# Medical-image-based aorta modeling with zero-stress-state estimation

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## Abstract

Because the medical-image-based geometries used in patient-specific arterial fluid–structure interaction computations do not come from the zero-stress state (ZSS) of the artery, we need to estimate the ZSS required in the computations. The task becomes even more challenging for arteries with complex geometries, such as the aorta. In a method we introduced earlier the estimate is based on T-spline discretization of the arterial wall and is in the form of integration-point-based ZSS (IPBZSS). 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 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. The method has two main components. 1. An iteration technique, which starts with a calculated ZSS initial guess, is used for computing the IPBZSS such that when a given pressure load is applied, the medical-image-based target shape is matched. 2. A design procedure, which is based on the Kirchhoff–Love shell model of the artery, is used for calculating the ZSS initial guess. Here we increase the scope and robustness of the method by introducing a new design procedure for the ZSS initial guess. The new design procedure has two features. (a) An IPB shell-like coordinate system, which increases the scope of the design to general parametrization in the computational space. (b) Analytical solution of the force equilibrium in the normal direction, based on the Kirchhoff–Love shell model, which places proper constraints on the design parameters. This increases the estimation accuracy, which in turn increases the robustness of the iterations and the convergence speed. To show how the new design procedure for the ZSS initial guess performs, we first present 3D test computations with a straight tube and a Y-shaped tube. Then we present a 3D computation where the target geometry is coming from medical image of a human aorta, and we include the branches in the model.

## Keywords

Patient-specific arterial FSI Medical-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

Space–time (ST) computational methods [1, 2], with all the desirable features of moving-mesh methods, have a relatively long track record in arterial fluid–structure interaction (FSI) analysis, starting with computations reported in [3, 4, 5, 6]. These were among the earliest arterial FSI computations, and the core method was the early version of the Deforming-Spatial-Domain/Stabilized ST (DSD/SST) method [1, 2], now called “ST-SUPS.” The acronym “SUPS” indicates the stabilization components, the Streamline-Upwind/Petrov-Galerkin (SUPG) [7] and Pressure-Stabilizing/Petrov-Galerkin (PSPG) [1].

The ST computations have been a small part of the many cardiovascular fluid mechanics and FSI computations reported 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] for computations with other methods), with the Arbitrary Lagrangian–Eulerian (ALE) method having the largest share. Still, many ST computations 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 [30], carotid artery [30] and cerebral aneurysms [31, 32, 33, 34, 35, 36, 37]. 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 [38, 39], stent treatment of cerebral aneurysms [40, 41, 42, 43, 44], heart valve flow computation [45, 46, 47, 48, 49, 50], aorta flow analysis [50, 51, 52, 53], and coronary arterial dynamics [54]. The computational challenges encountered were addressed by the advances in the core methods for moving boundaries and interfaces (MBI) and FSI (see, for example, [20, 21, 39, 45, 46, 48, 49, 55, 56, 57, 58, 59, 60, 61, 62, 63] and references therein) and in the special methods targeting cardiovascular MBI and FSI (see, for example, [20, 37, 43, 44, 47, 48, 49, 50, 53, 64] and references therein). For an overview of the ST MBI and FSI computations in general, see [65].

A challenge specific to patient-specific arterial FSI computations is how to use the medical-image-based arterial geometry, which does not come from the zero-stress state (ZSS) of the artery. The special methods targeting cardiovascular MBI and FSI include those intended to account for that. The task becomes even more challenging for arteries with complex geometries, such as the aorta. 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 [31], as “a rudimentary technique” for addressing the issue. It was pointed out in [31, 66] that quite often the medical-image-based geometries were used as arterial geometries corresponding to zero blood pressure, and that it would be more realistic to use the medical-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. The special methods developed to address the issue include the newer EZP versions [20, 33, 36, 37, 64] and the prestress technique introduced in [16], which was refined in[18] and presented also in [20, 64].

A method for estimation of the element-based ZSS (EBZSS) was introduced in [67] in the context of finite element discretization of the arterial wall. The method has three main components. 1. An iteration technique, which starts with a calculated ZSS initial guess, is used for computing the EBZSS such that when a given pressure load is applied, the medical-image-based target shape is matched. 2. A technique 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 ZSS initial guess for the iteration technique that matches the medical-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 [54] in coronary arterial dynamics computations with medical-image-based time-dependent anatomical models.

The version of the EBZSS estimation method with isogeometric wall discretization, using NURBS basis functions, was introduced in [68]. 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. 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 fewer elements, and the NURBS basis functions enable 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], how the method can be used in a 3D computation where the target geometry is coming from medical image of a human aorta was also shown.

In the method introduced in [70], the estimate is based on T-spline discretization of the arterial wall and is in the form of integration-point-based ZSS (IPBZSS). 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. The method has two main components. 1. An iteration technique, which starts with a calculated ZSS initial guess, is used for computing the IPBZSS such that when a given pressure load is applied, the medical-image-based target shape is matched. 2. A design procedure, which is based on the Kirchhoff–Love shell model of the artery, is used for calculating the ZSS initial guess.

In this article we increase the scope and robustness of the method by introducing a new design procedure for the ZSS initial guess. The new design procedure has two features. (a) An IPB shell-like coordinate system, which increases the scope of the design to general parametrization in the computational space. (b) Analytical solution of the force equilibrium in the normal direction, based on the Kirchhoff–Love shell model, which places proper constraints on the design parameters. This increases the estimation accuracy, which in turn increases the robustness of the iterations and the convergence speed. To show how the new design procedure for the ZSS initial guess performs, we first present 3D test computations with a straight tube and a Y-shaped tube. Then we present a 3D computation where the target geometry is coming from medical image of a human aorta, and we include the branches in the model.

In Sect. 2, from [70], we describe the Element-Based Total Lagrangian (EBTL) method, including the EBZSS and IPBZSS concepts. The IPB shell-like coordinate system and the related mesh generation are described in Sect. 3. The design procedure for the ZSS initial guess, based on the Kirchhoff–Love shell model, is described in Sect. 4. The numerical examples are given in Sect. 5, and the concluding remarks in Sect. 6.

## 2 EBTL method

We first provide, from [70], an overview of the EBTL method [67], including the EBZSS and IPBZSS concepts 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, \(\mathbf {X}_\mathrm {REF}\in {\varOmega }_{\mathrm {REF}}\), 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 \(\left( (\mathbf {X}_\mathrm {REF})_B - (\mathbf {X}_\mathrm {REF})_A\right) \times \left( (\mathbf {X}_\mathrm {REF})_C - (\mathbf {X}_\mathrm {REF})_A\right) \).

## 3 Coordinate systems for the artery inner surface and wall

A geometrical relationship between the ZS and reference states, for a straight tube, was described in [70]. It was based on the shell model, where it is assumed that the inner-surface elements are extruded in the normal direction. Here we increase the scope of the method to general parametrization in the computational space. For each integration point of the computational space, we use a special shell-like coordinate system specific to that integration point. We will explain that coordinate system later in this section, after first explaining how the mesh is generated.

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 across the wall. Extending the method to multiple elements is straightforward.

### 3.1 Mesh generation

We again start with the artery inner surface, and build the wall in some fashion. Here we first build a T-spline inner-surface mesh. Then we expand that by an estimated thickness to obtain the outer-surface mesh. After that we modify the outer-surface mesh manually by moving the control points. When the thickness is larger than the radius of curvature, parts of the outer surface overlap. This might happen near the branches. Therefore, a simple extrusion does not work. Since the outer surface cannot be obtained from medical images, the design of the outer surface has to be based on other anatomical knowledge. This is the reason why currently meshes are generated manually rather than by an automated process. After defining the outer-surface mesh, which will have a control point corresponding to every control point on the inner surface, we add two control points for each pair. We use the four control points to form a cubic Bézier element across the wall. This is the way we obtain a T-spline volume mesh.

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

### 3.3 Inner-surface coordinates in the ZSS

### 3.4 Wall coordinates in the target state

### Remark 1

We note that even if \(\hat{\mathbf {g}}_\alpha = \mathbf {g}_\alpha \), \(\hat{\mathbf {g}}^\gamma \) and \(\mathbf {g}^\gamma \) are not the same in general, because \(\mathbf {g}_3\) is not perpendicular to \(\mathbf {g}_1\) and \(\mathbf {g}_2\) and \(\mathbf {g}^\gamma \) will have an out-of-plane component.

### Remark 2

The first normal-vector option is closer to the shell theory. The downside is that we may not always have a reasonable geometry, for example when the radius of curvature is low compared to the thickness. The second option does not suffer from that problem and gives us the possibility of coming up with an \(\hat{\mathbf {n}}\) design that would make the method better. However its quality depends on the mesh, such as the smoothness of the constant-\(\xi ^3\) surfaces.

### Remark 3

### 3.5 Wall coordinates in the ZSS

Once we define the shell-like coordinate system for *k*th integration point, we can use the corresponding ZSS coordinate system as described in [70]. From that we compute the components of the metric tensor corresponding to the natural coordinates.

## 4 Analytical expression and design for the ZSS initial guess

As explained in [70], the design parameters are the principal curvatures \(\left( \hat{\kappa }_0\right) _1\) and \(\left( \hat{\kappa }_0\right) _2\), and the stretches \(\hat{\lambda }_1\) and \(\hat{\lambda }_2\) for each principal-curvature direction. However, they are not a set of independent values because there are constraints for the ZSS to give us the target shape for the given load, and we try to get close to that even for the initial guess. To that end, we use the force equilibrium in the normal direction by using a local analytical solution for each integration point. After that, we describe the design for the ZSS initial guess.

### 4.1 Analytical solution based on the Kirchhoff–Love shell model

We use the Kirchhoff–Love shell model with plane-stress condition to obtain the analytical solution. That is the generalized version of the solutions given in [71] for pressurized sphere and cylinder.

To deal with an arbitrary geometry, we assume the shape is given in terms of the principal curvatures. That is the reason we use only the equilibrium equation in the normal direction and focus on a small surface area \(\delta \overline{{\varGamma }}\). From that surface area, we extrude in the normal direction by \(h_\mathrm {th}\) to define a volume: \(\delta {\varOmega }= \int _{0}^{h_\mathrm {th}} \delta {\varGamma }(\vartheta ) \mathrm {d}\vartheta \).

*p*are the Cauchy stress tensor and pressure load. Now we express this by using the shell-coordinate system. Using the plane-stress condition, we can write

### 4.2 The design for the ZSS initial guess

## 5 Computations

All computations are based on the Fung’s model (see Appendix C) with \(D_1 = 2.6447\,{\times }\,10^3~\mathrm {Pa}\), \(D_2 = 8.365\), and the Poisson’s ratio \(\nu = 0.49\). The load is \(p = 92~\mathrm {mm~Hg}\). We use the normal-vector definition of Eq. (42).

### 5.1 Analytical solution for constant stretch on the inner surface

### Remark 4

We note that the stretch value 1.05 gives \(\phi _1 = 410^\circ \) for a straight tube with \(\hat{\kappa }_1h_\mathrm {th}= 0.16\), and \(\phi _1 = 310^\circ \) with \(\hat{\kappa }_1 h_\mathrm {th}= 0.20\).

We use this map for generating the initial guess. When the geometry is out of scope, we set \(\phi _\alpha =0\).

### 5.2 Straight tube

### 5.3 Y-shaped tube

### 5.4 Patient-specific aorta

From the results of the Y-shaped tube we learned that the initial guess should have had, on the outer surface, less variation in the stretch in \(\hat{\mathbf {n}}\) direction. Based on that, we improve the initial guess in parts of the region where we do not have both curvatures positive. For a straight tube with \(\hat{\kappa }_1 h_\mathrm {th}= 0.18\), on the outer surface, the stretch in \(\hat{\mathbf {n}}\) direction is 0.80. We target that value in improving the initial guess.

- 1.
\(\hat{\kappa }_1 h_\mathrm {th}< - 0.6\) or \(\hat{\kappa }_2 h_\mathrm {th}< - 0.6\)

- 2.
\(\hat{\kappa }_1 h_\mathrm {th}< - 0.4\) or \(\hat{\kappa }_2 h_\mathrm {th}< - 0.4\)

- 3.
\(\hat{\kappa }_1 h_\mathrm {th}+ \hat{\kappa }_2 h_\mathrm {th}< 0\)

- 4.
Elsewhere

- 1.
Set \(\phi _1=\phi _2 = 0\) and \(\hat{\lambda }_1 = \hat{\lambda _2} = 1\).

- 2.
Set \(\phi _1 = \phi _2 = 0\). Assume that \(\hat{\lambda }_1\) and \(\hat{\lambda }_2\) are the same. Determine them in such a way that, on the outer surface, the stretch in \(\hat{\mathbf {n}}\) direction is 0.80.

- 3.
Leave \(\hat{\lambda }_1=\hat{\lambda }_2 = 1.05\) unchanged. Assume that \(\phi _1\) and \(\phi _2\) are the same. Determine them in such a way that, on the outer surface, the stretch in \(\hat{\mathbf {n}}\) direction is 0.80.

- 4.
No modification.

## 6 Concluding remarks

We have increased the scope and robustness of the method we introduced earlier for estimating the ZSS required in patient-specific arterial FSI computations, where the medical-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 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 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.

The method has two main components. 1. An iteration technique, which starts with a calculated ZSS initial guess, is used for computing the IPBZSS such that when a given pressure load is applied, the medical-image-based target shape is matched. 2. A design procedure, which is based on the Kirchhoff–Love shell model of the artery, is used for calculating the ZSS initial guess.

We increased the scope and robustness of the method by introducing a new design procedure for the ZSS initial guess. The new design procedure has two features. (a) An IPB shell-like coordinate system, which increases the scope of the design to general parametrization in the computational space. (b) Analytical solution of the force equilibrium in the normal direction, based on the Kirchhoff–Love shell model, which places proper constraints on the design parameters. This increases the estimation accuracy, which in turn increases the robustness of the iterations and the convergence speed.

To show how the new design procedure for the ZSS initial guess performs, we first presented 3D test computations with a straight tube and a Y-shaped tube. After that we presented a 3D computation where the target geometry is coming from medical image of a human aorta, and we included the branches in the model. The computations show that the new design procedure for the ZSS initial guess is reaching the design targets well.

## 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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