Abstract
The imagebased arterial geometries used in patientspecific arterial fluid–structure interaction (FSI) computations, such as aorta FSI computations, do not come from the zerostress state (ZSS) of the artery. We propose a method for estimating the ZSS required in the computations. Our estimate is based on Tspline discretization of the arterial wall and is in the form of integrationpointbased 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 imagebased 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 Tspline discretization enables dealing with complex arterial geometries, such as an aorta model with branches, while retaining the desirable features of isogeometric discretization. With higherorder 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 largersize and much fewer elements. In addition, the higherorder 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 Tspline 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 Yshaped 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.
Introduction
The patientspecific 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 DeformingSpatialDomain/Stabilized Space–Time (DSD/SST) method [5, 6], which is now called the “STSUPS.” The acronym “SUPS” indicates the stabilization components, the StreamlineUpwind/Petrov–Galerkin (SUPG) [7] and PressureStabilizing/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 patientspecific arterial FSI computations, such as patientspecific aorta FSI computations, is how to use the imagebased arterial geometry. The imagebased geometry does not come from the zerostress 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 zeropressure (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 imagebased geometries were used as arterial geometries corresponding to zero blood pressure, and that it would be more realistic to use the imagebased geometry as the arterial geometry corresponding to the timeaveraged value of the blood pressure. Given the arterial geometry at the timeaveraged 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 elementbased 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 imagebased target shape is matched. (2) A method for straighttube 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 elementbased mapping between the artery and straighttube is extracted from the mapping between the artery and straighttube segments. This provides the mapping from the arterial configuration to the straighttube configuration, and from the estimated EBZSS of the straighttube configuration back to the arterial configuration, to be used as the initial guess for the iterative method that matches the imagebased target shape. Test computations with the method were also presented in [67] for straighttube configurations with single and three layers, and for a curvedtube configuration with single layer. The method was used also in [55] in coronary arterial dynamics computations with medicalimagebased timedependent 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 elementbased mapping directly, instead of extracting it from the mapping between the artery and straighttube segments. That is because all we need for the elementbased 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 largersize and much fewer elements, and the NURBS basis functions allow representation of more complex shapes within an element. The 2D test computations with straighttube 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 Tspline discretization of the arterial wall and is in the form of integrationpointbased 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 imagebased 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 Tspline 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 Tspline 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 Yshaped 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 ElementBased 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.
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.
Let \(\varOmega _0\subset \mathbb {R}^{n_{\mathrm {sd}}}\) be the material domain of a structure in the ZSS, where \({n_{\mathrm {sd}}}\) is the number of space dimensions, and let \(\varGamma _0\) be its boundary. Let \(\varOmega _t\subset \mathbb {R}^{n_{\mathrm {sd}}}\), \(t \in (0,T)\), be the material domain of the structure in the deformed state, and let \(\varGamma _{t}\) be its boundary. The structural mechanics equations based on the total Lagrangian formulation can be written as
Here, \(\mathbf {y}\) is the displacement, \(\mathbf {w}\) is the virtual displacement, \(\delta \mathbf {E}\) is the variation of the Green–Lagrange strain tensor, \(\mathbf {S}\) is the second Piola–Kirchhoff stress tensor, \(\rho _0\) is the mass density in the ZSS, \(\mathbf {f}\) is the body force per unit mass, and \(\mathbf {h}\) is the external stress vector applied on the subset \(\left( \varGamma _t\right) _\mathrm {h}\) of the boundary \(\varGamma _{t}\).
EBZSS
In the EBTL method the ZSS is defined with a set of positions \(\mathbf {X}_0^e\) for each element 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:
where \(\mathbf {x}\) is the position in the deformed configuration. The deformation gradient tensors for different elements are on different states, but the terms in Eq. (1), including the second term, do not depend on the orientation. Therefore the rest of the process is the same as it is in the total Lagrangian formulation.
IPBZSS
The key idea behind the EBZSS method was that, due to the objectivity, all the quantities seen in Eq. (1) can be computed with any orientation of the ZSS. We can extend the way we see \(\mathbf {X}_0^e\) to integrationpoint counterpart of \(\mathbf {X}_0^e\). As we did with the EBZSS, we work with the reference domain. With the reference Jacobian
Eq. (1) can be rearranged as
In our implementation, we use the natural coordinates, with covariant basis vectors
where \(\xi ^I\) is the parametric coordinate, and \(I = 1, \ldots , {n_{\mathrm {pd}}}\), with \({n_{\mathrm {pd}}}\) being the number of parametric dimensions. The contravariant basis vectors can be calculated with the metric tensor components as
where
With those vectors, we can express the deformation gradient tensor:
and the Cauchy–Green deformation tensor:
The Jacobian, \(J= \det \mathbf {F}\), can be expressed as
We can write \(\det \mathbf {C}\) as
and from that we obtain
We define the covariant basis vectors corresponding to \(\mathbf {X}_\mathrm {REF}\):
and the components of the metric tensor are
In Eq. (20), we replace \(\mathbf {g}_I\) and \(\mathbf {g}_J\) with \(\left( \mathbf {G}_\mathrm {REF}\right) _I\) and \(\left( \mathbf {G}_\mathrm {REF}\right) _J\) and obtain an alternative to the expression given by Eq. (4):
The Green–Lagrange strain tensor,
where \(\mathbf {I}\) is the identity tensor, can be expressed with the contravariant basis vectors as
The second Piola–Kirchhoff tensor can be expressed with the covariant basis vectors as
where \(S^{IJ}\) can be expressed with the components of the metric tensors. Thus, the inner product \(\delta \mathbf {E}: \mathbf {S}\), and all the other quantities, in the weak form given by Eq. (5) can be evaluated without actually using the basis vectors \(\mathbf {G}_I\). This justifies using \(\left( G_{IJ}\right) _k\) as the integrationpoint counterpart of \(\mathbf {X}_0^e\), with \(k=1, \ldots , n_\mathrm {int}\), where \(n_\mathrm {int}\) is the number of integration points. Note that \(G_{IJ}\) is symmetric, and therefore the IPBZSS representation will in 3D have \(6{\times }n_\mathrm {int}\) parameters for each element.
EBZSS to IPBZSS
Converting the EBZSS representation to IPBZSS representation is straightforward. From given \(\mathbf {X}_0^e\) we can calculate the covariant basis vectors at each integration point \(\pmb {\xi }_k\):
and obtain the components of the metric tensor from Eq. (11).
IPBZSS to EBZSS
Converting the IPBZSS representation to EBZSS representation, which we might need for visualization purposes, will, in general, not be exact because the IPBZSS has more parameters than the EBZSS. Given \(\left( G_{IJ}\right) _k\), we solve a steadystate elementbased problem (with \(\mathbf {f}=\mathbf {0}\) and \(\mathbf {h}= \mathbf {0}\)):
and the solution to that, in the form \(\mathbf {X}_\mathrm {REF} + \mathbf {y}\), will be the EBZSS representation. If the stress calculated from the solution is zero, then the conversion will be exact. We note that to obtain a steadystate solution, we need to preclude translation and rigidbody rotation by imposing 6 appropriate constraints. To do that we first select three control points: 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 \(\left( (\mathbf {X}_\mathrm {REF})_B  (\mathbf {X}_\mathrm {REF})_A\right) \times \left( (\mathbf {X}_\mathrm {REF})_C  (\mathbf {X}_\mathrm {REF})_A\right) \).
An iterative method
Here we assume that we have a reasonably good initial guess for the IPBZSS (see Sect. 3). The iterative method below is used in calculating the IPBZSS that results in the target state associated with the given load. In the iterative method, we estimate \(\mathbf {F}\) from the ith solution. We simply assume
which means
The inverse of the deformation gradient tensor from the ith solution can be written as
Similarly, the inverse of the target deformation gradient tensor is
Thus, we obtain the following equation:
Innerproducting both sides of this equation from the right with the covariant basis vectors corresponding to \(\mathbf {X}_\mathrm {REF}\), we obtain
which results in
The components of the metric tensor are
Rearranging the terms, we obtain
Thus, we update the components of the metric tensor without actually knowing the basis vectors \(\mathbf {G}_I\).
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.
Initial guess based on the shell model of the artery
An analytical relationship between the ZS and reference states of straighttube segments was given in [67]. The relationship was called “straighttube 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 Tspline 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 innersurface mesh with Tsplines. Then we build a Tspline 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.
Innersurface coordinates in the target state
The coordinate system we have here is similar to the one used for the shell modeling in [70]. We note that the “midsurface” of the shell formulation has been shifted to the inner surface here, and \(\overline{\bullet }\) indicates the inner surface. The basis vectors are
where \(\alpha = 1, \ldots , {n_{\mathrm {sd}}}1\), and the third direction is
The second fundamental form is defined as
and the curvature tensor is
Clearly, \(\hat{\pmb {\kappa }}\) is symmetric.
For a given unit vector \(\mathbf {t}\) on the surface, we obtain the curvature \(\hat{\kappa }\) as
If \(\mathbf {t}\) is a principal direction,
and \(\hat{\kappa }\) is the corresponding principal curvature. The eigenvector can be expressed as
Substituting this into Eq. (43) and innerproducting with \(\overline{\mathbf {g}}^\alpha \), we obtain
where the mixed components indicate
Since the inverse of \(\left[ \overline{g}_{\gamma \alpha }\right] \) exists,
\(\hat{\kappa }\) is an eigenvalue of the matrix defined by the mixed components \(\hat{\kappa }^\alpha _{\bullet \beta }\), and we will call the two eigenvalues \(\hat{\kappa }_1\) and \(\hat{\kappa }_2\). For \(\hat{\kappa }_1 > \hat{\kappa }_2\) we obtain the corresponding eigenvectors:
where \(\mathbf {t}_1\) and \(\mathbf {t}_2\) are unit vectors. From Eq. (43), we write
Since \(\hat{\pmb {\kappa }}\) is symmetric,
Substituting Eqs. (50) and (51) into this, we obtain
Thus, the two vectors are orthonormal, and we can express the curvature tensor as
When \(\hat{\kappa }_1 = \hat{\kappa }_2\), an arbitrary orthonormal set of \(\mathbf {t}_1\) and \(\mathbf {t}_2\) can be used in the above equation.
For more details on calculating the eigenvalues and eigenvectors, see Appendix A.
Innersurface coordinates in the ZSS
Since the principal curvature directions \(\mathbf {t}_1\) and \(\mathbf {t}_2\) of the target shape are orthogonal to each other, we can build the ZSS shape using those directions. The basis vectors on the inner surface in the ZSS are
and the third direction is
The stretches corresponding to those directions will be \(\hat{\lambda }_1\) and \(\hat{\lambda }_2\). Then the ZSS basis vectors are calculated from
That is
Because the third direction is orthogonal to \(\mathbf {t}_1\) and \(\mathbf {t}_2\), we can reduce these equations to
Substituting Eqs. (48) and (49) into these, we get
This can also be written as
and from that we calculate the basis vectors as
Wall coordinates in the target state
The position in the target configuration is
where \(0 \le \vartheta \le h_\mathrm {th}\), and \(h_\mathrm {th}\) is the wall thickness in the target configuration. The basis vectors will vary along the thickness direction as
The second and third lines are explained in Appendix 1. The third coordinate is mapped as
where \(1 \le \xi ^3 \le 1\). The basis vector in the third direction is constant as
and
With that, the components of the metric tensor are
Wall coordinates in the ZSS
The position in the ZSS configuration is
where \(0 \le \vartheta _0 \le \left( h_\mathrm {th}\right) _0\), and \(\left( h_\mathrm {th}\right) _0\) is the wall thickness in the ZSS configuration. The basis vectors will vary along the thickness direction as
The curvature tensor in the ZSS configuration is
From that,
Similar to what we had for \(\mathbf {t}_1\) and \(\mathbf {t}_2\),
which becomes
We substitute Eq. (73) into this and obtain
and
Calculating the components of the ZSS metric tensor at each integration point
For an integration point \(\pmb {\xi }\), we can obtain the components of the metric tensor as
The third coordinate can be obtained from
Assuming incompressible material, \(J=1\),
where
The components of the matrix tensors are given by Eqs. (74) and (89).
Design of the ZSS
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. Those parameters can be determined from \(\hat{\kappa }_1\) and \(\hat{\kappa }_2\) of the target configuration.
As proposed in [69], the two principal directions are seen as circumferential and longitudinal directions, and \(\hat{\kappa }_1\) is in the circumferential direction, giving us
Here \(\phi \) is the opening angle, which is seen after a longitudinal cut, based on artery experimental data [71]. The stretch in that direction, \(\hat{\lambda }_1\), corresponds to \(\lambda _\theta \) in [69] and that is determined from the 2D computations in [69]. We assume that in the longitudinal direction the ZSS configuration has zero curvature, \(\left( \hat{\kappa }_0\right) _2 = 0\). The stretch in that direction, \(\hat{\lambda }_2\), corresponds to \(\lambda _z\) in [69]. If at an integration point \(\hat{\kappa }_2 \approx \hat{\kappa }_1\), we set \(\left( \hat{\kappa }_0\right) _2 = \left( \hat{\kappa }_0\right) _1\) and \(\hat{\lambda }_2 = \hat{\lambda }_1\). That makes the assignment of the principal directions less consequential.
Numerical examples
In the numerical examples we use the Fung’s model [20, 38, 70] with \(D_1 = 2.6447{\times }10^3~\mathrm {Pa}\), \(D_2 = 8.365\), and the Poisson’s ratio \(\nu = 0.45\). The energydensity function we use is in the form
where
We assume the pressure associated with the target shape is 92 mm Hg.
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.
Yshaped tube
The target state of the Yshaped tube is shown in Fig. 1. The end diameters of the tube are 20, 14 and 10 mm. Figure 2 shows the Tspline mesh. The mesh is based on a mixture of cubic and quartic Tsplines. 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.
Since the IPBZSS cannot be visualized, we show it using the EBZSS representation. Figure 4 shows the initial guess for the IPBZSS. Figure 5 shows an element in the target state and the corresponding IPBZSS initial guess.
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.
Patientspecific aorta geometry
The target state of the patientspecific geometry is shown in Fig. 11. Figure 12 shows the Tspline mesh.
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.
We again compute \(\mathbf {y}\) after converting the converged IPBZSS to EBZSS. Figure 17 shows \(\Vert \mathbf {y}\Vert \) computed that way. The maximum value of \(\Vert \mathbf {y}\Vert \) is \(1.057{\times }10^{1}~\mathrm {mm}\). Figure 18 shows the von Mises strain. These results indicate that the EBZSS obtained by conversion from the IPBZSS is also a reasonable representation of the aorta ZSS.
Concluding remarks
We have introduced a new method for estimating the ZSS required in patientspecific arterial FSI computations, where the imagebased arterial geometries do not come from the ZSS of the artery. The estimate is based on Tspline 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 imagebased 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 Tspline 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 higherorder basis functions of isogeometric discretization include being able to achieve a similar level of accuracy as with the linear basis functions, but using largersize 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 Tspline 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 Yshaped 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.
References
Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2004) Computation of cardiovascular fluid–structure interactions with the DSD/SST method. In: Proceedings of the 6th world congress on computational mechanics (CDROM), Beijing, China
Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2004) Influence of wall elasticity on imagebased blood flow simulations. Trans Jpn Soc Mech Eng Ser A 70:1224–1231. https://doi.org/10.1299/kikaia.70.1224 in Japanese
Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2006) Computer modeling of cardiovascular fluid–structure interactions with the deformingspatialdomain/stabilized space–time formulation. Comput Methods Appl Mech Eng 195:1885–1895. https://doi.org/10.1016/j.cma.2005.05.050
Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2006) Fluid–structure interaction modeling of aneurysmal conditions with high and normal blood pressures. Comput Mech 38:482–490. https://doi.org/10.1007/s0046600600656
Tezduyar TE (1992) Stabilized finite element formulations for incompressible flow computations. Adv Appl Mech 28:1–44. https://doi.org/10.1016/S00652156(08)701534
Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43:555–575. https://doi.org/10.1002/fld.505
Brooks AN, Hughes TJR (1982) Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 32:199–259
Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid–structure interaction analysis with applications to arterial blood flow. Comput Mech 38:310–322
Bazilevs Y, Calo VM, Tezduyar TE, Hughes TJR (2007) YZ\(\beta \) discontinuitycapturing for advectiondominated processes with application to arterial drug delivery. Int J Numer Methods Fluids 54:593–608. https://doi.org/10.1002/fld.1484
Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid–structure interaction: theory, algorithms, and computations. Comput Mech 43:3–37
Isaksen JG, Bazilevs Y, Kvamsdal T, Zhang Y, Kaspersen JH, Waterloo K, Romner B, Ingebrigtsen T (2008) Determination of wall tension in cerebral artery aneurysms by numerical simulation. Stroke 39:3172–3178
Bazilevs Y, Gohean JR, Hughes TJR, Moser RD, Zhang Y (2009) Patientspecific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device. Comput Methods Appl Mech Eng 198:3534–3550
Bazilevs Y, Hsu MC, Benson D, Sankaran S, Marsden A (2009) Computational fluid–structure interaction: methods and application to a total cavopulmonary connection. Comput Mech 45:77–89
Bazilevs Y, Hsu MC, Zhang Y, Wang W, Liang X, Kvamsdal T, Brekken R, Isaksen J (2010) A fullycoupled fluid–structure interaction simulation of cerebral aneurysms. Comput Mech 46:3–16
Sugiyama K, Ii S, Takeuchi S, Takagi S, Matsumoto Y (2010) Full Eulerian simulations of biconcave neoHookean particles in a Poiseuille flow. Comput Mech 46:147–157
Bazilevs Y, Hsu MC, Zhang Y, Wang W, Kvamsdal T, Hentschel S, Isaksen J (2010) Computational fluid–structure interaction: methods and application to cerebral aneurysms. Biomech Model Mechanobiol 9:481–498
Bazilevs Y, del Alamo JC, Humphrey JD (2010) From imaging to prediction: emerging noninvasive methods in pediatric cardiology. Progr Pediatr Cardiol 30:81–89
Hsu MC, Bazilevs Y (2011) Blood vessel tissue prestress modeling for vascular fluid–structure interaction simulations. Finite Elem Anal Des 47:593–599
Yao JY, Liu GR, Narmoneva DA, Hinton RB, Zhang ZQ (2012) Immersed smoothed finite element method for fluid–structure interaction simulation of aortic valves. Comput Mech 50:789–804
Bazilevs Y, Takizawa K, Tezduyar TE (2013) Computational fluid–structure interaction: methods and applications. Wiley, New York ISBN 9780470978771
Bazilevs Y, Takizawa K, Tezduyar TE (2013) Challenges and directions in computational fluid–structure interaction. Math Models Methods Appl Sci 23:215–221. https://doi.org/10.1142/S0218202513400010
Long CC, Marsden AL, Bazilevs Y (2013) Fluid–structure interaction simulation of pulsatile ventricular assist devices. Comput Mech 52:971–981. https://doi.org/10.1007/s0046601308583
EsmailyMoghadam M, Bazilevs Y, Marsden AL (2013) A new preconditioning technique for implicitly coupled multidomain simulations with applications to hemodynamics. Comput Mech 52:1141–1152. https://doi.org/10.1007/s0046601308681
Long CC, EsmailyMoghadam M, Marsden AL, Bazilevs Y (2014) Computation of residence time in the simulation of pulsatile ventricular assist devices. Comput Mech 54:911–919. https://doi.org/10.1007/s004660130931y
Yao J, Liu GR (2014) A matrixform GSMCFD solver for incompressible fluids and its application to hemodynamics. Comput Mech 54:999–1012. https://doi.org/10.1007/s0046601409908
Long CC, Marsden AL, Bazilevs Y (2014) Shape optimization of pulsatile ventricular assist devices using FSI to minimize thrombotic risk. Comput Mech 54:921–932. https://doi.org/10.1007/s004660130967z
Hsu MC, Kamensky D, Bazilevs Y, Sacks MS, Hughes TJR (2014) Fluid–structure interaction analysis of bioprosthetic heart valves: significance of arterial wall deformation. Comput Mech 54:1055–1071. https://doi.org/10.1007/s0046601410594
Hsu MC, Kamensky D, Xu F, Kiendl J, Wang C, Wu MCH, Mineroff J, Reali A, Bazilevs Y, Sacks MS (2015) Dynamic and fluid–structure interaction simulations of bioprosthetic heart valves using parametric design with Tsplines and Fungtype material models. Comput Mech 55:1211–1225. https://doi.org/10.1007/s004660151166x
Kamensky D, Hsu MC, Schillinger D, Evans JA, Aggarwal A, Bazilevs Y, Sacks MS, Hughes TJR (2015) An immersogeometric variational framework for fluid–structure interaction: application to bioprosthetic heart valves. Comput Methods Appl Mech Eng 284:1005–1053
Hughes TJR, Liu WK, Zimmermann TK (1981) Lagrangian–Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29:329–349
Tezduyar TE, Sathe S, Cragin T, Nanna B, Conklin BS, Pausewang J, Schwaab M (2007) Modeling of fluid–structure interactions with the space–time finite elements: arterial fluid mechanics. Int J Numer Methods Fluids 54:901–922. https://doi.org/10.1002/fld.1443
Tezduyar TE, Sathe S, Schwaab M, Conklin BS (2008) Arterial fluid mechanics modeling with the stabilized space–time fluid–structure interaction technique. Int J Numer Methods Fluids 57:601–629. https://doi.org/10.1002/fld.1633
Tezduyar TE, Schwaab M, Sathe S (2009) Sequentiallycoupled arterial fluid–structure interaction (SCAFSI) technique. Comput Methods Appl Mech Eng 198:3524–3533. https://doi.org/10.1016/j.cma.2008.05.024
Takizawa K, Christopher J, Tezduyar TE, Sathe S (2010) Space–time finite element computation of arterial fluid–structure interactions with patientspecific data. In J Numer Methods Biomed Eng 26:101–116. https://doi.org/10.1002/cnm.1241
Tezduyar TE, Takizawa K, Moorman C, Wright S, Christopher J (2010) Multiscale sequentiallycoupled arterial FSI technique. Comput Mech 46:17–29. https://doi.org/10.1007/s0046600904232
Takizawa K, Moorman C, Wright S, Christopher J, Tezduyar TE (2010) Wall shear stress calculations in space–time finite element computation of arterial fluid–structure interactions. Comput Mech 46:31–41. https://doi.org/10.1007/s0046600904250
Takizawa K, Moorman C, Wright S, Purdue J, McPhail T, Chen PR, Warren J, Tezduyar TE (2011) Patientspecific arterial fluid–structure interaction modeling of cerebral aneurysms. Int J Numer Methods Fluids 65:308–323. https://doi.org/10.1002/fld.2360
Tezduyar TE, Takizawa K, Brummer T, Chen PR (2011) Space–time fluid–structure interaction modeling of patientspecific cerebral aneurysms. Int J Numer Methods Biomed Eng 27:1665–1710. https://doi.org/10.1002/cnm.1433
Takizawa K, Brummer T, Tezduyar TE, Chen PR (2012) A comparative study based on patientspecific fluid–structure interaction modeling of cerebral aneurysms. J Appl Mech 79:010908. https://doi.org/10.1115/1.4005071
Takizawa K, Bazilevs Y, Tezduyar TE, Hsu MC, Øiseth O, Mathisen KM, Kostov N, McIntyre S (2014) Engineering analysis and design with ALEVMS and space–time methods. Arch Comput Methods Eng 21:481–508. https://doi.org/10.1007/s1183101491130
Takizawa K, Schjodt K, Puntel A, Kostov N, Tezduyar TE (2012) Patientspecific computer modeling of blood flow in cerebral arteries with aneurysm and stent. Comput Mech 50:675–686. https://doi.org/10.1007/s0046601207604
Takizawa K, Schjodt K, Puntel A, Kostov N, Tezduyar TE (2013) Patientspecific computational fluid mechanics of cerebral arteries with aneurysm and stent. In: Li S, Qian D (eds) Multiscale simulations and mechanics of biological materials, Chapter 7. Wiley, New York, pp 119–147 ISBN 9781118350799
Takizawa K, Schjodt K, Puntel A, Kostov N, Tezduyar TE (2013) Patientspecific computational analysis of the influence of a stent on the unsteady flow in cerebral aneurysms. Comput Mech 51:1061–1073. https://doi.org/10.1007/s004660120790y
Takizawa K, Bazilevs Y, Tezduyar TE, Long CC, Marsden AL, Schjodt K (2014) ST and ALEVMS methods for patientspecific cardiovascular fluid mechanics modeling. Math Models Methods Appl Sci 24:2437–2486. https://doi.org/10.1142/S0218202514500250
Takizawa K, Bazilevs Y, Tezduyar TE, Long CC, Marsden AL, Schjodt K (2014) Patientspecific cardiovascular fluid mechanics analysis with the ST and ALEVMS methods. In: Idelsohn SR (ed) Numerical simulations of coupled problems in engineering, volume 33 of computational methods in applied sciences, chapter 4. Springer, Berlin, pp 71–102. https://doi.org/10.1007/9783319061368_4 ISBN 9783319061351
Takizawa K, Tezduyar TE, Buscher A, Asada S (2014) Space–time interfacetracking with topology change (STTC). Compu Mech 54:955–971. https://doi.org/10.1007/s0046601309357
Takizawa K (2014) Computational engineering analysis with the newgeneration space–time methods. Comput Mech 54:193–211. https://doi.org/10.1007/s004660140999z
Takizawa K, Tezduyar TE, Buscher A, Asada S (2014) Space–time fluid mechanics computation of heart valve models. Comput Mech 54:973–986. https://doi.org/10.1007/s0046601410469
Takizawa K, Tezduyar TE, Terahara T, Sasaki T (2018) Heart valve flow computation with the space–time slip interface topology change (STSITC) method and isogeometric analysis (IGA). In: Wriggers P, Lenarz T (eds) Biomedical technology: modeling, experiments and simulation, Lecture notes in applied and computational mechanics. Springer, Berlin, pp 77–99. https://doi.org/10.1007/9783319595481_6 ISBN 9783319595474
Takizawa K, Tezduyar TE, Terahara T, Sasaki T (2017) Heart valve flow computation with the integrated space–time VMS, slip interface, topology change and isogeometric discretization methods. Comput Fluids 158:176–188. https://doi.org/10.1016/j.compfluid.2016.11.012
Takizawa K, Tezduyar TE, Uchikawa H, Terahara T, Sasaki T, Shiozaki K, Yoshida A, Komiya K, Inoue G (2018) Aorta flow analysis and heart valve flow and structure analysis. In: Tezduyar TE (ed) Frontiers in computational fluid–structure interaction and flow simulation: research from Lead Investigators under Forty—2018, modeling and simulation in science, engineering and technology. Springer, Berlin, pp 29–89. https://doi.org/10.1007/9783319964690_2 ISBN 9783319964683
Suito H, Takizawa K, Huynh VQH, Sze D, Ueda T (2014) FSI analysis of the blood flow and geometrical characteristics in the thoracic aorta. Comput Mech 54:1035–1045. https://doi.org/10.1007/s0046601410171
Suito H, Takizawa K, Huynh VQH, Sze D, Ueda T, Tezduyar TE (2016) A geometricalcharacteristics study in patientspecific FSI analysis of blood flow in the thoracic aorta. In: Bazilevs Y, Takizawa K (eds) Advances in computational fluid–structure interaction and flow simulation: new methods and challenging computations, modeling and simulation in science, engineering and technology. Springer, Berlin, pp 379–386. https://doi.org/10.1007/9783319408279_29 ISBN 9783319408255
Takizawa K, Tezduyar TE, Uchikawa H, Terahara T, Sasaki T, Yoshida A (2018) Mesh refinement influence and cardiaccycle flow periodicity in aorta flow analysis with isogeometric discretization. Comput Fluids. https://doi.org/10.1016/j.compfluid.2018.05.025
Takizawa K, Torii R, Takagi H, Tezduyar TE, Xu XY (2014) Coronary arterial dynamics computation with medicalimagebased timedependent anatomical models and elementbased zerostress state estimates. Comput Mech 54:1047–1053. https://doi.org/10.1007/s0046601410496
Takizawa K, Tezduyar TE (2012) Space–time fluid–structure interaction methods. Math Models Methods Appl Sci 22(supp02):1230001. https://doi.org/10.1142/S0218202512300013
Takizawa K, Bazilevs Y, Tezduyar TE, Hsu MC, Øiseth O, Mathisen KM, Kostov N, McIntyre S (2014) Computational engineering analysis and design with ALEVMS and ST methods. In: Idelsohn SR (ed) Numerical simulations of coupled problems in engineering, volume 33 of computational methods in applied sciences, chapter 13. Springer, Berlin, pp 321–353. https://doi.org/10.1007/9783319061368_13 ISBN 9783319061351
Bazilevs Y, Takizawa K, Tezduyar TE (2015) New directions and challenging computations in fluid dynamics modeling with stabilized and multiscale methods. Math Models Methods Appl Sci 25:2217–2226. https://doi.org/10.1142/S0218202515020029
Takizawa K, Tezduyar TE, Mochizuki H, Hattori H, Mei S, Pan L, Montel K (2015) Space–time VMS method for flow computations with slip interfaces (ST–SI). Math Models Methods Appl Sci 25:2377–2406. https://doi.org/10.1142/S0218202515400126
Takizawa K, Tezduyar TE (2016) New directions in space–time computational methods. In: Bazilevs Y, Takizawa K (eds) Advances in computational fluid–structure interaction and flow simulation: new methods and challenging computations, modeling and simulation in science, engineering and technology. Springer, Berlin, pp 159–178. https://doi.org/10.1007/9783319408279_13 ISBN 9783319408255
Takizawa K, Tezduyar TE, Otoguro Y, Terahara T, Kuraishi T, Hattori H (2017) Turbocharger flow computations with the space–time isogeometric analysis (STIGA). Comput Fluids 142:15–20. https://doi.org/10.1016/j.compfluid.2016.02.021
Takizawa K, Tezduyar TE, Asada S, Kuraishi T (2016) Space–time method for flow computations with slip interfaces and topology changes (STSITC). Comput Fluids 141:124–134. https://doi.org/10.1016/j.compfluid.2016.05.006
Otoguro Y, Takizawa K, Tezduyar TE (2017) Space–time VMS computational flow analysis with isogeometric discretization and a generalpurpose NURBS mesh generation method. Comput Fluids 158:189–200. https://doi.org/10.1016/j.compfluid.2017.04.017
Otoguro Y, Takizawa K, Tezduyar TE (2018) A generalpurpose NURBS mesh generation method for complex geometries. In: Tezduyar TE (ed) Frontiers in computational fluid–structure interaction and flow simulation: research from lead investigators under Forty—2018, modeling and simulation in science, engineering and technology. Springer, Berlin, pp 399–434. https://doi.org/10.1007/9783319964690_10 ISBN 9783319964683
Takizawa K, Bazilevs Y, Tezduyar TE (2012) Space–time and ALEVMS techniques for patientspecific cardiovascular fluid–structure interaction modeling. Arch Comput Methods Eng 19:171–225. https://doi.org/10.1007/s1183101290713
Tezduyar TE, Cragin T, Sathe S, Nanna B (2007) FSI computations in arterial fluid mechanics with estimated zeropressure arterial geometry. In: Onate E, Garcia J, Bergan P, Kvamsdal T (eds) Marine 2007. CIMNE, Barcelona, Spain
Takizawa K, Takagi H, Tezduyar TE, Torii R (2014) Estimation of elementbased zerostress state for arterial FSI computations. Comput Mech 54:895–910. https://doi.org/10.1007/s0046601309197
Takizawa K, Tezduyar TE, Sasaki T (2018) Estimation of elementbased zerostress state in arterial FSI computations with isogeometric wall discretization. In: Wriggers P, Lenarz T (eds) Biomedical technology: modeling, experiments and simulation, lecture notes in applied and computational mechanics. Springer, Berlin, pp 101–122. https://doi.org/10.1007/9783319595481_7 ISBN 9783319595474
Takizawa K, Tezduyar TE, Sasaki T (2017) Aorta modeling with the elementbased zerostress state and isogeometric discretization. Comput Mech 59:265–280. https://doi.org/10.1007/s0046601613445
Takizawa K, Tezduyar TE, Sasaki T (2018) Isogeometric hyperelastic shell analysis with outofplane deformation mapping. Comput Mech. https://doi.org/10.1007/s0046601816163
Holzapfel GA, Sommer G, Auer M, Regitnig P, Ogden RW (2007) Layerspecific 3D residual deformations of human aortas with nonatherosclerotic intimal thickening. Ann Biomed Eng 35:530–545
Scott MA, Simpson RN, Evans JA, Lipton S, Bordas SPA, Hughes TJR, Sederberg TW (2013) Isogeometric boundary element analysis using unstructured Tsplines. Comput Methods Appl Mech Eng 254:197–221. https://doi.org/10.1016/j.cma.2012.11.001
Acknowledgements
This work was supported in part by JSTCREST; GrantinAid for Scientific Research (S) 26220002 from the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT); GrantinAid 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 GrantinAid 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 W911NF1710046 and Top Global University Project of Waseda University.
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Appendices
A Eigenvalues and eigenvectors of the curvature tensor
In calculating the eigenvalues from the quadratic equation
we obtain
This can also be written as
with \(l=1, 2\). In calculating the eigenvectors, we work with \(\left( v_1\right) ^\alpha \) and \(\left( v_2\right) ^\alpha \), the contravariant components of the nonnormalized eigenvectors. The vectors are normalized with the scaling factors \(\tau _1\) and \(\tau _2\):
With that, we obtain the relationship
which yields
Given \((v_1)^\alpha \), which we will explain later how to calculate, from Eqs. (103) and (105) we obtain
We calculate \(\mathbf {t}_2\) from the orthogonality between \(\mathbf {t}_1\) and \(\mathbf {t}_2\):
With the substitutions \(\mathbf {t}_1 = \left( {t}_1\right) ^\alpha \overline{\mathbf {g}}_\alpha \) and \(\mathbf {t}_2 = \left( {t}_2\right) _\beta \overline{\mathbf {g}}^\beta \), we obtain
The expression
where
satisfies Eq. (108). To calculate \(\tau _2\), we start from the covariant version of (105):
Substituting Eq. (109) into Eq. (111), we obtain
By expanding the matrices and rearranging, we obtain the relationship
Combining Eqs. (112) and (114), we obtain
and then
Using Eqs. (109) and (116) in the expression
we obtain
Now we explain how to calculate \(\left( v_1\right) ^{\alpha }\). From Eqs. (43) and (100), we write the equations that \(\left( v_1\right) ^\alpha \) and \(\hat{\kappa }_1\) satisfy:
If \(\hat{\kappa }^1_{\bullet 2}\) or \(\hat{\kappa }_1  \hat{\kappa }^1_{\bullet 1}\) is nonzero, we can write, from Eq. (119),
If \(\hat{\kappa }_1  \hat{\kappa }^2_{\bullet 2}\) or \(\hat{\kappa }^2_{\bullet 1}\) is nonzero, we can write, from Eq. (120),
If both \(\hat{\kappa }_1  \hat{\kappa }^1_{\bullet 1}\) and \(\hat{\kappa }^2_{\bullet 1}\) are zero, we can write,
Taking into account the numerical stability issues related to dealing with very small numbers, we select one of these three equations in the order given below.

1.
If \(\left \hat{\kappa }_1\hat{\kappa }^1_{\bullet 1}\right > \varepsilon \left \hat{\kappa }^1_{\bullet 2} \right \), where \(\varepsilon = 10^{6}\), then Eq. (122) is selected.

2.
If \(\left \hat{\kappa }_1\hat{\kappa }^1_{\bullet 1}\right \le \varepsilon \left \hat{\kappa }^1_{\bullet 2} \right \) and if \(\left \hat{\kappa }^2_{\bullet 1} \right > \varepsilon \left \hat{\kappa }_1  \hat{\kappa }^2_{\bullet 2} \right \), then Eq. (123) is selected.

3.
Otherwise Eq. (124) is selected.
B Derivative of the normal vector
Derivative of the normal vector with respect to \(\xi ^\alpha \) can be obtained as follows:
In the derivation, we used the following relationships, which generally hold:
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Sasaki, T., Takizawa, K. & Tezduyar, T.E. Aorta zerostress state modeling with Tspline discretization. Comput Mech 63, 1315–1331 (2019). https://doi.org/10.1007/s0046601816510
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DOI: https://doi.org/10.1007/s0046601816510
Keywords
 Patientspecific arterial FSI
 Imagebased geometry
 Aorta
 Zerostress state
 Isogeometric wall discretization
 Tspline basis functions
 Integrationpointbased zerostress state
 Shellmodelbased initial guess