A Computational Model of Ventricular Dimensions and Hemodynamics in Growing Infants

Abstract

Previous computer models have successfully predicted cardiac growth and remodeling in adults with pathologies. However, applying these models to infants is complicated by the fact that they also undergo normal, somatic cardiac growth and remodeling. Therefore, we designed a computational model to predict ventricular dimensions and hemodynamics in healthy, growing infants by modifying an adult canine left ventricular growth model. The heart chambers were modeled as time-varying elastances coupled to a circuit model of the circulation. Circulation parameters were allometrically scaled and adjusted for maturation to simulate birth through 3 yrs of age. Ventricular growth was driven by perturbations in myocyte strain. The model successfully matched clinical measurements of pressures, ventricular and atrial volumes, and ventricular thicknesses within two standard deviations of multiple infant studies. To test the model, we input 10th and 90th percentile infant weights. Predicted volumes and thicknesses decreased and increased within normal ranges and pressures were unchanged. When we simulated coarctation of the aorta, systemic blood pressure, left ventricular thickness, and left ventricular volume all increased, following trends in clinical data. Our model enables a greater understanding of somatic and pathological growth in infants with congenital heart defects. Its flexibility and computational efficiency when compared to models employing more complex geometries allow for rapid analysis of pathological mechanisms affecting cardiac growth and hemodynamics.

1 Introduction

Congenital heart defects (CHDs), failures of the heart to develop properly in utero, are the most common birth defect, occurring in approximately 1 in 100 births [1]. A quarter of those infants will have a critical CHD, requiring surgery in their first year of life [2]. Fortunately, advances in care over the past 50 yrs have increased survival rates enormously; over 85% of children born with CHDs today can expect to live into adulthood [3].

As children live and grow with CHDs, their cardiovascular system also grows. In adults, cardiac hypertrophy is largely a function of pathology, occurring in response to hemodynamic overload from conditions such as valvular disease, hypertension, and myocardial infarction. In children, however, hypertrophy is a normal part of development. Complicating matters further, children with CHDs experience this somatic growth alongside pathologic growth, making it difficult to estimate how disease progression or intervention will change hemodynamics or heart size. Separating these growth mechanisms in vivo is extremely challenging, limiting our understanding of their interplay and our ability to predict changes in cardiac dimensions in children with CHDs.

Computational models are useful tools for uncoupling processes that are inextricably linked in the body so they can be modified and studied in isolation. For example, Burkhoff and Tyberg investigated proposed sources of the rise in pulmonary venous pressure after ventricular dysfunction using a simple zero-dimensional model [4]. Other cardiovascular models have evaluated hemodynamically induced stresses on aneurism development and rupture, oxygen transport, and atherosclerosis [5–8]. Computer models have become powerful aids in surgical planning for CHD patients [9–11]. Image-based computational fluid dynamics simulations have been produced for many different congenital defects including tetralogy of Fallot [12], Kawasaki disease [13], and coarctation of the aorta (CoA) [14]. The application of these models for “virtual surgery” in patients with single ventricle physiology has been the most active area of research, including in silico assessment of different surgical approaches and patient-specific modeling [15,16].

One of the most exciting new areas for computational modeling is the simulation of growth and remodeling. These models predict changes in the mass and structure of cardiovascular tissues over time, usually in response to acute alterations in hemodynamics. They utilize growth laws that relate stress and strain changes in the myocardium due to perturbances in hemodynamics with cardiomyocyte lengthening and thickening. In our recent review [17], we evaluated the ability of a number of published growth laws to reproduce the characteristic growth patterns resulting from strains typical of ventricular pressure and volume overload in adults. By altering hemodynamic inputs consistent with pathology, the models can predict the resultant pathologic growth and remodeling [18,19].

One such model, previously developed by our group [17,20], predicted the time-course of ventricular hypertrophy in adult canines under volume and pressure overload as well as myocardial infarction. In this study, we adapted this model for human infants and successfully predicted changes in ventricular thickness and volume for healthy infants from birth to 3 yrs of age. To demonstrate the model's capacity for pathologic growth in infants, we simulated CoA, a defect which generally results in left ventricular (LV) pressure overload. The CoA simulation led to expected increases in LV thickness, end-diastolic volume, and systemic blood pressure.

2 Methods

A previously published computational model [17] capable of predicting LV growth in adult canines in response to hemodynamic overload was modified for human infants. For more flexibility in future simulations of pathology, additional compartments were created and right ventricular (RV) growth was implemented into the simulation. As explained in detail below, a strain-based growth law determined ventricular growth in both the circumferential and radial directions. Changes in ventricular geometry throughout growth were computed using analytic expressions relating strain and volume as well as stress and pressure in a thin-walled sphere. Allometric scaling was used to adjust input parameters from an adult canine model [17,20] and an adult human model [21] according to mean infant body mass. A maturation approach was used to adjust specific parameters in order to capture reported hemodynamic trends associated with birth. To test the predictive capacity of the model, infant weights at the 10th and 90th percentiles were inputted. Lastly, to evaluate the model's capacity to predict pathologic hypertrophy, CoA was simulated.

2.1 Expanding the Model to Simulate Infant Physiology.

The original adult canine model [17,20] was composed of six compartments: LV, RV, systemic veins (SV), systemic arteries (SA), pulmonary veins (PV), and pulmonary arteries (PA). Briefly, the ventricles were simulated using time-varying elastances, systemic and pulmonary arterial behavior were simulated by three-element Windkessels, distal vessels were represented by capacitors in parallel with resistors, and valves were represented by pressure-sensitive diodes. To better replicate infant physiology, particularly for cases of congenital heart defects, an additional six compartments were added, for a total of 12 compartments (Fig. 1). First, left and right atria were added using identical time-varying elastances, along with respective resistances for the mitral and tricuspid valves [21]. Parameter values for this eight-compartment model matching the hemodynamics of a healthy adult human are detailed in Table S1 (available in the Supplemental Materials on the ASME Digital Collection).

Fig. 1.

Schematic of the circuit model used to simulate the pressure–volume behavior of the cardiovascular system. The left and right atria (LA and RA) were simulated using time-varying elastances. The LV and RV were simulated using time-varying elastances, and growth relied on strains computed assuming a thin-walled spherical geometry. The ascending aorta and main pulmonary artery behavior were simulated by three-element Windkessels consisting of a characteristic resistance (R_SC and R_PC), a capacitance (C_AA and C_MPA), and an arterial resistance (R_AA and R_MPA). Distal vessels were represented by capacitors in parallel with resistors for the pulmonary arteries (C_PA and R_PA), pulmonary veins (C_PV and R_PV), lower body systemic arteries (C_SA_LB and R_SA_LB), upper body systemic arteries (C_SA_UB and R_SA_UB), lower body systemic veins (C_SV_LB and R_SV_LB), and upper body systemic veins (C_SV_UB and R_SV_UB). Pressure sensitive diodes (TV, PV, MV, and AV) represented the tricuspid, pulmonary, mitral, and aortic valves, respectively. Arrows indicate the direction of blood flow and blood oxygenation levels.

To model the main pulmonary artery (MPA), the pulmonary arterial pool was split by adding an additional series resistor and parallel capacitor. The total pulmonary arterial compliance and resistance were

$$C_{\mathrm{total},\mathrm{PA}}=C_{\mathrm{MPA}}+C_{\mathrm{PA}}\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}{R}_{\mathrm{total},\mathrm{PA}}=R_{\mathrm{MPA}}+R_{\mathrm{PA}}$$ (1)

respectively. C _MPA and R _MPA indicate the capacitance and resistance of the main pulmonary artery, respectively. C _PA and R _PA indicate the capacitance and resistance of the remaining distal pulmonary arteries, respectively. The value of C _MPA /C _PA was set to 8 such that the mean pressure and pulse pressure in the main pulmonary artery were ≈1.5 mm Hg and 2.5 mm Hg higher than the distal vessels, respectively [22,23]. The ratio of main pulmonary artery resistance to that of the distal arteries

$$R_{\mathrm{MPA}}/R_{\mathrm{PA}}={\left(C_{\mathrm{MPA}}/C_{\mathrm{PA}}\right)}^{\left(-3/4\right)}$$ (2)

was determined using the dependence of vascular resistances and capacitances on vessel dimensions (R ≈ lr^{− 4} and C ≈ lr ³, where r and l are the radius and length of the vessel, respectively) [24,25]. The systemic arterial pool was split in a similar manner to model the ascending aorta (AA) and the distal systemic arteries. However, to simulate the upper and lower body separately the distal arteries were further divided. The new upper and lower body systemic arterial resistors were placed in parallel and the capacitors were also in parallel. Thus, the total systemic arterial compliance and resistance were

$$C_{\mathrm{total},\mathrm{SA}}=C_{\mathrm{AA}}+C_{\mathrm{SA},\mathrm{UB}}+C_{\mathrm{SA},\mathrm{LB}}$$ (3)

and

$$R_{\mathrm{total},\mathrm{SA}}=R_{\mathrm{AA}}+\frac{R_{\mathrm{SA},\mathrm{UB}}{R}_{\mathrm{SA},\mathrm{LB}}}{R_{\mathrm{SA},\mathrm{UB}}+R_{\mathrm{SA},\mathrm{LB}}}\hspace{0.17em}$$ (4)

respectively. C _AA and R _AA indicate the capacitance and resistance of the ascending aorta, C _SA,UB and R _SA,UB indicate the capacitance and resistance of the systemic arteries in the upper body, and C _SA,LB and R _SA,LB indicate the capacitance and resistance of the systemic arteries in the lower body. The value of C _AA/(C _SA,UB + C _SA,LB) was set to 16 such that the mean and pulse pressure in the ascending aorta were ≈4.4 mm Hg and 2.7 mm Hg higher than the distal vessels, respectively [26]. Again, using the dependence of vascular resistances and capacitances on vessel dimensions, the ratio of the ascending aortic resistance to that of the distal arteries was

$$R_{\mathrm{AA}}\left(\frac{R_{\mathrm{SA},\mathrm{UB}}+R_{\mathrm{SA},\mathrm{LB}}}{R_{\mathrm{SA},\mathrm{UB}}{R}_{\mathrm{SA},\mathrm{LB}}}\right)={\left(\frac{C_{\mathrm{AA}}}{C_{\mathrm{SA},\mathrm{UB}}+C_{\mathrm{SA},\mathrm{LB}}}\right)}^{(-3/4)}$$ (5)

The values of C _SA,LB /C _SA,UB and R _SA,LB /R _SA,UB were set to 1 based on the ratio of flow in the superior vena cava to the main pulmonary artery reported for infants by Salim et al. [27].

Finally, the upper and lower systemic veins were divided in the same way as the upper and lower systemic arteries. The total systemic venous compliance and resistance were

$$C_{\mathrm{total},\mathrm{SV}}=C_{\mathrm{SV},\mathrm{UB}}+C_{\mathrm{SV},\mathrm{LB}}$$ (6)

and

$$R_{\mathrm{total},\mathrm{SV}}=\frac{R_{\mathrm{SV},\mathrm{UB}}{R}_{\mathrm{SV},\mathrm{LB}}}{R_{\mathrm{SV},\mathrm{UB}}+R_{\mathrm{SV},\mathrm{LB}}}\hspace{0.17em}$$ (7)

respectively. C _SV,UB and R _SV,UB and C _SV,LB and R _SV,LB indicate the capacitances and resistances of the systemic veins in the upper and lower body, respectively. As in the arteries, the values of C _SV,LB /C _SV,UB and R _SV,LB /R _SV,UB were set to 1.

2.2 Scaling Model Parameters.

To adapt the model to human infants, weight-based allometric scaling was employed for ventricular pressure–volume and timing parameters, capacitances, and the majority of resistances such that

$$Y_i=Y_0{(M_i/M_0)}^b$$ (8)

where Y is the parameter under consideration, M is body weight, and b is a scaling factor [28]. The subscripts i and 0 indicate current and reference values, respectively. Reference parameters (Y ₀ and M ₀) were set to those of a healthy 70 kg adult human (See Table S1, available in the Supplemental Materials on the ASME Digital Collection). M_i was a vector of infant weights for ages 0–3 yrs, linearly interpolated from the Centers for Disease Control and Prevention's weight-for-age charts for 50th percentile male and female infants [29]. Based on theoretical and empirical measurements, resistances were scaled by b = −3/4 and capacitances were scaled linearly [28,30]. Other atrial and ventricular parameters were scaled using unit analysis. The scaling factor for the timing parameter indicating the percentage of the cardiac cycle spent in systole, b = −0.07, was computed from reported percentages in rats [31] and dogs [20]. See Table S2 (available in the Supplemental Materials) for a full list of scaling factors.

2.3 Customizing Model Parameters for Infant Development.

Some parameters were not scaled throughout the entire age range of 0–3 yrs. While in utero, pulmonary vascular resistance (PVR) is high due to the fetus's lack of respiration. It plummets sharply after birth with ventilation, when the lungs open and alveolar PO2 increases dramatically [32]. Systemic vascular resistance (SVR) follows an opposite trend: the removal of the placenta and umbilical cord eliminates a low-resistance source of blood flow, thus SVR rises modestly postbirth [32]. To replicate these unique circumstances associated with birth, scaling was combined with an age-based function, sometimes called a “maturation approach” in pharmacology [33,34]. Therefore, at age in days, t

$$\mathrm{PVR}\left(t\right)=cQ\left(t\right)+{\mathrm{PVR}}_i\left(1-Q\left(t\right)\right)$$ (9)

where

$$Q\left(t\right)=\frac{1}{1+{\left(\frac{t}{c_1}\right)}^{c_2}}\hspace{0.17em}$$ (10)

PVR _i indicates the scaled value of pulmonary vascular resistance computed from Eq. (8) based on the infant's weight at age t. Their associated capacitance

$$\mathrm{PVC}\left(t\right)={\mathrm{PVC}}_0{\left(\frac{\mathrm{PVR}\left(t\right)}{{\mathrm{PVR}}_0}\right)}^{-4/3}\hspace{0.17em}$$ (11)

was computed according to the dependencies of vascular resistances and compliances on vessel dimensions [24,25]. SVR(t) and its associated capacitance, SVC(t), were determined in the same manner. Maturation parameters c, c ₁, and c ₂ were fit using the built-in matlab function fminsearch to minimize the sum squared error between model outputs for mean pulmonary artery pressure (MPAP) and mean arterial pressure (MAP) and the mean values of reported measurements from ages 0 to 120 days [35–37]. Figure 2 compares resistances determined using the maturation function to those from allometric scaling alone, along with their resultant trajectories of MAP and MPAP. In the model, both MAP and MPAP were computed as the sum of one-third of the systolic pressure and two-thirds of the diastolic pressure, i.e., MAP = (SBP + 2 × DBP)/3. For PVR the fitted values of $c,c_1$ , and $c_2$ were 3.7 mm Hg × s/mL, 1.3 days and 1.6, respectively, and for SVR they were 5.0 mm Hg × s/mL, 29.9 days, and 4.1, respectively. Heart rate was originally computed via weight-based allometric scaling (using b = −1/4 and adult reference heart rate = 70 beats/min). Though the weight-based scaled heart rate values were within the range reported for infants between 0 and 36 months, they peaked slightly early (at approximately 1 week of age), whereas heart rates reported by Fleming et al. [41] peaked at approximately 4 weeks (See Figure S1(a), available in the Supplemental Materials). Therefore, to capture the observed trend, the measured 50th percentile curve from Fleming et al. was inputted directly into the model.

Fig. 2.

Scaled pulmonary and systemic vascular resistances and resistances produced via a maturation approach ((a) and (c), respectively) as well as their resultant mean pulmonary and systemic pressures ((b) and (d), respectively) for a median-weight infant in the four months of life, compared to measured data [35–40]. Error bars on data indicate 10th and 90th percentiles [38], 5th and 95th percentiles [39,40], or one standard deviation [35–37]. Measurements from Qi et al. were from the sea-level cohort only. PVR, pulmonary vascular resistance; MPAP, mean pulmonary arterial pressure; SVR, systemic vascular resistance; MAP, mean arterial pressure.

With the exception of heart rate, PVR, and SVR, all model input parameters were determined by infant weight. The ability of the model to produce dimensions, namely, LV and RV end-diastolic and end-systolic volumes and thicknesses as well as left atrial volume, within reported bounds at ages 0–3 yrs was evaluated for model validation. Infant body masses at the 10th and 90th percentiles were also considered for further model validation. In these cases, PVR and SVR were computed using Eqs. (9) and (10), and the constants (c, c ₁, and c ₂) were kept at the values fitted based on the median weight. Heart rate was adjusted upward or downward according to weight (See Figure S1(b), available in the Supplemental Materials). Lastly, all other model parameters were scaled using the smaller or larger weight trajectories (Eq. (8)).

2.4 Simulating Somatic Growth.

A previously published computational model that reproduced pathologic ventricular growth in response to pressure overload, volume overload, and myocardial infarction in canines was modified to reproduce both pathologic and somatic ventricular growth in infants [17,20]. A flowchart describing the model is shown in Fig. 3. As discussed in Secs. 2.2 and 2.3, a combination of allometric scaling and a maturation approach were utilized to compute all model circulatory parameters for ages 0–3 yrs.

Fig. 3.

Flowchart of model development. Input parameters were first determined using either allometric scaling or maturation functions, including all resistances and capacitances, stressed blood volume, and ventricular parameters A, B, Ees, and V ₀. In the healthy simulation, the full time course of A, B, Ees, and V ₀ was generated. In the pathologic simulation, only the first day (pre-growth) was generated. From there, the unloaded ventricular radius and thickness (r _{0_ s} and h _{0_ s} for the healthy simulation, r ₀ and h ₀ for the pathologic simulation) were determined and used to find the loaded radius and thickness (r and h) throughout the cardiac cycle. Strains were then calculated and used as homeostatic setpoints (in the healthy simulation) or to determine growth stretch tensor F_g (in the pathologic simulation). In the pathologic simulation, F_g was used to determine the new unloaded radius and thickness, which in turn gave new values for A, B, Ees, and V ₀. The cycle repeated until the maximum growth step was reached at the three year time point.

Ventricular end-diastolic and end-systolic pressure–volume relationships were defined by

$$P_{\mathrm{ED}}=B{e}^{A\left(V_{\mathrm{ED}}-V_0\right)}-B\hspace{0.17em}$$ (12)

and

$$P_{\mathrm{ES}}=\mathrm{Ees}(V_{\mathrm{ES}}-V_0)\hspace{0.17em}$$ (13)

respectively, where Ees was the end-systolic elastance of the ventricle, V ₀ was the unloaded volume of the ventricle, and A and B were coefficients describing the exponential shape of the end-diastolic pressure–volume relationship. For healthy infants at three different birth weights (10th, 50th, and 90th percentiles), allometric scaling was utilized to determine values of $V_0$ , $A$ , $B$ , and $\mathrm{Ees}$ (see Sec. 2.2 and Table S2 available in the Supplemental Materials on the ASME Digital Collection) throughout the entire three year time-course. To account for changes in geometry with growth, analytic expressions based on the relationships between strain and volume and stress and pressure in a thin-walled sphere were developed [17,20] that relate material parameters (a, b, and e) to compartmental parameters (A, B, and Ees):

$$a=A\frac{4\pi }{3}{\left(F_{g,c}{r}_{0\_s}\right)}^3$$ (14)

$$b=\frac{B}{2}\frac{F_{g,c}{r}_{0\_s}}{F_{g,r}{h}_{0\_s}}$$ (15)

and

$$e=\frac{2\pi }{3}\mathrm{E}\mathrm{es}\frac{{\left(F_{g,c}{r}_{0\_s}\right)}^4}{F_{g,r}{h}_{0\_s}}$$ (16)

r _{0 s} and h _{0 s} indicate the ventricle's homeostatic unloaded radius and thickness, respectively. The growth stretches, $F_{g,c}$ and $F_{g,r}$ (discussed in Sec. 2.5 in more detail) indicate pathologic changes in the unloaded circumferential and radial dimensions of the ventricle, respectively. They were set to 1 at all times for the healthy simulations. In the previous adult canine simulations [17,20], $r_{0\_s}$ and $h_{0\_s}$ were constant, as the unloaded homeostatic ventricular dimensions are unchanged with time in healthy adults. To simulate growth in healthy infants, these parameters must change with age. $r_{0\_s}$ was computed for each weight trajectory for each age in days as

$$r_{0\_s}=\frac{1}{F_{g,c}}{\left(\frac{3}{4\pi }{V}_0\right)}^{1/3}$$ (17)

Then, a single value of a (1.56 for the 50th percentile weight trajectory) was determined from Eq. (14) for all ages. This was well within the range of a values from our previous studies [17,20]. b was set to 0.14, the average value from our previous studies [17,20], and Eq. (15) was rearranged to compute $h_{0s}$ for each age:

$$h_{0\_s}=\frac{B}{2}\frac{F_{g,c}{r}_{0\_s}}{F_{g,r}b}\hspace{0.17em}$$ (18)

Then, from Eq. (16) a single value for e (240 mm Hg for the 50th percentile weight trajectory) was computed. Though compartmental parameters change during growth, myocardial properties are particularly difficult to measure and there is no consensus on changes during normal somatic development, therefore the material parameters were kept constant throughout the three year period.

For each growth step (corresponding to one day), the circulation portion of the model was run until it reached steady-state, defined as compartmental volumes being within 0.0001 mL of each other at the beginning and end of the cardiac cycle. The loaded LV and RV radii and thicknesses, r and h, were calculated throughout the cardiac cycle from the ventricular volumes.

In the previous adult canine model [17,20], pathologic growth in the circumferential and radial directions was driven by changes in the maximum circumferential and radial strain from homeostasis, respectively. However, in a growing infant, the ventricles' homeostatic unloaded ventricular dimensions, and thus their homeostatic strains, change with age. Circumferential strain

$$E_c=0.5{\left(\frac{r}{r_{0\_s}{F}_{g,c}}\right)}^2-0.5$$ (19)

and radial strain,

$$E_r=0.5{\left(\frac{h}{h_{0\_s}{F}_{g,r}}\right)}^2-0.5$$ (20)

were computed for each ventricle from the loaded radii and thicknesses. Then, other model outputs, including ventricular volumes and thicknesses as well as atrial volumes and arterial pressures were computed and compared to clinical measurements.

2.5 Simulating Pathologic Growth: Coarctation of the Aorta.

To explore the capacity of the model to produce simultaneous somatic and pathologic ventricular growth, CoA was simulated. This congenital heart defect presents as a narrowing of the descending aorta distal to the left subclavian artery, resulting in increased systolic blood pressure and LV hypertrophy [42,43]. In general, systemic vascular resistance drops as infants age and their vessels become larger. To simulate unrepaired CoA, the resistance of the lower body arteries was gradually increased from its peak somatic value at 23 days to a maximum of 25% over baseline (Fig. 4(a)). The compliance of the lower body arteries was determined from the resistance via their dependence on vessel dimensions [24,25]. The gradual increase in resistance over the first year of life was simulated to replicate the time-course in the presentation of coarctation observed by Eerola et al. [42] and Vogt et al. [43], wherein systemic blood pressures in CoA patients are similar to control soon after birth before becoming significantly larger as the child grows. In previous simulations of pressure overload [17,44], it was necessary to increase stressed blood volume to capture increases in ventricular filling pressure. Therefore, stressed blood volume was increased proportionally with lower body artery resistance (Fig. 4(b)).

Fig. 4.

Resistance for the lower body arterial compartment (a)and stressed blood volume and (b) for the healthy and coarctation simulations for a median-weight infant in the first three years of life. Resistance was gradually increased from its peak somatic value at 23 days to a maximum of 25% over baseline and stressed blood volume was increased 14% over baseline.

The pathologic growth mechanism follows the same path as somatic growth, with a few exceptions (see Fig. 3). Simulating pathologic growth relied on iteration. Rather than generating the entire three year time-course of ventricular parameters V ₀, A, B, and Ees from allometric scaling, only the initial day 0 set was determined via scaling. No pathologic growth was assumed in utero, therefore the initial values of r ₀ and h ₀ were equivalent to their somatic, healthy counterparts, r _{0 _s} and h _{0 _s} .

Next, the circulation portion of the model was run with the CoA parameters, and r and h were calculated throughout the cardiac cycle from the ventricular volumes. As lower body arterial resistance and stressed blood volume began to differ from their somatic values, so did the loaded radii and thicknesses, r and h, and therefore so did the resulting strain values from Eqs. (19) and (20).

Hill-type functions, described in detail in Ref. [17], determined the rate of pathologic circumferential and radial ventricular growth (ΔF_g,c and ΔF_g,r , respectively) based on these deviations in circumferential and radial strains from their somatic values. The pathologic growth rate was governed by three parameters: the first parameter limited the maximum growth rate per day, the second dictated the slope of the sigmoid or the change in rate with strain difference, and the third defined a quiescent zone. These six parameters (three for the circumferential direction and three for the radial direction) were unchanged from the previous model of myocardial growth in adult canines [17,20]. Inputting the difference between the homeostatic strain and the pathologic strain into the Hill-type functions produced new values of F_g,c and F_g,r , the pathologic growth amounts in the circumferential and radial directions of the ventricle, respectively. At each time-step, F_g,c and F_g,r were then multiplied by the previous step's values of r ₀ and h ₀, respectively, to give the current r ₀ and h ₀ (see Fig. 5 for a comparison of somatic and pathologic unloaded radii and thicknesses). New values of the ventricular compartment parameters were then computed by rearranging Eqs. (14)–(17) and solving for A, B, Ees, and V ₀, respectively, for the current time-step. Pathology was assumed to only alter ventricular mass, therefore material parameters a, b, and e were held constant throughout the simulation at their somatic values. The circulation portion of the model was run until it reached steady-state, and the loaded LV and RV radii and thicknesses, r and h, were calculated throughout the cardiac cycle from the ventricular volumes. Growth proceeded iteratively until the maximum growth step of 1095 days was reached, at which point all hemodynamic and geometric outputs were plotted. The ability of the model to produce dimensions, namely, LV end-diastolic volume and thickness, was evaluated for model validation.

Fig. 5.

Top: comparison of unloaded radii for somatic and CoA cases (r _{0_ s} and r ₀, respectively). Bottom: comparison of unloaded thicknesses for somatic and CoA cases (h _{0_ s} and h ₀, respectively). Left-hand panels indicate the left ventricle, right-hand indicates the right ventricle. Unloaded radii and thicknesses were used to find loaded radii and thicknesses r and h, which were then used to compute strain at each time-step.

3 Results

The objective of this study was to modify a previously published computational model of left ventricular growth in adult canines for human infants. Homeostatic strain setpoints were fitted to capture somatic growth over time in median-weight infants. To test the maturation approach for parameter scaling, homeostatic strain setpoints were also fitted for infants in the 10th and 90th percentile of birth weight. Then, the ability of the model to predict pathologic growth in combination with somatic growth was tested by simulating coarctation of the aorta. Model volume and thickness outputs were validated against reported measurements for somatic and coarctation simulations.

3.1 Simulating Somatic Growth for the Median Infant.

Three years of growth were simulated for an infant with a median weight trajectory. This trajectory included a typical drop and regain of weight shortly after birth [45], resulting in a similar trend in model outputs. As shown in Fig. 6(a), simulated MAP closely matched reported measurements. Model MAP fell between the 10th and 90th percentiles [38], 5th and 95th percentiles [39,40], and within one standard deviation [35–37] of reported means throughout the entire three-year period. In these studies, MAP was measured using a blood pressure cuff, therefore comparable model pressures were from the upper body arteries compartment. Likewise, Fig. 6(b) shows the simulated MPAP was well within one standard deviation of the reported values throughout the entire three-year period. MPAP was measured in the main pulmonary artery by Doppler echocardiography [35,36] and cardiac catheterization [37], thus model pressures shown were from the main pulmonary artery compartment. Pulmonary and systemic systolic and diastolic pressures also matched reported trajectories (Figure S2, available in the Supplemental Materials on the ASME Digital Collection).

Fig. 6.

Simulated systemic (a) and pulmonary (b) mean arterial pressures throughout the first three years of life for a median weight infant compared to reported measurements [35–40]. In the studies referenced, MAP was measured using a blood pressure cuff, therefore comparable model pressures were from the upper body arteries compartment. MPAP was measured in the main pulmonary artery by Doppler echocardiography [35,36] and cardiac catheterization [37], thus model pressures shown were from the main pulmonary artery compartment. Measurements from Qi et al. were from the sea-level cohort only. Error bars on data indicate 10th and 90th percentiles [38], 5th and 95th percentiles [39,40], or one standard deviation [35–37].

Simulated chamber volumes agreed well with reported echocardiographic measurements. Maximum left atrial volume followed reported trends [46,47] and was well within the 10th and 90th percentiles reported by Linden et al. [48] (Figure S6, available in the Supplemental Materials). Model LV end-diastolic and end-systolic volumes (Figs. 7(a) and 7(b)) were within two standard deviations of reported measurements. Simulated RV end-diastolic volumes were also within two standard deviations [50] and 10th and 90th percentiles [51] of reported measurements (Fig. 7(c)) and simulated RV end-systolic volumes fell between measurements reported by Thilenius and Arcilla [52], Buechel et al. [53], and Lange et al. [54] (Fig. 7(d)).

Fig. 7.

Simulated left and right ventricular volumes at end diastole ((a) and (c), respectively) and end systole ((b) and (d), respectively) for the first three years of life for a median-weight infant compared to reported measurements [49–56]. Dashed lines indicate mean values for reported data. Dotted lines indicate ±two standard deviations [49,50], ±one standard deviation [52], 10th and 90th percentiles [51], or 95% confidence interval [53]. LV, left ventricle; RV, right ventricle; ED, end diastole; ES, end systole.

Simulated ventricular thicknesses also matched reported echocardiographic measurements. LV wall thickness at end diastole and end systole followed trends reported by Akiba et al. [55] and was within the 10th and 90th percentiles reported by Kampmann [57] (Figs. 8(a) and 8(b)). RV wall thickness at end diastole was within one standard deviation of measurements reported by Qi et al. [36] (Fig. 8(c)). No measurements of RV wall thickness at end systole were available for healthy infants below 3 yrs of age.

Fig. 8.

Simulated left and right ventricular thicknesses at end diastole ((a) and (c), respectively) and end systole ((b) and (d), respectively) for the first three years of life for a median-weight infant. Simulated left ventricular thicknesses are compared to posterior wall measurements and simulated right ventricular end diastolic thickness is compared to anterior wall measurements [36,55,57,58]. No data were available for right ventricular thicknesses at end systole during the first three years of life. Measurements from Qi et al. were from the sea-level cohort only. Dashed lines indicate mean values for reported data. Dotted lines indicate 10th and 90th percentiles [57] or 5th and 95th percentiles [58]. Error bars on data indicate ±one standard deviation [36].

3.2 Simulating Somatic Growth for Larger and Smaller Infants.

To validate the model for infants at weights other than the 50th percentile, the 10th and 90th percentile weight curves were also used to simulate somatic growth. Maturation constants for pulmonary and systemic vascular resistance were not refitted. Heart rates computed from the 10th and 90th percentile weight curves input into the model are shown in Figure S1(b) (available in the Supplemental Materials on the ASME Digital Collection). There was negligible difference in simulated MAP and MPAP (Figure S3, available in the Supplemental Materials) and in systemic and pulmonary systolic and diastolic pressures (not shown). Simulated end-diastolic LV volume and thickness increased with weight but were within the bounds of reported values (Fig. 9). Other chamber volumes and thicknesses were also within the bounds of reported values (Figures S4 and S5, available in the Supplemental Materials).

Fig. 9.

Comparison of 50th, 10th, and 90th weight percentile simulations for left ventricular (a) volume and (b) thickness at end diastole compared to bounds of reported measurements [36,49,50,57,58]. Measurements from Qi et al. were from the sea-level cohort only. Dotted lines indicate 10th and 90th percentiles [57], 5th and 95th percentiles [58], or ± two standard deviations [49,50]. Error bars on data indicate ± one standard deviation [36]. LV, left ventricle; ED, end diastole.

3.3 Simulating Coarctation of the Aorta.

In a simulation of CoA, the model predicted a substantial increase in end-diastolic LV thickness, comparable with the significant increase in both the end-diastolic LV posterior wall thickness and the interventricular septal thickness observed by Eerola et al. [42] (Fig. 10(a)). It also predicted an increase in LV end-diastolic volume comparable with their reported significant increase in LV end-diastolic diameter (Fig. 10(b)). An increase in systemic systolic and diastolic pressures in the upper body arteries (Figs. 10(c) and 10(d)) was also produced, consistent with observed increases measured using a blood pressure cuff [42,43].

Fig. 10.

Comparison of normal and coarctation simulations for (a) left ventricular thickness at end diastole, (b) left ventricular volume at end diastole, (c) systolic, and (d) diastolic systemic pressures compared to reported control and coarctation measurements [42,43]. Error bars on data indicate ± one standard deviation. LV, left ventricle; ED, end diastole; SBP, systolic blood pressure; DBP, diastolic blood pressure.

4 Discussion

Our goal was to create and implement a predictive cardiac growth model for normal healthy infants that can be further modified to simulate pathology. To accomplish this, somatic growth was uncoupled from pathologic growth. We simulated somatic growth by adapting the input parameters of a model of adult canine LV growth for human infants using a combination of allometric and maturation approaches. The ventricular dimensions and hemodynamics produced by the model matched reported measurements for healthy infants aged 0–3 yrs. Simultaneously, the model produced time-varying homeostatic strain setpoints later used to determine pathologic growth in the case of CoA. Our model's speed and flexibility allow it to be used as a tool to quickly analyze pathological mechanisms affecting cardiac growth and hemodynamics in growing infants, and its modular nature makes it a potential base for more sophisticated approaches aimed at clinical integration and translation.

4.1 Adjusting Model Parameters for Infants.

A major challenge in simulating the hemodynamics of infants and children arises from needing to change all parameters simultaneously to capture somatic growth, as dimensionalized parameters must change with body size. Furthermore, many model parameters (e.g., stressed blood volume, venous compliances, and resistances) are not measurable clinically and require optimization from other measurable quantities (e.g., end-diastolic volume). Obtaining sufficient clinical data to optimize all parameters for infants from 0 to 3 yrs of age over a range of weights is not possible. Therefore, in this study we used allometric scaling and a maturation approach to determine parameter sets for infants at different weights and ages. Allometric scaling relates a biological variable to body mass [28]. It has been widely used to translate computational models between species [30] and even to simulate somatic development [24,25]. For example, Pennati and Fumero created a computer model of fetal development from 20 to 40 weeks of gestation by assuming a set weight at each gestational age and fitting allometric scaling exponents for each vascular structure and organ [25]. A maturation function, used in anesthesiology for predicting drug clearance in pediatric patients, takes both weight and age into consideration when calculating output to account for differences between neonates and adults that cannot be explained solely by size [33,34]. In this study, the maturation approach enabled separation of postnatal deviations in resistance associated with birth from those occurring due to increasing size. Using this approach, the model produced reasonable values for chamber dimensions as well as arterial pressures for infants of different weights across a wide range of ages (See Fig. 9 and Figures S3-5, available in the Supplemental Materials on the ASME Digital Collection). While early values of MAP and MPAP in normal infants with mean body weight trajectories were initially necessary to fit maturation constants, these constants were then used successfully for infants with different weight trajectories and CoA. Thus, with these constants, weight and heart rate alone are the necessary inputs to the model.

4.2 Predicting Ventricular Dimensions.

We utilized alterations in homeostatic strains across time to simulate somatic heart growth (r _{0 _s} and h _{0 _s} in Fig. 5). The computed strain setpoints changed rapidly in the first 10 days, however, this variation was largely driven by age—when the 10th and 90th percentile weight trajectories were input along with the base fitted maturation constants, there was little change to the homeostatic setpoint trajectories. Therefore, this strategy is well-suited to a wide range of birth weights, indicating promise for future simulations of pediatric patients with complex CHDs, who tend to weigh less than their healthy peers. For example, using this approach, homeostatic setpoints could be established for an infant with normal anatomy but abnormally low weight trajectory. Then, a simulation including the anatomic features of the congenital defect could be run with the established setpoints. Resulting deviations in myocardial strain would predict pathologic changes in unloaded heart radius and thickness. In this way, we could uncouple deficiencies in overall growth (weight) and somatic ventricular development from pathologic ventricular hypertrophy.

Our model predictions of ventricular dimensional changes were within empirical ranges for healthy infants and those with CoA from 0 to 3 yrs of age. Notably, when capturing the pathologic growth associated with CoA, it was unnecessary to alter the growth parameters that relate the rates of pathologic circumferential and radial growth with deviations in circumferential and radial strain. Fitted homeostatic maximum circumferential strain was within the range of previously fitted values for adult canines (0.44 to 0.80) throughout the entire three-year period despite the change in species and age [17,20]. Fitted homeostatic maximum radial strain was within the range of previously fitted values (−0.11 to −0.01) throughout the entire three-year period for the LV and after day 3 for the RV. Thus, differences in strain produced reasonable growth rates regardless of their application to developing children or adult canines.

In all simulations the active and passive intrinsic material properties of the myocardium were constant at all times. Changes in these parameters can be prescribed easily if warranted. In a previous simulation of myocardial infarction, when the passive stiffness of the ventricle was increased at a rate matching increased collagen content, the difference between predicted and observed ventricular volumes narrowed [17]. Fibrosis is commonly observed clinically in patients with aortic stenosis [59–61] as well as in animal experiments of aortic banding [62–64]. However, the clinical studies on aortic coarctation referenced [42,43] did not comment on LV myocardial fibrosis. When we simulated a 20% increase in b, the parameter dictating the passive stiffness of the LV in Eq. (15), there were minimal changes in arterial pressure or LV end-diastolic thickness, however there was substantially less LV dilation (Figure S7, available in the Supplemental Materials on the ASME Digital Collection).

Our rationale for using CoA to model pathologic growth was twofold: first, CoA is a relatively simple defect in comparison to more complex CHDs, such as hypoplastic left heart syndrome or tetralogy of Fallot. Simulating it does not require extensive changes to our model, only an increase in lower body arterial resistance. Second, uncorrected CoA results in LV pressure overload, which we have successfully modeled previously in adult canines [17]. Pressure overload results in predictable and measurable changes in LV geometry, namely, increased LV thickness and larger end-diastolic diameters (and, relatedly, end-diastolic volume) than healthy controls [42]. Though the cause of this diameter increase is unclear for coarctation, in other types of hypertension presenting with reduced arterial compliance, vasodilation and congestion have been implicated (rather than ventricular growth and remodeling) for increased end-diastolic LV volume [65,66]. Thus, consistent with previous models, stressed blood volume was increased proportionally with increased arterial resistance to capture this change in hemodynamic compensation.

4.3 Limitations.

We assumed a simplified spherical geometry when representing the ventricles in our model. While finite element models can generate undoubtedly more accurate representations of the anatomy derived from high-resolution imaging, children rarely receive multiple cardiac MRIs or CTs, making it impractical to fit and validate spatially varying growth predictions and supporting a simpler analytic approach. Additionally, simulating growth is numerically intensive and current finite element models are already computationally expensive. Though advances in computing strategies are being developed to reduce computation time [67,68], current models often require hours to run a single cardiac cycle and thus weeks to simulate a month of growth, even when utilizing computing clusters [69]. In contrast, our matlab model predicts three years of growth in 40.5 min on a desktop computer with 16 GB RAM, a 64-bit operating system, and a 3.0 GHz Intel Core i7-9700 CPU. In our previous usages of the thin-walled spherical geometry, we successfully predicted relative changes in radius and wall thickness following pressure overload, volume overload, and myocardial infarction [17,20]. Others have also successfully simulated changes in ventricular dimensions due to both overload [70] and dyssynchrony [71,72] with similar geometries, implying a similar effect of alterations in ventricular pressure and volume changes on myocardial stress as in more complex geometries. It is possible more intricate geometries will be needed in the future to replicate more complex CHDs. In particular, the RV has a more irregular shape than the LV and has generally received less attention by the modeling community [73]. Thus, the spherical assumption may impact the predictions of RV growth under an RV-specific pathology such as hypoplastic left heart syndrome. In similar complex cases, however, lower order models have proven to be critical tools in determining boundary conditions, initial conditions, and parameter optimization for higher order models, suggesting continued utility of our approach [24].

As mentioned in Sec. 4.2, the growth parameters relating rates of pathologic growth to deviations in strain were unchanged with time or from previous simulations. Thus, we assumed that a myocyte's abilities to sense and respond to stretch do not depend on developmental stage or species. Situations where the inherent responsiveness of the myocytes to stretch is abnormal will likely require modified parameters. For example, they would likely need to be altered for patients taking pharmacologic agents known to modulate intracellular signaling (e.g., beta-blockers), or in the case of genetic abnormalities such as mutations in beta-cardiac myosin, which can cause hypercontractility [74]. A systems biology approach explicitly modeling cell signaling could potentially capture the detailed interactions driving stretch-responsiveness in the setting of medical treatment or genetic disorders.

In both healthy and pathologic simulations, circulatory parameters were prescribed based on weight via allometric scaling, weight and age via a maturation function (pulmonary and vascular resistance), or based on reported data (heart rate). Allometric scaling might not be appropriate for all of the input parameters. We suspect other parameters in our model that are difficult or impossible to measure might behave similarly to heart rate, varying from their scaled values somewhat in magnitude but generally following a similar trend. Pennati and Fumero [25] approached this issue by comparing model outputs to measured data and adjusting their scaling factors. Ultimately, a more mechanistic approach, such as that employed by Beard et al. [75] is likely necessary to capture long-term interactions between pathologic ventricular hypertrophy and circulatory hemodynamic compensation. This may be particularly relevant to simulate treatments involving medications, as they often affect these mechanisms (e.g., prostaglandin).

5 Conclusion

Treatment and management of pediatric cardiovascular pathology often involves estimating how hemodynamics and ventricular dimensions will change as a child grows. While computational modeling has become a powerful tool for clinical decision-making in adults, growth remains a barrier in adapting these approaches to children. In particular, models of the cardiovascular system typically require several parameters that are not measurable clinically and change with body size—even our simple model required over 30 parameters. Using a combined allometric scaling and maturation approach, we were able to estimate these parameters and their evolution for a range of weights. The final computational growth model presented here was able to capture the time-course of circulatory hemodynamics and ventricular hypertrophy in both healthy infants and those with aortic coarctation, a common congenital defect, from birth through three years of life with simulation times of less than one hour. In addition, the model was able to replicate the pathologic hypertrophic pattern associated with aortic coarctation, demonstrating that our previous approach for pathologic LV hypertrophy [17,20] was successful in this new context without reparameterization. In the future, our model could serve as a foundation for predictive approaches to customize surgical timing by enabling clinicians to project the future extent and rate of ventricular dilation and thickening for a range of congenital defects.

Supplementary Material

Supplementary Material

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Funding Data

• American Heart Association (Grant No. #20CDA35210754; Funder ID: 10.13039/100000968).

• National Science Foundation Graduate Research Fellowship Program (Funder ID: 10.13039/100000082).