Estimating external tissue support parameters with fluid–structure interaction models from 4D ultrasound of murine thoracic aortae

Abstract

Modeling of fluid–structure interactions (FSIs) between the deformable arterial wall and blood flow is necessary to obtain physiologically realistic computational models of cardiovascular systems. However, lack of information on the nature of contact between the outer vessel wall and surrounding tissue presents challenges in prescribing appropriate structural boundary conditions. Imaging techniques used to visualize wall deformation in vivo may be useful for estimating simulation parameters that capture the effects of both vascular composition and surrounding tissue support on the vessel wall displacement. Here, we present a method to calibrate external tissue support parameters in FSI simulations against four-dimensional ultrasound (4DUS) of the murine thoracic aortae. We collected ultrasound, blood pressure, and histological data from several mice infused with angiotensin II ($n=4$) and created a representative model of healthy and diseased (at 28 days post-angiotensin II infusion) murine aortae. We ran pulsatile FSI simulations after accounting for increased arterial wall stiffness with varying levels of tissue support, which demonstrated non-trivial variation in not only structural quantities, such as vessel wall deformation, but also hemodynamic quantities, such as wall shear stress across simulations. Furthermore, we compared simulation results with in vivo 4DUS imaging data and observed that the suitable range of the tissue support spring parameter was identical for both healthy and diseased states. This indicated that the same tissue support parameter estimates could be used for modeling the healthy and diseased states of the vessel, provided that changes in arterial wall stiffness had been considered. We anticipate this technique and the tissue support estimates reported herein will help inform computational models of blood flow and vasculature that incorporate the influence of external tissue.

Graphical abstract

Appendices

Appendix 1: Grid independence

To ensure that computational quantities reported, such as pressure, velocity and wall shear stress, were independent of the grid resolution of the fluid domain’s mesh, a grid sensitivity analysis was performed. The pertinent details for each mesh are shown in Table 6. A two-step approach was used to establish grid independence. First, a core mesh resolution was determined such that pressure and velocity were independent of the core mesh resolution. Second, varying degrees of mesh refinement close to the fluid–solid interface were implemented on top of the chosen core mesh resolution from the previous step, to ensure that the computed wall shear stress was independent of the near-wall mesh refinement resolution. In Fig. 6c–f, the area-averaged and point-wise pressure and velocity magnitude at the inlet plane, as well as at an arbitrary point located in the interior of the ascending region of the aorta (see Fig. 6a) are plotted over a single cardiac cycle.

Fig. 6

Pressure and velocity data over a cardiac cycle at the inlet plane (panels c and e) and at a point (panels d and f) in the interior of the ascending aorta (shown in a) for different mesh resolutions (shown in b). The error bars of each plot point show a deviation of 5% from the corresponding value on the finest mesh ($\Delta x = 0.008\,{{\hbox {cm}}}$). Abbreviations used—$\vert {\mathbf {v}}\vert$ Velocity magnitude, R Right, L Left, A Anterior, P Posterior. Based on the above plots, $\Delta x = 0.01\,{\hbox {cm}}$ was chosen as the optimal core mesh resolution

Based on the plots in Fig. 6c–f, we observed that the pressure and velocity magnitude values computed on both the coarse and medium grid (i.e. with $\Delta x = 0.015\,{\hbox {cm}}$ and $\Delta x = 0.01\,{\hbox {cm}}$) lie within a 5% margin of the values computed on the fine grid. However, in Fig. 6d, the velocity magnitude for the coarse grid ($\Delta x = 0.015\,{\hbox {cm}}$) lies beyond this tolerance margin. Therefore, $\Delta x = 0.01\,{\hbox {cm}}$ was determined to be the core mesh resolution of choice.

Next, Fig. 7 shows the x, y, and z components of the WSS (wall shear stress) computed at a point on the surface of the ascending aorta. Here, the core mesh resolution was identical in all cases ($\Delta x = 0.01\,{\hbox {cm}}$). However, close to the fluid–solid interface, different number of layers of mesh refinement (0, 3, 4, and 5) were considered (see Fig. 7a) From Fig. 7b–d, we observed a non-trivial difference ($> 5\%$) between the surface shear stress values computed on meshes with and without mesh refinement. Furthermore, meshes with different levels of mesh refinement ($N_{{\mathrm{BL}}}=3,4$, and 5) yield shear stress values within the above tolerance limit with minor differences in the computation time. Therefore, we proceeded with a mesh refinement level of $N_{{\mathrm{BL}}} = 4$, to balance the need for increased resolution with the corresponding computational cost.

Fig. 7

Components (b–d) of the WSS over a cardiac cycle at a point on the interior surface of the ascending aorta (shown by a dot in the model geometry in a), for different number of boundary layers each. $N_{{\mathrm{BL}}}$ represents the number of layers of boundary layer elements. Here, $N_{{\mathrm{BL}}} = 0$ represents a mesh without boundary layer refinement. The error bars on each plot point show a deviation of 5% from the corresponding value on the mesh with the largest number of boundary layer refinements (i.e. $N_{{\mathrm{BL}}} = 5$). Based on the above plots, the boundary layer mesh resolution corresponding to $N_{{\mathrm{BL}}} = 4$ was chosen as for the FSI simulations. Abbreviations used—R Right, L Left, S Superior, I Inferior

A constant time step of $\Delta t=10^{-5}\,{\hbox {s}}$ was used for all cases. Table 6 reports an estimate of the maximum cell-based Courant number computed for each of the meshes used, over a single cardiac cycle. The Courant number was computed as:

$$\begin{aligned} {{\mathrm{CFL}}} = \frac{\vert {\mathbf {v}}\vert \Delta t}{\Delta x}, \end{aligned}$$

(14)

where $\vert {\mathbf {v}}\vert$ is the velocity magnitude at the cell center, $\Delta t$ is the time step size, and $\Delta x$ is a length scale computed for each cell as $\Delta x = \mathcal {V}^{1/3}$, where $\mathcal {V}$ is the cell volume.

Table 6 Mesh details for grid optimization

We observed that, for cases for which the maximum ${{\mathrm{CFL}}} > 1$, only a few cells (${<} \,5$) outside the region of interest (viz. the ascending aorta) exceeded the threshold. This observation, together with the fact that the time integration scheme implemented in svFSI is an implicit scheme [45], allowed us to use the same time step size of $\Delta t=10^{-5}\,{\hbox {s}}$ for the subsequent FSI simulations as well.

Appendix 2: Material properties

The Young’s moduli for the Day 0 and 28 time points were estimated using circumferential stress-stretch data for wildtype C57BL/6J and AngII-infused apolipoprotein E$^{-/-}$ mice, respectively, as reported by Bellini et al. [35]. For a biaxial state of stress of an incompressible neo-Hookean material, the theoretical relationship between circumferential stress $\sigma _{\theta \theta }$ and circumferential stretch ratio $\lambda _{\theta \theta }$ is:

$$\begin{aligned} \sigma _{\theta \theta } = -p + \frac{E}{3}\lambda ^2_{\theta \theta }, \end{aligned}$$

(15)

where p is the Lagrange multiplier that enforces the incompressibility constraint. Therefore, using the biaxial stress-stretch data reported in [35], the Young’s modulus was estimated to be three times the slope of the best fit line to $\sigma _{\theta \theta }$ versus $\lambda _{\theta \theta }$ (see Fig. 8). The values are reported in Table 2.

Fig. 8

Experimental circumferential stress-stretch-squared data from Bellini et al. [35] along with best fit lines and corresponding best fit equations (with units implied). The Young’s modulus (in kPa) was estimated to be three times the fitted slope (colour figure online)

Appendix 3: Comparison of other cross-sections

This appendix provides plots of the effective diameter and non-overlapping area (see Sect. 4.1) at the other two cross-sections for the Day 0 and Day 28 time points (Figs. 9 and 10). Overall, the observations regarding these cross-sections are consistent with the data obtained for the cross-section reported in Sect. 4.1.

Fig. 9

Quantitative metrics comparing segmentations from 4DUS and FSI simulations for different values of k at peak systole. a, b Show the location of the cross-section being considered. Red squares in c, d show the plot of effective diameter of the cross-section, obtained from FSI simulations (calculated using Eq. (12)) as a function of tissue support parameter k. The solid red line represents the effective diameter of the same cross-section obtained from segmentations of 4DUS imaging data. e, f Show the variation of non-overlapping area at the cross-section, calculated using Eq. (13) as a function of the varying tissue support parameter k (colour figure online)

Fig. 10

Quantitative metrics comparing segmentations from 4DUS and FSI simulations for different values of k at peak systole. a, b Show the location of the cross-section being considered. Red squares in c, d show the plot of effective diameter of the cross-section, obtained from FSI simulations (calculated using Eq. (12)) as a function of tissue support parameter k. The solid red line represents the effective diameter of the same cross-section obtained from segmentations of 4DUS imaging data. e, f Show the variation of non-overlapping area at the cross-section, calculated using Eq. (13) as a function of the varying tissue support parameter k (colour figure online)