Model Development

A computational model was first constructed for a normal biventricular circulation with the lumped parameter modeling method. The resulting model consisted of a limited number of parameters but carried sufficient details to account for the key hemodynamic behaviors, such as cardiac dynamics and its interactions with preload and afterload. A Fontan circulation model was obtained later by incorporating a total cavopulmonary connection (TCPC), which bypasses the right atrium and ventricle, into the biventricular circulation model. As a consequence, the left ventricle is the unique normally functioning ventricle in the modeled Fontan circulation. An electric circuit analogy of the biventricular circulation model is shown in Fig. 1A and that of the Fontan circulation model is shown in Fig. 1B.

Modeling of the pulmonary and systemic vascular systems.

The pulmonary vascular system was represented by three serially arranged compartments that represent the arterial, capillary, and venous vascular portions, respectively. The systemic vascular system was divided into aorta, vena cava, and two parallel-arranged upper-body and lower-body subsystems (see Fig. 1). Each vascular portion was represented by a Windkessel model consisting of three parameters [namely, resistance (R), compliance (C), and inertance (L)] that account for the viscous resistance to blood flow, wall deformability, and blood inertia of the vascular portion, respectively. If we name the C elements as “P” nodes and the nodes between R and L elements as “Q” nodes along the pathway of blood flow (see Fig. 2), the governing equations for blood flow can be formulated by imposing mass or momentum conservation at the nodes.

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At a P node, formulating mass conservation yields

$\frac{{\text{dP}}_j}{\text{d}t}=\frac{{\text{Q}}_j-{\text{Q}}_{j+1}}{C_j},$(1)

and at a Q node, momentum conservation reads

$\frac{{\text{dQ}}_j}{\text{d}t}=\frac{{\text{P}}_{j-1}-{\text{Q}}_j{\text{R}}_j-{\text{P}}_j}{L_j},$(2)

where P_j and P_j−1 denote the blood pressures at the C_j and C_j−1 elements, respectively; t is time; Q_j and Q_j+1 represent, respectively, the inflow to and outflow from the C_j element; and R_j and L_j are the resistance and inertance that link the C_j and C_j−1 elements.

Modeling of the heart.

MODELING OF CARDIAC CONTRACTION/RELAXATION.

The heart was modeled as a four-chamber organ according to its anatomical structure. The contracting/relaxing action of each cardiac chamber was represented by a time-varying elastance function derived from the well-known myocardial elastance theory (81). In previous studies (34, 74), blood pressure (P_h) in each cardiac chamber was defined as a function of elastance (E) and chamber volume (V). In the present study, we further took into account the viscoelastic property of cardiac walls to more reasonably describe cardiac hemodynamics

${\text{P}}_{\text{h}}\left(t\right)=E\left(t\right)\left(\text{V}-{\text{V}}_0\right)+S\frac{\text{dV}}{\text{d}t},$(3)

where V₀ refers to the unstressed volume (here set to be zero) and S is the viscoelasticity coefficient of the cardiac walls, which is herein taken to be a function of P_h (84).

The elastance function [E(t)] was used to account for the dynamic contraction-relaxation behavior together with the passive mechanical properties of the cardiac chamber, which was herein defined as the sum of a time-varying component and a constant in light of the models adopted in previous studies (34, 47)

$E\left(t\right)=E_{\text{a}}e\left(t\right)+E_{\text{p}}.$(4)

Here, the first term on the right-hand side describes the time-varying chamber elastance resulting from the active stimulation of myocardial fibers, with a peak value of E_a, which represents the contractility of the myocardium, while the second term (E_p, herein termed passive elastance) represents the baseline stiffness of the cardiac chamber in the absence of active stimulation, reflecting the passive mechanical properties of myocardial fibers. E_a and E_p are denoted, respectively, by E_lva and E_lvp for the left ventricle, E_laa and E_lap for the left atrium, E_rva and E_rvp for the right ventricle, and E_raa and E_rap for the right atrium. e(t) is a normalized time-varying function of the active elastance and was expressed by a piecewise function. The function takes different forms for the ventricles and atria.

For the left and right ventricles, we modified the elastance function used in Refs. 34, 47 to further account for the effect of the time constant of isovolumic relaxation on diastolic function,

$e_{\text{v}}\left(t\right)=\{\begin{array}{ll}0.5\left[1-\cos \left(\pi \text{t}/T_{\text{vcp}}\right)\right]+\exp \left[-\left(T_0-T_{\text{vcp}}\right)/{\tau }_{\text{au}}\right]\cos \left(\pi \text{t}/2T_{\text{vcp}}\right)\hfill & 0\le t\le T_{\text{vcp}}\hfill \\ \exp \left[-\left(t-T_{\text{vcp}}\right)/{\tau }_{\text{au}}\right]\hfill & T_{\text{vcp}}<t\le T_{\text{0}}\hfill \end{array},$(5)

and for the left and right atria, the function proposed in our previous studies (47) was adopted,

$e_{\text{a}}\left(t\right)=\{\begin{array}{ll}0.5\left\{1+\cos \left[\pi \left(t+T_0-t_{\text{ar}}\right)/T_{\text{arp}}\right]\right\}\hfill & 0\le t\le t_{\text{ar}}+{\mathit{\text{T}}}_{\text{arp}}-T_0\hfill \\ 0\hfill & t_{\text{ar}}+T_{\text{arp}}-T_0<t\le t_{\text{ac}}\hfill \\ 0.5\left\{1-\cos \left[\pi \left(t-t_{\text{ac}}\right)/T_{\text{acp}}\right]\right\}\hfill & t_{\text{ac}}<t\le t_{\text{ac}}+T_{\text{acp}}\hfill \\ 0.5\left\{1+\cos \left[\pi \left(t-t_{\text{ar}}\right)/T_{\text{arp}}\right]\right\}\hfill & t_{\text{ac}}+T_{\text{acp}}<t\le T_0\hfill \end{array}.$(6)

Here, the subscript “v” denotes the ventricles and “a” the atria, T₀ is the duration of a cardiac cycle, T_vcp and T_acp refer, respectively, to the durations of ventricular contraction and atrial contraction, T_arp is the duration of atrial relaxation, τ_au is the time constant of ventricular isovolumic relaxation, and t_ac and t_ar denote, respectively, the times when the atria begin to contract and relax.

It is worth noting that the mathematical expressions of cardiac elastance function present significant variations in the literature (77), although most of them have been constructed based on the elastance theory (81). In the present study, the ventricular elastance function has been formulated to clearly parameterize the different aspects of ventricular function so that the hemodynamic effects of ventricular function can be quantitatively studied by varying a limited number of model parameters (i.e., E_a, E_p, and τ_au) in the simulations.

MODELING OF CARDIAC VALVES.

The pressure gradient (ΔP_cv) across a cardiac valve was related to transvalve flow rate (Q_cv) by (47, 84).

$\Delta {\text{P}}_{\text{cv}}=R_{\text{cv}}{\text{Q}}_{\text{cv}}+B_{\text{cv}}{\text{Q}}_{\text{cv}}\left|{\text{Q}}_{\text{cv}}\right|+L_{\text{cv}}\frac{{\text{dQ}}_{\text{cv}}}{\text{d}t}+R_{\text{do}}{\text{Q}}_{\text{cv}},$(7)

where R_cv, B_cv, and L_cv represent, respectively, the transvalve viscous resistance, Bernoulli's resistance, and blood inertance when the valve is opened; and R_do is a nominal resistance used to control the opening and closing of the cardiac valve so that ventricle-to-atrium or artery-to-ventricle reverse flow does not occur. Here, R_do plays the role of a diode, with its value being dynamically switched between infinity and zero depending on time-dependent hemodynamic conditions. Taking the aortic valve as an example,

$R_{\text{do}}=\{\begin{array}{cc}0& {\text{P}}_{\text{lv}}-{\text{P}}_{\text{ao}}>0\hfill \\ \infty & {\text{P}}_{\text{lv}}-{\text{P}}_{\text{ao}}<0\&{\text{Q}}_{\text{av}}\to 0\hfill \end{array},$(8)

where P_lv and P_ao refer, respectively, to the blood pressures in the left ventricle and the aorta and Q_av is the blood flow rate across the aortic valve. When the value of R_do is switched from ∞ to 0, the valve is opened; otherwise, the valve is closed.

RELATIONSHIP BETWEEN VENTRICULAR SYSTOLIC DURATION AND HEART RATE.

Heart rate (HR) is an important physiological factor involved in the regulation of CO. A change in HR alters not only the length of ventricular systolic duration but also the proportion of systolic duration to the entire cardiac cycle. Although many formulas have been used to account for the variation of systolic duration with HR (34, 60), so far, no formula has been confirmed to be valid over a wide range of HR. Fortunately, a large amount of in vivo data on the relationship between ECG QT interval and RR interval (=60/HR) has been reported in the literature (1, 8, 15, 20, 37, 69, 70). These data allowed us to construct a theoretical formula for the QT-RR relationship by means of data fitting. Herein, a second-degree polynomial function was used to fit the in vivo data (see Fig. 3).

$\text{QT}=-0.33{\text{RR}}^2+0.69\text{RR}+\mathrm{0.029.}$(9)

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Here, QT (with its unit being second) represents the ventricular electrical systole ranging from the beginning of the QRS complex to the end of the T wave on an ECG. QT is closely related to but not a strict measure of the systolic duration (T_vcp) defined in our ventricular model (Eq. 5). Herein, we assumed that T_vcp and QT are linearly related (T_vcp = αQT). The constant α (= 0.714) was determined by fitting the modeled ventricular elastance curve (at HR = 80 beats/min) to measured data (74) (see Fig. 4). For other time parameters used in the atrial model (Eq. 6), since reports on their relations to HR are rare, we assumed them to be functions of T_vcp: T_acp = 0.6T_vcp, T_arp = T_acp, t_ac= T₀ − T_acp − 0.05, and t_ar = t_ac + T_acp.

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RELATIONSHIP BETWEEN τ_AU AND HR.

The time constant (τ_au) of ventricular isovolumic relaxation varies with HR (7, 22, 48, 79). Their relationship was approximated by an exponential function derived from data fitting (see Fig. 5).

${\tau }_{\text{au}}=30.2e^{-\text{HR}/81.2}+\mathrm{31.4.}$(10)

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Here, the unit of τ_au is taken to be millisecond in line with the measured data but will be converted to second when used in Eq. 5.

Numerical Methods

The models of the biventricular circulation and Fontan circulation were each governed by an ordinary differential equation system (see appendix for the details of the governing equations) coupled with algebraic equations that describe the nonlinear elements (i.e., cardiac elastances). The equation system was solved in an explicit time-marching manner using a multivariable fourth-order Runge-Kutta method (45). The numerical time step was fixed at 0.0002 s, which was found to enable a stable convergence of all the simulations carried out in the present study. A computer code was written in FORTRAN language for each model and run on a personal computer installed with a Linux OS. The inputs to the codes consist of values of model parameters and initial values of model state variables. Since the assigned initial conditions may deviate from the converged state of the models, each simulation was repeated for several cardiac cycles until a periodic solution (the maximum percentage change in simulated blood pressures between the last 2 cardiac cycles is <0.01%) was obtained. Convergence of the models is guaranteed by the closed-loop nature of the models in which any perturbations introduced by the initial conditions will be swept out when a global hemodynamic balance over the cardiovascular system is achieved after repeated simulations. Our numerical experiments proved that the simulations converged rapidly (usually converged within 100 cardiac cycles) in spite of random changes in assigned initial conditions. However, it should be noted that the initial conditions can significantly affect the results of simulation. For instance, a variation in initial total blood volume (sum of the blood volumes assigned to the vascular “C” components and cardiac chambers) can induce considerable changes in simulated central venous pressure and CO. With these in mind, we have adjusted the initial conditions together with model parameters to allow simulated hemodynamic variables to fall within the ranges of measured data.

Parameter Assignment

Fontan model.

When a Fontan circulation model (herein named baseline Fontan model) was initially created by modifying the biventricular circulation model, the values of model parameters were maintained unchanged. The new parameters involved in the Fontan model are those used to describe the lateral atrial tunnel (or extracardiac conduit). These parameters are not derivable from the literature, although there exist a number of studies addressing energy loss or flow resistance associated with TCPC (32, 85). We herein estimated the parameters from the geometric data of atrial tunnel or extracardiac conduit measured in Fontan patients (2, 91). Based on the measured data, the diameter and length of the atrial tunnel in an adult Fontan patient were assumed to be 25 and 63.7 mm, respectively. Further assuming the intratunnel blood flow to be laminar and Newtonian, we calculated the flow resistance (R_ten in Fig. 1B) of the tunnel to be 2.93E-6 mmHg·s/ml according to Poiseuille's Law. Blood inertance (L_ten in Fig. 1B) in the tunnel was obtained by integrating the flow momentum equation along the tunnel, which was 0.001mmHg·s²/ml. Due to the lack of data on the mechanical properties of the tunnel wall, the compliance of the tunnel (C_ten in Fig. 1B) could not be derived theoretically and was herein assumed to be 0.1 ml/mmHg. Hemodynamic simulation for the baseline Fontan model shows that the biventricular-to-Fontan circulation conversion induces a significant reduction in the cardiac index (by 45.5%) and a marked increase in venous pressure (by 78.7%; column 5). However, the baseline Fontan model is found not able to reasonably reproduce the in vivo data measured in Fontan patients without severe clinical symptoms (39, 58, 59; column 4). Under real physiological conditions the cardiovascular properties of a Fontan circulation may differ from those of a normal biventricular circulation due to the onset of adaptive responses to impaired hemodynamic conditions after the Fontan operation. For instance, a Fontan circulation usually has increased peripheral vascular resistance and reduced venous compliance (39, 43). Therefore, we adjusted the model parameters (by increasing the peripheral vascular resistance by 22.6%, reducing the systemic venous compliance by 36% and reducing resting HR from 75 to 60 beats/min based on the data reported in Ref. 39) to build a physiological Fontan circulation model. Moreover, pulmonary vascular compliance was reestimated based on the in vivo data reported in (75): C_pua = 0.289 ml/mmHg, C_puc = 2.347 ml/mmHg, and C_puv = 0.975 ml/mmHg. As can be seen from column 6 in Table 2, after parameter adjustment, the simulated CO, stroke volume, central venous pressure and aortic pressure get closer to the in vivo data (column 4).

To further examine the ability of our models to capture the hemodynamic differences between the biventricular circulation and the Fontan circulation, we investigated the transient pulmonary arterial flow, which is expected to be significantly influenced by bypass of the right heart. Figure 6A shows the pulmonary arterial flow waveforms simulated using the biventricular model and the physiological Fontan model (under resting conditions). Figure 6B shows the in vivo data measured in control subjects (17 healthy subjects with a normal biventricular circulation) and Fontan patients (5 Fontan patients with TCPC) (31), where the illustrated flow waveforms have been obtained by superimposing the waveforms measured in different subjects. To facilitate comparison, each flow waveform has been normalized by the mean flow rate averaged over a cardiac cycle. The simulations are comparable to the measurements in terms of the primary differences in the contour of flow waveform between the biventricular circulation and the Fontan circulation. For instance, both our results and the in vivo data demonstrate that the pulsatility of pulmonary arterial flow is significantly reduced in the Fontan circulation compared with the biventricular circulation and that the flow waveform of the Fontan circulation exhibits a typical biphasic pattern as has been observed in previous in vivo studies (35). It should be noted, however, that the ability of our models to simulate pulmonary arterial flow waveform is limited due to the lack of a detailed description of the pulmonary arterial network and the intrinsic deficiency of the employed lumped parameter modeling method in accounting for wave propagation phenomena.

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Sensitivity Analysis

The models used in this study, although being simple in structure, still contain >50 parameters. To simplify our analysis, it is necessary to select a limited number of model parameters that are most relevant to the hemodynamic variables of interest [herein including mean central venous pressure (mP_V), mean left atrial pressure (mP_LA), mean aortic pressure (mP_AO), and CO]. For this purpose, a global sensitivity analysis method was employed to extensively evaluate the sensitivity of model output to parameters in a wide range of cardiovascular conditions. The sensitivity analysis started with creating a stratified sampling space (comprised by 3,000 sampling points) with the Latin Hypercube method (89) based on a random variation of all the model parameters (between 20 and 200% relative to their default values). Subsequently, the local impacts of each model parameter on model-predicted hemodynamic variables were evaluated at each sampling point by

Here, S_i,j is the percentage change in a hemodynamic variable (P_j) induced by a small change in a model parameter (x_i), reflecting the sensitivity of P_j to x_i. S_i,j has been nondimensionalized to enable a cross comparison between sensitivity indexes calculated for different P_j, x_i pairs. X = (x₁, x₂…x_i, …x_n) represents the parameter vector at a sampling point (n indicates the number of model parameters).

Finally, S_i,j obtained at all sampling points was statistically analyzed by calculating the mean (S̄_i,j) and standard deviation (σ_i,j) according to the following equations

where M represents the total number of sampling points (herein = 3,000) and S_i,j^k is the sensitivity index calculated at the no. k sampling point.

The results of sensitivity analysis performed on the baseline Fontan model are reported in form of means ± SD in Table 3. Here, the larger the mean value of the sensitivity index, the stronger the influence of the parameter on the corresponding hemodynamic variable is. A large SD indicates the existence of strong correlation of the studied parameter with the remaining parameters in determining the sensitivity index. It is noted that the pulmonary vascular system has been evaluated as a whole with respect to total pulmonary resistance (R_pul), compliance (C_pul), and inertance (L_pul) by varying the R, C, or L of different pulmonary vascular portions simultaneously in our sensitivity analysis. A similar evaluation was performed for the systemic vascular system as well.

From Table 3, seven model parameters were found to dominate the model-simulated results of the hemodynamic variables of interest [i.e., mean central venous pressure (mP_v), mean left atrial pressure(mP_la), mean aortic pressure (mP_ao), and CO], which include the maximum active elastance (E_lva) and baseline passive elastance (E_lvp) of the left ventricle, the total pulmonary vascular resistance (R_pul), the total pulmonary vascular compliance (C_pul), the total systemic vascular compliance (C_sys), the total systemic vascular resistance (R_sys), and the time constant of ventricular isovolumic relaxation (τ_au). Here, E_lva is a measure of left ventricular systolic function and E_lvp and τ_au together characterize left ventricular diastolic function.

Computation Conditions

Based on the results of sensitivity analysis, the six most influential model parameters (E_lva, E_lvp, τ_au, C_sys, R_sys, and R_pul) were selected to further study their hemodynamic effects. In addition, HR, as an important physiological factor, was studied as well. The effects on CO and central venous pressure of each parameter were quantified through model-based computer simulations where the value of each parameter is varied in a given physiological range. In the study, two models were used, namely the physiological Fontan model and the normal biventricular model. The ranges of parameter variation assigned for the two models are given in Table 4. Note that when one model parameter was studied, other parameters were fixed at their default values. Other computation conditions assigned to investigate the combined effects of several parameters will be described later. In the simulations, the initial conditions assigned to each model were fixed to highlight the effects of variations in model parameters.

Effects of Cardiac Parameters (HR, E_lva, E_lvp, and τ_au)

Figure 7 shows the comparisons between the Fontan and biventricular circulations regarding the effects of cardiac parameters (HR, E_lva, E_lvp, and τ_au) on CO and mean central venous pressure (mP_v). In Fig. 7, the vertical axis shows the simulated changes in CO and mP_v relative to their default values (simulated with all parameters being fixed at their default values), while the lateral axis shows the variations in model parameters.

In the case of biventricular circulation, increasing HR is accompanied by a progressive increase in CO until HR reaches a high value (∼140 beats/min) beyond which the effect is blunted. Increasing HR, however, induces much less increase in CO in the case of Fontan circulation. The effects of E_lva on CO are much greater in the biventricular circulation than in the Fontan circulation. E_lvp significantly affects the CO of both circulations, with the effects being more evident in the Fontan circulation. The effects of τ_au on CO are moderate, although slightly enhanced effects are observed for the Fontan circulation.

In comparison with CO, mP_v is only moderately affected by the cardiac parameters [as indicated by the small ranges (<3 mmHg) of changes under all the simulated conditions]. The largest change in mP_v is induced by variations in E_lvp in the Fontan circulation (see Fig. 7G). Noticeably, the responses of mP_v to variations in cardiac parameters differ significantly between the Fontan and biventricular circulations, implying the potential roles of the right ventricle in regulating venous pressure.

Effects of Total Systemic Vascular Compliance (C_sys) and Resistance (R_sys)

A great influence of C_sys on CO is predicted for both the Fontan and biventricular circulations, with the Fontan circulation showing a steeper change in CO with C_sys (see Fig. 8A). Variations in C_sys are accompanied by considerable changes in mP_v in the Fontan circulation but only induce slight changes in mP_v in the biventricular circulation (see Fig. 8C). Compared with C_sys, R_sys has less influence on CO and mP_v in both circulations, although its influence seems to be more evident in the biventricular circulation (see Fig. 8, B and D).

Effects of Total Pulmonary Vascular Resistance (R_pul)

The effects of R_pul on CO are pronounced, especially in the Fontan circulation (see Fig. 9A). Similar to C_sys, variations in R_pul induce significant changes in mP_v only in the Fontan circulation (see Fig. 9B).

Changes in CO and mP_v Induced by ±50% Changes in Model Parameters

Table 5 summarizes the simulated changes in CO and mP_v induced, respectively, by a 50% reduction and a 50% increase (relative to its default value) in each parameter for the Fontan and biventricular circulations. The results show that both CO and mP_v in the Fontan circulation are strongly affected by C_sys, followed by R_pul and E_lvp. When the two circulations are compared, it is observed that changes in CO and mP_v with C_sys, E_lvp, τ_au, and R_pul are more pronounced, whereas changes in CO and mP_v with HR and E_lva are less evident in the Fontan circulation than in the biventricular circulation.

Responses of CO to Variations in Cardiovascular Properties

When a normal biventricular circulation is converted into a Fontan circulation by bypassing the right ventricle, our results show that CO becomes less sensitive to HR and ventricular contractility. The results are consistent with existing clinical findings. For instance, increasing HR or enhancing ventricular contractility has been found to induce less improvement of CO in Fontan patients compared with normal biventricular subjects (6, 27, 28, 78). In addition to HR and ventricular contractility, the effects of ventricular diastolic function, which have been difficult to be addressed in clinical studies, were investigated as well in the present study. Our results demonstrate that CO is more severely affected by ventricular diastolic dysfunction [characterized by increased diastolic stiffness (higher E_lvp) or prolonged isovolumetric relaxation (larger τ_au)] in the Fontan circulation. Compared with E_lvp, the effects of τ_au are moderate. It should be noted, however, that the effects of τ_au have been studied by fixing HR at 60 beats/min in the simulations. To study whether the effects of τ_au are dependent on HR, we re-ran simulations under different HR conditions (HR = 60, 80, and 100 beats/min). The results revealed that the effects of τ_au on CO become more pronounced at a higher HR, whereas the effects of E_lvp are less affected by HR (see Fig. 10).

The effects on CO of systemic/pulmonary vascular properties also differ between the two circulations. The Fontan circulation exhibits an enhanced CO response to variations in systemic vascular compliance (C_sys) and pulmonary vascular resistance (R_pul). Interestingly, the influence of systemic vascular resistance (R_sys) on CO is slightly impaired in the Fontan circulation. A potential cause of the phenomenon may be the increased baseline value of R_sys in the Fontan circulation.

It should be stressed that the relationships between CO and model parameters are nonlinear, with the slope being steeper in the low HR, low E_lva, low E_lvp, long τ_au, small C_sys, and low R_pul regions. The nonlinear nature of the relationships implies that the effectiveness of changing a cardiovascular property to improve CO is dependent on the baseline status of the property. Moreover, the effects of a cardiovascular property on CO may be coaffected by other cardiovascular properties. To test the hypothesis, we carried out additional simulations to study the changes of CO with various combinations of R_pul and E_lvp using the Fontan circulation model. The results show that the steepness of the relationship between CO and any of the two parameters depends strongly on the value of the other parameter (see Fig. 11).

Responses of Central Venous Pressure to Variations in Cardiovascular Properties

The responses of central venous pressure to variations in cardiovascular properties differ significantly between the Fontan circulation and the biventricular circulation. For example, venous pressure is highly sensitive to C_sys and R_pul in the Fontan circulation but only slightly affected by them in the biventricular circulation (see Figs. 8C and 9B). Deterioration in left ventricular diastolic function induces an increase in venous pressure in the Fontan circulation but a decrease in venous pressure in the biventricular circulation (see Fig. 7G). These results indicate that the right ventricle plays an important role not only in providing sufficient preload for the left ventricle but also in stabilizing venous pressure under various venous return conditions or when abnormalities develop in the pulmonary circulation or the left ventricle. The absence of a functional right (subpulmonary) ventricle in the Fontan circulation not only reduces CO but also significantly increases the variability of venous pressure.

Clinical Implications

Reduced CO and increased central venous pressure are the major manifestations of hemodynamic abnormalities in Fontan patients and have been found associated closely with some systemic complications, such as exercise intolerance, hepatic dysfunction, and protein-losing enteropathy (17). In particular, the status of hemodynamic abnormalities has a tendency to deteriorate progressively over time after the Fontan operation (56). This highlights the importance of active hemodynamic management in the care of long-term survivors. Although some studies have reported the beneficial roles of some medications [e.g., administration of sildenafil (30), amrinone (78), or sodium nitroprusside (93)] in increasing resting CO or improving ventricular performance, to date, evidence supporting the long-term effectiveness of certain interventions in systemically improving hemodynamic conditions remains absent. The present theoretical study demonstrates that the mechanisms underlying the regulation of CO and central venous pressure differ significantly between the Fontan circulation and the biventricular circulation, which implies that some conventional medications for treating patients with biventricular physiology might not be applicable to Fontan patients. According to our results, medications (for instance, administration of catecholamines) aimed to enhance ventricular contractility cannot be expected to significantly increase CO in Fontan patients. Comparatively, treatments targeted on improving the diastolic function of the systemic ventricle and reducing the pulmonary vascular resistance may be more effective. In addition to the maintenance of CO, the control of central venous pressure is another crucial issue in the care of Fontan patients. Our results show that diastolic dysfunction [characterized by increased ventricular diastolic stiffness (E_lvp) and/or prolonged τ_au] and increased pulmonary vascular resistance both lead to a marked increase in central venous pressure in the Fontan circulation, as is profoundly different from the phenomena observed in the biventricular circulation. This further stresses the clinical significance of improving ventricular diastolic function and reducing pulmonary vascular resistance. This viewpoint is consistent with the findings of some clinical trials. For instance, reducing pulmonary vascular resistance has been found to significantly improve hemodynamic conditions (78, 93), and ventricular diastolic dysfunction has been found to be an independent predictor of increased risk of perioperative morbidities and longer hospital stay for Fontan patients (9, 26, 88). In view of the importance of venous pressure control, reducing C_sys may not be a good choice for improving CO in Fontan patients since it increases CO at the expense of elevating central venous pressure. Moreover, the findings regarding the variability in the effects of τ_au with HR might provide some, although possibly limited, insights into the clinical observation that Fontan patients are usually affected by τ_au-dominated diastolic dysfunction early after the Fontan operation when the patients are younger with a higher baseline HR but subsequently affected primarily by chamber stiffness-dominated diastolic dysfunction at mid-term follow-up when the baseline HR has decreased with age (68).

Another insight from our study is that CO and central venous pressure are codetermined by multiple cardiovascular factors. For instance, the effectiveness of reducing pulmonary vascular resistance to increase CO is significantly blunted in the presence of severe ventricular diastolic dysfunction and vice versa (see Fig. 11). In this sense, a full understanding of the cardiovascular function of a Fontan patient is a precondition for making effective patient-specific therapeutic plans.

Increased systemic vascular resistance (R_sys) has been frequently found in postoperative Fontan patients (39, 43). This has naturally motivated the use of angiotensin-converting enzyme inhibitors (ACEI) to reduce systemic vascular resistance with an expectation to improve CO; however, the clinical benefits of the medication remain to be proven (68). According to our results, R_sys has a much greater effect on aortic blood pressure than on CO, e.g., a 30% reduction in R_sys induces a <5% increase in CO while causing an ∼20% drop in aortic pressure (see Fig. 12). In view of the importance of arterial pressure maintenance, the role of reducing R_sys in improving CO is likely to be limited in Fontan patients. This is supported by previous clinical findings that ACEI did not significantly increase CO or improve exercise capacity in postoperative Fontan patients (39, 44). Nevertheless, it has been speculated that reducing R_sys using ACEI might be beneficial for patients with severe ventricular systolic dysfunction (68). To test the speculation, we simulated the changes of CO and aortic pressure with R_sys under various ventricular systolic/diastolic conditions using the Fontan model. It is noted that in the simulations when E_lva is varied E_lvp is held at its default value and vice visa. Figure 13 shows that the increase in CO induced by a reduction in R_sys is larger with a small E_lva (impaired systolic function) than with a large E_lva (enhanced systolic function). Accordingly, for a small E_lva, aortic pressure decreases to a less degree when R_sys is reduced. Differently from systolic function, the status of diastolic function does not significantly affect the strength of CO response to variations in R_sys, although with diastolic dysfunction (large E_lvp) the steepness of aortic pressure-R_sys relation is observed to be slightly reduced as well (see Fig. 14). These simulation results partly support the speculation regarding the potential role of ACEI in improving CO in patients with severe systolic dysfunction.

Limitations

In the present study, the Fontan circulation has been modeled by representing the major cardiovascular properties with lumped parameters. The resulting model allowed us to quantitatively study the hemodynamic effects of any cardiovascular properties of interest by varying the corresponding model parameters. Despite the advantages, the model is inherently a low-order approximation of the circulatory system, in which many details of anatomy and hemodynamic phenomena have been simplified or omitted. For example, the present model does not include a detailed description of the arterial system and hence cannot reasonably account for arterial wave propagation phenomenon and its interaction with left ventricular dynamics. Since the present study focuses on hemodynamic variables averaged over a cardiac cycle (such as CO, mean venous pressure, and mean aortic pressure), the simplification would introduce negligible errors. However, when ventricular-arterial coupling is the primary interest [for instance, when studying the energetics of the heart in the Fontan circulation (86)], arterial wave propagation should be carefully addressed by developing a more detailed model for the arterial system. When modeling the heart, we did not account for the details of cardiac wall motion, which has been found to influence intraventricular flow and early diastolic filling (66). According to previous studies (9, 26, 74), the peak active elastance (E_lva), the baseline passive chamber elastance (E_lvp), and the time constant of isovolumic relaxation (τ_au) are the most representative indexes of ventricular function and hence the omitted details of cardiac wall motion are secondary for the description of ventricular function, at least in the scope of the present study. Another limitation associated with the modeling of the heart is that although we have assumed the left ventricle to be the systemic ventricle of the Fontan circulation, there are a number of Fontan patients with the right ventricle serving as a systemic pumping chamber (17). Since a systemic right ventricle may differ significantly from the left ventricle in both anatomical and functional properties, our results on the hemodynamic effects of cardiac parameters might fail to reflect the characteristics of a Fontan circulation with a systemic right ventricle. Further studies will be necessary to address this issue.

Each model parameter has been varied separately with other parameters being held at their default values to provide insight into the mechanisms underlying hemodynamic regulation. In real physiological conditions, however, several parameters may deviate simultaneously from their reference values, and, moreover, hemodynamic perturbations induced by a parameter variation should be always accompanied by regulatory responses acting to restore arterial pressure or tissue/organ perfusion. Therefore, the results obtained from a single parameter-variation study would provide a trend rather than an accurate prediction of in vivo hemodynamic changes. It is stressed that our model can readily be applied to study the hemodynamic effects of multiple parameters. Moreover, the present study has focused on quantifying the general hemodynamic responses to variations in cardiovascular properties using a generalized model of the Fontan circulation. To predict the hemodynamic impacts of certain operations in a specific patient, the model should be tuned to the patient by optimizing model parameters based on patient-specific clinical data. This issue has been addressed elsewhere, such as in recent studies on surgical planning (40, 63) and those from our group (46, 82).

The influence of cardiac valvular diseases on CO has not been addressed in the present study. In fact, patients with single-ventricle physiology are among the patient cohorts most susceptible to cardiac valvular diseases. A recent clinical study revealed that ∼30% of patients undergoing Fontan procedure were affected by moderate to severe atrioventricular valve regurgitation (AVVR) (3). AVVR significantly deteriorates the pumping performance of the ventricle, and its close association with increased morbidity and mortality has been demonstrated (57). Therefore, incorporating AVVR in the modeling of the Fontan circulation would be necessary to more extensively study the hemodynamic characteristics of the Fontan circulation. The issue will be addressed in our future studies.

Conclusions

The present study analyzed the hemodynamic performance of the Fontan circulation compared with the normal biventricular circulation under various cardiovascular conditions using computational models. Obtained results demonstrate the inherent differences between the two types of circulation in the regulation of CO and central venous pressure. The main findings of the study include the following: 1) while increasing HR or ventricular contractility contributes significantly to the increase of CO in the normal biventricular circulation, the contribution is significantly reduced in the Fontan circulation; 2) the sensitivity of CO to pulmonary vascular resistance, ventricular diastolic function, and systemic vascular compliance is enhanced in the Fontan circulation; 3) the amount of CO change induced by a variation in cardiovascular parameter is dependent on the status of the entire cardiovascular system owing to the nonlinear nature of the CO-parameter relationship and the interaction among different parameters; 4) reducing systemic vascular resistance insignificantly contributes to the improvement of CO in the Fontan circulation unless ventricular systolic function is severely impaired; and 5) the absence of a subpulmonary ventricle in the Fontan circulation significantly increases the variability of the central venous pressure. The findings may serve as a useful theoretical reference for clinical doctors to identify the major cardiovascular factors underlying patient-specific hemodynamic characteristics so as to implement efficacious hemodynamic management.