A multiscale cardiovascular system model can account for the loaddependence of the endsystolic pressurevolume relationship.  
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PMID: 23363818 Owner: NLM Status: Publisher 
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ABSTRACT: BACKGROUND: The endsystolic pressurevolume relationship is often considered as a loadindependent property of the heart and, for this reason, is widely used as an index of ventricular contractility. However, many criticisms have been expressed against this index and the underlying timevarying elastance theory: first, it does not consider the phenomena underlying contraction and second, the endsystolic pressure volume relationship has been experimentally shown to be loaddependent. METHODS: In place of the timevarying elastance theory, a microscopic model of sarcomere contraction is used to infer the pressure generated by the contraction of the left ventricle, considered as a spherical assembling of sarcomere units. The left ventricle model is inserted into a closedloop model of the cardiovascular system. Finally, parameters of the modified cardiovascular system model are identified to reproduce the hemodynamics of a normal dog. RESULTS: Experiments that have proven the limitations of the timevarying elastance theory are reproduced with our model: (1) preload reductions, (2) afterload increases, (3) the same experiments with increased ventricular contractility, (4) isovolumic contractions and (5) flowclamps. All experiments simulated with the model generate different endsystolic pressurevolume relationships, showing that this relationship is actually loaddependent. Furthermore, we show that the results of our simulations are in good agreement with experiments. CONCLUSIONS: We implemented a multiscale model of the cardiovascular system, in which ventricular contraction is described by a detailed sarcomere model. Using this model, we successfully reproduced a number of experiments that have shown the failing points of the timevarying elastance theory. In particular, the developed multiscale model of the cardiovascular system can capture the loaddependence of the endsystolic pressurevolume relationship. 
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Antoine Pironet; Thomas Desaive; Sarah Kosta; Alexandra Lucas; Sabine Paeme; Arnaud Collet; Christopher G Pretty; Philippe Kolh; Pierre C Dauby 
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Type: JOURNAL ARTICLE Date: 2013130 
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Title: Biomedical engineering online Volume: 12 ISSN: 1475925X ISO Abbreviation: Biomed Eng Online Publication Date: 2013 Jan 
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Journal Information Journal ID (nlmta): Biomed Eng Online Journal ID (isoabbrev): Biomed Eng Online ISSN: 1475925X Publisher: BioMed Central 
Article Information Download PDF Copyright ©2013 Pironet et al.; licensee BioMed Central Ltd. openaccess: Received Day: 9 Month: 11 Year: 2012 Accepted Day: 17 Month: 1 Year: 2013 collection publication date: Year: 2013 Electronic publication date: Day: 30 Month: 1 Year: 2013 Volume: 12First Page: 8 Last Page: 8 PubMed Id: 23363818 ID: 3610305 Publisher Id: 1475925X128 DOI: 10.1186/1475925X128 
A multiscale cardiovascular system model can account for the loaddependence of the endsystolic pressurevolume relationship  
Antoine Pironet1  Email: a.pironet@ulg.ac.be 
Thomas Desaive1  Email: tdesaive@ulg.ac.be 
Sarah Kosta1  Email: sarah.kosta@ulg.ac.be 
Alexandra Lucas1  Email: lucasalexandra@hotmail.com 
Sabine Paeme1  Email: sabine.paeme@ulg.ac.be 
Arnaud Collet1  Email: a.collet@ulg.ac.be 
Christopher G Pretty1  Email: c.pretty@ulg.ac.be 
Philippe Kolh1  Email: philippe.kolh@chu.ulg.ac.be 
Pierre C Dauby1  Email: pc.dauby@ulg.ac.be 
1University of Liege (ULg), GIGACardiovascular Sciences, Liege, Belgium 
Since the experiments of Suga and Sagawa [^{1}], the concept of timevarying elastance (also termed “timevarying pressurevolume ratio”) has been extensively used by clinicians and engineers to simply and accurately represent ventricular function. This powerful concept states that ventricular pressure and volume can be related at any moment of the cardiac cycle by means of an activation function. This function, once normalized with respect to time and amplitude, is able to represent any loading condition of the ventricle. At the end of cardiac ejection (systole), this pressurevolume ratio is called the “endsystolic pressurevolume relationship” (ESPVR). The slope of the ESPVR has been widely used as a loadindependent index of ventricular contractility.
Thanks to its simplicity, the timevarying elastance theory has been used in many lumped mathematical models of the cardiovascular system. This theory has been extended to better reproduce the diastolic properties of the heart [^{2},^{3}]. A great advantage of this concept is that it allows fast model simulations, enabling the large number of model runs needed to identify model parameters and design patientspecific models for use at the bedside [^{4},^{5}].
However, many criticisms have been leveled against the timevarying elastance concept. First, its biggest advantage, namely that it allows a simple relationship between ventricular pressure and volume, is also its biggest drawback. Indeed, this ad hoc approach does not consider the fact that cardiac muscle contraction begins at a microscopic scale. Second, more recent experiments have shown the endsystolic pressurevolume relationship to be more parabolic than linear in shape [^{6}^{}^{8}]. Some researchers subsequently modified the timevarying elastance theory to include various nonlinear pressurevolume relationships [^{9},^{10}]. Third, instantaneous ventricular pressure has also been shown to be negatively dependent on instantaneous flow out of the ventricle, an effect that has been termed the “internal resistance” of the ventricle [^{11}^{}^{13}]. These authors also added their ad hoc modifications to the timevarying elastance theory to account for this resistive effect by including a flow term in the pressurevolume relationship. Finally, the relationship between ventricular pressure and volume has been demonstrated to depend on the mechanical load exerted on the ventricle [^{14}]. This result implies the loaddependence of the ESPVR, i.e. that the ESPVR is not unique. This effect cannot be accounted for by any modification of the timevarying elastance theory.
These objections have lead many authors to gain deeper knowledge of the fundamental mechanisms underlying cardiac contraction. For example, Burkhoff [^{8}] described a ventricular model, with its contraction based on chemical mechanisms initiated by a timevarying intracellular calcium concentration. Negroni and Lascano [^{15}] conceived a muscle model based on the same type of chemical pathways. This muscle model has been inserted into different ventricle models [^{15}] and even full cardiovascular system (CVS) models [^{16}]. In these studies, attention was more focused on the microscopic events happening during contraction while, to our knowledge, no study investigated the influence of such microscopic models on macroscopic hemodynamic variables such as volume and pressure. However, these two variables are the ones that have been used by experimental researchers to underline the limitations of the timevarying elastance theory.
The goal of the present work is thus to implement a multiscale CVS model, taking into account the physiological origin of cardiac contraction at the cellular level, that will provide answers to the objections formulated against the timevarying elastance theory. This implementation will allow a critical comparison of this model with the timevarying elastance model.
In the following sections, we describe how we assembled the multiscale CVS model from existing micro and macroscopic models of myocyte contraction and the circulatory system. Then, the model parameters are identified to reproduce the hemodynamics of a normal dog. Finally, experimental protocols that have shown the limitations of the timevarying elastance theory, including the loaddependence of the ESPVR, are reproduced with the model, demonstrating that these effects are correctly captured.
The multiscale cardiovascular system model developed in this work was assembled from three existing models: (1) a cardiac sarcomere model, (2) a ventricle model and (3) a CVS model (Figure 1). This section describes how the models are interconnected, while the following subsections contain descriptions of the models.
In this work, cellular electrophysiology is not described in detail. Instead, a typical experimental curve representing how intracellular calcium concentration varies with time was used as input for a cardiac sarcomere model. This cardiac sarcomere model involves chemical equations depicting how variations in calcium concentration modify force generation. The model also describes how muscle length influences force generation. A geometrical ventricle model was built by assembling sarcomere units around a sphere. The output force of the sarcomere was then used to compute ventricular pressure, which allowed it to be inserted into a cardiovascular circulation model. Closing the loop, the circulation model dictates how much volume goes in and out of the ventricle, thereby dictating length of the individual sarcomere units. The models interconnections are presented in Figure 1.
Intracellular calcium concentration has been derived from previously published data [^{17}] (see Figure 2). Two cosine branches were fit to the experimental profile, yielding the following expression for Ca^{2 +}(t):
(1)
Ca2+t={Camax21−cosπtT1if0≤t<T1Camax21+cosπt−T1T2−T1ifT1≤t<T20otherwise 
where T_{1} is the time at which maximum calcium concentration occurs, T_{2}, the time at which calcium concentration goes back to zero and Ca_{max} is the maximum value of the calcium concentration. The fitting result is displayed in Figure 2 with corresponding parameter values displayed in Table 1.
The fourstate model of Negroni and Lascano [^{15}] was used to compute active force generated by sarcomeres from intracellular calcium concentration. The functioning of this model is briefly described in this section. Originally, this model did not use an input calcium concentration as described in the previous section but instead described calcium dynamics. For simplicity and rapidity of model simulation, we chose to use a calcium driver function, as done by others [^{8}], including Negroni and Lascano [^{15}].
The model considers only a halfsarcomere of length L, as shown in Figure 3. This halfsarcomere consists of an elastic element, responsible for passive force (F_{p}) generation, according to the following equation:
(2)
Fp=KL−L05, 
where K is the stiffness of the element and L_{0} is its resting length.
This elastic element is in parallel with a thick (myosin) filament that can bind a thin (actin) filament. The binding takes place through one “equivalent” crossbridge (representing the average of all actual crossbridges), whose mobile end (the black dot in Figure 3) binds actin. If the halfsarcomere shortens (L decreases), the equivalent crossbridge also does (h decreases), and is not at equilibrium anymore. Hence, the crossbridge will stretch, which means its length h will increase to reach its equilibrium value h_{c}, as dictated by the following equation:
(3)
dXdt=Bh−hc=BL−X−hc 
where X = L − h and B is the rate at which h reaches its equilibrium value h_{c}.
The mechanism of active force generation involves troponin (T), which, in the halfsarcomere model, goes through four different states during one calcium cycle. First, when calcium is released, it can bind troponin to form a complex, denoted TCa. This allows troponin to bind myosin, located on the thin filaments (TCa*). Afterwards, calcium can be released, while troponin is still bound to myosin (T*). Finally, troponin and myosin detach, and troponin goes back to its initial state T.
The chemical kinetics behind all these transformations are given by the following equations (see [^{15}] for details):
(4)
dTCadt=QbQadTCa*dt=QaQrQd1dT*dt=QrQdQd2 
The rates involved in the previous equations are given by:
(5)
Qa=Y2·TCaeffZ2·TCa*Qb=Y1·Ca2+·TZ1·TCaQr=Y3·TCa*Z3·T*·Ca2+Qd=Y4·T*Qd1=Yd·dXdt2·TCa*Qd2=Yd·dXdt2·T* 
where TCaeff=TCae−RL−La2 is the effective concentration of calcium bound to troponin. This effective concentration is introduced to account for the overlap between thin and thick filaments. Overlap is maximal when L = L_{a}. R is a parameter controlling the curvature of the function and parameters Y_{i} and Z_{i} are reaction rates of the previous chemical reactions. Values for these parameters are displayed in Table 1. They are either taken from [^{15}] or identified (see section “Parameter adjustment to canine data”).
Since troponin concentration does not vary with time, concentration of troponin in its initial state can be deduced from concentration of troponin in modified states ([TCa*], [TCa] and [T*]) by:
(6)
T=Tt−TCa*−TCa−T* 
where T_{t} is total troponin concentration.
In the equations describing the chemical kinetics of the sarcomeres, reaction rates can be seen to depend on L, the length of a halfsarcomere (through TCa_{eff}), and dL/dt, the change of L with time (through dX/dt that influences Q_{d1} and Q_{d2}). This dependence modifies the reaction rates if the sarcomere is stretched and how it is stretched, accounting for wellknown forcelength and forcevelocity relationships [^{20}]. Finally, active force, F_{b}, is related to the concentration of troponin attached to myosin via:
(7)
Fb=ATCa*+T*L−X 
where A is a lumped constant establishing the bridge between forces generated by a single halfsarcomere and the whole muscle unit.
Total muscle force is given by the sum of active and passive forces:
(8)
F=Fb+Fp. 
A series elastic element was added to model the effects of myocyte compliance. The force produced by this elastic element was defined to be:
(9)
Fs=αeβLs−1. 
where α and β are parameters describing the shape of this passive force. The force, F_{s}, and length, L_{s}, of this elastic element are linked to the force, F, and length, L, of the muscle unit by:
(10)
F=Fs 
(11)
Lt=L+Ls 
Equation (10) has to be solved numerically to find the value of L.
The spherical ventricle model of Negroni and Lascano [^{15}] was used to describe the left ventricle. This model consists of an arrangement of N_{c} halfsarcomeres on the circumference of a sphere. The number, N_{c}, of halfsarcomeres can be obtained by the formula
(12)
Vmw=KvLt3 
where V_{mw} is midwall volume, the logarithmic mean of inner and outer ventricle volumes [^{21}] and K_{v} = N_{c}^{3}/6π^{2}.
The pressure inside the spherical left ventricle can be obtained by the following relationship, derived from energetic considerations [^{21}]:
(13)
Plv=5σVwVmw 
where σ is the average fiber stress and V_{w} is the volume of the ventricular wall. Note that the numerical factor 5 in this equation is different from the one published by Regen [^{21}] because pressure has been converted from mN/mm^{2} to mmHg for comparison with physiological data (1 mN/mm^{2} ≈ 7.5 mmHg).
Since the force, F, computed in the previous section is a force per unit of muscle surface, a total force for the whole muscle unit can be computed by multiplying F by the reference area (a_{r}) of the muscle unit:
(14)
FM=Far. 
Then, the average fiber stress, σ, can be computed by dividing this total force by the current area (a) of the muscle unit:
(15)
σ=FMa=Fara. 
Under the assumption of constant muscle unit volume, we have:
(16)
Lrar=Lta. 
where L_{r} is the reference length of the muscle unit. Finally, this yields:
(17)
σ=FLtLr. 
This, combined with Equation 13, gives the following expression for left ventricular pressure:
(18)
Plv=5FLrVwKvLt2. 
Originally, a passive term was added to this equation to more correctly account for diastolic properties, but we chose to neglect it here after verifying that it did not greatly affect the results of the model simulations.
To derive the total length of the muscle unit from the left ventricle volume V_{lv}, the fraction of wall volume enclosed in the midwall volume is assumed to be a constant [^{15}], denoted f:
(19)
f=Vmw−VlvVw. 
This equation is supposed to be valid for any values of the left ventricle volume, V_{lv}, thus making it possible to derive the length of the muscle fiber inverting the previous equation and using Equation (12):
(20)
Lt=Vwf+VlvKv3. 
The ventricular model introduced in the previous section is then inserted into a closedloop CVS model. Our approach is based on a model developed by Smith [^{2}], that describes cardiac contraction by the means of the timevarying elastance theory applied to the left and right ventricles. For completeness, this initial model will be briefly described below. Then, the ventricular model presented above will replace the timevarying elastance model for the left ventricle.
The CVS model of Smith [^{2}] is a lumpedparameter model consisting of six elastic chambers. These chambers represent the left (lv) and right (rv) ventricles, the aorta (ao), the vena cava (vc), the pulmonary artery (pa) and the pulmonary veins (pu). Vessels with modelled flow resistance link the six chambers. Those vessels are, respectively, the systemic (sys) and pulmonary (pul) circulations and the cardiac valves: the mitral (mt), aortic (ao), tricuspid (tc) and pulmonary (pv) valves (Figure 4). Valvular behavior is modelled with elements analogous to ideal diodes in series with additional flow resistance.
The initial formulation of the model also included inertances, elements accounting for the inertia of the blood going through the valves [^{2}]. We choose not to include these elements in the model, as they have been shown to have small values and to weakly affect model dynamics. Furthermore, neglecting these inertial parameters reduces uncertainty in the parameter estimation process. Finally, these parameters are difficult to measure and not well defined [^{22}].
The model chambers are characterized by two variables: their volume (V) and the pressure (P) inside of the chamber. The two elastic chambers representing the ventricles are said to be active, which means that the relationship between the pressure and the volume is variable. More precisely, it varies between the enddiastolic pressurevolume relationship (EDPVR) and the ESPVR, respectively:
(21)
EDPVR:PiVi=P0,ieλiVi−V0,i−1 
(22)
ESPVR:PiVi=EiVi−Vd,i 
where E_{i} (i being either “lv” or “rv”) is the endsystolic elastance, V_{d,i}, the endsystolic volume at zero pressure, V_{0,i}, the enddiastolic volume at zero pressure and P_{0,i} and λ_{i} are parameters of the nonlinear relationship (21). For simplicity, we assume V_{d,i} and P_{0,i} to be zero, such that the EDPVR is coincident with the volume axis and the ESPVR goes through the origin.
Transition between these two extreme relationships is mediated by an activation function, varying between 0 and 1 and denoted e(t). Consequently, the pressure in a ventricle at any moment t of the cardiac cycle is linked to the volume by:
(23)
Pit,Vit=etEiVit−Vd,i+1−etP0,ieλiVit−V0,i−1. 
The shape of the activation function for a dog has been described [^{3}] as:
(24)
et=∑j=13Aje−Bjt−Cj2. 
Values of the constants A_{j}, B_{j} and C_{j} are taken from [^{3}] and are displayed in Table 1.
The other elastic chambers are passive, which means that their volumes and pressures are linked by a constant, the elastance, E.
(25)
P=EV 
The volume change in the six elastic chambers can be derived from the continuity equation:
(26)
dVdt=Qin−Qout 
where Q_{in} and Q_{out} are, respectively, flow coming in and going out of the compartment. Equation (26) does not consider the heart valves, whose role is to prevent backwards flow. Hence, to correctly model the effect of the valves, a negative flow has to be replaced by a zero flow. Mathematically, it can be easily done by replacing each flow, Q(t), controlled by a valve by r(Q(t)), where r denotes the ramp function, defined as
(27)
rx={0,ifx<0x,ifx≥0 
Then, a positive flow is unchanged and a negative flow is replaced by 0.
A vessel element links each pair of elastic compartments of the model in Figure 4. Each of these vessels is characterized by an hydraulic resistance, denoted R. The relationship between flow, pressure and resistance is given by Poiseuille’s law:
(28)
Q=Pup−PdownR 
where P_{up} and P_{down} represent the pressure up and downstream of the chamber, respectively. Note that the original model also takes into account the effect of the pericardium and the septum [^{2}]. For simplicity, these effects were neglected here.
The complete cardiovascular system model of Smith et al. has been previously shown to be able to reproduce the major features of the cardiovascular system. It has been validated in silico[^{23}], in several animal model studies [^{4},^{24},^{25}] and is currently applied in one human study.
In our approach, and for the corresponding results discussed below, this original model is modified by considering for the left ventricle the model described in the previous section and based on the microscopic behavior of the equivalent halfsarcomere, instead of the timevarying elastance concept. This amounts to replace the description of the left ventricle by Equation (23) with Equations (1) to (11), (18) and (20). A summary of all the model equations is given in Additional file 1.
The right ventricle is still described by Equation (23), but values of C_{j} have been shifted so that left and right ventricles contract at the same time, as is the case physiologically.
The complete model of the CVS presented above involves a large number of parameters, which are all summarized in Table 1. The hemodynamic parameters of our model and the geometrical properties of the ventricle (K_{v}, V_{w} and f) were numerically adjusted to match pressure and volume data from two canine experimental studies [^{6},^{18}]. These are displayed in Table 2. We chose to identify these parameters because they directly influence ventricular pressure and volume (cf. Equations (18) and (20)) and have a macroscopic relevance. Other microscopic parameters related to the biochemical reactions and sarcomere length were kept at values obtained from literature as no data was available to adjust them.
The experimental canine pressurevolume loops published in the previous studies were taken as reference for parameter identification. More precisely, four characteristic points were manually identified on these pressurevolume loops, namely points of beginning and end of systole and diastole. This yielded a total of eight pressurevolume points that were used for parameter computation.
Four parameters were directly computed from these pressurevolume points, while others were iteratively adjusted. These steps of the identification process are described in the following sections and are summarized in in Figure 5.
Four parameters can be directly computed from the reference pressurevolume loops, the first of which is the volume of the left ventricular wall, V_{w}. The enddiastolic volume makes it possible to derive the radius of the left ventricle, R_{lv}, which can then be used to compute the outer ventricular volume by taking into account the ventricular wall thickness, t:
(29)
Vlv,ed+Vw=43πRlv+t3. 
where thickness was set at t = 0.9 cm [^{15}]. Finally, Equation (29) can be used to compute V_{w}.
Second, we simplified Equation (23) for the right ventricle by assuming V_{d,rv} and P_{0,rv} to be zero, which is a common assumption [^{5}]. Equation (23) then becomes:
(30)
Prv=etErvVrv. 
Using the point of endsystole detected on the right ventricular pressurevolume loop and knowing that e(t) = 1 at endsystole (es) gives a direct way of computing E_{rv}:
(31)
Erv=Vrv,esPrv,es. 
Finally, flow resistances of the systemic and pulmonary circulations, R_{sys} and R_{pul}, can be directly computed using the definition of systemic vascular resistance [^{20}] and the pulmonary equivalent, namely:
(32)
Rsys=PÂ¯ao−PÂ¯vcCO 
(33)
Rpul=PÂ¯pa−PÂ¯puCO 
where the bar is used to denote the mean and CO represents cardiac output (the volume of blood ejected by the heart per unit time). All the righthand side elements of Equations (32) and (33) can be estimated by inspection of the pressurevolume loops. First, cardiac output is computed by dividing stroke volume (equal to the difference between enddiastolic and endsystolic volumes) by cardiac period, which is a fixed parameter of the model. Mean arterial pressures were assumed equal to endsystolic ventricular pressures. Finally, mean vena cava pressure was assumed to be equal to the average of right ventricular pressure at the beginning and end of diastole. The same approach was applied to mean pulmonary vein pressure and left ventricular pressure.
Direct computation of the four parameters V_{w}, E_{rv}, R_{sys} and R_{pul} carried out as described above yielded the values displayed in Table 1.
An iterative process, described in the next section, was used to adjusted other hemodynamic parameters and geometric features of the left ventricle. Before describing this process, we will first discuss how initial values for these identified parameters were obtained.
Initial values for identified hemodynamic parameters, namely elastances of arterial and venous chambers (E_{ao}, E_{pa}, E_{vc} and E_{pu}) and valve resistances (R_{mt}, R_{av}, R_{tc} and R_{pv}) were taken from previously published values for pigs [^{5}]. This initialization process is termed as “Step 2” in Figure 5.
Initial values for parameters describing geometric features of the left ventricle (K_{v} and f) were obtained by performing simulations of isovolumic contractions. In these experiments, the ventricle contracts while being submitted to a constant volume. An example of such simulations can be found in section "Loaddependence of the ESPVR". The interest of performing isovolumic contractions simulations is that a constant volume input suppresses the influence of the circulatory system and the related parameters, which allows the ventricular parameters to be estimated separately. Parameters K_{v} and f were iteratively adapted so that endsystolic and enddiastolic pressures for four different isovolumic contractions match corresponding points on the (ejecting) pressurevolume loops published by Kass [^{6}]. The loaddependence of the ESPVR implies that maximum pressure for ejecting and isovolumic beats are different [^{11},^{12}], this is why this technique was only used to get initial parameter values. The step described in this paragraph is referred to as “Step 3” in Figure 5.
First, the CVS model was split into systemic and pulmonary circulations, by assuming constant systemic and pulmonary venous pressures [^{26}]. (Constant venous pressures were computed as the mean values of begin and end diastolic ventricular pressures, see above.) The vena cava and pulmonary veins compartments become points with constant pressures, implying that the two subsystems, each composed of a ventricle and an arterial compartment, become independent. The two subsystems must share the same stroke volume, thus single stroke volume value in Table 2. Furthermore, mean ventricular volumes were assumed to be similar, hence the unique value for mean ventricular volume in Table 2.
Step 4 of the identification process was to identify parameters of the systemic subsystem (R_{mt}, R_{av}, E_{ao}, K_{v} and f) with the left ventricle described by the physiological model detailed in the previous sections. Reference data are those displayed in the first 6 rows of Table 2 (mean and amplitude of left ventricle volume and pressure, plus aortic pressures at aortic valve opening and closing).
Step 5 was equivalent to step 4 applied to the pulmonary subsystem, where cardiac contraction is dictated by a timevarying elastance. Parameters E_{rv}, R_{tc} and R_{pv} were adjusted so that simulated data matched reference values displayed in the 4 bottom rows of Table 2 (mean and amplitude of right ventricle volume and pressure, plus mean pulmonary artery pressure).
Finally, for step 6, the two subsystems were put back together, with venous pressures allowed to vary. The last step of the identification process was to adjust parameters E_{vc} and E_{pu} using the whole set of reference values displayed in Table 2.
In all the steps of this process, parameters were adjusted using Nelder and Mead’s simplex method [^{27}]. All the computations were carried out using MATLAB (2011a, MathWorks, Natick, MA).
With the adjusted model parameters, we will first show (1) that the model is able to correctly reproduce the sequence of events occurring during a cardiac cycle. Then, some experimental protocols that have proven the limitations of the timevarying elastance theory will be reproduced with the model: (2) preload reduction simulations through atrial hemorrhage and increase of mitral valve resistance, (3) afterload increase simulations via increase of aortic elastance or systemic vascular resistance, (4) the same simulations with increased ventricular contractility, (5) isovolumic contraction simulations, and (6) flowclamp simulations. These simulations allow us to investigate how well the model reproduces experimental reality. Specifically, we investigated if the model reproduced the loaddependence of the ESPVR.
After parameter adjustment, simulated pressures and volumes were of the same order of magnitude as reference experimental results displayed in Table 2. As an example, the pressurevolume loop of the left ventricle is shown in Figure 6. Figure 7 represents simulated temporal pressure curves in the pulmonary veins, left ventricle and aorta during one heartbeat.
One goal of our work was to numerically investigate the loaddependence of the endsystolic pressurevolume relationship. To do so, we numerically reproduced the experiments of Kass et al. [^{6}]. These authors recorded left ventricular pressurevolume loops in six openchest dogs with rapidly varying preload. Preload reduction was achieved by left atrial hemorrhage: a cannula was inserted in the left atrium and connected to a 1 l reservoir that was lowered to decrease left ventricle filling pressure. The preload reduction experiment consisted of lowering the reservoir for 10 to 12 s and simultaneously recording the pressurevolume loops. This technique allowed them to create a broad range of preloads, and thus to more completely characterize the ESPVR. One of their observations was that the shape of the ESPVR curve was more parabolic than linear.
To remain as close as possible to such experimental protocols in our CVS model, an outward flow of 50 ml/s was inserted in the circulation model, at the level of the pulmonary veins. (Since the atria are not explicitly included in the model, they are merged with the pulmonary vein compartment). The model was then simulated for a further 4.8 s (i.e. 8 cycles). Resulting pressurevolume loops are shown in Figure 8.
To evaluate the linearity of the endsystolic pressurevolume relationship, we reproduced the computations carried out by Kass et al. [^{6}]. Simulated pressurevolume loops were divided into two sets (one containing the first four loops, the other containing the last four) and a linear ESPVR of the form
(34)
Pes=EesVes−V0 
was computed for the two sets with an iterative process [^{28}]. This process involved fitting a straight line to the endsystolic points, defined as points with the maximal P/(V − V_{0}) ratio. During the first step of the process, V_{0} was fixed at zero. Then, coefficients of the linear regression yielded new estimates for E_{es} and V_{0} and the process was repeated until convergence was achieved.
As suggested by Kass et al. [^{6}], a parabolic regression of the form
(35)
Pes=aVes−V0'2+bVes−V0' 
was also computed to fit the points of endsystole. To perform this regression, the previous function was fit to points of endsystole with a nonlinear leastsquares algorithm. We performed such a computation, the result of which is shown in Figure 9 and in Table 3. The curvature parameter a in Equation (35) was statistically different from zero with a p = 0.0383.
In this section, we compare the previously derived ESPVR to four other ESPVRs: one resulting from another preload reduction, two coming from afterload variations and another one resulting from isovolumic contraction simulations.
In the previous section, preload was varied by the means of a simulated atrial hemorrhage. Here, we simulated another preload reduction experiment by a tenfold increase of the mitral valve resistance. The result of this simulation is shown in Figure 10. Points of endsystole were computed as previously described and a parabolic ESPVR was fit to these points (white dots in Figure 10).
Afterload can be defined as the pressure the ventricle has to overcome to eject. Doubling aortic elastance, which has the effect of increasing aortic pressure, increases afterload. The pressurevolume loops obtained by such a simulation are shown in Figure 11. Once again, we computed the ESPVR as a parabolic fit of the points of endsystole. In this case, the ESPVR was parabolic, but with a slightly positive concavity (a = 0.2426 mmHg/ml^{2}, p < 0.001). This ESPVR is represented in dashdotted line in Figure 11, along with ESPVRs obtained with preload reduction simulations.
Afterload can also be increased by doubling the value of the systemic vascular resistance in the model [^{23}]. The simulation result is shown in Figure 12. In this figure, the ESPVR computed from points of endsystole is also displayed (blackdotted line). It was not significantly different from a straight line. (The pvalue of the quadratic parameter a was p = 0.662.) .
To numerically reproduce the experiments of Suga and Sagawa [^{1}] and Burkhoff [^{14}], we also performed simulations of isovolumic contraction experiments. In an isovolumic contraction, the ventricular volume remains constant and thus only depends on the fixed volume.
The isovolumic contraction simulations were repeated for different constant volumes, ranging from 10 ml to 25 ml. The developed pressure for each of the constant volumes is plotted in bold line in Figure 13. For comparison, the ESPVRs that were computed from the four previous load variation simulations are also shown.
As previously explained, the ESPVR was initially proposed as a loadindependent index of contractility [^{29}]. After having examined the loadindependence of the ESPVR in the previous section, we will now focus on the variations of the ESPVR due to changes in ventricular contractility.
Contractility was increased in the model by doubling the peak value of intracellular calcium concentration. In their experimental protocol, Suga et al.[^{29}] increased contractility by epinephrine infusion, the effect of which is to promote calcium release by the sarcoplasmic reticulum [^{20}]. To model this effect, we thus choose to double the value of the parameter Ca_{max} (Equation (1)), hence doubling the peak value of intracellular calcium concentration in the model.
The previous preload reduction and afterload increase simulations were repeated with increased contractility. The result of these simulations is shown in Figure 14. Each one of the four ESPVRs presented in the previous section is displayed in black in this figure. The four ESPVRs obtained after increasing contractility and repeated the load variation simulations are displayed in red for comparison.
Shroff [^{12}] performed flow clamp experiments on an isolated canine heart. These experiments consisted in imposing linear ramps of ventricular volume (and consequently, constant flows out of the ventricle) and observing the effects of their timing and magnitude on ventricular pressure. The main conclusion of these authors was that, for a given ventricular volume, the bigger the flow out of the ventricle, the lower the pressure. They found that this relation between ventricular pressure and flow was linear and called the slope of the linear relation the “internal resistance” of the ventricle. This resistance was found to be approximately equal to 0.0799 mmHg s/ml (in absolute value) for a normal contractile state.
The ability of the model of Negroni and Lascano [^{15}] to correctly reproduce various flow clamp experiments has been extensively described by these authors and hence, will not be described in detail here. In short, we also simulated flowclamp experiments in our left ventricle model with adjusted parameters and observed, as expected, a decrease in ventricular pressure as flow out of the ventricle increased.
The model developed in this work accounts for the physiological, microscopic origin of cardiac contraction, making it a much more realistic model of cardiac contraction than the timevarying elastance model. But this improved realism comes with a cost: to be used, the timevarying elastance concept only requires two parameters (maximum elastance and time to peak) and one input function (the activation function) [^{29}]. On the other hand, the model developed here requires 22 parameters (see sections “Chemical kinetics”, “Crossbridge parallel and elastic forces” and “Forcelength to pressurevolume conversion” of Table 1) and one input function (the intracellular calcium concentration). Values have to be assigned to all of these parameters, which required the development of a numerical parameter estimation scheme, as described in section “Parameter adjustment” and in Figure 5. Due to the large number of parameters, a large amount of data was required, which could not be obtained for a single animal. However, to remain as close as possible to the experiments we intended to simulate, we only used reference data coming from experiments on dogs.
After parameter adjustment, the model was able to correctly simulate the normal succession of events during a cardiac cycle. The four phases of cardiac contraction, namely filling, isovolumic contraction, ejection and isovolumic relaxation can clearly be distinguished on the pressurevolume loop of Figure 6. The timing of these phases was also physiologically correct, as shown in Figure 7. The first phase represented in Figure 7 is isovolumic contraction: after mitral valve closing, ventricular pressure increased until the aortic valve opened (crossover of ventricular and aortic pressures). Then, during ejection, the aorta filled up with blood, increasing its pressure until it exceeded the ventricular pressure. At this moment, the aortic valve closed, denoting the beginning of isovolumic relaxation. When ventricular pressure dropped below pulmonary venous pressure, the mitral valve opened, allowing filling of the ventricle. Mitral valve closure occurred when ventricular pressure rose at the initiation of a new contraction, and the cycle repeats.
Since one of the main objectives of this article was to evaluate the loaddependence of the ESPVR, this concept had to be clearly defined. We made the choice to compute ESPVRs during transient variations in ventricular volume and pressure following abrupt changes of load exerted on the ventricle. We chose this method to better agree with experimental reality. However, this method is different than some experimental procedures where ESPVR is computed only from stabilized pressures and volumes [^{30}] or from isovolumic pressurevolume relationships [^{14}]. Since the ESPVR is loaddependent, it is important to describe precisely how it was derived to enable correct comparisons between our model simulations and previously published experimental results.
As can be seen in Figure 8 and in Table 3, the two linear ESPVRs computed from the atrial hemorrhage experiment yielded different slopes and volume axis intercepts for each one of the two sets of loops. The same observation was made experimentally by Kass et al. [^{6}], who suggested that the ESPVR could more accurately be described by a quadratic curve. The result of this computation is shown in Figure 9. As can be seen in this figure, the parabolic ESPVR obtained by atrial hemorrhage is close to linear. This may come from the insufficient variation in ventricular volume. Indeed, we had to limit our atrial hemorrhage simulation to an outward flow of 50 ml/s during 4.8 s to avoid complete emptying of the pulmonary veins. Kass et al. [^{6}] used an outward flow of 80 to 100 ml/s during 10 to 12 s. Still, our parameter a, describing the concavity, is negative, meaning that the ESPVR is concave towards the volume axis, as experimentally determined [^{6},^{7}].
The biggest assumption of the timevarying elastance theory is that the ESPVR is loadindependent. It is thus assumed to be unique and to only depend on the contractile state of the heart, which makes it a powerful index of contractility. We numerically reproduced experiments that have shown the opposite, i.e. that the ESPVR depends on the load imposed on the ventricle. To do so, we showed that the model could exhibit many different ESPVRs. The simulation of a preload reduction experiment by atrial hemorrhage allowed us to trace a first ESPVR that was slightly parabolic and concave towards the volume axis (Figure 9). A second preload reduction experiment by increase of the mitral valve resistance yielded a clearly different ESPVR (Figure 10). Yet, the result of the two preload reduction experiments was, as expected, a shift of the pressurevolume loops towards the lower volumes. The same observation can be formulated from the results of the two afterload increase simulations (Figures 11 and 12). The ESPVRs resulting from these two simulations were similar but different from one another. They were also clearly different from the ESPVRs computed by the preload reduction simulations, reproduced in dashed and whitedotted lines in Figures 11 and 12. Also, as physiologically expected, the result of the preload increase was an increase of ventricular pressure and a decrease of stroke volume.
It has been experimentally shown that the endsystolic pressure for an ejecting beat is higher than for an isovolumic beat [^{14}]. This is called the positive effect of ejection. This effect does not appear on our simulations since the maximal isovolumic pressures are higher than the one obtained by simulating load variations. However, as can be seen in Figure 13, one ESPVR was higher than pressures generated by isovolumic contraction simulations, at least for some values of volume. Hence, this positive effect of ejection is also present in the model. Negroni and Lascano [^{15}] performed the same kind of experiments and were able to reproduce this experimental finding that ventricular pressure during ejection is higher than during isovolumic contraction. However, because they had not introduced their ventricle model in a model of the circulatory system, they had to create a physiological approximation of a ventricular volume curve. In our model, the ventricular volume is not imposed; it is a consequence of pressures in the other model compartments. Finally, it is also clear in Figure 13 that the pressurevolume points resulting from isovolumic contraction simulations do not fall on any of the four previously computed ESPVRs. This shows that, if one defines the ESPVR from isovolumic contraction experiments [^{14}], the result is again different than what would be obtained by load variations.
The effect of increased contractility on the ESPVRs is a shift towards higher volumes and lower pressures (Figure 14), which is what has been experimentally observed [^{6},^{7}]. Indeed, the effect of increased contractility is an increased developed pressure and a reduction in enddiastolic volume [^{20}]. We also notice an increase in the curvature of the ESPVRs derived from preload reduction simulations. It can be seen in Figure 14 that the dashed and whitedotted ESPVRs become more curved after increasing the contractility. The same observation has been made experimentally in dogs [^{6}]. Interestingly, we find a similar change in curvature of ESPVRs derived by afterload increases, i.e. the ESPVRs become more curved when contractility is increased. We found no experimental study assessing contractilitydependence of ESPVRs derived from afterload variations to compare this finding with.
Shifting of the ESPVR can be used to track variations of contractility, irrespective of how the ESPVR was derived. However, since the ESPVR is not unique, it cannot be assumed to represent an absolute measure of cardiac contractility.
As a concluding remark, please note that figures extracted from simulations and those published from experiments are not supposed to match exactly, since model parameters were adjusted to match two different experimental studies (see section “Parameter adjustment”). Furthermore, the CVS model used here is a very simple one in which the lumped properties cannot account for the rich features of experimentally measured waveforms. Additionally, only the contraction of the left ventricle was described from a microscopic point of view. This work is a first step into the microscopic description of cardiac contraction and its repercussion on hemodynamics. Hence, the goal was only for the model to be able to reproduce experimental trends, which it has achieved successfully.
The model presented in this work is able to overcome the drawbacks of the timevarying elastance theory, namely its lack of physiological foundations and the loaddependence of the ESPVR. A large number of experiments that have proven the flaws of the timevarying elastance theory have been reproduced in silico, namely preload and afterload variation experiments, isovolumic contraction experiments, flowclamp experiments and investigation of the effect of contractility on the ESPVR. Conclusions derived from model simulations were the same as these coming from canine experiments, i.e. the ESPVR is loaddependent. Consequently, when describing an ESPVR, it is essential to explain how it was derived to allow for comparison between ESPVRs.
Because the ESPVR depends on the load exerted by the vasculature on the ventricle, it cannot be considered as an intrinsic ventricular property. Additionally, the ESPVR cannot represent an absolute measure of cardiac contractility, because the shape of the ESPVR depends on how it was derived. But, instead of focusing on the absolute value of the contractility by the means of the slope of the ESPVR, it would be more reliable to speak in terms of variations of contractility from a reference state.
The work that has been done here for the left ventricle could easily be adapted to the right ventricle and, with some further modeling work, to the atria. Such a representation of the atria could be useful, since, to our knowledge, there exists no accurate timevarying elastance model applicable to the atria. Another possible improvement would be to introduce the effects of the septum and the pericardium in the circulation model. These effects have been neglected here for simplicity, but the model could easily be adapted to take them into account. With these proposed ameliorations, this microscopic contraction model would make it possible to fully describe a complete heart, composed of its four interacting chambers.
In conclusion, the multiscale cardiovascular model developed in this work overcomes the lack of physiological grounds of the timevarying elastance theory. In addition, model simulations successfully replicate trends observed in experiments that showed the limitations the timevarying elastance theory.
The authors declare that they have no competing interests.
AP performed the model simulations and drafted the manuscript, AL, AC and SP carried out the literature review, studied the Burkhoff (1994) model and helped writing the manuscript, SK also carried out the literature review and studied the Negroni and Lascano (1996, 1999 and 2008) models, TD and PD conceived the study and participated in its design and coordination and helped to draft the manuscript, PK provided useful support about cardiovascular physiology and CP helped writing the manuscript. All authors read and approved the final manuscript.
Model Equations.
This work was supported by the Fédération WallonieBruxelles (Actions de Recherches Concertées – Académie WallonieEurope), the F.R.I.A. (Belgium) and the FNRS (Belgium).
References
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Figures
Tables
Values of the model parameters
Parameter  Value  Units  Source 

Intracellular calcium concentration


T_{1}

40.6

ms

Fit from [^{17}]

T_{2}

130.2

ms

Fit from [^{17}]

Ca_{max}

1.47

μM

Fit from [^{17}]

Chemical kinetics


Y_{1}

39

μM s^{–1}

[^{15}]

Z_{1}

30

s^{–1}

[^{15}]

Y_{2}

1.3

s^{–1}

[^{15}]

Z_{2}

1.3

s^{–1}

[^{15}]

Y_{3}

30

s^{–1}

[^{15}]

Z_{3}

1560

μM s^{–1}

[^{15}]

Y_{4}

40

s^{–1}

[^{15}]

Y_{d}

8

s μm^{–2}

[^{15}]

T_{t}

70

μM

[^{15}]

B

800

s^{–1}

[^{15}]

h_{c}

0.005

μm

[^{15}]

L_{a}

1.17

μm

[^{15}]

R

20

μm^{–2}

[^{15}]

Crossbridge parallel and elastic forces


A

1800

mN mm^{–2} μm^{–1} μM^{–1}

[^{15}]

K

140 000

mN mm^{–2} μm^{–5}

[^{15}]

L_{0}

0.97

μm

[^{15}]

α

0.5

mN mm^{–2}

[^{15}]

β

75

μm^{–1}

[^{15}]

Forcelength to pressurevolume conversion


K_{v}

29.0

ml μm^{–3}

Adjusted

L_{r}

1.05

μm

[^{15}]

V_{w}

60.6

ml

Computed from [^{6}]

f

0.217



Adjusted

Hemodynamic parameters


E_{pa}

0.953

mmHg ml^{–1}

Adjusted

E_{pu}

0.0302

mmHg ml^{–1}

Adjusted

E_{ao}

0.965

mmHg ml^{–1}

Adjusted

E_{vc}

0.0107

mmHg ml^{–1}

Adjusted

E_{rv}

2.06

mmHg ml^{–1}

Computed from [^{6},^{18}]

R_{pul}

2.74

mmHg s ml^{–1}

Computed from [^{6},^{18}]

R_{sys}

5.65

mmHg s ml^{–1}

Computed from [^{6},^{18}]

R_{av}

0.152

mmHg s ml^{–1}

Adjusted

R_{mt}

0.0313

mmHg s ml^{–1}

Adjusted

R_{pv}

0.0269

mmHg s ml^{–1}

Adjusted

R_{tc}

0.459

mmHg s ml^{–1}

Adjusted

Total blood volume

1500

ml

[^{19}]

Right ventricle driver function


A_{1}

0.956



[^{3}]

A_{2}

0.625



[^{3}]

A_{3}

0.0180



[^{3}]

B_{1}

255

s^{–2}

[^{3}]

B_{2}

225

s^{–2}

[^{3}]

B_{3}

4230.0

s^{–2}

[^{3}]

C_{1}

0.431

s

Adapted from [^{3}]

C_{2}

0.328

s

Adapted from [^{3}]

C_{3}

0.374

s

Adapted from [^{3}]

Cardiac period  0.6  s  [^{3}] 
These values have been taken as such from the literature or have been adjusted by an iterative process (see text for details).
Reference values for parameter identification
Measurement  Value  Units  Source 

LV pulse pressure

119

mmHg

[^{6}]

Mean LV pressure

66

mmHg

[^{6}]

Aortic pressure at valve opening

115

mmHg

[^{6}]

Aortic pressure at valve closing

121

mmHg

[^{6}]

Mean ventricular volume

20.8

ml

[^{6}]

Stroke volume

11.7

ml

[^{6}]

RV pulse pressure

45.8

mmHg

[^{18}]

Mean RV pressure  22.9  mmHg  [^{18}] 
Coefficients of the linear and parabolic ESPVRs

Linear ESPVR

Parabolic ESPVR



First loops  Last loops  

E_{es}

V_{0}

E_{es}

V_{0}

a

p*

b

V_{0}^{’}

Experiments

6.49

−5.70

23.30

4.00

−2.68

0.005

30.00

3.90

Simulations  1.74  −42.2  2.33  −27.4  −0.0678  0.0383  5.62  −10.4 
*Statistical significance of parameter a.
Experimental values come from [^{6}].
Article Categories:
Keywords: Endsystolic pressurevolume relationship, Timevarying elastance, Cardiovascular system. 
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