Modeling hemodynamic forces in carotid artery based on local geometric features

Abstract

Hemodynamic wall shear stress (WSS) plays an important role in the initiation and progression of carotid atherosclerosis. This study aims at developing a technique to model WSS distribution based on point-wise geometric features that can be efficiently computed. Computational fluid dynamic analysis was performed for ten subjects. Surface curvatures, vascular radius, rate of change of radius along the longitudinal direction and standardized longitudinal/circumferential coordinates were computed on a point-wise basis for the arteries. Each of these point-wise geometric parameters was transformed to maximize the adjusted correlation coefficient. The transformed geometric parameters subsequently served as input variables of a multiple regression model. Multiple regression analysis revealed a significant relationship ($p<0.0001$) between WSS and three geometric parameters in internal and external carotid arteries (ICA and ECA). These three geometric parameters include vascular radius (ICA: $\beta = 0.50$, ECA: $\beta = 0.23$), standardized longitudinal/circumference coordinates (ICA: $\beta = 0.16$, ECA: $\beta = 0.27$) and Gaussian curvature (ICA: $\beta = -0.24$, ECA: $\beta = -0.19$). The results suggest that the proposed geometric parameters can serve as risk indicator in large-scale clinical studies aiming at elucidating the roles of local geometric risk of atherosclerosis.

Appendix: Relationship between patch-wise RMSE and point-wise RMSE

Suppose the carotid surface is divided into p patches, and each of the patches consists of $T/p = n$ points, where T is the total number of points on a carotid surface. With the following notations defined,

$d_{ij} \triangleq$ :

WSS error on point j of patch i

$d_{i,rms} \triangleq$ :

point-wise RMS deviation within patch i

$\overline{d_{i}} \triangleq$ :

average error in patch i

$\sigma _{d,i}^2 \triangleq$ :

variance of point-wise error in patch i

it can be shown that the square of the point-wise RMS difference is the sum of the square of patch-wise difference and the variance of the point-wise difference [5]:

$$\begin{aligned} d_{i, \mathrm{rms}}^2 = \overline{d_{i}}^2 + \sigma _{d,i}^2. \end{aligned}$$

(12)

The square of the point-wise RMSE over the whole carotid, denoted by $d_{\mathrm{rmse}}^2$, can be expressed as:

$$\begin{aligned} d_{\mathrm{rmse}}^2 = \frac{1}{T}\sum _{i=1}^{p}\sum _{j=1}^{n}d_{ij}^2 = \frac{n}{T} \sum _{i=1}^{p}d_{i, \mathrm{rms}}^2 = \frac{1}{p} \sum _{i=1}^{p}d_{i, \mathrm{rms}}^2 \end{aligned}$$

(13)

The second equality holds because $d_{i, \mathrm{rms}}^2 = (1/n)\sum _{j=1}^{n}d_{ij}^2$. The square of the patch-wise RMSE over the whole carotid, denoted by $P_{\mathrm{rmse}}^2$, can be expressed as:

$$\begin{aligned} P_{\mathrm{rmse}}^2 = \frac{1}{p} \sum _{i=1}^{p}\overline{d_{i}}^2 \end{aligned}$$

(14)

Rearranging Eqs. 13 and 14 yields

$$\begin{aligned} \begin{aligned} d_{\mathrm{rmse}}^2&= P_{\mathrm{rmse}}^2 + \frac{1}{p} \sum _{i=1}^{p} \left( d_{i, \mathrm{rms}}^2 -\overline{d_{i}}^2\right) \\&= P_{\mathrm{rmse}}^2 + \frac{1}{p} \sum _{i=1}^{p} \sigma _{d,i}^2. \qquad \text{ by } \text{ Eq. } 12 \end{aligned} \end{aligned}$$

(15)