Simulation of Left Atrial Function Using a Multi-Scale Model of the Cardiovascular System

During a full cardiac cycle, the left atrium successively behaves as a reservoir, a conduit and a pump. This complex behavior makes it unrealistic to apply the time-varying elastance theory to characterize the left atrium, first, because this theory has known limitations, and second, because it is still uncertain whether the load independence hypothesis holds. In this study, we aim to bypass this uncertainty by relying on another kind of mathematical model of the cardiac chambers. In the present work, we describe both the left atrium and the left ventricle with a multi-scale model. The multi-scale property of this model comes from the fact that pressure inside a cardiac chamber is derived from a model of the sarcomere behavior. Macroscopic model parameters are identified from reference dog hemodynamic data. The multi-scale model of the cardiovascular system including the left atrium is then simulated to show that the physiological roles of the left atrium are correctly reproduced. This include a biphasic pressure wave and an eight-shaped pressure-volume loop. We also test the validity of our model in non basal conditions by reproducing a preload reduction experiment by inferior vena cava occlusion with the model. We compute the variation of eight indices before and after this experiment and obtain the same variation as experimentally observed for seven out of the eight indices. In summary, the multi-scale mathematical model presented in this work is able to correctly account for the three roles of the left atrium and also exhibits a realistic left atrial pressure-volume loop. Furthermore, the model has been previously presented and validated for the left ventricle. This makes it a proper alternative to the time-varying elastance theory if the focus is set on precisely representing the left atrial and left ventricular behaviors.

Supplementary Text S1: Parameter estimation Step 1: direct parameter computation from data Manipulating the CVS model equations and using reference data on dogs [1][2][3], we could directly compute the values of the hemodynamic parameters R prox , R mt , R av , R sys , E ao and E pa . The data published by Gare et al. [1] provide curves of left atrial pressure, left ventricular pressure and aortic pressure for one cardiac cycle. To compute the model parameters, we also needed the value of the stroke volume SV. It was computed from a canine pressure-volume loop published by Kass et al. [2] as: SV ≈ 11.67 ml Mean vena cava pressureP vc was computed from a previously published right ventricular canine pressurevolume loop [3], as:P vc ≈ P lv (t T V O ) + P lv (t T V C ) 2 ≈ 11.5 mmHg where P lv is left ventricular pressure and t T V O and t T V C denote opening and closing times of the tricuspid valve. First, if pulmonary vein pressure is assumed constant atP pu , it is clear that, during atrial filling, atrial pressure will converge to this value ofP pu . The maximum value of atrial pressure during atrial filling (i.e. the peak of the v wave) can be considered as an estimation ofP pu . This yields, from data published by Gare et al. [1]:P pu ≈ 14.3 mmHg Second, by integrating Equation (18), one can see that model resistances R prox and R sys can be computed as: where P up and P down respectively denote pressures up and downstream the resistance. The integral is computed during one cardiac period T and the bar denotes the mean on one cardiac period. Using the previously mentioned reference data, this gives: In case there is a valve in series with the resistance, the integral is only computed for positive values of the integrand. This allows to directly compute R mt and R av as: and t AV C denote opening and closing times of the mitral and aortic valves. Finally, during diastole, Equation (21) applied to the aorta reduces to: In the previous equation, vena cava pressure has been assumed to be constant. Since P ao (t) = E ao ·V ao (t), this equation can be integrated to find a closed-form expression for P ao (t) during diastole: This equation, evaluated at t = t AV O , can be solved to compute E ao .
Since not so many data was available for the right side of the cardiovascular system, we computed pulmonary artery elastance using the following approximation: The first equality is the definition of elastance, i.e. the pressure variation caused by a volume variation divided by this volume variation. In the second equality, the pressure variation has been taken as pulmonary artery pulse pressure P P pa and the corresponding volume variation has been estimated equal to the stroke volume. (This is an overestimation, as only a fraction of the stroke volume is causing an increase of arterial pressure, while the rest goes directly into the pulmonary circulation.) Pulmonary artery pulse pressure has been estimated from right ventricular pressure as: where t P V C and t P V O respectively denote closing and opening times of the pulmonary valve. We thus assume that maximum and minimum pulmonary artery pressure occur at closing and opening of the pulmonary valve, respectively. Equation (10) describes the passive force generated by the stretching of a sarcomere unit as Using equations (12) and (14) linking force to pressure and sarcomere length to chamber radius, this passive force-length relationship can be converted to a passive pressure-radius relationship as: where P p,i is passive pressure. Finally, using the equation V = 2πR 3 /3 to convert chamber radius to chamber volume, we obtain the following passive pressure-volume relationship: This relationship is equivalent to the notion of end-diastolic pressure-volume relationship. Using reference pressure-volume loops for the left ventricle and atrium, we could directly compute values of R 0,i and the product t i · K i for both the left ventricle and atrium: Parameter s representing the shift between maximal left atrial and left ventricular calcium concentrations was inferred from the shift between maximal left atrial and left ventricular pressures.
Parameters of the right ventricle time-varying elastance were compressed and shifted from Chung et al. [4] to account for a heart period of 0.45 s and to synchronize beats of the left and right ventricles.
An important but often neglected parameter of cardiovascular system models is the total stressed blood volume (SBV) in the model. More explanations on this concept can be found in [5]. In the same article, the authors state that for a human, total SBV represents 750/5500 of total blood volume. The dogs used in the experiments from which reference data was taken weighed 25 kg on average. According to [6], a 25-kg dog has a total blood volume of 25 × 80 = 2000 ml. If we assume that the ratio of SBV to total blood volume is the same as for a human, SBV for a 25-kg dog can be assumed to be 2000 × 750/5500 = 273 ml.
Step 2: iterative parameter adjustment for the systemic submodel To simplify the parameter estimation process, the cardiovascular system model we used was split into systemic and pulmonary circulations, by assuming constant systemic and pulmonary venous pressures atP vc andP pu , respectively. The vena cava and pulmonary veins compartments thus become points with constant pressures, implying that the two remaining subsystems, one composed of the left atrium, left ventricle and aorta and the second composed of the right ventricle and pulmonary artery, become independent. The two subsystems still have to share the same stroke volume.
The remaining four parameters of the systemic submodel, namely the parameters describing active chamber contraction (A i ) and the slopes of the passive force-length relationships (K i ) have been iteratively adjusted to match reference data coming from the previously mentioned studies. For further details, the reference data is displayed in  Step 3: iterative parameter adjustment for the pulmonary submodel Because less data was available for the pulmonary submodel we could not directly compute all model parameters from reference data, as done in step 1. This was only possible for pulmonary artery elastance E pa . To iteratively adjust the parameters R tc , R pul and E rv of the pulmonary submodel, we used the proportional method developed by Hann et al. [7]. In short, this method consists in using proportional relations existing between model parameters and available measurements. First, and to exemplify the method, for sufficiently large values of the tricuspid valve resistance R tc , the following inversely proportional relation exists between R tc and stroke volume SV : (This is related to Equation S.2 applied to tricuspid valve resistance.) Consequently, R tc can be iteratively updated as where R n tc denotes the previous value of R tc , SV (R tc,n ) is the simulated stroke volume with R tc being set at R tc,n and SV ref is the reference value of stroke volume (given in Table S.1).
Second, the following proportionality relation has been proposed [7] to iteratively estimate the value of the pulmonary resistance R pul : R pul ∝P pa .
(This can be seen to be valid from Equation S.1 applied to pulmonary resistance.) As this mean pulmonary artery pressureP pa was not available in the reference data we used, we based ourselves on the following approximation:P This approximation is generally used to compute mean aortic pressure [8], but we assumed it to be also true for mean pulmonary artery pressure. Then, as stated before, we could estimate P pa (t P V O ) as since a heart valve opens when the downstream pressure equals the upstream pressure. Consequently, the relation that was used to adjust R pul is Third, the model considers that tricuspid valve opens and closes when left ventricular pressure equals (fixed) vena cava pressure, thus we have: Using Equation 16, this gives: From the previous equation, it can be seen that end-systolic and end-diastolic ventricular volumes, respectively V rv (t T V O ) and V rv (t T V C ) are both inversely proportional to E rv . This suggests the following proportionality relation: E rv ∝V −1 rv . The reference value of mean ventricular volumeV rv that was used is given in Table S Parameter R pv was kept constant at a reference value because information needed to estimate it (the derivative of the pulmonary artery waveform [7]) was not available.
Step 4: iterative parameter adjustment for venous elastances To close the loop, the systemic and pulmonary submodels are put back together, and the venous elastances E vc and E pu are iteratively adjusted so that mean vena cava pressure and mean pulmonary vein pressure match the (previously constant) values computed at step 1.