# Blood Flow Simulations of the Renal Arteries - Effect of Segmentation and Stenosis Removal

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## Abstract

Patient specific based simulation of blood flows in arteries has been proposed as a future approach for better diagnostics and treatment of arterial diseases. The outcome of theoretical simulations strongly depends on the accuracy in describing the problem (the geometry, material properties of the artery and of the blood, flow conditions and the boundary conditions). In this study, the uncertainties associated with the approach for *a priori* assessment of reconstructive surgery of stenoted arteries are investigated. It is shown that strong curvature in the reconstructed artery leads to large spatial- and temporal-peaks in the wall shear-stress. Such peaks can be removed by appropriate reconstruction that also handles the post-stenotic dilatation of the artery. Moreover, it is shown that the effects of the segmentation approach can be equally important as the effects of using advanced rheological models. This fact has not been recognized in the literature up to this point, making patient specific simulations potentially less reliable.

## Keywords

Atherosclerotic indicators Stenosis Segmentation Non-Newtonian Hemodynamics CFD## 1 Introduction

Advances in medical imaging technologies have led to the possibility to study *patient specific* blood flow in more detail by utilizing computational fluid dynamics (CFD) models in geometries derived from medical images. The advantages of using patient specific data in order to assess the details of blood flow in the vessel of interest also enables assessment of the merits of different possible surgical measures. Computed Tomography Angiography (CTA) with improved spatial resolution has become an important tool in diagnosis of cardiovascular diseases. The diagnostic quality of the non-invasive CTA technique is now almost on par with invasive (catheter) angiography for detecting for example coronary artery diseases (c.f. [1, 2]). CTA enables measurements of not only the shape of the lumen of the vessel but also the thickness of its wall. However, the accuracy of such measurements depends on the accuracy of the segmentation method used to delineate the arterial wall. Significant differences in results among different algorithms were demonstrated in a large study comparing different segmentation algorithms, as reported by Kirişli et al. [3]. In the literature, there are several review papers describing the most common approaches for vessel lumen segmentation (c.f. [4, 5, 6, 7]).

The lumen segmentation of a blood vessel may act as a first step in several different applications such as better diagnosis of stenosis, evaluation of possible of surgery alternatives or, as in our case, reconstruction of the vessel to the shape prior to being affected by atherosclerosis. In all these cases, the required level of accuracy of the lumen segmentation may be different; the accuracy required for determining the presence of a stenosis is much lower as compared to the accuracy needed for a reconstruction that would resemble the shape of a healthy vessel.

Atherosclerosis is a disease that develops over a long period of time (decades). The process is slow and manifests in form of morphological changes in the arterial wall at several common locations (branches of certain arteries). A hypothesis that has been around for decades states that the blood flow in these locations is a major factor in the formation of stenotic lesions. It has also been hypothesized that the process is related to the Wall Shear Stress (WSS) [8]. Yet, it has been claimed that both high WSS [9] as well as low WSS [10] can play a role in the processes. In recent years, the temporal- and spatial gradients of the WSS has been recognized as more important as compared to the absolute value of the WSS itself [10]. Several measures for defining the accumulated effect of the WSS has been proposed (e.g. [11, 12, 13]). The most common criteria used nowadays are related to averages of the WSS. To this group of measures one may include the Time-Averaged WSS (TAWSS), the Oscillatory Shear Index (OSI) (c.f. [14, 15, 16]), and the Relative Residence Time (RRT) (cf. [17]). In order to assess such prognostic criteria for the development of atherosclerosis, the logical starting point is the affected artery as it looked like at an earlier time, i.e. at its healthy state. To achieve this, CTA data needs to be converted to a three-dimensional structure and thereafter “reconstructed” into the smooth shape that the vessel originally had. The main issues are how to reconstruct the vessel as well as the needed accuracy for the predictive purpose that we seek. The latter question can be expressed in terms of sensitivity of vessel pathology characterizing parameters, such as TAWSS, OSI and RRT to errors in the reconstruction. In recent years, patient specific geometries for numerical simulations has also been used where the effects of the blood rheology models were considered (c.f. [16]). Blood rheology depends, among other factors, primarily on the local hematocrit (Red Blood Cell, RBC, concentration) and the local shear-stress. The effects of rheological model are non-negligible and should be taken into account. In this work, it will be shown that the treatment of segmentation is equally important as the rheological approaches we applied.

In the following, the computational framework for arterial blood flow simulations is described where after methods for reconstructions are addressed. The sensitivity of flow parameters to the reconstruction is detailed in the result section.

## 2 Computational Methods

The computational methods used to analyze blood flow in the segmented arteries are presented below. First, the set of model equations are discussed, followed by the methods used to reconstruct the diseased artery into a healthy state.

### 2.1 Governing equations and boundary conditions

*α*).

*= (∇*

**γ****u**+ (∇

**u**)

^{T})/2 is the shear rate tensor,

*ρ*= (1 −

*α*)

*ρ*

_{plasma}+

*α*

*ρ*

_{rbc}the density of the mixture of plasma and RBCs,

*D*the diffusivity of the RBC phase, and

*μ*the mixture viscosity. Two different viscosity models are considered; a Newtonian blood analog fluid and a non-Newtonian Quemada model. For the Newtonian model, a constant value of

*μ*= 3.38 ⋅ 10

^{− 3}Pa⋅s is used. In the Quemada model [18, 19], the viscosity of blood depends on the local RBC concentration and the local shear rate. This viscosity model has the following form:

*μ*

_{plasma}= 1.32 ⋅ 10

^{− 3}Pa⋅s. The viscosity coefficients,

*k*

_{0}and \(k_{\infty }\), as well as the critical shear rate,

*γ*

_{c}, have been fitted to a large range of hematocrits and shear rates by Cokelet [20]:

*p*= 0) is imposed at the left and right renal artery outlets, allowing the flow to redistribute between the arteries. For the other outlets, the flow rate is set to a fraction of the instantaneous inflow. The fraction is 1/3 for the descending aorta, and 1/9 each for the superior mesenteric artery and the celiac artery. The flow rate in the different artery branches are depicted in Fig. 1b.

*α*

_{min}= 0.25, Δ

*α*= 0.25,

*a*= 4000/m,

*d*the wall distance and

*d*

_{min}= 0.8 mm. The resulting distribution is shown in Fig. 2b.

The walls of the arteries are assumed to be solid, hence the no-slip condition is imposed (**u** = **0**). This assumption is justified for atherosclerotic arteries. For the RBC concentration, no flux of cells into the wall is assumed. Hence, the wall-normal gradient of the hematocrit is assumed to vanish (i.e. **n**_{wall} ⋅∇*α* = 0). A zero-gradient condition is applied for the hematocrit at all outlets.

### 2.2 Flow solver

*t*is the timestep and

*ϕ*is the considered field. The time step is adapted throughout the simulations to keep the Courant number close to, but below 1. The time step size ranges between 10

^{− 4}and 2 ⋅ 10

^{− 6}s, where the smaller timestep is applied during peak systole for the stenoted geometry, and the higher during diastole for the non-stenoted case.

The discretized equations are solved sequentially, starting with the hematocrit (3), coupled to the flow equations through the density and viscosity fields. The momentum equation is solved component by component, after which three pressure correction steps with the PISO algorithm are performed in order for the velocity field to fulfill (1).

The initial value of the velocity and pressure fields are set to zero, whereas the hematocrit is *α* = 0.45. The simulations are carried out over a few heart cycles such that the effects of the initial conditions diminish. This is carried out by monitoring the cycle-to-cycle variation. Once the effect of the initial conditions is small, statistics of the relevant parameters is evaluated for one heart-cycle.

## 3 Vessel Reconstruction

A patient specific vessel geometry, including the descending aorta, the superior mesenteric artery, the celiac artery, and the renal arteries is considered. The left renal artery has a stenosis where the vessel area abruptly reduces by approximately 90%. Segmentation of CTA data was performed from which the vessel surface was extracted. The degree of smoothing applied to the geometry was varied in order to enable for the sensitivity to geometry perturbations to be assessed.

### 3.1 Stenosis removal

- 1.
Compute the centerline of the stenoted vessel

- 2.
Cut out the stenosis from the surface and the centerline

- 3.
Recompute the centerline in the removed part of the vessel

- 4.
Interpolate the surface along the reconstructed centerline

- 5.
Stitch together the meshes

### 3.2 Centerline computation

Following the work presented in [24], the centerline of a vessel is defined as the path between the centerpoints of the inlet and the outlet section that maximizes the distance to the boundary. The curves are obtained with the vmtk library [25], according to the algorithm proposed in the aforementioned reference.

### 3.3 Voronoi diagram based reconstruction

Two cutting points, placed upstream and downstream of the stenosis, are chosen, and the intermedial centerline points and Voronoi vertices are removed (Fig. 3b). The now missing part of the centerline is replaced by a cubic spline (Fig. 3c), used in order to guide an interpolation of the Voronoi diagram along the gap. For the interpolation, a set of Voronoi vertices close to each cutting point is chosen and vertex start-end pairs are formed. The associated sphere radius as well as distance to and rotation angle around the centerline is interpolated along the line (Fig. 3d). The interpolated Voronoi cells are mapped to a grid from which the surface is reconstructed through the Marching Cube algorithm [27] (Fig. 3e).

### 3.4 Mesh sweep reconstruction

**r**

_{1}(

*s*),

**r**

_{2}(

*s*), of the form,

*s*is 0 at the start point of the spline and 1 at the end, and

**a**

_{i},

**b**

_{i},

**c**

_{i}and

**d**

_{i}are constants.

**t**

^{F}, the normal,

**n**

^{F}, and the binormal,

**b**

^{F}. These are defined as

*s*.

**p**

_{0}and

**p**

_{2}) of the cutplanes used to remove the stenosis, and with tangent vectors coinciding with the plane normals (

**n**

_{0}and

**n**

_{2}). This gives four conditions for the spline constants:

**p**

_{1}) and the last four conditions can be derived by requiring the position, as well as the Frenet tangent and normal to be continuous at the knot point:

Once the corrected centerline has been computed (Fig. 4c), the non-stenoted geometry can be obtained by sweeping the mesh along the line. The only known information is the surface curve at the start and the end point. A transformation is thus required that smoothly morphs the surface between these curves. The surface curves are transformed into their local Frenet frame, and an optimal pairing of points is obtained by finding start-end point pairs minimizing the Frenet distance. The Frenet coordinates are interpolated linearly along the centerline, shown in Fig. 4d. A transformation to world coordinates gives the points on the surface (Fig. 4e). Triangular elements are created connecting the surface points, yielding the final reconstructed surface (Fig. 4f).

### 3.5 Vessel area evaluation

**p**

_{i}, and the tangent vector,

**t**

_{i}are known. The local area is defined as the area of the intersection between the vessel volume,

*V*

_{vessel}and a cutplane having an origin at

**p**

_{i}and normal

**t**

_{i}. If this plane is denoted

*P*

_{i}, the local area can be written as

**n**is the normal of the surface (here, the centerline tangent,

**t**

_{i}).

**F**⋅

**n**= 1, and Eq. 15 can be restated as,

The procedure applied to compute the vessel area distribution can be summarized as:

*i*:

- 1.
Find the intersection between the vessel surface and a plane with origin

**p**_{i}and normal**t**_{i}, producing an ordered list of points {**r**_{j}}_{j}making up the perimiter of the cutsurface. - 2.Compute the area by approximating the integral in Eq. 17 as$$ A_{i} \approx \frac{1}{2} \sum\limits_{j} \left( \mathbf{t}_{i} \times \frac{\mathbf{r}_{j}+\mathbf{r}_{j + 1}}{2} \right) \cdot (\mathbf{r}_{j + 1} - \mathbf{r}_{j}) $$(18)

## 4 Results

The effects of the methods used for the segmentation of the entire geometry and reconstruction of the left renal artery are considered by studying different combination of segmentation approaches and rheological models. Such a comparison provides insight into the relative importance and uncertainty associated with the treatment of the CTA data. Similarly, by comparing the effect of the segmentation/reconstruction and rheology on common parameters for characterizing risk for atherosclerosis, the possibility of using these parameters as risk for disease indicators can be assessed.

### 4.1 Atherosclerotic indicators

## 5 Summary

In order to be able to carry out reliable *patient specific* blood flow simulations of artery reconstruction, one has to be aware of the effects due to the segmentation method used for the original CTA images. The level of uncertainty due to segmentation techniques is of the same order as uncertainties and errors due to using rheological models neglecting temporal- and spatial-variations in the concentration of RBC. Commonly used parameters for characterizing atherosclerosis, such as the level of WSS, its temporal and/or spatial averages, or parameters as the OSI, RRT and the WSS streamwise gradient, are all sensitive to different degree to the segmentation parameters. Thus, these parameters should be used with caution when different segmentation methods are used.

## Notes

### Acknowledgements

Prof. Örjan Smedby at KTH, School of Technology and Health is gratefully acknowledged for providing the vessel geometry. The authors would like to thank Dr. Stevin van Wyk for his contribution during the initial stages of the project. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the High Performance Computing Center North (HPC2N).

### Compliance with Ethical Standards

### **Conflict of interests**

The authors declare that they have no conflict of interest.

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