv − Vid

, i = l, r.

Diastolic ventricular pressures are set using literature values [29]. The subject studied has CHF andis thus expected to have an enlarged left ventricle, while the size of the right ventricle is closer tonormal. We estimate these volumes using results for CHF patients without sleep-apnea reported byTkacova et al. [59].

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CARDIOVASCULAR & RESPIRATORY DYNAMICS IN CONGESTIVE HEART FAILURE 9

End systolic elastances are estimated using a similar relation, but as functions of systolic ven-tricular pressure psys, end systolic volume ESV, and zero pressure volume Vd. Systolic ventricularpressure is obtained from the maximum measured arterial pressure. End systolic volume is obtainedby subtracting stroke volume (SV) from the end diastolic volume: ESV = EDV−SV. Stroke volumeis estimated from end left ventricular diastolic volume and ejection fraction (EF, measured 26% forthe subject studied): SV = EF · EDVlv.

The timing fractions TM,frac and TR,frac were estimated from literature values suggested byOttesen and and Danielsen [47] and Heldt [30]. We assume that the ejection of the left and the rightsides of the heart are synchronous, thus we use the same values for both ventricles.

Initial values for blood pressures were obtained partly from the data and partly from literatureestimates. All arterial pressures were scaled relative to the measured arterial pressure, whereasvenous pressures were set using standard literature values. Initial blood pressures were used bothin computation of initial resistances and compliances and as initial conditions for the differentialequations.

Initial flows were scaled to cardiac output, which was computed from stroke volume SV and meanheart rate HR, i.e., CO = SV · HR. Mean heart rate was obtained from the measurements. Flowsin the circuit was distributed to let 20% cardiac output go to the brain, while 80% was directed tothe systemic arteries.

Resistors were predicted using Ohm’s law as

R =pi − pi+1

qi

and compliances were predicted using the pressure volume relation (1). Estimates of total bloodvolume was computed as a function of body surface area (BSA, m2), estimated from Mosteller’sformula [42]

BSA =√

(W · H)/3600,

where W is the subjects weight in kg and H is the subjects height in cm. Total blood volume (ml)was computed using Baker’s formula [3] as

Vt = (23.9 · BSA − 1.229) · 1000.

Distributions of volume and prediction of unstressed volumes were obtained using values fromBeneken and DeWitt [6].

4.2. Respiratory parameters. Nominal values for metabolic rates and tissue volumes are givenin Table 2. As is standard practice, all tissue gas volumes and blood gas concentrations are givenin STPD, thus units are consistent. Metabolic rates were set using standard allometric scalingproportional to body mass by the power of 3/4 [61, 62] using values in Table 2. We used metabolicrates given in [4, 34, 27] for total systemic (including both the body and the brain) CO2 (MCO2

) andO2 (MO2

) combined with a metabolic rate for the brain for CO2 (MB,CO2). To compute metabolic

rates for oxygen in the brain MB,O2we assumed that the ratio of metabolism between oxygen and

carbon dioxide in the brain was approximately equal to one given that brain tissue burns primarilycarbohydrate (glucose) ([27]). Finally, we used these four metabolic rates to adjust metabolic ratesin the body. Similarly, for volumes of the gases we used total systemic volumes and brain volumesto get the volumes in the body, based on values given by Batzel [4]. Alveolar volume was a dynamicquantity as shown in (13). Dead space volume was set proportional to the body weight in pounds(the subject weight 82.3 kg or 181.3 lbs). Finally, the coefficients of the dissociation equations areassumed to be independent of the size of the subject.

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10 L.M. ELLWEIN, S.R. POPE, A. XIE, J. J. BATZEL, C.T. KELLEY, M.S. OLUFSEN

Table 1. Nominal values for all cardiovascular parameters.

Parameter Physiologic description Value Reference

EDVlv [ml] End diastolic volume (left ventricle) 312 [59]EDVrv [ml] End diastolic volume (right ventricle) 100 [8, 29]EF Ejection fraction 0.26 measuredSV [ml] Stroke volume EF · EDVl

ESVl End systolic volume (left ventricle) EDVl − SVESVr End systolic volume (right ventricle) EDVr − SV

HR [bpm] mean heart rate measured

CO [l/min] Cardiac output SV · HRpd

sa [mmHg] Measured arterial pressure measuredplv,sys [mmHg] Systolic left ventricular pressure max(pd

sa) measuredprv,sys [mmHg] Systolic right ventricular pressure 30 [8, 29]plv,dia [mmHg] Diastolic left ventricular pressure 3 [8, 29]plv,dia [mmHg] Diastolic right ventricular pressure 6 [8, 29]ppa [mmHg] Pulmonary arterial pressure 20 [8, 29]ppv [mmHg] Pulmonary venous pressure 3.3 [8, 29]psa [mmHg] Systemic arterial pressure pd

sa measuredpsv [mmHg] Systemic venous pressure 6.6 [8, 29]pca [mmHg] Cerebral arterial pressure psa · 0.99 measuredpcv [mmHg] Cerebral venous pressure 7 [8, 29]qs [ml/sec] Systemic flow 0.8 CO [8, 29]qca, qc, qcv [ml/sec] Flow in cerebral circulation 0.2 CO [8, 29]qpa, qpv [ml/sec] Flow in pulmonary circulation [8, 29]Rmv , Rav [mmHg sec/ml] Mitral and aortic valve resistances 0.001 estimatedRtv, Rpv [mmHg sec/ml] Tricuspid and pulmonary valve resistances 0.001 estimatedRs [mmHg sec/ml] Systemic resistance (psa − psv)/qs

Rca [mmHg sec/ml] Cerebral arterial resistance (psa − pca)/qca

Rc [mmHg sec/ml] Cerebral resistance (pca − pcv)/qc

Rcv[mmHg sec/ml] Cerebral venous resistance (pcv − psv)/qcv

Rp [mmHg sec/ml] Pulmonary resistance (ppa − ppv)/qp

Vsa [ml] Systemic artrial blood volume 0.1178 · Vt [6]Vsv [ml] Systemic venous blood volume 0.6091 · Vt [6]Vca [ml] cerebral arterial blood volume 0.237 · Vt [6]Vcv [ml] Cerebral venous blood volume 0.936 · Vt [6]Vpa [ml] Pulmonary arterial blood volume 0.0288 · Vt [6]Vpv [ml] Pulmonary arterial blood volume 0.1243 · Vt [6]Csa [ml/mmHg] Systemic arterial compliance 0.3 · Vsa/psa [6]Csv [ml/mmHg] Systemic venous compliance 0.08 · Vsv/psv [6]Cca [ml/mmHg] Cerebral arterial compliance 0.22 · Vca/pca [6]Ccv [ml/mmHg] Cerebral venous compliance 0.08 · Vcv/pcv [6]Cpa [ml/mmHg] Pulmonary arterial compliance 0.58 · Vpa/ppa [6]Cpv [ml/mmHg] Pulmonary venous compliance 0.11 · Vpv/ppv [6]Ac [cm2] Cerebral scaling factor 0.3 estimated

4.3. Parameter estimation. Model parameters discussed in the parameter above are obtainedusing physiological considerations and allometric scaling, so even though they to some extend havebeen adapted to the patient with CHF studied here, significant variations from these standardvalues is to be expected. Below we discuss how to estimate model parameters though the solu-tion to the inverse problem: given data for cerebral blood flow velocity vc, arterial blood pressurepsa, end tidal partial pressure of expiratory CO2 pexp,CO2

, and cardiac ejection fraction EF findthe set of parameters that minimize the difference between computed and measured values of theobserved quantities. The coupled model uses heart rate and airflow as inputs to predict these quan-tities. However, the model contains a large number of parameters (37) including 24 cardiovascularparameters and 13 respiratory parameters. The cardiovascular parameters include 9 resistances

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CARDIOVASCULAR & RESPIRATORY DYNAMICS IN CONGESTIVE HEART FAILURE 11

Table 2. Nominal values for metabolic rates, tissue volumes, and gas dissociation constants. Allvalues are adapted from [4, 34, 27].

Parameter Physiologic description Value

MCO2[mlSTPD/sec] Systemic tissue metabolic rate of CO2 4.333

MO2[mlSTPD/sec] Systemic tissue metabolic rate of O2 5.167

MB,CO2[mlSTPD/sec] Cerebral tissue metabolic rate of CO2 0.875

MB,O2[mlSTPD/sec] Cerebral tissue metabolic rate of O2 MB,CO2

MS,CO2[mlSTPD/sec] Systemic (body) tissue metabolic rate of CO2 MCO2

− MB,CO2

MS,O2[mlSTPD/sec] Systemic (body) tissue metabolic rate of O2 MO2

− MB,O2

VT,CO2[mlSTPD] Systemic tissue volume of CO2 15000

VT,O2[mlSTPD] Systemic tissue volume of O2 6000

VB,CO2[mlSTPD] Cerebral tissue volume of CO2 900

VB,O2[mlSTPD] Cerebral tissue volume of O2 1000

VS,CO2[mlSTPD] Systemic tissue volume of CO2 VT,CO2

− VB,CO2

VS,O2[mlSTPD] Systemic tissue volume of O2 VT,O2

− VB,O2

VA,CO2[mlBTPS ] Alveolar tissue volume of CO2 3200

VA,O2[mlBTPS ] Alveolar tissue volume of O2 2500

VD [mlBTPS ] Total dead space volume 181.1K1 [mlSTPD/ml] Dissociation coefficient for O2 0.200K2 [mmHg−1 ] Dissociation coefficient for O2 0.046KCO2

[mlSTPDmmHg/ml] Dissociation coefficient for CO2 0.0065kCO2

[mlSTPD/ml] Dissociation coefficient for CO2 0.244

R = {Rs,Rca,Rc,Rcv,Rp,Rav,o,Rmv,o,Rpv,o,Rtv,o}, 6 compliances C = {Csa,Cca,Ccv,Csv, Cpa,Cpv},8 heart parameters θheart = {Vrd,Vld,Emr,Eml,EMr ,EMl,TM,frac, TR,frac}, and a scaling factor Ac.The respiratory parameters include 3 metabolic rates M = {MCO2

,MO2,MB,CO2

}, 4 gas tissue vol-umes V = {VT,CO2

, VT,O2, VB,CO2

, VB,O2}, the lung dead space volume VD, 4 dissociation constants

K = {K1,K2,KCO2,kCO2

, }, and a fraction indicating alveolar air in the exiting air stream falv.Solving this inverse problem uniquely is not possible using this complex model, since it is likely thatmany combinations of these parameters can give rise to the same solution. Thus, the goal here isto identify a limited set of uncorrelated parameters that can be estimated reliably. To do so, weuse sensitivity analysis to rank parameters from the most to the least sensitive, and we use subsetselection to select a set of uncorrelated parameters. Subsequently, we use gradient based nonlinearoptimization to estimate this reduced set of parameters, while the remaining parameters will befixed at their nominal values.

The data analyzed in this study comprise baseline values, thus it is essential that the solutionof the system allows all states to reach “steady state” (in this study steady state means that themodel should display steady oscillatory behavior). Timescales in the cardiovascular model are short,and this model will reach steady state within a few cycles. However, the respiratory model operateson much slower timescales. To analyze how long the model should run to reach steady state weanalyzed timescales for each of the two systems. Time constants in the cardiovascular model are allof the same order of magnitude, while time constants in the respiratory model vary significantly. Thelargest time constant in the respiratory model is associated with the effective systemic tissue CO2

volume, which is approximately 14.1 liter. To compare the two models, we look at time constantsrelated to cerebral arterial pressure pca and concentration of systemic tissue CO2 (cS,CO2

). Thesetime constants appear in the equations

dpca

dt=

1

Cca

(

psa − pca

Rca

− pca − pcv

Rc

)

,

dcS,CO2

dt=

MS,CO2+ qs(ca,CO2

− cS,CO2)

VS,CO2

.

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12 L.M. ELLWEIN, S.R. POPE, A. XIE, J. J. BATZEL, C.T. KELLEY, M.S. OLUFSEN

0 1000 2000 3000 4000 5000 6000 70000.54

0.545

0.55

0.555

0.56

0.565

0.57

time [sec]

cS

,CO

2 [

ml S

TP

D/m

l]

Figure 2. Dynamics of concentration of CO2 in the systemic tissue computed using nominal pa-rameter value, sequence between the two vertical lines denotes the last sequence of the data.

The time constant for these two equations are approximately (CcaRcaRc)/(Rca + Rc) ≈ 0.0167seconds and VS,CO2

/qs ≈ 257 seconds, respectively. These estimates indicate that we need longsequences of baseline data to reach steady state dynamics. In this study we analyze 172 seconds ofdata, thus to ensure steady state dynamics we repeat the data-segment 40 times to reach an endtime of approximately 7000 seconds. Figure 2 show dynamics of the tissue CO2 concentration.

Another important issue is that each respiratory cycle contain approximately 4-5 cardiac cycles,requiring significantly smaller time steps to resolve cardiovascular dynamics than respiratory dynam-ics. Consequently, computation time for one simulation with the coupled cardiovascular-respiratorymodel is approximately 23 min on a 3 GHz Mac Pro with 4 Dual-Core Intel processors and 12 GBmemory, rendering it computationally intensive to solve the parameter estimation problem using thecoupled model.

The cardiovascular and respiratory systems are weakly coupled. The respiratory model dependson blood flow through the pulmonary, systemic and brain tissues, while as stated the cardiovascularmodel does not depend on quantities from the respiratory model. Physiologically, the thoracicarteries and veins depend on respiration since change of volume in the lungs imposes oscillation ofthe external pressure in the tissue surrounding the thoracic arteries and veins. Such oscillations canbe observed in measurements of arterial pressure, even on the finger pressure measurements usedfor analysis in this study. However, this slow oscillation is a secondary effect not included in thepresent study. Furthermore, when control of the system is included respiration will impact cerebralvascular resistance and compliance through autoregulation.

Based on the above observations, we decoupled the models. The parameter estimation problemwas solved by first estimating parameters for the cardiovascular model. Using the optimized car-diovascular parameters, systemic, pulmonary and cerebral mean blood flows were computed andused as inputs to the respiratory model. Finally, respiratory model parameters were estimated. Asolution to the original coupled model was found using the optimized cardiovascular and respiratoryparameter values and this solution was compared with the experimental data. This was done toensure that the optimized parameters from the decoupled models could predict the dynamics of thecomplete model.

4.4. Least squares cost. As mentioned earlier, model parameters are estimated by minimizing theleast squares error between computed and measured values of arterial pressure, cerebral blood flowvelocity, and left ventricular ejection fraction (for the cardiovascular model) and end tidal partial

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CARDIOVASCULAR & RESPIRATORY DYNAMICS IN CONGESTIVE HEART FAILURE 13

pressure of expiratory CO2 for the respiratory model. Similar to previous studies [50, 15] we notethat to accurately predict cardiovascular dynamics it is important to account for changes in systolicand diastolic values of arterial pressure and cerebral blood flow velocity. For each timeseries, thedata are measured at a fixed frequency (128 Hz); thus the quantities (pressure, velocity, partialpressure of CO2) are observed at N equally spaced times, ti. The timeseries analyzed contain agiven number of cardiac cycles (M), thus for arterial pressure and cerebral blood flow velocity wehave M observations of systolic and diastolic values. In addition, we have an average value forejection fraction (EF). Since the study is done during steady state, we assume that the ejectionfraction should be the same for each of the M cardiac cycles. In summary, we have the followingobservations:

• Arterial blood pressure psa(ti), i = 1 : N [mmHg]• Cerebral blood flow velocity vc(ti), i = 1 : N [cm/sec]• Systolic arterial blood pressure psa,sys,j , j = 1 : M [mmHg]• Diastolic arterial blood pressure psa,dia,j, j = 1 : M [mmHg]• Systolic cerebral blood flow velocity vc,sys,j , j = 1 : M [cm/sec]• Diastolic cerebral blood flow velocity vc,dia,j , j = 1 : M [cm/sec]• Left ventricular ejection fraction EFj , j = 1 : M = 0.26• End tidal partial pressure of expiratory CO2 pexp,CO2

(ti), i = 1 : N [mmHg].

Each of these data vectors has an associated predicted vector, which we denote by a superscript (p).The predicted vectors are pp

sa, vpc , pp

sa,sys, ppsa,dia, vp

c,sys, vpc,dia, EF p, and pp

exp,CO2.

Since the parameter estimation problem is solved separately for the cardiovascular and the respi-ratory models, we define two residual vectors Rcar and Rresp, as

Rcar =[

Rpsa, Rvc

, Rpsa,sys, Rpsa,dia

, Rva,sys, Rva,dia

, REF

]T,

Rresp =[

Rpexp,CO2

]T.

where each component is scaled to account for the number of elements in the vector and relative tothe data, i.e.,

Ri =1√K

[

ppi (t1) − pi(t1)

pi(t1), · · · ,

ppi (tN ) − pi(tN )

pi(tN )

]T

,

where i = psa, vc, psa,sys, psa,dia, . . . , EF , and K = N, M , respectively.Thus, the vector Rcar has 2N + 5M entries, while Rresp has N entries. For each of the two

models, the least squares cost J is defined by

J = ‖Ri‖22 = RT

i Ri, i = car, resp. (14)

Minimizing the cost function is a nonlinear least squares problem. Solutions were found using theLevenberg-Marquardt variant of the Gauss-Newton optimization method [32]. Before solving theleast squares problem we used sensitivity analysis and subset selection to prune the set of parameters.

4.5. Sensitivity analysis. Sensitivity analysis gives a measure of how much the output of a modelis affected by changes in the model parameters. For the cardiovascular part of the model, sensitivitieswere computed for arterial blood pressure, pp

sa, cerebral blood flow velocity vpc , and ejection fraction

EF p. For each quantity x, the sensitivity to a parameter θ is a time-varying quantity, given by

dxi

θ=θ0

θ0

xi

for i = 1, . . . , N.

To remove the effects of poorly scaled parameters and output, each component of the derivative isweighted by the parameter value and the reciprocal of predicted pressure. This gives sensitivities asdimensionless quantities, allowing them to be compared amongst each other.

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14 L.M. ELLWEIN, S.R. POPE, A. XIE, J. J. BATZEL, C.T. KELLEY, M.S. OLUFSEN

To rank parameters from the most to the least sensitive we use a weighted 2-norm to obtain asensitivity for each quantity. For the cardiovascular model, we compute sensitivities with respect tothree quantities (psa, vc, and EF), then take the mean over the three quantities, i.e,

Scar,θ = mean{‖Sppsa,θ‖2/

√N, ‖Sv

pc ,θ‖2/

√N‖SEF p,θ‖2/

√M}.

However, the respiratory model is only validated against one quantity (end tidal partial pressure ofexpiratory CO2, i.e., the sensitivity is given by

Sresp,θ = ‖Spexp,CO2‖2/

√N.

If the model is insensitive to a parameter, it should be not be considered during an optimization.

4.6. Subset Selection. Let θ be the vector of P model parameters. Let R be the model residual.Subset selection looks at the Jacobian, dR/dθ, and attempts to find a subset of columns that are“maximally independent”. We employ a method from [24] which also appears in [25]. The methodwas also used in [50, 15]. The method includes the following steps

• Compute a Jacobian matrix at nominal parameter values, R′(θ0) = dR/dθ|θ=θ0, with error

estimate ε.• Find a singular value decomposition of R′(θ0) = UΣV T with

Σ = diag(σ1, . . . , σP )

and singular values σ1 ≥ σ2 ≥ · · · ≥ σP .• Determine the number of columns, k, by counting the number of singular values larger than ε.• Use QR factorization with column pivoting [25] on V T

k where Vk is the matrix consisting ofthe first k columns of V . The factorization is

V Tk P = QR.

• The first k components of θT P = [θn1· · · θnP

] should be used in the optimization.

Label the vector of k chosen parameters θk and the vector of remaining parameters θP−k. These stepsensure that the condition number of the reduced Jacobian, κ(dR/dθk) = ‖dR/dθk‖2‖(dR/dθk)†‖2

remains small, and that dR/dθk does a good job of representing dR/dθP−k [25].

5. Results. We first present results from the sensitivity analysis, then we discuss the subset selec-tion algorithm and present simulation results obtained from the nonlinear least squares optimization.

5.1. Sensitivity analysis. For both the cardiovascular and the respiratory models we computedand ranked relative sensitives as described above. Figure 3 shows that both models contain both sen-sitive and insensitive parameters, but no clear jump separates the two groups. For both models we de-fined all parameters with a sensitivity greater than 10−2 as sensitive. This choice allows us to includeall parameters that we can account for using physiological arguments. For the cardiovascular modelthis leaves 16 parameters including xcar,sens = {Vld,Emr,Rc,Eml,TM,frac,Rp,Csv,Csa,Cpv,EMl,EMr ,Rs,Vrd,Rca,Rcv,Cca}. Note that all heart valve resistances are insensitive. From previous work [50,15] we know that Ac is correlated with Rc, thus we keep this parameter constant and do not includethis parameter in our list of parameters that we seek to estimate using nonlinear optimization. Forthe respiratory model, including parameters with sensitivity larger than 10−2 allows us to pick 6parameters including xresp,sens = {MCO2

,VD,KCO2,kCO2

,VT,CO2,VB,CO2

,MB,CO2}. Note, we do not

include any data predicting partial pressure of O2, thus all parameters related to prediction of O2

partial pressures/concentrations are insensitive.

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CARDIOVASCULAR & RESPIRATORY DYNAMICS IN CONGESTIVE HEART FAILURE 15

Figure 3. Ranked sensitivities, top graph shows ranking for the cardiovascular model and bottomgraph shows ranking for the respiratory model. For both graphs, blue squares denote sensitivitiescomputed using the initial (nominal) parameter values, and the red diamonds denote sensitivitiescomputed using the optimized parameters. Note, that the y-axis is a log scale.

5.2. Subset selection. For each subsystem, we investigated correlation between parameters usingsubset selection. For the cardiovascular model, subset selection allowed us to extract five uncorrelatedparameters including xcar,sub = {Rc, Rs, Ca,TM,frac, EMl}. It should be noted that these are thesame cardiovascular parameters found to be uncorrelated in our previous studies [50, 15]. Estimationof these five parameters using nonlinear optimization did not allow us to accurately predict theobserved data. An analysis of the remaining parameters (not picked by subset selection) revealedthat including a sixth parameter Cca allowed us to better predict the data. We remind the readerthat subset selection was based on initial parameter values. We repeated the subset selection withoptimized parameter values, and at this stage Cca was included in the subset. Furthermore, itshould be noted, that if we ran the models using data from a young subject, the same cardiovascularparameters were identified, but for the young subject it was not necessary to also estimate arterialcerebral compliance.

Before predicting sensitivities for the respiratory model we used optimized cardiovascular pa-rameters to compute the mean flow in the systemic qs, cerebral qc, and pulmonary qp circulations.Along with airflow data these flows were used as inputs to the model. These quantities were keptfixed during the subset selection. Results from subset selection allowed us to estimate 4 parame-ters including xresp,sub = {KCO2

, kCO2, MS,CO2

, VD}. It should be noted that none of the selectedparameters were insensitive.

5.3. Nonlinear optimization. For both models parameters were estimated using the Levenberg-Marquardt variant of the Gauss-Newton optimization method [32]. Optimized parameters are givenin Table 3. For the cardiovascular model data were subsampled at 64 Hz to speed up simulations,and computations settled at steady state within the data analyzed. For the respiratory model datawere subsampled at 4.2667 Hz to speed up simulations, furthermore to ensure that steady state wasreached respiratory data were repeated 40 times. The comparison used to compute the least squareserror were done over the last repeat of the data. Results of optimizations are shown in Figures 4

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16 L.M. ELLWEIN, S.R. POPE, A. XIE, J. J. BATZEL, C.T. KELLEY, M.S. OLUFSEN

Table 3. Optimized model parameters, note the value for TM,frac is the fraction relative to thelength of the cardiac cycle. Averaged model quantities include ejection fraction, cardiac output,mean heart rate, as well as systemic, cerebral, and pulmonary flow.

Parameter Description Optimized Unit

Rs Systemic resistance 1.28 mmHg sec/mlRc Cerebral resistance 2.54 mmHg sec/mlCsa Systemic arterial compliance 0.536 mmHg/mlCca Cerebral arterial compliance 2.96 mmHg/mlEMl Maximum left ventricular elastance 0.721 ml/mmHgTM,frac Time for maximum ventricular elastance 0.139

EF Ejection fraction 0.262CO Cardiac output 4.92 l/minHR mean heart rate 62.5 beats/minqs Mean systemic flow 55.0 ml/secqc Mean cerebral flow 26.9 ml/secqp Mean pulmonary flow 82.0 ml/sec

KCO2Dissociation constant 0.00688 mlSTPD mmHg/ml

kCO2Dissociation constant 0.261 mlSTPD mmHg/ml

MS,CO2Metabolic rate systemic tissue 4.20 mlSTPD mmHg/sec

VD Total deadspace volume 151 ml

and 5. The cardiovascular model used heart rate as an input, and the respiratory model used airflowand mean blood flows computed from the cardiovascular model with optimized parameters.

6. Discussion. Results from the cardiovascular and respiratory models (Figures 4, 5, and 6) showedthat the modular model approach used in this study enable prediction of both cardiovascular andrespiratory quantities. Results from the coupled model were indistinguishable from the resultsobtained with each of the two models indicating that using mean values for flows do not impact therespiratory model significantly, which is to be expected since respiration occurs over a much slowertimescale as the cardiovascular oscillations.

In addition to estimating observed quantities, we also computed all internal states and mostof these were within physiological bounds as discussed below. It should be noted that techniquesused for parameter estimation do not guarantee that internal states cannot fluctuate away fromphysiological values, even though initial parameter estimates were within physiological bounds. Thisis one of the main problems with the proposed method. Besides including bounds on all modelparameters, we did not impose additional constraints limiting the states within certain bounds.Consequently some states did drift away from expected values.

In regards to cardiovascular characteristics as reflected by the model for this subject with CHF,we note that approximately 30% of cardiac output was utilized as cerebral blood flow. This islarge, however, cerebral blood flow velocity measured for this subject was significantly larger thanmean velocities reported in the literature [28]. This could be due to poor circulation in the bodyor exceptionally high values from the transcranial Doppler measurements. This subject exhibitsa severely enlarged left ventricle (consistent with left ventricular failure) and reduced pumpingeffectiveness, with an ejection fraction of 26%. Neverless, the cardiac output (modeled) is close tonormal. This type of behavior is seen in approximately half the patients suffering from CHF [43].

With optimized parameters the maximum left ventricular volume was approximately 317 ml,which is close to initial volume chosen as 312 ml. The minimum ventricular volume was approxi-mately 238 ml, yielding a stroke volume of 79 ml and with the mean heart rate of 62.5 beats/minthis gives an approximate CO of 4.9 computed from the flows in the model (see Table 3). Systemicvenous pressures were approximately 7.2 mmHg somewhat higher than normal, which is expectedfor a patient with CHF (see Table 4). Pulmonary arterial pressures were approximately 23 mmHg

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CARDIOVASCULAR & RESPIRATORY DYNAMICS IN CONGESTIVE HEART FAILURE 17

0 50 100 15060

80

100

120

p sa [m

mH

g]

0 50 100 15040

60

80

100

time [sec]

v c [cm

/sec

]

DataModel

DataModel

0 50 100 15060

80

100

120

p sa [m

mH

g]

0 50 100 150

80

90

100

time [sec]

v c [cm

/sec

]

DataModel

DataModel

150 152 154 156 158 16060

80

100

120

p sa [m

mH

g]

150 152 154 156 158 16040

60

80

100

time [sec]

v c [cm

/sec

]

DataModel

DataModel

150 152 154 156 158 16060

80

100

120

p sa [m

mH

g]

150 152 154 156 158 160

80

90

100

time [sec]

v c [cm

/sec

]

DataModel

DataModel

50 100 1500.25

0.26

0.27

0.28

0.29

0.3

0.31

0.32

Heartbeat

Eje

ctio

n F

ract

ion

(EF

)

ModelData

50 100 1500.25

0.255

0.26

0.265

0.27

0.275

0.28

Heartbeat

Eje

ctio

n F

ract

ion

(EF

)

ModelData

Figure 4. Simulation results cardiovascular model. Top panel shows arterial pressure and cerebralblood flow velocity for the full dataset, second panel shows a zoom for 150 ≤ t ≤ 160 seconds.Bottom panel shows computed and expected ejection fraction (set at 26%). Left graphs give resultswith initial (nominal) parameter values, while the right graphs show results with the optimizedvalues.

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18 L.M. ELLWEIN, S.R. POPE, A. XIE, J. J. BATZEL, C.T. KELLEY, M.S. OLUFSEN

0 50 100 1500

5

10

15

20

25

30

35

40

time [sec]

p

[mm

Hg]

exp,

CO

2

0 50 100 1500

5

10

15

20

25

30

35

40

time [sec]

p

[mm

Hg]

exp,

CO

2

100 110 120 130 140 1500

5

10

15

20

25

30

35

40

time [sec]

p

[mm

Hg]

exp,

CO

2

100 110 120 130 140 1500

5

10

15

20

25

30

35

40

time [sec]

p

[mm

Hg]

exp,

CO

2

Figure 5. Estimation of respiratory parameters minimizing difference between measured and com-puted values of partial pressure of CO2. Left graphs show computations with initial parameters andright graphs show results with optimized parameters.

100 110 120 130 140 1500

5

10

15

20

25

30

35

40

time [sec]

pCO

2 [m

mH

g]

arterialexp

Figure 6. Model simulation of psa,CO2and pexp,CO2

showing relation between these quantitiesduring inhalation and exhalation

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CARDIOVASCULAR & RESPIRATORY DYNAMICS IN CONGESTIVE HEART FAILURE 19

and pulmonary venous pressures were approximately 5.5 mmHg. This is also to be expected for apatient with left ventricular CHF (see Table 4). Heart rate (not modeled) was within normal range.For this model both cerebral and systemic vascular resistance and compliance were kept constant.The total resistance RT = (1/RB + 1/Rs) = 0.86, where RB = Rca + Rc + Rcv is fairly low, againthis is to be expected in CHF where CO is lower than normal, and for this subject in particular,where a large quantity of the blood goes to the brain. Moreover, the somewhat large proportion offlow to the brain could also be a result of the somewhat simplified modeling included for the cerebralarteries. The blood flow velocity measured is from the middle cerebral arteries, this has then beenscaled (assuming constant area of the major cerebral arteries) to compute a total flow to the brain.Without information about anatomical quantities for the size of the cerebral arteries, this scalingfactor cannot be computed accurately, and the magnitude of the scaling factor Ac would impact theoptimized values for the cerebrovascular resistance Rc since these two parameters are correlated asstated earlier. However, if e.g., the area of all major cerebral vessels (PCA, MCA and ACA) weremeasured e.g., with an angiogram, then this scaling parameter could be computed from data.

For the respiratory quantities values for dead space pD,CO2and pD,O2

, arterial values pa,CO2

and pa,O2, and systemic values pS,CO2

and pS,O2(calculated from dissociation laws) are reasonable.

Dynamics of pA,CO2(equivalent with pa,CO2

) are shown in Figures 5 and 6. Dead space valuesfor CO2 oscillate between zero (inhaled air) and end-expiratory air approximately the same asalveolar air. Oxygen values follow similar reasonable patterns. It should be noted that brain partialpressures pB,CO2

and pB,O2did not match typical values so well in the optimized case. Estimates

of the dissociation constants varied from the initial guesses which resulted in pB,CO2approximately

43 mmHg (too low) and pB,O2approximately 48 mmHg (too high). There are likely several reasons

for this. Firstly, the data for cerebral blood velocity for this subject was rather high, which wouldtend to depress brain pB,CO2

and raise pB,O2. Secondly, we did not have data on O2 which might

impact the results. Further, research indicates that normal cerebral metabolism is distorted in CHFand cerebral energy deficits can exist in certain cerebral tissue [35]. This complicates the picture forthe estimation process.

These observations lead us to conclude that the proposed methodology and model can be used forquantitative prediction of changes within several subjects with CHF. To use this model to comparequantities from a larger population it should be emphasized that the model should be used with care.It is important that nominal parameters reflect known characteristics from a subject with CHF, inparticular since only a subset of the parameters will be estimated. Conclusions based on analysis ofparameters obtained from several subjects should all be interpreted related to the nominal parametervalues used. Furthermore, additional measurements that would increase predictability of the modelparameters include measurements of cardiac output and estimation of diameters of major cerebralvessels. Knowing these quantities would allow better prediction of systemic and arterial flow aswell as help to get better estimation of the scaling factor needed, which would help in prediction ofthe scaling factor relating MCA CBFV to CBF. Furthermore, if constants used in dissociation laws(optimized with our respiratory model) are fixed, subset selection allows estimation of the tissue andbrain CO2 metabolic rates as well as the deadspace volume. The parameter values for deadspacevolume were identical to simulation reported here, while parameters for tissue metabolic rate wentdown slightly (from 4.20 to 4.16) and the parameter for brain CO2 metabolic rate was reducedfrom its initial value (from 1.04 to 0.87). Furthermore, simulations with this set of parametersleads to similar dynamics that were observed in results presented above. Again, it should be notedthat it is essential that caution is used when choosing parameters fed to the subset selection. Subsetselection will give a subset of uncorrelated parameters that can be estimated given particular nominalparameter values and initial conditions for the differential equations.

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20 L.M. ELLWEIN, S.R. POPE, A. XIE, J. J. BATZEL, C.T. KELLEY, M.S. OLUFSEN

Table 4. Generic state changes in left systolic CHF.

State Value Source

Mean pap [mmHg] increase [44]Mean pvp [mmHg] increase [40, 31]Mean pas [mmHg] decrease or stable [49]Mean pvs [mmHg] increase [31]Mean plv [mmHg] steady to increase [41]Mean pla [mmHg] increase [49]LVEDP [mmHg] increase [41]LVESP [mmHg] steady or increase [41]

LVEDV [ml] increase [10, 49]LVESV [ml] increase [10, 49]LVEF [ml] decrease [10, 20]CO [mlmin] decrease [49]

SV [ml] decrease [49]HR [beat/min] increase [49, 20]

Systemic resistance [mmHg · s /ml] increase [31]Fluid retention [ml] increase [31, 19]

Acknowledgments. Olufsen and Ellwein were partially supported by National Science Founda-tion under grant DMS-0616597 and OISE-524249, Batzel was suported in part by FWF (AustrianResearch Funds) project P18778-N13, Pope and Kelley was supported in part by National ScienceFoundation under grant DMS-0707220, and Xie was supported in part by the American Lung As-sociation of Wisconsin.

Appendix.

6.1. Cardiovascular model. Blood pressures, ventricular volumes, and intrathoracic pressure.

dppa

dt=

(

prv − ppa

Rpv

− ppa − ppv

Rp

)

/Cpa

dppv

dt=

(

ppa − ppv

Rp

− ppv − plv

Rmv

)

/Cpv

dpsa

dt=

(

plv − psa

Rav

− psa − psv

Rs

− psa − pca

Rca

)

/Csa

dpsv

dt=

(

psa − psv

Rs

+pcv − psv

Rcv

− psv − prv

Rtv

)

/Csv

dpca

dt=

(

psa − pca

Rca

− pca − pcv

Rc

)

/Cca

dpcv

dt=

(

pca − pcv

Rc

− pcv − psv

Rcv

)

/Ccv

dVlv

dt=

ppv − plv

Rmv

− plv − psa

Rav

dVrv

dt=

psv − prv

Rtv

− prv − ppa

Rpv

.

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CARDIOVASCULAR & RESPIRATORY DYNAMICS IN CONGESTIVE HEART FAILURE 21

Table 5. Initial values for the cardiovascular model. Note initial values for the cardiovascularsystem are set using the corresponding nominal parameter values given in Table 1.

Variable Physiologic description Value Reference

ppa [mmHg] Pulmonary arterial pressure 20 [8, 29]ppv [mmHg] Pulmonary venous pressure 3.3 [8, 29]pd

sa [mmHg] Systemic arterial pressure pdsa measured

psv [mmHg] Systemic venous pressure 6.6 [8, 29]pca [mmHg] Cerebral arterial pressure psa · 0.99 measuredpcv [mmHg] Cerebral venous pressure 7 [8, 29]

Vlv [ml] Left ventricular volume EDVlv = 312 [59]Vrv [ml] Right ventricular volume EDVrv = 100 [8, 29]

Note the airflow VIE is defined below. Heart valves

Rav = min[

Rav,o + e−2(plv−psa), Rav,c

]

Rmv = min[

Rmv,o + e−2(ppv−plv), Rmv,c

]

Rpv = min[

Rpv,o + e−2(prv−ppa), Rpv,c

]

Rtv = min[

Rtv,o + e−2(psv−prv), Rtv,c

]

.

Ventricular pressures

plv(t) = Elv(t)[Vlv(t) − Vld]

prv(t) = Erv(t)[Vrv(t) − Vrd].

where

Elv(t) =

Eml + (EMl − Eml)

[

1 − cos

(

πt

TM

)]

/2 0 ≤ t ≤ TM

Eml + (EMl − Eml)

[

cos

(

π(t − TM )

TR

)

+ 1

]

/2 TM ≤ t ≤ TM + TR

Eml TM + TR ≤ t ≤ T,

Erv(t) =

Emr + (EMr − Emr)

[

1 − cos

(

πt

TM

)]

/2 0 ≤ t ≤ TM

Emr + (EMr − Emr)

[

cos

(

π(t − TM )

TR

)

+ 1

]

/2 TM ≤ t ≤ TM + TR

Emr TM + TR ≤ t ≤ T.

Note, the timing parameters TM and TR are the same for both the left and right ventricles. Inputto this model is heart rate HR. Output from the model is arterial blood pressure psa(t) and cerebralblood flow velocity vc = (pca − pcv)/Rc. Initial values for these differential equations are listed inTable 5 (below), while nominal values for all model parameters are listed in Table 1 and optimizedvalues for estimated parameters are given in Table 3.

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22 L.M. ELLWEIN, S.R. POPE, A. XIE, J. J. BATZEL, C.T. KELLEY, M.S. OLUFSEN

6.2. Respiratory model.

dcS,CO2

dt= MS,CO2

+ qs(ca,CO2− cS,CO2

)/VS,CO2

dcS,O2

dt= −MS,O2

+ qs(ca,O2− cS,O2

)/VS,O2

dcB,CO2

dt= MB,CO2

+ qc(ca,CO2− cB,CO2

)/VB,CO2

dcB,O2

dt= −MB,O2

+ qc(ca,O2− cB,O2

)/VB,O2.

Inspiration

dpD1,CO2

dt= Vie(pi,CO2

− pD1,CO2)/VD1

dpD1,O2

dt= Vie(pi,O2

− pD1,O2)/VD1

dpD2,CO2

dt= Vie(pD1,CO2

− pD2,CO2)/VD2

dpD2,O2

dt= Vie(pD1,O2

− pD2,O2)/VD2

dpD3,CO2

dt= Vie(pD2,CO2

− pD3,CO2)/VD3

dpD3,O2

dt= Vie(pD2,O2

− pD3,O2)/VD3

dpa,CO2

dt= 863 · 0.98 · qp(cv,CO2

− ca,CO2) + Vie(pD3,CO2

− pa,CO2)/VA

dpa,O2

dt= 863 · 0.98 · qp(cv,O2

− ca,O2) + Vie(pD3,O2

− pa,O2)/VA.

Expiration

dpD1,CO2

dt= Vie(pD1,CO2

− pD2,CO2)/VD1

dpD1,O2

dt= Vie(pD1,O2

− pD2,O2)/VD1

dpD2,CO2

dt= Vie(pD2,CO2

− pD3,CO2)/VD2

dpD2,O2

dt= Vie(pD2,O2

− pD3,O2)/VD2

dpD3,CO2

dt= Vie(pD3,CO2

− pa,CO2)/VD3

dpD3,O2

dt= Vie(pD3,O2

− pa,O2)/VD3

dpa,CO2

dt= 863 · 0.98 · qp(cv,CO2

− ca,CO2)/VA

dpa,O2

dt= 863 · 0.98 · qp(cv,O2

− ca,O2)/VA.

Note, instantaneous airflow is obtained from integrating the air flow velocity data as described inequation (13).

Inputs to the stand alone respiratory model are airflow velocity, average systemic, pulmonary, andcerebral blood flow obtained by solving the cardiovascular model. For the coupled model, these flows

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CARDIOVASCULAR & RESPIRATORY DYNAMICS IN CONGESTIVE HEART FAILURE 23

are computed at any instant in time. Initial values for the respiratory model are given in Table 6,while nominal parameters for all respiratory parameters are listed in Table 2, and optimized valuesfor estimated parameters are given in Table 3.

Table 6. Initial values for the respiratory model.

Variable Physiologic description Value Reference

pD1,CO2[mmHg] CO2 partial pressure lung deadspace 1 5 [27, 33]

pD1,O2[mmHg] O2 partial pressure lung deadspace 1 159 [27, 33]

pD2,CO2[mmHg] CO2 partial pressure lung deadspace 2 6 [27, 33]

pD2,O2[mmHg] O2 partial pressure lung deadspace 2 158 [27, 33]

pD3,CO2[mmHg] CO2 partial pressure lung deadspace 3 7 [27, 33]

pD3,O2[mmHg] O2 partial pressure lung deadspace 3 157 [27, 33]

pa,CO2[mmHg] Systemic arterial CO2 partial pressure 40 [58]

pa,O2[mmHg] Systemic arterial O2 partial pressure 100 [18]

cS,CO2[mlSTPD/ml] Systemic tissue CO2 concentration 0.543 [18]

cS,O2[mlSTPD/ml] Systemic tissue O2 concentration 0.128 [18]

cB,CO2[mlSTPD/ml] Cerebral tissue CO2 concentration 0.569 [27, 33]

cB,O2[mlSTPD/ml] Cerebral tissue O2 concentration 0.112 [27, 33]

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