A design-based model of the aortic valve for fluid-structure interaction

Abstract

This paper presents a new method for modeling the mechanics of the aortic valve and simulates its interaction with blood. As much as possible, the model construction is based on first principles, but such that the model is consistent with experimental observations. We require that tension in the leaflets must support a pressure, then derive a system of partial differential equations governing its mechanical equilibrium. The solution to these differential equations is referred to as the predicted loaded configuration; it includes the loaded leaflet geometry, fiber orientations and tensions needed to support the prescribed load. From this configuration, we derive a reference configuration and constitutive law. In fluid-structure interaction simulations with the immersed boundary method, the model seals reliably under physiological pressures and opens freely over multiple cardiac cycles. Further, model closure is robust to extreme hypo- and hypertensive pressures. Then, exploiting the unique features of this model construction, we conduct experiments on reference configurations, constitutive laws and gross morphology. These experiments suggest the following conclusions: (1) The loaded geometry, tensions and tangent moduli primarily determine model function. (2) Alterations to the reference configuration have little effect if the predicted loaded configuration is identical. (3) The leaflets must have sufficiently nonlinear material response to function over a variety of pressures. (4) Valve performance is highly sensitive to free edge length and leaflet height. These conclusions suggest appropriate gross morphology and material properties for the design of prosthetic aortic valves. In future studies, our aortic valve modeling framework can be used with patient-specific models of vascular or cardiac flow.

Appendices

Appendix

A Pressure differences across periodic domains

Here, we show that prescribing a uniform body force on a periodic domain is equivalent to prescribing the value of pressure at the top and bottom of the domain via a change of variables. Let H denote the domain height and P(t) denote the desired pressure difference at time t. Define a body force

$$\begin{aligned} {\mathbf {d}} = \left( 0, 0, \frac{P(t) }{H} \right) . \end{aligned}$$

(35)

Consider a solution of the Navier Stokes momentum equation on a domain periodic in z subject to this body force with modified pressure $p_{mod}$.

$$\begin{aligned} \rho ( {\mathbf {u}}_{t} + {\mathbf {u}} \cdot \nabla {\mathbf {u}} )&= - \nabla p_{mod} + \mu \varDelta {\mathbf {u}} + {\mathbf {d}} . \end{aligned}$$

(36)

By periodicity, the pressure at the bottom of the domain $z_{min}$ and top of the domain $z_{max}$ are equal, or

$$\begin{aligned} p_{mod}(x,y,z_{min}) = p_{mod}(x,y,z_{max}) . \end{aligned}$$

(37)

Now consider the pressure p defined as

$$\begin{aligned} p = p_{mod} - \frac{P(t)}{H} z . \end{aligned}$$

(38)

The identical velocity field ${\mathbf {u}}$ as in Eq. (36) and the pressure field p then solve the Navier Stokes momentum equation on a nonperiodic domain

$$\begin{aligned} \rho ( {\mathbf {u}}_{t} + {\mathbf {u}} \cdot \nabla {\mathbf {u}} )&= - \nabla p + \mu \varDelta {\mathbf {u}} , \end{aligned}$$

(39)

and further

$$\begin{aligned} p(x,y,z_{min}) = p(x,y,z_{max}) + P(t) . \end{aligned}$$

(40)

Thus, the pressure p has the desired pressure difference across the domain.

B Convergence

Here, we discuss convergence and periodicity of the simulations. We show results at two resolutions. The fine fluid resolution is $\varDelta x_{fine} = 0.046875$ cm, and the fine structure resolution is targeted to half the fluid resolution, or $\varDelta s_{fine} = 0.023$ cm. The coarse fluid resolution is twice that of the fine resolution, or $\varDelta x_{coarse} = 0.09375$ cm, and the coarse structure resolution is targeted to $\varDelta s_{coarse} = 0.047$ cm. Resolution double the fine resolution is prohibitively expensive to run. Simulations are run for four cardiac cycles, rather than three as in the remainder of the paper.

Fig. 19

Pressure and flow rates at fine resolution (as used throughout the paper) and coarse resolution

Fig. 20

Slice view of the z component of velocity during closure and forward flow with fine and coarse resolution

Precise convergence in IB simulations is challenging to achieve for the following reasons, though we achieve some qualitative and quantitative agreement across cycles and resolutions. Since the IB method uses diffuse interfaces, the structure interacts with fluid points in the support of the discrete delta function, up to $2.5 \varDelta x$ away. This causes a decrease in the effective orifice area of the valve, even with the same configuration of the structure itself, and thus a slightly lower flow rate. Thus, the fluid/structure domain has a resolution-dependent resistance to forward flow. The aortic pressure is determined by the lumped parameter network, which in turn depends on the flow rate through the three-dimensional model. Precise, simultaneous periodicity of the ODE and three-dimensional flow dynamics is thus challenging to achieve. Additionally, the velocity field has physical instabilities since Reynolds numbers are much larger than one, so precise correspondence of flow fields from cycle to cycle is not expected.

The driving pressure and flow rates are shown in Fig. 19. After the first cardiac cycle, the fine resolution has aortic minimum and maximum pressures of 75–79 and 119–121 mmHg, respectively. The total flow per cycle ranges from 70.7 to 75.5 ml. Due to lack of dramatic changes in flow or pressure from the second through fourth cycles, all other simulations are stopped after three cycles and results are presented during the third cycle. The coarse resolution, due to increased resistance and lower flow rates driving the lumped parameter network, shows a decreasing trend in aortic pressure and a corresponding increase in flow. Thus, at fine resolution the results are closer to periodic in time than at coarse resolution.

The velocity field at fine and coarse resolution during two points in the cardiac cycle is shown in Fig. 20. The valve and flow fields during closure appear similar at both resolutions. The leaflets are closer together in the fine version, and about twice as far apart given the thicker discrete delta function in the coarse version. During forward flow, both flows show a strong jet. Vortices and flow structure appear in the fine resolution jet. The coarse resolution valve opens slightly less, and the effective thickening from the IB method is greater, so the coarse resolution jet is narrowed and has a more uniform appearance. Despite substantially more visible structure in the fine resolution version, both resolutions produce qualitatively similar flow fields.