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In this paper, we develop a pulsatile compartmental model of the Fontan circulation and use it to explore the effects of a fenestration added to this physiology. A fenestration is a shunt between the systemic and pulmonary veins that is added either at the time of Fontan conversion or at a later time for the treatment of complications. This shunt increases cardiac output and decreases systemic venous pressure. However, these hemodynamic benefits are achieved at the expense of a decrease in the arterial oxygen saturation. The model developed in this paper incorporates fenestration size as a parameter and describes both blood flow and oxygen transport. It is calibrated to clinical data from Fontan patients, and we use it to study the impact of a fenestration on several hemodynamic variables, including systemic oxygen availability, effective oxygen availability, and systemic venous pressure. In certain scenarios corresponding to highrisk Fontan physiology, we demonstrate the existence of a range of fenestration sizes in which the systemic oxygen availability remains relatively constant while the systemic venous pressure decreases.
Single ventricle physiology corresponds to a spectrum of congenital heart defects in which there is only one functioning ventricular chamber. Patients with this type of condition require complex medical and surgical interventions to ensure survival. The typical course of treatment for these defects is a sequence of surgeries during the first several years of life, ending with a procedure that establishes an abnormal physiology known as the Fontan circulation. This physiology was conceived in 1971 and is characterized by the systemic organs and lungs in series, as in a normal circulation (
Sketches of the Fontan circulation in panel
Fontan patients may experience severe complications because of their abnormal physiology, including proteinlosing enteropathy, ventricular and hepatic dysfunction, and plastic bronchitis (
In practice, this shunt is established between the systemic veins and atria as a “sidebyside” connection. In our model, the atria and venous compartments are lumped together, so the fenestration here is described by a shunt between the systemic and pulmonary veins. Furthermore, the nonlinear resistance model used for the fenestration in this paper is derived assuming two compartments are connected by a hole with a predetermined crosssectional area. This setup considered here is also consistent with the typical creation of such a connection. For a schematic of the Fontan circulation with a fenestration, refer to panel (B) of
A fenestration has been introduced into the Fontan physiology both at the time of Fontan conversion and also later in the patient’s life for the treatment of complications (
The complexity of the Fontan circulation has motivated the development of many mathematical models that focus on various aspects of this physiology. For an extensive review of modeling efforts devoted to the Fontan physiology, see Degroff et al. (
We develop a pulsatile compartmental model of the Fontan physiology and use it to study the impact of a fenestration on blood flow and oxygen transport. The model, with the fenestration closed, is calibrated to clinical data in order to provide a baseline set of parameters. We vary both the pulmonary vascular resistance and fenestration size, with respect to the baseline model, to study the fenestration’s impact on several important variables, including systemic venous pressure, cardiac output, oxygen saturation, systemic oxygen availability, and effective oxygen availability. Our model demonstrates that in certain scenarios which correspond to highrisk Fontan physiology, a range of fenestration sizes exists in which the systemic oxygen availability remains relatively constant. In these cases, the systemic venous pressure decreases, which could be of substantial benefit to the patient.
Our model for the Fontan circulation contains two main parts: 1) a blood flow model that incorporates a nonlinear resistance for the fenestration and 2) an oxygen transport model. The blood flow model is presented in the first subsection along with details for modeling the fenestration. The oxygen transport model is described in the second subsection. Methods for numerically approximating these models are detailed in the final subsection. The models and methods presented here closely follow those given in related work by Han et al., and additional details may be found in that paper (
The blood flow model used here describes the circulation as a series of compartments that are either compliance chambers or resistor elements. Our approach is derived from earlier work by Peskin and Tu, with modifications to account for the Fontan physiology and fenestration (
The timedependent ventricular compliance, denoted
The resistor elements in our model describe connections between compliance chambers. In this framework, compliance chambers
The equations of motion for the blood flow model follow from conservation of volume in each compliance chamber. Using our notation given above and defining
The fenestration between the systemic and pulmonary veins is taken to be a nonlinear resistor that depends on its crosssectional area and the magnitude of flow through it. Let
In our model for oxygen transport, each compliance chamber has a timedependent oxygen concentration denoted
In this section, we describe the numerical approximation schemes for the blood flow and oxygen transport models. Let
In this section, we present and discuss results from our numerical simulations. As mentioned in
We first describe the calibration of our model using hemodynamic data from Fontan patients, with the goal of deriving a baseline set of parameters. In this case, we use 1,000 time steps per cardiac cycle, and simulations are run for 40 cardiac cycles. This duration is long enough to ensure periodic steady states are reached for all hemodynamic variables, which are needed for the model calibration. The model is calibrated, with the fenestration closed, to a data set provided by Liang et al. (
Parameters for the circulation model. Abbreviations: s, systemic organs; p, pulmonary; ov, outflow valve; av, atrium/ventricle valve; fo, Fontan connection; sa, systemic arteries; pa, pulmonary arteries; sv, systemic veins; pv, pulmonary veins.
Parameters  Resistance ( 
Dead Volume ( 
Compliance ( 

Units  mmHg min L^{−1}  L  L mmHg^{−1} 
s  20.78  —  — 
p  0.5517  —  — 
ov  0.20  —  — 
av  0.01  —  — 
fo  0.01  —  — 
sa  —  0.7051  7.333 × 10^{–4} 
pa  —  0.0930  0.00412 
sv  —  2.869  0.0990 
pv  —  0.1475  0.01 
Parameters for the time varying ventricular compliance in the heart model.
Parameters  Symbol  Units  Ventricle 

Minimal elastance 

mmHg L^{−1}  79.52 
Maximal elastance 

mmHg L^{−1}  5,232 
systolic exponent 

—  1.32 
diastolic exponent 

—  27.4 
Systolic time constant 

min  0.269 × 
Diastolic time constant 

min  0.452 × 
Dead volume 

L  0.028 
Period of heartbeat 

min  0.016 
Hemodynamic variables calculated from our calibrated model with a closed fenestration, compared to the clinical data reported by Liang et al. (
Variable  Our model  Clinical data reported in ( 

cardiac index (L min^{−1} m^{−2})  2.683  2.9, 2.1 
stroke volume index (ml m^{−2})  42.93  39,40 
end diastolic volume index (ml m^{−2})  75.91  72, 76 
end systolic volume index (ml m^{−2})  32.98  33, 36 
end systolic pressure (mmHg)  111.8  — 
end diastolic pressure (mmHg)  6.828  6.6 
vena cava mean pressure (mmHg)  9.347  8 
pulse pressure (mmHg)  56.52  54 
systemic artery systolic pressure (mmHg)  118.2  124 
systemic artery diastolic pressure (mmHg)  61.68  70 
systemic artery mean pressure (mmHg)  92.99  88 
pulmonary artery mean pressure (mmHg)  9.306  9 
Based on a report by Ohuchi, the calibrated variables from our model correspond to a “latesurviving” and “excellentsurviving” Fontan patient (
Results from our calibrated model with a closed fenestration. Three pressure waveforms from the end of the simulation are shown in panel
In this section, we explore the effect of an open fenestration on both hemodyamic and oxygen transport variables. In these cases, we use 100 time steps per cardiac cycle and simulations are run for 4,000 cardiac cycles. This duration is long enough to ensure periodic steady states are reached for all hemodynamic and oxygen transport variables. As described in
Fenestration flow waveforms for two different fenestration sizes and our baseline pulmonary vascular resistance value of
Results corresponding to oxygen consumption equal to −0.3236 L min^{−1}. Pulmonary resistance values
Results corresponding to oxygen consumption equal to −0.3236 L min^{−1}. Pulmonary resistance values
High pulmonary vascular resistance and low cardiac output have been identified as important risk factors for Fontan patients (
Results corresponding to oxygen consumption equal to −0.2 L min^{−1}. Pulmonary resistance values
Results corresponding to oxygen consumption equal to −0.2 L min^{−1}. Pulmonary resistance values
In general, the fenestration has a more substantial impact on hemodynamics and oxygen transport for larger pulmonary vascular resistances. The trends seen here are similar to those in
The effect of the fenestration on hemodynamic and oxygen transport variables, as predicted by our models, is consistent with trends seen in clinical case reports (
The previous section considered variations in several hemodynamic and oxygen transport variables as functions of the fenestration size, pulmonary vascular resistance, and oxygen consumption parameters. In reality, the introduction of a fenestration may induce physiologic changes in many of the parameters in the model. To quantify the impact these changes might have on important variables, we perform a sensitivity analysis in three distinct cases: 1) with our baseline model corresponding to
For a variable
Sensitivities for cases 1, 2, and 3 computed for the systemic venous pressure.








Case 1  −5.6e02  3.9e02  4.5e04  −2.2e04  8.7e04  −5.8e02 
Case 2  −5.7e02  3.4e02  3.9e04  −2.2e04  7.6e04  −5.9e02 
Case 3  −4.4e02  3.8e02  9.0e06  −1.8e04  2.1e04  −4.0e02 






Case 1  −3.3e02  −7.4e01  −6.4e02  5.0e02  7.3e03  
Case 2  −3.3e02  −7.4e01  −6.7e02  5.3e02  7.1e03  
Case 3  −3.5e02  −7.9e01  −4.5e02  4.2e03  1.3e02 
Sensitivities for cases 1, 2, and 3 computed for the systemic oxygen availability.








Case 1  −2.1e01  −2.3e01  −3.7e03  −1.1e03  −4.1e03  −3.1e02 
Case 2  −2.2e01  −2.5e01  −4.5e03  −1.2e03  −4.4e03  −3.4e02 
Case 3  −1.7e01  −5.6e01  −3.0e03  −9.3e04  −2.0e03  −2.7e02 






Case 1  −3.3e02  −7.4e01  −4.7e02  −6.1e01  1.8e01  
Case 2  −3.5e02  −7.9e01  −5.9e02  −6.7e01  2.0e01  
Case 3  −4.3e02  −9.7e01  −8.3e03  −5.2e01  1.5e01 
Sensitivities for cases 1, 2, and 3 computed for the effective oxygen availability.








Case 1  −2.1e01  −2.3e01  −3.7e03  −1.1e03  −4.1e03  −3.1e02 
Case 2  −2.2e01  −2.9e01  −5.7e03  −1.2e03  −5.2e03  −3.3e02 
Case 3  −1.5e01  −5.7e01  −3.4e03  −8.2e04  −2.1e03  −2.4e02 






Case 1  −3.3e02  −7.4e01  −4.7e02  −6.1e01  1.8e01  
Case 2  −3.4e02  −7.7e01  −7.4e02  −6.5e01  2.0e01  
Case 3  −3.8e02  −8.5e01  −1.4e02  −4.6e01  1.4e01 
In all cases, the systemic oxygen availability and effective oxygen availability also appear to be most sensitive to the systemic venous compliance. Note that for case 1, the systemic and effective oxygen availabilities and their corresponding sensitivities are equal since the shunt flow is zero. Both variables are very sensitive to the systemic and pulmonary resistances, likely due to the significant impact of these parameters on cardiac output. The sensitivities with respect to the minimum and maximum elastances of the single ventricle are also notable. Notice that the sensitivities of the oxygen availabilities with respect to
In summary, the sensitivity analysis of our models indicates the importance of the systemic and pulmonary resistances as well as the single ventricle contraction parameters on oxygen availabilities and the systemic venous pressure. Although care was taken to calibrate the model with a closed fenestration to a clinical data set from Fontan patients, results for the open fenestration cases will depend on the chosen parameters. The pulmonary vascular resistance has been cited as important parameter in determining Fontan hemodynamics (
In this paper, we developed a pulsatile compartmental model of the Fontan circulation that describes blood flow and oxygen transport. It also incorporates a fenestration between the systemic and pulmonary veins. The model was calibrated to clinical data with the fenestration closed in order to create a baseline set of parameters.
We then studied the impact of an open fenestration on several hemodynamic and oxygen transport variables. An open fenestration decreased the pulmonary artery pressure and increased cardiac output, with larger impacts on these variables seen for larger values of the pulmonary vascular resistance. For our baseline oxygen consumption value and for pulmonary vascular resistances close to the baseline value, systemic oxygen availability monotonically decreased as a function of the fenestration diameter. For pulmonary vascular resistances that are typical of atrisk patients, however, the systemic oxygen availability for small fenestration sizes remained relatively constant until a critical size was reached, at which point the availability dropped quickly. This trend was accompanied by a decrease in systemic venous (≈ pulmonary arterial) pressure. Furthermore, as the fenestration size increased, the effective oxygen availability decreased in all cases. Thus, the reason for fenestrating the Fontan circulation may not so much be to affect systemic or effective oxygen availability, but rather to ensure that systemic oxygen availability is not reduced by an intervention with a different benefit–the reduction of systemic venous pressure.
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.
All authors contributed to conception and design of the study. CP, ZA, and LJ executed simulations and wrote the first draft of the manuscript. All authors edited the manuscript.
This work was supported in part by the Research Training Group in Modeling and Simulation funded by the National Science Foundation via grant RTG/DMS1646339.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
The authors thank the reviewers for many helpful comments that greatly improved the manuscript.
PVRI calculated here does not directly correspond to the