# Aorta zero-stress state modeling with T-spline discretization

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## Abstract

The image-based arterial geometries used in patient-specific arterial fluid–structure interaction (FSI) computations, such as aorta FSI computations, do not come from the zero-stress state (ZSS) of the artery. We propose a method for estimating the ZSS required in the computations. Our estimate is based on T-spline discretization of the arterial wall and is in the form of integration-point-based ZSS (IPBZSS). The method has two main components. (1) An iterative method, which starts with a calculated initial guess, is used for computing the IPBZSS such that when a given pressure load is applied, the image-based target shape is matched. (2) A method, which is based on the shell model of the artery, is used for calculating the initial guess. The T-spline discretization enables dealing with complex arterial geometries, such as an aorta model with branches, while retaining the desirable features of isogeometric discretization. With higher-order basis functions of the isogeometric discretization, we may be able to achieve a similar level of accuracy as with the linear basis functions, but using larger-size and much fewer elements. In addition, the higher-order basis functions allow representation of more complex shapes within an element. The IPBZSS is a convenient representation of the ZSS because with isogeometric discretization, especially with T-spline discretization, specifying conditions at integration points is more straightforward than imposing conditions on control points. Calculating the initial guess based on the shell model of the artery results in a more realistic initial guess. To show how the new ZSS estimation method performs, we first present 3D test computations with a Y-shaped tube. Then we show a 3D computation where the target geometry is coming from medical image of a human aorta, and we include the branches in our model.

## Keywords

Patient-specific arterial FSI Image-based geometry Aorta Zero-stress state Isogeometric wall discretization T-spline basis functions Integration-point-based zero-stress state Shell-model-based initial guess## 1 Introduction

The patient-specific arterial fluid–structure interaction (FSI) computations reported in [1, 2, 3, 4] were among the earliest of its kind. The core method in these computations was the early version of the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) method [5, 6], which is now called the “ST-SUPS.” The acronym “SUPS” indicates the stabilization components, the Streamline-Upwind/Petrov–Galerkin (SUPG) [7] and Pressure-Stabilizing/Petrov–Galerkin (PSPG) [5] stabilizations.

The ST computations have been only a small part of the large number of cardiovascular fluid mechanics and FSI computations seen in the last 15 years (see, for example, [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]), with the Arbitrary Lagrangian–Eulerian (ALE) method [30] having the largest share in the computations reported. Still, a large number of computations with the ST methods were also reported in the last 15 years. In the first 8 years of that period the ST computations were for FSI of abdominal aorta [31], carotid artery [31] and cerebral aneurysms [32, 33, 34, 35, 36, 37, 38]. In the last 7 years, the ST computations focused on even more challenging aspects of cardiovascular fluid mechanics and FSI, including comparative studies of cerebral aneurysms [39, 40], stent treatment of cerebral aneurysms [41, 42, 43, 44, 45], heart valve flow computation [46, 47, 48, 49, 50, 51], aorta flow analysis [51, 52, 53, 54], and coronary arterial dynamics [55]. The large number of computational challenges encountered were addressed by the advances in core methods for moving boundaries and interfaces (MBI) and FSI (see, for example, [20, 21, 40, 46, 47, 49, 50, 56, 57, 58, 59, 60, 61, 62, 63, 64] and references therein) and in special methods targeting cardiovascular MBI and FSI (see, for example, [20, 38, 44, 45, 48, 49, 50, 51, 54, 65] and references therein).

A challenge very specific to patient-specific arterial FSI computations, such as patient-specific aorta FSI computations, is how to use the image-based arterial geometry. The image-based geometry does not come from the zero-stress state (ZSS) of the artery. Special methods targeting cardiovascular MBI and FSI include those designed to account for that. The attempt to find a ZSS for the artery in the FSI computation was first made in a 2007 conference paper [66], where the concept of estimated zero-pressure (EZP) arterial geometry was introduced. The method introduced in [66] for calculating an EZP geometry was also included in a 2008 journal paper on ST arterial FSI methods [32], as “a rudimentary technique” for addressing the issue. It was pointed out in [32, 66] that quite often the image-based geometries were used as arterial geometries corresponding to zero blood pressure, and that it would be more realistic to use the image-based geometry as the arterial geometry corresponding to the time-averaged value of the blood pressure. Given the arterial geometry at the time-averaged pressure value, an estimated arterial geometry corresponding to zero blood pressure needed to be built. Special methods developed to address the issue include the newer EZP versions [20, 34, 37, 38, 65] and the prestress technique introduced in [16], which was refined in[18] and presented also in [20, 65].

We introduced in [67] a method for estimation of the element-based ZSS (EBZSS) in the context of finite element discretization of the arterial wall. The method has three main components. (1) An iterative method, which starts with a calculated initial guess, is used for computing the EBZSS such that when a given pressure load is applied, the image-based target shape is matched. (2) A method for straight-tube segments is used for computing the EBZSS so that we match the given diameter and longitudinal stretch in the target configuration and the “opening angle.” (3) An element-based mapping between the artery and straight-tube is extracted from the mapping between the artery and straight-tube segments. This provides the mapping from the arterial configuration to the straight-tube configuration, and from the estimated EBZSS of the straight-tube configuration back to the arterial configuration, to be used as the initial guess for the iterative method that matches the image-based target shape. Test computations with the method were also presented in [67] for straight-tube configurations with single and three layers, and for a curved-tube configuration with single layer. The method was used also in [55] in coronary arterial dynamics computations with medical-image-based time-dependent anatomical models.

In [68], we introduced the version of the EBZSS estimation method with isogeometric wall discretization, using NURBS basis functions. With isogeometric discretization, we can obtain the element-based mapping directly, instead of extracting it from the mapping between the artery and straight-tube segments. That is because all we need for the element-based mapping, including the curvatures, can be obtained within an element. With NURBS basis functions, we may be able to achieve a similar level of accuracy as with the linear basis functions, but using larger-size and much fewer elements, and the NURBS basis functions allow representation of more complex shapes within an element. The 2D test computations with straight-tube configurations presented in [68] showed how the EBZSS estimation method with NURBS discretization works. In [69], which is an expanded, journal version of [68], we also showed how the method can be used in a 3D computation where the target geometry is coming from medical image of a human aorta.

In this article we are introducing a new method for estimating the ZSS. The estimate is based on T-spline discretization of the arterial wall and is in the form of integration-point-based ZSS (IPBZSS). The method has two main components. (1) An iterative method, which starts with a calculated initial guess, is used for computing the IPBZSS such that when a given pressure load is applied, the image-based target shape is matched. (2) A method, which is based on the shell model of the artery, is used for calculating the initial guess. The T-spline discretization enables dealing with complex arterial geometries, such as an aorta model with branches, while retaining the desirable features of isogeometric discretization. The IPBZSS is a convenient representation of the ZSS because with isogeometric discretization, especially with T-spline discretization, specifying conditions at integration points is more straightforward than imposing conditions on control points. Calculating the initial guess based on the shell model of the artery results in a more realistic initial guess. To show how the new method for estimating the ZSS performs, we first present 3D test computations with a Y-shaped tube. Then we show a 3D computation where the target geometry is coming from medical image of a human aorta, and we include the branches in our model.

In Sect. 2, we describe the Element-Based Total Lagrangian (EBTL) method, including the EBZSS and IPBZSS concepts. How the initial guess is calculated based on the shell model of the artery is described in Sect. 3. The numerical examples are given in Sect. 4, and the concluding remarks in Sect. 5.

## 2 EBTL method

In this section we provide an overview of the EBTL method [67], including the EBZSS concept, and describe the IPBZSS concept and the conversion between the two ZSS.

### 2.1 EBZSS

*e*. Positions of nodes from different elements mapping to the same node in the mesh do not have to be the same. In the reference state, \(\mathbf {X}_\mathrm {REF}\), all elements are connected by nodes, and we measure the displacement \(\mathbf {y}\) from that connected state. The implementation of the method is simple. The deformation gradient tensor \(\mathbf {F}\) is evaluated for each element:

### 2.2 IPBZSS

### 2.3 EBZSS to IPBZSS

### 2.4 IPBZSS to EBZSS

*A*,

*B*and

*C*. We set all three components of \(\mathbf {y}_A\) to be zero and constrain \(\mathbf {y}_B\) to be in the direction \((\mathbf {X}_\mathrm {REF})_B - (\mathbf {X}_\mathrm {REF})_A\). The last constraint is \(\mathbf {y}_C\) to be on the plane defined by the vector \(\left( (\mathbf {X}_\mathrm {REF})_B - (\mathbf {X}_\mathrm {REF})_A\right) \times \left( (\mathbf {X}_\mathrm {REF})_C - (\mathbf {X}_\mathrm {REF})_A\right) \).

### 2.5 An iterative method

*i*th solution. We simply assume

*i*th solution can be written as

Once we obtain the IPBZSS at the end of the iterations, we will have the option to convert it to EBZSS and use that in the subsequent \(\mathbf {y}\) computations.

## 3 Initial guess based on the shell model of the artery

An analytical relationship between the ZS and reference states of straight-tube segments was given in [67]. The relationship was called “straight-tube ZSS template” in [69] and was extended to curved tubes. These were for the EBZSS. Here we directly build the IPBZSS instead of building the EBZSS first. This is simpler because with isogeometric discretization, especially with T-spline discretization, specifying conditions at integration points is far more straightforward than imposing conditions on control points.

We start with the artery inner surface, which is what the medical images show. Typically, we cannot discern the wall thickness from the medical image. Therefore we first build the inner-surface mesh with T-splines. Then we build a T-spline volume mesh by extruding the surface elements by an estimated thickness.

In our notation here, \(\mathbf {x}\) will now imply \(\mathbf {X}_\mathrm {REF}\), which is our “target” shape, and \(\mathbf {X}\) will imply \(\mathbf {X}_0\). We explain the method in the context of one element in the thickness direction. Extending the method to multiple elements is straightforward.

### 3.1 Inner-surface coordinates in the target state

For more details on calculating the eigenvalues and eigenvectors, see Appendix A.

### 3.2 Inner-surface coordinates in the ZSS

### 3.3 Wall coordinates in the target state

### 3.4 Wall coordinates in the ZSS

### 3.5 Calculating the components of the ZSS metric tensor at each integration point

### 3.6 Design of the ZSS

## 4 Numerical examples

The initial guess for the iterations is determined as described in Sect. 3, with \(\phi = \frac{5}{2}\pi \) and \(\hat{\lambda _2} = 1.05\). After the iterations, explained in Sect. 2.5, for comparison purposes, we convert the IPBZSS representation to EBZSS representation, with the method described in Sect. 2.4. With the EBZSS, we compute \(\mathbf {y}\) again, and compare that to what we obtained directly from the IPBZSS.

### 4.1 Y-shaped tube

The target state of the Y-shaped tube is shown in Fig. 1. The end diameters of the tube are 20, 14 and 10 mm. Figure 2 shows the T-spline mesh. The mesh is based on a mixture of cubic and quartic T-splines. The wall thickness distribution is smooth, outcome of solving the Laplace’s equation over the inner surface, with Dirichlet boundary conditions at the tube ends, where the value specified is 0.1 times the end diameter. Figure 3 shows the thickness distribution. The volume mesh is built with one element (cubic Bézier element) in the thickness direction. The number of control points and elements are 5, 180 and 2, 592.

We iterate and obtain the converged IPBZSS. Figure 6 shows \(\Vert \mathbf {y}\Vert \) computed from that. The maximum value of \(\Vert \mathbf {y}\Vert \) is \(1.279{\times }10^{-14}~\mathrm {mm}\). Figure 7 shows the von Mises strain, computed from the IPBZSS initial guess and from the converged IPBZSS. The converged IPBZSS element is shown in Fig. 8.

### Remark 1

Figure 8 shows that the opening angle and the longitudinal stretch do not change much between the IPBZSS initial guess and the converted IPBZSS. However, the circumferential stretches are somewhat different. The initial guess for the circumferential stretch was based on the 2D computations reported in [69]. Alternatively, we can estimate the stretch by an analytical solution, similar to the one described in [70].

We also compute \(\mathbf {y}\) after converting the converged IPBZSS to EBZSS. Figure 9 shows \(\Vert \mathbf {y}\Vert \) computed that way. The maximum value of \(\Vert \mathbf {y}\Vert \) is \(1.626{\times }10^{-2}~\mathrm {mm}\). Figure 10 shows the von Mises strain. There is no visible difference between the strains obtained from the IPBZSS directly and after conversion to EBZSS.

### 4.2 Patient-specific aorta geometry

The wall thickness distribution is smooth, outcome of solving the Laplace’s equation over the inner surface, with Dirichlet boundary conditions at the tube ends, where the value specified is 0.08 times the end diameter. In some parts of the branched area the thickness exceeds the radius of curvature, and there we reduce the thickness to 0.8 times the radius of curvature. Figure 13 shows the thickness distribution.

The volume mesh is built again with one element (cubic Bézier element) in the thickness direction. The number of control points and elements are 9, 244 and 4, 360.

Figure 14 shows the initial guess for the IPBZSS. We again iterate and obtain the converged IPBZSS. Figure 15 shows \(\Vert \mathbf {y}\Vert \) computed from that. The maximum value of \(\Vert \mathbf {y}\Vert \) is \(1.163{\times }10^{-13}~\mathrm {mm}\). Figure 16 shows the von Mises strain, computed from the IPBZSS initial guess and from the converged IPBZSS.

## 5 Concluding remarks

We have introduced a new method for estimating the ZSS required in patient-specific arterial FSI computations, where the image-based arterial geometries do not come from the ZSS of the artery. The estimate is based on T-spline discretization of the arterial wall and is in the form of IPBZSS. The method has two main components. (1) An iterative method, which starts with a calculated initial guess, is used for computing the IPBZSS such that when a given pressure load is applied, the image-based target shape is matched. (2) A method, which is based on the shell model of the artery, is used for calculating the initial guess. The T-spline discretization enables dealing with complex arterial geometries, such as an aorta model with branches, while retaining the desirable features of isogeometric discretization. The desirable features of higher-order basis functions of isogeometric discretization include being able to achieve a similar level of accuracy as with the linear basis functions, but using larger-size and much fewer elements, and being able to represent more complex shapes within an element. The IPBZSS is a convenient representation of the ZSS because with isogeometric discretization, especially with T-spline discretization, specifying conditions at integration points is more straightforward than imposing conditions on control points. Calculating the initial guess based on the shell model of the artery results in a more realistic initial guess. To show how the new ZSS estimation method performs, we first presented 3D test computations with a Y-shaped tube. Then we presented a 3D computation where the target geometry was coming from medical image of a human aorta, and the model included the aorta branches.

## Notes

### Acknowledgements

This work was supported in part by JST-CREST; Grant-in-Aid for Scientific Research (S) 26220002 from the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT); Grant-in-Aid for Scientific Research (A) 18H04100 from Japan Society for the Promotion of Science; and Rice–Waseda research agreement. This work was also supported (first author) in part by Grant-in-Aid for JSPS Research Fellow 18J14680. The mathematical model and computational method parts of the work were also supported (third author) in part by ARO Grant W911NF-17-1-0046 and Top Global University Project of Waseda University.

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