The impact of blood viscosity modeling on computational fluid dynamic simulations of pediatric patients with Fontan circulation

Abstract

For univentricular heart patients, the Fontan circulation presents a unique pathophysiology due to chronic non-pulsatile low-shear-rate pulmonary blood flow, where non-Newtonian effects are likely substantial. This study evaluates the influence of non-Newtonian behavior of blood on fluid dynamics and energetic efficiency in pediatric patient-specific models of the Fontan circulation. We used immersed boundary-lattice Boltzmann method simulations to compare Newtonian and non-Newtonian viscosity models. The study included models from twenty patients exhibiting a low cardiac output state (cardiac index of 2 L/min/m²). We quantified metrics of energy loss (indexed power loss and viscous dissipation), non-Newtonian importance factors, and hepatic flow distribution. We observed significant differences in flow structure between Newtonian and non-Newtonian models. Specifically, the non-Newtonian simulations demonstrated significantly higher local and average viscosity, corresponding to a higher non-Newtonian importance factor and larger energy loss. Hepatic flow distribution was also significantly different in a subset of patients. These findings suggest that non-Newtonian behavior contributes to flow structure and energetic inefficiency in the low cardiac output state of the Fontan circulation.

I. INTRODUCTION

The Fontan procedure is the final stage of surgical palliation for patients born with a univentricular heart defect.¹ This procedure completes a total cavopulmonary connection (TCPC), where all systemic venous blood bypasses the heart and drains directly into the lungs.¹ The result is a non-physiologic circulation characterized by low-shear non-pulsatile pulmonary blood flow, central venous hypertension, and decreased cardiac output.²

Due to the abnormal hemodynamics of the Fontan circulation, patients commonly develop multiorgan dysfunction and a progressively increasing risk of Fontan circulation failure, requirement for heart transplant, or premature death.³ In an effort to decrease these complications, many researchers have used computational fluid dynamic (CFD) modeling to better understand the pathophysiology of the Fontan circulation,^4–6 test new surgical approaches,^7–10 and design novel devices to augment blood flow through the circulation.^11,12

The majority of prior CFD studies of the Fontan circulation have assumed blood to be a Newtonian fluid with constant viscosity.^{4,7–9,11,12} While a Newtonian viscosity model may be a suitable assumption in large systemic arteries where shear rates are sufficiently high (above 100 s⁻¹),^13,14 blood is most accurately modeled as a shear-thinning non-Newtonian fluid in which viscosity increases exponentially at shear rates less than 100 s⁻¹.^15–18 Since shear rates in the Fontan pulmonary vasculature fall in this range, it is likely that non-Newtonian effects on blood viscosity significantly influence fluid dynamics.

We previously demonstrated that non-Newtonian behavior is an important determinant of circulatory efficiency in simplified in vitro and in silico models of the Fontan circulation.^19–21 In these studies, we observed significant differences in qualitative flow patterns, particularly areas of flow stagnation and recirculation, between Newtonian and non-Newtonian viscosity models. When we evaluated a simplified vascular geometry consisting of straight cylinders both in vitro and in silico, measures of circulation efficiency (viscous dissipation and indexed power loss) were worse with non-Newtonian viscosity models. However, when we evaluated two patient-specific vascular geometries (in vitro only), we observed a complex relationship between efficiency metrics, viscosity model, vascular geometry, and cardiac output. Given these complex interactions and the significant heterogeneity in vascular geometry among patients with Fontan circulation, we sought to evaluate the impact of non-Newtonian behavior on Fontan circulation performance more extensively in a larger number of diverse patient-specific models.

The objective of this study was to evaluate the extent to which non-Newtonian behavior contributes to Fontan circulation performance in pediatric patient-specific models under a low cardiac output state. In this study, we used immersed boundary-Lattice Boltzmann (IB-LBM) based CFD coupled with a physiologically accurate non-Newtonian blood model to simulate the flow in patient-specific Fontan circulations. We hypothesized that non-Newtonian characteristics increase in blood viscosity at low shear rates would increase power loss and affect hepatic blood flow distribution to the lungs.

II. METHODS

A. Patient-specific geometry

Cardiac magnetic resonance imaging (MRI) scans were performed on a 1.5 T Philips Achieva system (Philips Healthcare, Best, The Netherlands) for clinical indications. Images of the thoracic vasculature and heart were obtained using a 3-dimensional steady state free precession sequence with respiratory navigator gating and electrocardiogram triggering. We used 3 D Slicer 5.6 (https://www.slicer.org/) to segment the Fontan pulmonary vasculature from the MRI images and create a patient-specific vasculature geometry of the vascular geometry. We then used MeshLab to refine the mesh. Figure 1 shows the workflow from MRI images to patient specific CFD models.

FIG. 1.

Workflow for creating CFD simulations from MRI images.

B. Numerical methods

We employed a computational approach using a Casson viscosity model combined with IB-LBM to accurately capture the non-Newtonian characteristics of blood flow.^20,22 LBM uses simplified kinetic equations combined with a modified molecular-dynamics approach to simulate fluid flows as an alternative method to conventional CFD methods that use Navier–Stokes equations. The accuracy and usefulness of LBM were demonstrated in various fluid dynamics problems including bio-inspired fluid-structure interactions,^23,24 porous media,²⁵ thermal flow,²⁶ non-Newtonian flow,^21,27 and hemodynamics.^28,29

1. 3D Lattice Boltzmann equations

We used a single-relaxation-time incompressible LBM formulation to obtain the pressure and flow fields.^30,31 In such a method, the synchronous motions of the particles on a regular lattice are enforced through a particle distribution function that inherently satisfies mass and momentum conservations. This distribution function also ensures that the fluid is Galilean invariant and isotropic.^30,31 The evolution of the distribution functions on the lattice is governed by the discrete Boltzmann equation with the Bhatnagar–Gross–Krook (BGK) collision model as

$$\begin{array}{c}{f}_i\left(\mathit{x}+{\mathit{e}}_i\Delta t,\hspace{0.17em}t+\Delta t\right)-f_i\left(\mathit{x},\hspace{0.17em}t\right)=-\frac{1}{\tau }\left[f_i\left(\mathit{x},\hspace{0.17em}t\right)-f_i^{eq}\left(\mathit{x},\hspace{0.17em}t\right)\right]+\Delta t{\mathit{F}}_{\mathit{i}},i=0,\dots ,N_0-1,\end{array}$$ (1)

where $f_i\left(x,\hspace{0.17em}t\right)$ is the distribution function for particles with velocity $e_i$ at position $x$ and time t and $F_i$ is the forcing term to couple the fluid and solid domains. $\Delta t$ and $\Delta x$ are the time step and lattice space, respectively, that are assigned as equal to each other in the simulations. $\tau$ is a dimensionless relaxation time constant, which is associated with fluid viscosity in the form of $\mu =\rho \vartheta =\rho c_s^2(\tau -\frac{1}{2})\Delta t$, where $\vartheta$ is the kinematic viscosity, $\rho$ is the fluid density and $c_s=\frac{1}{\sqrt{3}}$ is the lattice sound speed. In this model, a 19 discrete velocity vector stencil (D3Q19) is used; hence, $N_0=19.$^27,32 The local equilibrium distributions for the incompressible LBM and the forcing term are defined as $f_i^{eq}={\omega }_i\frac{p}{c_s^2}+\hspace{0.17em}{\omega }_i\rho \left[\frac{e_i·v}{{\hspace{0.17em}{c}_s}^2}+\frac{{\left(e_i·v\right)}^2}{{\hspace{0.17em}2c_s}^4}-\frac{v^2}{{\hspace{0.17em}2c_s}^2}\right]$, and ${\mathit{F}}_i=\left(1-\frac{1}{2\tau }\right){\omega }_i\left(\frac{e_i-v}{{\hspace{0.17em}{c}_s}^2}+\frac{e_i·v}{{c_s}^4}{e}_i\right){\mathit{b}}_{\mathit{f}}$, where ${\omega }_i$ is the weighting factor, ${\mathit{b}}_{\mathit{f}}$ is the force density at Eulerian grid, $p$ is the pressure, and $v$ is the velocity vectors. The macroscopic variables are calculated by $p=c_s^2{\sum }_i\hspace{0.17em}{f}_i$ and $\rho \mathit{v}=\frac{1}{2}{\mathit{b}}_{\mathit{f}}\Delta t+{\sum }_i{\mathit{e}}_i{f}_i$. We used an immersed boundary algorithm at the interface of the fluid and solid domains to obtain the force density, ${\mathit{b}}_{\mathit{f}}$, and bounce-back conditions at the rigid inlets and outlets.

2. Non-Newtonian fluid modeling

Non-Newtonian properties of blood were modeled by applying the modified Casson model to the fluid domain.^19,33–35 This model has been widely used to mimic the shear-thinning behavior in hemodynamic simulations.^19,33 In this study, the Casson parameters were obtained by curve-fitting to in vitro viscosity data from a historical cohort of 61 Fontan patients (Fig. 2).^19,20 The historical cohort was a mean age of 15.2 years (SD 5.3 years) and was 41% female.

FIG. 2.

Viscosity of shear-thinning Casson model based on patient-specific data. Red circles are mean viscosity measurements from 61 Fontan patients.

The apparent viscosity of the modified Casson model is given by

$$\mu \left(\dot{\gamma }\right)={\left(\sqrt{{\tau }_0\frac{\left(1-e^{-m\dot{\gamma }}\right)}{\dot{\gamma }}}+\sqrt{{\mu }_{\infty }}\right)}^2,$$ (2)

where $\dot{\gamma }$ is the shear rate, ${\tau }_0$ is the yield stress, and ${\mu }_{\infty }$ is the infinite-shear plateau viscosity. The model parameters were empirically determined from patient data as ${\mu }_{\infty }=3.149\hspace{0.17em}\mathit{\text{mPa}}·s,\hspace{0.17em}{\tau }_0=12.97\hspace{0.17em}\mathit{\text{mPa}},m=3.913\hspace{0.17em}s$.

The shear rate ( $\dot{\gamma }$) is computed from the second invariant of the rate-of-strain tensor ( $D_{II}$): $\dot{\gamma }=\sqrt{2D_{II}}$, where the $D_{II}$ is defined as $D_{II}={\sum }_{\alpha ,\beta =1}^3{S}_{\alpha \beta }{S}_{\alpha \beta }$. In LBM, the strain tensor $S_{\alpha \beta }$ can be calculated locally at each node using:³³ $S_{\alpha \beta }=-\frac{3}{2\tau }{\sum }_i\left(f_i-f_i^{eq}\right)e_{i\alpha }{e}_{i\beta }$. Thus, the shear strain is obtained locally without using the derivatives of the velocity.³⁶ For the baseline Newtonian cases, the dynamic viscosity is set to be the infinite shear viscosity of the Casson model: ${\mu =\mu }_{\infty }=3.149\hspace{0.17em}\mathit{\text{mPa}}·s$.

3. Fluid–structure interactions

For coupling the fluid and patient-specific structure systems, an explicit velocity correction-based IB method is used.^37,38 This method has been extensively applied to cardiovascular biomechanics problems to capture the fluid-solid interactions.^27,39–41 In this method, the body force, $f$, enforces the no-slip velocity boundary condition at the interface of the fluid and solid domains by introducing velocity correction, $\delta \mathit{v}$:

$$v\left(x,t\right)=v^{*}\left(x,t\right)+\delta v\left(x,t\right),$$ (3)

where the uncorrected velocity is ${\mathit{v}}^{*}=\frac{1}{\rho }{\sum }_i{\mathit{e}}_i{f}_i$.

In the velocity correction-based immersed boundary approach, $\delta v$ term at the Eulerian point (fluid domain) is obtained by the following Dirac delta interpolation function⁴² as

$$\delta \mathit{v}\left(\mathit{x},t\right)={\int }_{\Gamma }\delta \mathit{V}\left(s,t\right)\delta \left(\mathit{x}-\mathit{X}\left(s,t\right)\right)ds,$$ (4)

where $\delta (x-X(s,\hspace{0.17em}t))$ is smoothly approximated by a continuous kernel distribution and $\delta V(s,t)$ is the unknown velocity correction vector at every Lagrangian point s of the solid boundary.⁴² Note that $x$ denotes the Eulerian coordinates related to the fluid field, while $X$ stands for Lagrangian coordinates of the solid domain. In order to meet the non-slip boundary condition, the fluid velocity must be equal to the wall velocity $V$ at the same position that can be described as

$$\mathit{V}\left(s,t\right)={\int }_{\Omega }\mathit{v}\left(\mathit{x},t\right)\delta \left(\mathit{x}-\mathit{X}\left(s,t\right)\right)d\mathit{x}.$$ (5)

Substituting Eqs. (3) and (4) into Eq. (5), we can have the following equation:

$$\begin{array}{c}\mathit{V}\left(s,t\right)={\int }_{\Omega }{\mathit{v}}^{*}\left(\mathit{x},t\right)\delta \left(\mathit{x}-\mathit{X}\left(s,t\right)\right)d\mathit{x}\\ +{\int }_{\Omega }\left[{\int }_{\Gamma }\delta \mathit{V}\left(s,t\right)\delta \left(\mathit{x}-\mathit{X}\left(s,t\right)\right)ds\right]\delta \left(\mathit{x}-\mathit{X}\left(s,t\right)\right)d\mathit{x}.\end{array}$$ (6)

After obtaining $\delta \mathit{V}\left(s,t\right)$ via solving Eq. (6), the body force density, $f$, is obtained by using the following relations: $\delta \mathit{v}\left(x,t\right)=\frac{1}{2\rho }f\left(x,t\right)\delta t$. These IB equations are solved explicitly with the efficient boundary condition enforced method, whose details can be found elsewhere.⁴³

4. Concentration field

Dye concentration evolution $C(x,t)$ subjected to a velocity field $v(x,t)$ is governed by the homogeneous advection-diffusion equation,⁴⁴ i.e.,

$$\frac{\partial C}{\partial t}+v\cdot \nabla C-D{\nabla }^{2}C=0.$$ (7)

In this study, we used an additional distribution function, $g_i\left(x,\hspace{0.17em}t\right)$, in our LBM model to solve the advection-diffusion equation.²⁶ This distribution function is also governed by BGK collision model⁴⁵ with 19 discrete velocity vectors. The Boltzmann equation for the concentration field can be written as

$$\begin{array}{c}{g}_i\left(x+e_i\Delta t,\hspace{0.17em}t+\Delta t\right)-g_i\left(x,\hspace{0.17em}t\right)=-\frac{1}{{\tau }_c}\left[g_i\left(x,\hspace{0.17em}t\right)-g_i^{eq}\left(x,\hspace{0.17em}t\right)\right]+\Delta t{G}_i,i=1,\dots ,19.\end{array}$$ (8)

Here, ${\tau }_C$ is dimensionless relaxation time constant associated with diffusion coefficient, $D=c_s^2({\tau }_C-1/2)\Delta t$; the local equilibrium distribution $(g_i^{eq})$ and immersed boundary force term for the concentration field $(G_i$) can be written as⁴⁶ $g_i^{eq}={\omega }_{i}C\left[1+\frac{e_i·v}{{c_s}^2}+\frac{{\left(e_i·v\right)}^2}{2c_s^2}-\frac{v^2}{2c_s^2}\right]$, and $G_i=\left(1-\frac{1}{2{\tau }_c}\right){\omega }_i{b}_c$, where $b_c$ is the immersed boundary force for concentration field in Eulerian coordinates. Macroscopic dye concentration can be calculated by $\rho C\left(\mathit{x},t\right)=\frac{\Delta t}{2}{b}_c+{\sum }_{i=1}^{19}{g}_i\left(\mathit{x},t\right)$.

C. Simulation conditions

The mean cardiac index measured by MRI was 2.5 L/min/m² (interquartile range (IQR) 2–2.8 L/min/m²). In all simulations, we applied patient-specific flow rates by multiplying the patient-specific body surface area by a constant cardiac index of 2 L/min/m² to model a low cardiac output state ( $\mathit{\text{cardiac}}\hspace{0.17em}\mathit{\text{output}}=\mathit{\text{cardiac}}\hspace{0.17em}\mathit{\text{index}}\times \hspace{0.17em}\mathit{\text{body}}\hspace{0.17em}\mathit{\text{surface}}\hspace{0.17em}\mathit{\text{area}}$). Reynolds number (defined as $Re=\rho U_{\mathit{\text{ave}}}{D}_{\mathit{\text{ave}}}/{\mu }_{\infty }$, where $\rho$ is the blood density, $U_{\mathit{\text{ave}}}$ is the average inlet velocity, and $D_{\mathit{\text{ave}}}$ is the average inlet diameter) has been calculated for all patients using the cardiac index of 2 L/min/m². The Reynolds numbers for the patients range from 327 to 1119, with a median value of 596 and the first and third quartile values of 502 and 865, respectively. The values for each patient are provided in Supplementary Table 3. We assigned the flow distribution of the inlets as 30% from superior vena cava (upper body) and 70% from inferior vena cava (lower body). The inlet flow profile was assigned by applying a uniform velocity profile (plug flow) with an inlet flow extension (plug flow conditions at the inflow of the extension).^47,48 A pressure boundary condition (BC) is set at the outlets without imposing specific outflow conditions. In this study, we consider steady flow conditions within the Fontan circuit, as the specific characteristics of this circulation justify the assumption of steady-state flow.^47,48 However, previous work has explored the pressure BC implementation for a non-Newtonian model by a 1 D network of complex impedances to link outlet pressure to flow.⁴⁹ Such a study was performed for pulsatile flow conditions in coronary arteries. Given the steady flow nature of Fontan circuit, the outlet pressure BCs can be simplified using a linear relationship where the pressure is linked to outlet flow through pulmonary resistance. Specifically, the outlet pressures at the left and right pulmonary arteries (P_{out, LPA/RPA}) are calculated using the following relationship: $P_{\mathit{\text{out}},\hspace{0.17em}\mathit{\text{LPA}}/\mathit{\text{RPA}}}-P_0=R_{PA}·\hspace{0.17em}{Q}_{\mathit{\text{LPA}}/\mathit{\text{RPA}}},$ where the pulmonary vascular resistance (R_PA) is set to a constant value of 2 Wood Units · m², ensuring consistency in calculating the outlet pressures. These simulation parameters were kept consistent for both Newtonian and non-Newtonian fluid models in order to observe the isolated effect of non-Newtonian behavior on Fontan hemodynamics. A fluid mesh cardinality of $N_{\mathit{\text{mesh}}\hspace{0.17em}}=2000/ml$ was used for all simulations (corresponding to a total number of fluid mesh $N_{\mathit{\text{total}}\hspace{0.17em}}=3.4\times {10}^6$ for patient 1). The solid mesh size ( $\Delta s$) and fluid element size ( $\Delta x$) employed here satisfy the comparability requirement ( $\Delta s<2\Delta \mathrm{x}$) to ensure the stability of the fluid-solid coupling by immersed boundary method.⁴³ The patient-specific Fontan meshes were generated with open-source software, MeshLab.⁵⁰

D. Hemodynamic analysis and metrics

1. Power loss

We quantified power loss in the overall flow field by calculating the indexed power difference between the inlets and outlets,^20,48,51,52

$$\mathit{\text{iPL}}=\frac{{\sum }_{\mathit{\text{Inlets}}}Q\left(P+\frac{1}{2}\rho u^2\right)-{\sum }_{\mathit{\text{Outlets}}}Q\left(P+\frac{1}{2}\rho u^2\right)}{\rho Q^3/A^2},$$ (9)

where $\rho$ is the fluid density, Q is total inlet volume flow rate, and A is the average cross-sectional area of the inlets.

2. Viscous dissipation rate

We computed total viscous dissipation ( ${\Phi }_{VD}$) using

$$\begin{array}{c}{\Phi }_{VD}=\underset{V}{\iiint }2\mu \left[{\left(\frac{\partial u_x}{\partial x}\right)}^2+{\left(\frac{\partial u_y}{\partial y}\right)}^2+{\left(\frac{\partial u_z}{\partial z}\right)}^2\right]\\ +\mu \left[{\left(\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}\right)}^2+\mu {\left(\frac{\partial u_y}{\partial z}+\frac{\partial u_z}{\partial y}\right)}^2+{\mu \left(\frac{\partial u_z}{\partial x}+\frac{\partial u_x}{\partial z}\right)}^2\right]dV,\end{array}$$ (10)

where $V$ is the volume for the fluid domain. Accordingly, the indexed viscous dissipation was calculated as

$$i{\Phi }_{VD}=\frac{{\Phi }_{VD}}{\rho Q^3/A^2},$$ (11)

where Q is the total inlet volume flow rate and A is the average cross-sectional area of the outlets.

3. Non-Newtonian importance factor

We quantified the importance of shear-thinning effects using the non-Newtonian importance factor^21,53 to have an overall indication of non-Newtonian effects in the flow field.⁵⁴ This metric is defined as

$$I_L=\frac{\mu }{{\mu }_{\mathit{\text{Newtonian}}}}=\frac{\mu }{{\mu }_{\infty }}.$$ (12)

Global and average non-Newtonian importance factors were computed as

$$\overline{I_A}=\frac{{\sum }_N\left(\frac{\mu }{{\mu }_{\infty }}\right)}{N},\hspace{1em}\text{and}\hspace{1em}\overline{I_G}=\frac{1}{N}\frac{\sqrt{{\sum }_N{\left(\mu -{\mu }_{\infty }\right)}^2}}{{\mu }_{\infty }}\times 100,$$ (13)

where $N$ is the total element number in the fluid domain. Overall, $I_L$ highlights the areas where the non-Newtonian effects are more substantial. $\overline{I_A}$ and $\overline{I_G}$ shows the average and deviation from the Newtonian viscosity in the fluid domain.

4. Hepatic blood flow distribution

We calculated the hepatic blood flow (HBF) at the outlets (left and right pulmonary arteries) by tracking the concentration field with a threshold ( $C_0=0.001$) to avoid numerical error

$${\mathit{\text{HBF}}}_i={\int }_{\Omega \in \hspace{0.17em}\mathit{\text{outlet}}\hspace{0.17em}i}\left\{\begin{array}{l}u,\hspace{0.17em}C\ge 0.001\\ 0,\hspace{0.17em}\mathit{\text{otherwise}}\end{array}\right.dA,\hspace{1em}i=\mathit{\text{Left}},\mathit{\text{Right}}.$$ (14)

We calculated hepatic blood distribution (HFD) as the ratio of HBF at the left pulmonary artery (LPA) to the total HBF at the outlets^5,6

$$\mathit{\text{HFD}}=\frac{{\mathit{\text{HBF}}}_{\mathit{\text{Left}}}}{{\mathit{\text{HBF}}}_{\mathit{\text{Left}}}+{\mathit{\text{HBF}}}_{\mathit{\text{Right}}}}.$$ (15)

5. Statistical analysis

Categorical variables were summarized as counts and percentages. Continuous variables were summarized as mean with standard deviation for normally distributed variables or median with interquartile range for non-normally distributed variables. Normality was assessed by the Shapiro–Wilk test. We assessed differences in outcome measures between viscosity models using paired Student's t-test for normally distributed variables and Wilcoxon signed-rank test for non-normally distributed variables.

III. RESULTS

A. Demographics of Fontan patients

Twenty patients with Fontan circulation were included. MRI was performed at a mean age of 10.8 years (SD 4.4 years), which corresponded to a mean time of 6.1 years from the date of Fontan surgery (all non-fenestrated extracardiac Fontan). Patient demographics are shown in Table I.

TABLE I.

Characteristics of patient cohort. SD: standard deviation, IQR: interquartile range.

Number of patients	20
Age, years (SD)	10.8 (4.4)
Female, n (%)	7 (35%)
Body surface area, m2 (IQR)	1.12 (0.88, 1.54)
Race, n (%)
Hispanic	15 (75%)
White	3 (15%)
Asian	1 (5%)
Unknown	1 (5%)
Cardiac Diagnosis, n (%)
Hypoplastic left heart syndrome	7 (35%)
Double outlet right ventricle	3 (15%)
Atrioventricular septal defect	3 (15%)
Double inlet left ventricle	2 (10%)
Other	4 (25%)
Extracardiac Fontan, n (%)	20 (100%)

B. Non-Newtonian effects on flow patterns and energy loss

1. Flow structures

Flow structures of four patients with diverse vascular geometries are presented in Fig. 3. The main differences between viscosity models on the flow structures are observed in the TCPC region (denoted by red dashed boxes). In this low shear region, non-Newtonian viscosity leads to lower velocity values, decreased flow rotation, and more stagnant flow.

FIG. 3.

Streamlines and velocity magnitudes in four sample Fontan patients. Non-Newtonian models demonstrate lower velocity values, decreased flow rotation, and more stagnant flow in the TCPC region (dashed red boxes). SVC: superior vena cava, IVC: inferior vena cava, RPA: right pulmonary artery, LPA: left pulmonary artery.

2. Non-Newtonian importance factor

The local non-Newtonian importance factors for the four patients are shown in Fig. 4 only for the non-Newtonian cases [this value is equal to 1 for Newtonian simulations, cf. Eq. (13)]. The mean $I_A$ was 9.06 (standard deviation 2.88, range 4.18–13.18), and mean $I_G$ was 2.95 (standard deviation 1.79, range 1.13–5.22). Thus, the mean effective viscosity was nearly tenfold higher for the non-Newtonian simulations compared to Newtonian simulations.

FIG. 4.

Patient-specific variation in local non-Newtonian importance factor (note Patients 4 and 15 have a different color scale since the values are larger). SVC: superior vena cava, IVC: inferior vena cava, RPA: right pulmonary artery, LPA: left pulmonary artery.

3. Power loss and viscous dissipation rate

Figure 5(a) shows the indexed power loss for both Newtonian and non-Newtonian viscosity models. We observed a statistically significant increase in power loss with non-Newtonian models vs Newtonian models (median 1.96 [IQR 0.87, 3.98] vs 1.30 [IQR 0.47, 2.84], p < 0.0001). Indexed viscous dissipation rate for each case is shown in Fig. 5(b). Non-Newtonian models yielded significantly higher viscous dissipation rate vs Newtonian models (median 0.72 [IQR 0.38, 1.45] vs 0.38 [IQR 0.25. 0.81], p < 0.0001).

FIG. 5.

(a) Comparison of indexed power loss between Newtonian and non-Newtonian viscosity models. (b) Comparison of indexed viscous dissipation rate between Newtonian and non-Newtonian viscosity models. Patient-level differences are denoted with a black line.

C. Non-Newtonian effects on wall shear stress (WSS) and low WSS area

We computed the wall shear stress (WSS, ${\tau }_w=\mu \left({\dot{\gamma }}_{\mathit{\text{Wall}}}\right)·{\dot{\gamma }}_{\mathit{\text{Wall}}}$) distribution for all patients with both Newtonian and non-Newtonian models (see supplementary material Fig. 3). To quantify the changes in WSS with the viscosity model, we calculated the mean WSS across the total vascular geometry^47,55 and the low WSS area, defined as the proportion of wall area with WSS less than 0.4 Pa ( $A_{\mathit{\text{WSS}}<0.4Pa}/A_{\mathit{\text{Total}}}$).⁴⁷ Figure 6(a) shows the mean WSS for both Newtonian and non-Newtonian models. Mean WSS was significantly higher with the non-Newtonian models vs Newtonian models (median 3.19 [IQR 2.48 3.57] vs 1.26 [IQR 1.04 1.54], p < 0.0001). Figure 6(b) shows the low WSS area for each case. On average the non-Newtonian models had smaller low WSS area vs Newtonian models (median 1.89 [IQR 0.44 7.22] vs 15.52 [IQR 7.86 24.29], p < 0.0001). The mean shear rate (MSR) of the whole Fontan fluid domain (reported in Supplementary Table 2) ranged from 2.28 to 14.26 s⁻¹ (median MSR = 5.13 Std = 3.01) for Newtonian models, and from 1.59 to 11.22 s⁻¹ (median MSR = 3.81 Std = 2.35) for non-Newtonian models.

FIG. 6.

(a) Comparison of mean wall shear stress (WSS) between Newtonian and non-Newtonian viscosity models. (b) Comparison of low WSS area (as a percentage of total Fontan area) between Newtonian and non-Newtonian viscosity models. Patient-level differences are denoted with a black line.

D. Non-Newtonian effects on quantification of hepatic blood flow distribution

Figure 7 shows dye simulations and HFD in four diverse patients. When considering all cases together, there was not an overall statistically significant difference in HFD between Newtonian and non-Newtonian models (54.2% $\pm$ 17.3% vs 55.1% $\pm$ 18.4%, p = 0.36). However, we observed that patient geometry has non-negligible effects on HFD. As shown in Fig. 8, both the magnitude of difference between Newtonian and non-Newtonian models and which viscosity model yielded higher HFD was highly variable between patients.

FIG. 7.

Dye simulations for calculation of hepatic blood flow distribution (HFD). Dye was tracked from IVC to RPA and LPA. SVC: superior vena cava, IVC: inferior vena cava, RPA: right pulmonary artery, LPA: left pulmonary artery.

FIG. 8.

(a) Comparison of hepatic flow distribution between Newtonian and non-Newtonian fluid models for each patient. (b) Difference in hepatic flow distribution between Newtonian and non-Newtonian fluid models.

IV. DISCUSSION

Our study delves into the critical area of Fontan failure risk assessment through LBM based CFD, specifically focusing on the impact of non-Newtonian effects on hemodynamics. The results imply that there is a significant impact of blood viscosity models on Fontan hemodynamics, indicating that the Newtonian assumption might introduce considerably large errors into performance metrics. Importantly, we observed substantial variation between different vascular geometries, emphasizing the patient-specific nature of non-Newtonian effects. Although beyond the scope of this current work, it is also possible that patient-specific differences in blood viscosity could introduce additional variation in performance metrics.^56,57

Across all cases, substantial differences in blood flow structure, particularly near the center TCPC region, were clearly present and varied significantly between patients. For example, in patient 3 (Fig. 3), the streamlines became flatter for the non-Newtonian model and impacted a circular helical structure in the TCPC region. In contrast, non-Newtonian simulations for patient 4 revealed a larger stagnation area, corresponding to high local viscosity/non-Newtonian importance factors. The patient-specific effects imply that differences are not easily predictable from a Newtonian model with any simple corrections.

The non-Newtonian simulations consistently exhibited significantly higher viscosity compared to Newtonian models. In particular, the center TCPC region exhibited higher viscosity (because of low shear), corresponding to an area of stagnation. High global non-Newtonian importance factors were observed with a substantial average and global deviation, indicating that the Newtonian assumption significantly underestimates blood viscosity. These differences varied widely between different patient geometries.

Both indexed power loss, representing the energy difference between the inlets and outlets, and the indexed viscous dissipation rate, representing global energy loss, showed a substantial increase when shear-thinning effects were considered. This increase can be attributed to differences in flow and pressure distribution with the non-Newtonian viscosity model, as well as viscosity changes in non-Newtonian conditions.

In our simulations, the median shear rate was ∼5 s⁻¹ for Newtonian models and ∼4 s⁻¹ for non-Newtonian models. These shear rates fall within the range where viscosity is expected to exceed a critical threshold ( $\mu >{\mu }_{\mathit{\text{critical}}}=3{\mu }_{\infty },\hspace{0.17em}\mathit{\text{where}}\hspace{0.17em}\mathit{\text{shear}}\hspace{0.17em}\mathit{\text{rate}}\hspace{0.17em}\dot{\gamma }<7.69\hspace{0.17em}{s}^{-1}$) where non-Newtonian effects are expected to be substantial as defined by.^58–60 Therefore, the mean shear rate serves as additional evidence indicating that the non-Newtonian blood model has considerable effects in Fontan circulation hemodynamics. This finding aligns well with prior conclusions asserting the importance of the non-Newtonian effects in venous systems, particularly those characterized by lower flow pulsatility and velocities.⁵⁹

We investigated the impact of a non-Newtonian viscosity model on WSS, a clinically relevant hemodynamic parameter,^47,55,61,62 and low WSS area, a metric commonly reported in previous studies that is associated with atherosclerosis and thrombosis risk.^47,62 Given that the Fontan circulation exhibits lower flow pulsatility and velocity compared to the normal pulmonary circulation, our analysis revealed larger WSS values with non-Newtonian cases [as shown in Supplementary Fig. 2 and Fig. 7(a)], which correlated well with the global non-Newtonian importance factor $I_G$. This observation is consistent with previous research, suggesting that increased viscosity in non-Newtonian cases is significantly greater, consequently leading to elevated WSS, despite lower velocity gradients.⁴⁷ Interestingly, both mean WSS and low WSS area calculations demonstrated notable patient-specific variations [Figs. 7(a) and 7(b)].

Differences in HFD between Newtonian and non-Newtonian simulations also varied significantly between patients. Furthermore, neither viscosity model uniformly predicted higher HFD, again highlighting that differences are not easily predictable from a Newtonian model with simple corrections.

In summary, this study highlights the limitations of the Newtonian blood model in CFD simulations of the Fontan circulation and argues that non-Newtonian viscosity must be considered to accurately assess hemodynamics. Further investigation is needed to determine the extent of correlation between these and clinical outcomes.

V. LIMITATIONS

The scope of flow conditions analyzed in this study does not encompass the entirety of potential scenarios. We also did not account for the effects of respiration on pulmonary blood flow or the effects of vascular compliance. Further limitations of our study are simplifications of the simulation conditions, such as a constant cardiac index, rigid solid wall, and a simplified inlet profile. Although the plug profile with extension is an established assumption in patient-specific simulations,^48,62 the impact of such a simplification on the hemodynamic quantities may be significant in some scenarios.⁶³ Future investigations should aim to incorporate more dynamic and varying flow conditions by incorporating patient-specific flow rate waveform and profiles, consider non-rigid body dynamics, and thoroughly explore the influences of respiration to provide a more comprehensive understanding of the subject matter. Since the patients included in this study did not have blood viscosity measurements, we instead used average viscosity measurements from a historical cohort of Fontan patients who were on average 4 years older than the patients in this study. Among the historical cohort, there was a moderate correlation between viscosity and age. Therefore, the differences we detected in this study between the Newtonian and non-Newtonian simulations might have been slightly larger than what would have been measured had we been able to use patient-specific viscosity measurements. Although it is possible to estimate blood viscosity from other physiological parameters, such as hematocrit and fibrinogen,⁵⁶ we did not have these data for the patients included in this study.

VI. CONCLUSIONS

Our investigation illuminated significant variances in flow structures when comparing Newtonian and non-Newtonian models. The study demonstrates that non-Newtonian simulations consistently exhibit significantly higher viscosity compared to Newtonian models with potential implications for thrombosis in certain patients. Non-Newtonian effects also manifested in elevated power loss, indicating that using the conventional Newtonian blood assumption might lead to a significant underestimation of both power loss and viscous dissipation and, consequently, overestimation of exercise capacity. Furthermore, non-Newtonian behavior influenced hepatic blood flow distribution in a patient-specific manner. This comprehensive exploration underscores the imperative consideration of non-Newtonian effects in the optimization of the Fontan circulation.

In essence, our research substantiates the crucial need to transcend the confines of Newtonian assumptions, emphasizing the necessity for a more accurate approach that integrates non-Newtonian effects. Incorporating these factors into the optimization strategies for the Fontan procedure stands to significantly improve patient outcomes and refine our understanding of hemodynamics in complex circulatory systems.

SUPPLEMENTARY MATERIAL

See the supplementary material for Supplementary Table 1: Summary of hemodynamic metrics for all patients; Supplementary Fig. 1; Flow structures (streamlines and velocity magnitudes) for patients not shown in manuscript; Supplementary Fig. 2: Contours showing wall shear stress for Newtonian and non-Newtonian viscosity models and the difference in wall shear stress between the two viscosity models; Supplementary Table 2: Summary of mean wall shear stress (WSS), mean shear rate, and low WSS area for all patients; Supplementary Fig. 3: Fluid mesh and the Fontan geometry (patient 1); Supplementary Table 3: Reynolds number for all patients.

Ethics Approval

The study was approved by the Institutional Review Board (IRB) of Children's Hospital Los Angeles (CHLA-19–00385). The IRB determined that the study was exempt from the requirement for informed consent per federal regulations (45 CFR 46). Research was conducted in accordance with the Declaration of Helsinki.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Heng Wei: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Coskun Bilgi: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). Kellie Cao: Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). Jon A. Detterich: Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). Niema M. Pahlevan: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Writing – original draft (equal); Writing – review & editing (equal). Andrew L. Cheng: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.