Blood-Artery Interaction in Calcified Aortas and Abdominal Aortic Aneurysms

Abstract

A stabilized FSI method is presented for coupling shear-rate dependent model of blood with finitely deforming anisotropic hyperelastic model of arteries. The blood-artery coupling conditions are weakly enforced to accommodate non-matching blood-artery meshes which provides great flexibility in independent discretization of fluid and solid subdomains in the patient-specific geometric models. The variationally derived interface coupling terms play an important role in the concurrent solution of the nonlinear mixed-field problem across non-matching discretizations. Two test cases are presented that investigate the effect of growth of aortic aneurysm on local changes in blood flow and stress concentrations in calcified arteries under pulsating flows to highlight the clinical relevance of the proposed method for cardiovascular applications.

1. Introduction

Aortic aneurysm is an arterial disease that is characterized by decreased stiffness of the artery walls which leads to abnormal bulging of the aortic vessels. Tears in one or more of the layers of the wall of the aorta (aortic dissection) or a ruptured aortic aneurysm can lead to life-threatening internal bleeding. Small blood clots can develop inside the aortic aneurysm, and if a blood clot breaks loose from the inner wall of an aneurysm, it can block a blood vessel elsewhere in the body. The essential elements for accurate modeling of blood flow through aneurysms include non-Newtonian constitutive models of blood, the evolving mechanical properties of the artery walls, pulsatile nature of flow, and large motion of the confining arterial walls [1].

Likewise, percutaneous vascular access is an important determinant to assess the feasibility of complex structural heart and valvular interventions. Transcatheter Aortic Valve Replacement (TAVR) is an option for patients who are at intermediate to high morbidity and mortality risk from open heart surgery and surgical valve replacement [2]. A major difficulty in performing the procedure is the increased calcification and tortuosity of the femoral access route in the target TAVR population (70+ years old). Of particular concern is the risk of periprocedural vascular complications that can range from access site bleeding, failure of percutaneous vascular access due to severe calcification of the vessel wall, and rupture of the vessel from the inability to accommodate the sheath site. Modeling of flow and local stress concentrations around calcifications [3] can help visualize the blood artery interactions [4,5] in calcified and narrowed arteries and analyze different physiological parameters for assessing the safety of a particular vascular access to help reduce the risk of vascular complications.

A distinguishing feature of biological tissues is that they grow and remodel under mechanical and chemical stimuli by exchanging mass with their environment. In the modeling of flow through aneurysms and around calcified plaques, an important modeling consideration is the evolving mechanical properties of the arterial tissue that has profound impact on the local blood-artery interaction and flow features. Proliferation and atrophy of cells and their associated extracellular matrices in response to persistent mechanical stimuli have been documented for hard tissues [6], soft tissues [7,8], and tumors [9], and growth in the cardiovascular system [10–12]. A unified computational method is presented in [13] for the modeling of growth in hard as well as soft biological tissues using the theory of mixtures where relative motions and interactions of the constituents are treated through the incorporation of mass transfer and drag force terms.

In cardiovascular flows, the interactions between blood and artery walls produce unique interfacial physics, the effect of which is compounded due to the pulsatile nature of blood flow [14]. In addition, the intrinsic anisotropy of the tissue and the viscoelastic nature of blood lead to highly nonlinear coupled fields. Since interfacial phenomena are highly sensitive to small perturbations to the interfacial geometry, developing an appropriate computational grid requires a refined mesh next to the artery walls to capture the boundary layer because of its profound effect impact on the calculation of the wall shear stresses. The flexibility of not requiring nodally matched discretizations at blood-artery interaction surfaces to enforce continuity of primary fields is extremely beneficial in complex geometries arising in patient-specific models. A literature review reveals that Nitsche-type methods have also been employed to variationally coupled fluid and solid subdomain [15].

In this work we employ a stable interface method for enforcing interface conditions across non-matching interfacial meshes [16]. The method results in uniform coupling of a stabilized formulation for incompressible non-Newtonian model for blood and the finite-deformation formulation for anisotropic hyperelastic tissue for the arterial walls. The method also allows for weakly enforcing the Dirichlet boundary conditions in the variational formulation rather than prescribing the value of the Dirichlet data directly at the nodal points [17]and this helps with applying resistance boundary conditions at the outflows to replicate the distal part of the arterial system [18]. An important ingredient in the present paper is the use of shear-rate dependent non-Newtonian model for blood [19–26]. The flow of shear-thinning fluids generally gives rise to sharper and thinner boundary layersClick or tap here to enter text.. Therefore, the ability of the method to capture the boundary layer is of significant importance for the shear-thinning models than for the Newtonian models. The stabilized methods for this class of fluids were developed by the senior author, and the interested reader is referred to [21–26]. An overview of stabilized methods is presented in [27].

An outline of the paper is as follows: Section 2 presents the blood-artery interaction model. Section 3 presents the derivation of the stabilized formulation. Section 4 presents application to patient-specific arterial geometries of calcified aorta and abdominal aortic aneurysm to illustrate its clinical relevance. Conclusions are drawn in Section 5.

2. Blood-Artery Interaction Model

Let $\text{Ω}\subset {ℝ}^{n_{\text{sd }}}$ be an open bounded region with piecewise smooth boundary $\text{Γ}$. The number of spatial dimensions, n_sd , is equal to 2 or 3. The domain boundary assumes the usual split $\text{Γ}={\text{Γ}}_g\cup {\text{Γ}}_h$ and ${\text{Γ}}_g\cap {\text{Γ}}_h=\varnothing$, where ${\text{Γ}}_g$ and ${\text{Γ}}_h$ are parts of the boundary with essential and natural boundary conditions, respectively. We consider two subdomains comprised of fluid Ω₁ and solid Ω₂ representing blood and artery respectively, coupled at their common interface ${\text{Γ}}_I$. In the following sections, we present an interface coupling method for fluid-structure interaction (FSI), where (•)₁ denotes and (•)₂ denote quantities related to fluid and solid, respectively.

2.1. Modeling of non-Newtonian behavior of blood

The blood is modeled as incompressible non-Newtonian fluid. The system of equations is written in an Arbitrary Lagrangian-Eulerian (ALE) frame to accommodate the moving boundaries.

$$\begin{gathered} \rho \frac{\partial v}{\partial t}+{\rho }_1\left(v_1-v_m\right)\cdot \nabla v_1 \\ -\nabla \cdot {\mathbf{\sigma }}_v(v)+\nabla p=\rho f{\text{ in Ω}}_1\times ]0,\text{T}[ \end{gathered}$$ (1)

$$\nabla \cdot v_1=0{\text{ in Ω}}_1\times ]0,\text{T}[$$ (2)

$${\mathit{v}}_1={\mathit{g}}_1{\text{ on Γ}}_g\times ]0,\text{T}[$$ (3)

$${\mathit{\sigma }}_1{\mathit{n}}_1={\mathit{h}}_1{\text{ on Γ}}_h\times ]0,\text{T}[$$ (4)

$$\nabla \cdot \left(\left(1+{\alpha }^e\right)\nabla {\mathit{u}}_m\right)=0{\text{ in Ω}}_1\times ]0,T[$$ (5)

where v₁(x,t) and p(x,t) are the fluid velocity and pressure, respectively. We also append the initial conditions to the system. ρ is the density of the fluid, σ_v is the deviatoric stress tensor, σ is the Cauchy stress tensor defined as $\mathbf{\sigma }=-p\mathit{I}+{\mathit{\sigma }}_v$, I is the identity tensor, f is the body force per unit mass, g is the prescribed velocity on the boundary ${\text{Γ}}_g$, h is the prescribed traction on the boundary ${\text{Γ}}_h$, and n is the unit outward normal to the boundary of the domain Ω . Equations (1)–(4) represent balance of momentum, the continuity equation, the Dirichlet and Neumann boundary conditions, respectively. Equation (5) models the motion of the fluid mesh where u_m (x, t) is the mesh displacement field.

Blood shows non-Newtonian behavior that is characterized by the shear-thinning effect. The shear-rate dependent deviatoric stress tensor is defined as

$${\mathbf{\sigma }}_v=2\eta (\dot{\gamma })\mathbf{\epsilon }(v)$$ (6)

where $\mathit{\epsilon }\left(v\right)=\mathrm{½}\left(\nabla v+\nabla v^T\right)$ is the rate-of-deformation tensor, $\dot{\gamma }$ is the shear-rate defined as $\dot{\gamma }={(2\mathit{\epsilon }(\mathit{v}):\mathit{\epsilon }(\mathit{v}))}^{0.5}$, and $\eta \left(\dot{\gamma }\right)$ is the effective viscosity which is a function of the shear-rate. In this work, we have employed the Carreau-Yasuda model for shear-thinning behavior of blood, which is a commonly used model in computational hemodynamics [1,8,19,20,22]. In this model, the nonlinear function of the viscosity is given by

$$\eta (\dot{\gamma })={\mu }_{\infty }+\left({\mu }_0-{\mu }_{\infty }\right){\left[1+{(\lambda \dot{\gamma })}^a\right]}^{(n-1)/a}$$ (7)

where ${\mu }_0$ and ${\mu }_{\infty }$ are the asymptotic viscosities at zero and infinite shear-rate, respectively, and a , n and λ are empirically determined constitutive parameters. Coefficients a and n are non-dimensional parameters that control the shear-thinning or shear-thickening behavior of fluid between the two asymptotic viscosities. The model reverts to the Newtonian fluid model by setting ${\mu }_0={\mu }_{\infty }$.

2.2. Modeling of anisotropic hyperelastic arterial tissue

The deformation of arterial tissue is governed by nonlinear finite-strain elasticity. The momentum balance equations for the solid are written in terms of velocity as the primary unknown to develop a monolithic coupling method for FSI. The governing equations are written in the current configuration.

$${\rho }_2\frac{\partial v_2}{\partial t}-\nabla \cdot {\sigma }_2={\rho }_2{f}_2{\text{ in Ω}}_2\times ]0,\text{T}[$$ (8)

$$v_2=g_2{\text{ on Γ}}_2^g\times ]0,T[$$ (9)

$${\sigma }_2{n}_2=h_2{\text{ on Γ}}_2^h\times ]0,\text{T}[$$ (10)

$$u_2(x,t)=u_2(x,0)+{\displaystyle {\int }_0^f{v}_2(\tau )d\tau }{\text{ in Ω}}_2\times ]0,\text{T}[$$ (11)

where v₂ (x,t) and u₂(x,t) are the velocity and displacement fields in the solid, respectively, ρ₂ is the density of the solid, σ₂ is the Cauchy stress tensor, and f₂ is the body force vector. g₂ is the prescribed velocity at ${\text{Γ}}_2^g$, h₂ is the prescribed traction at ${\text{Γ}}_2^h$ and n₂ is the unit normal vector to the boundary ${\text{Γ}}_2$. We append $v_2(x,0)=v_{2,0}$ and $u_2(x,0)=u_{2,0}$ as the initial velocity and displacement conditions, respectively. Equations (8)–(10) represent the momentum balance equations, and the Dirichlet and Neumann boundary conditions, respectively. Equation (11) represents the relationship between the displacement and the velocity of the solid.

2.3. Evolving local arterial properties: Aneurysm (softening) and Calcification (stiffening)

In order to model biological processes at the tissue scale, some degree of homogenization is typically employed for the evolution and progression of disease in the arterial walls. In this work we have adopted a simplified continuum approach and treat the evolving material adaptation via a scalar function d to model softening (degradation of material properties) or stiffening (gain in stiffness) of the arterial wall. This function can vary locally in space to model the local change in the evolving properties of the arterial tissue.

$$W(C,d)\equiv (1-d)W^0(C)$$ (12)

where $W^0(C)$ is the potential energy function of the original (non-diseased) arterial tissue. The softening of the tissue leads to energy release rate that is computed based on this averaged free energy potential.

The notion of internal variable presented in (12) results in transition of mechanical properties from the healthy to the diseased tissue. This evolution of material parameter for softening or stiffening of artery is described by Equation (16) where E₁ and E₂ are the initial and final values for the parameter. t₁ and t₂ are the start and end points in time for the evolution of the biological processes.

$$E=\{\begin{array}{cc}E=E_1& t<t_1\\ \frac{\left(E_2-E_1\right)}{2}\sin \left(\pi \left(\frac{2t-\left(t_1+t_2\right)}{2\left(t_2-t_1\right)}\right)\right)& \\ +\frac{\left(E_1+E_2\right)}{2}& t_1\le t<t_2\\ E=E_1& t\ge t_2\end{array}$$ (13)

where it represents aneurysm if E₁ > E₂ , and represents calcification if E₁ < E₂.

3. The stabilized interface coupling method

Numerical methods that are not constrained by node-on-node match provide great flexibility in developing patient specific computational models. We employ the Variational Multiscale Discontinuous Galerkin (VMDG) method to develop a robust FSI algorithm for blood-artery interaction problem as described below.

3.1. VMDG Formulation for Blood-Artery Interaction

The coupling of models for blood and arterial tissue is achieved by imposing the following two interface conditions at the blood-artery interface ${\text{Γ}}_I$:

$$v_1=v_2$$ (14)

$${\sigma }_1{n}_1+{\sigma }_2{n}_2=0$$ (15)

where ${\sigma }_{\alpha }$ is the Cauchy stress tensor and $n_{\alpha }$ is the outward unit normal vector. The first interface condition in (14) is the kinematic constraint to satisfy the continuity of the velocity field. The second condition (15) enforces balance of tractions between the fluid and solid subdomains. In this work the fluid and solid subdomains are tied together using variationally derived interface coupling terms [16] that result in a monolithic FSI method for enforcing the two interfacial conditions on matching as well as on non-matching interfacial discretization. In the VMDG method, the continuity conditions are enforced via the numerical flux given in (16), which is derived by variationally embedding the fine-scale model (17) in the variational multiscale formulation.

$$\lambda =-\left\{t\right\}+{\tau }_S〚v〛$$ (16)

$${\tilde{v}}_{\alpha }=-{\delta }_S\left(t_1+t_2\right)+{(-1)}^{\alpha }{\delta }_{\alpha }^T〚v〛$$ (17)

where $〚·〛={(·)}_1-{(·)}_2$ is the jump operator and $\left\{\mathit{t}\right\}={\mathit{\delta }}_1{\mathit{t}}_1-{\mathit{\delta }}_2{\mathit{t}}_2$ is the weighted average operator for the Cauchy traction $\mathit{t}=\mathit{\sigma }\cdot \mathit{n}$ at the interface. In the above expressions, ${\tau }_S$ and ${\delta }_S$ are the stabilization tensors that are also defined in closed-form in [16].

Let ${\mathcal{V}}_{\alpha }={\left(H^1\left({\text{Ω}}_{\alpha }\right)\right)}^{n_{\text{sd }}}$, $\mathcal{P}=H^1\left({\text{Ω}}_{\alpha }\right)$ be the appropriate spaces of functions for the velocity and pressure fields such that $\left\{{\mathit{v}}_1,p\right\}\in {\mathcal{L}}_1\times \mathcal{P}$ and ${\mathit{v}}_2\in {\mathcal{V}}_2$ The weak form of the coarse-scale formulations for fluid and solid, that emanate from the application of the VMS method to the fluid and the solid subsystems, can be written as follows:

$$\begin{array}{l}{\left({\mathit{w}}_1,{\rho }_1\frac{\partial {\mathit{v}}_1}{\partial t}\right)}_{{\text{Ω}}_1}+{\left({\mathit{w}}_1,{\rho }_1\left({\mathit{v}}_1-{\mathit{v}}_m\right)\cdot \nabla \mathit{v}\right)}_{{\text{Ω}}_1}\hfill \\ +{\left(\nabla {\mathit{w}}_1,{\sigma }_{\mathit{v}}\left({\mathit{v}}_1\right)\right)}_{{\text{Ω}}_1}-{\left(\nabla \cdot {\mathit{w}}_1,p\right)}_{{\text{Ω}}_1}+{\left(q,\nabla \cdot {\mathit{v}}_1\right)}_{{\text{Ω}}_1}\hfill \\ +{\left(\mathit{\chi },\mathit{\tau }{\mathit{r}}_1\right)}_{{\text{Ω}}_1}+{\left({\mathit{w}}_1,\mathit{\lambda }\right)}_{{\text{Γ}}_I}\hfill \\ \begin{array}{l}+{\left({\rho }_1\left(\left({\mathit{v}}_1-{\mathit{v}}_m\right)\cdot {\mathit{n}}_1\right){\mathit{w}}_1+2\eta (\dot{\gamma })\mathit{\epsilon }\left({\mathit{w}}_1\right){\mathit{n}}_1+q{\mathit{n}}_1,\text{Δ}{\tilde{\mathit{v}}}_1\right)}_{{\text{Γ}}_I}\\ +{\left(4{\eta }^{\prime }(\dot{\gamma }){\dot{\gamma }}^{-1}\left(\nabla {\mathit{w}}_1:\mathit{\epsilon }\left({\mathit{v}}_1\right)\right)n_1\cdot \mathit{\epsilon }\left({\mathit{v}}_1\right),\text{Δ}{\tilde{\mathit{v}}}_1\right)}_{{\text{Γ}}_I}\end{array}\hfill \\ ={\left({\mathit{w}}_1,{\rho }_1{\mathit{f}}_1\right)}_{{\text{Ω}}_1}+{\left({\mathit{w}}_1,{\mathit{h}}_1\right)}_{{\text{Γ}}_1^{\mathit{h}}}\forall \left\{{\mathit{w}}_1,q\right\}\in {\mathcal{V}}_1\times \mathcal{P}{\text{ in Ω}}_1\times ]0,T[\hfill \end{array}$$ (18)

$$\begin{array}{l}{\left({\mathit{w}}_2,{\rho }_2\frac{\partial {\mathit{v}}_2}{\partial t}\right)}_{{\text{Ω}}_2}+{\left(\nabla {\mathit{w}}_2,{\mathit{\sigma }}_2\right)}_{{\text{Ω}}_2}-{\left({\mathit{w}}_2,\mathit{\lambda }\right)}_{{\text{Γ}}_I}\hfill \\ +{\left(\text{Δ}t\left[\left({\mathit{\sigma }}_2{\mathit{n}}_2\right)\cdot \nabla {\mathit{w}}_2+\left(\nabla {\mathit{w}}_2:\mathit{c}\right)\cdot {\mathit{n}}_2\right],\text{Δ}{\tilde{\mathit{v}}}_2\right)}_{{\text{Γ}}_I}\hfill \\ ={\left({\mathit{w}}_2,{\rho }_2{\mathit{f}}_2\right)}_{{\text{Ω}}_2}+{\left({\mathit{w}}_2,{\mathit{h}}_2\right)}_{{\text{Γ}}_h^h}\forall {\mathit{w}}_2\in {\mathcal{V}}_2{\text{ in Ω}}_2\times ]0,\text{T}[\hfill \end{array}$$ (19)

where (•,•) is the L₂ inner product δt is the variation of the traction. The first term in the third line in (18) is the stabilization term for the fluid part that addresses the issues of numerical instabilities associated with convective flow and mixed formulation. Incorporating the expressions for λ and ${\tilde{v}}_{\alpha }$ in the coarse-scale formulation yields the stabilized interface formulation for fluid-structure interaction which is now a function of only the coarse-scale fields. Let ${\mathcal{S}}_{1t}\times {\mathcal{P}}_t$ be the time-dependent counter parts of the spaces ${\mathcal{V}}_1\times \mathcal{P}$. The formal statement for finite deformation FSI problems can be written in a residual form as: Find $\left\{{\mathit{v}}_1,p\right\}\in {\mathcal{S}}_{1t}\times {\mathcal{P}}_t,{\mathit{v}}_2\in {\mathcal{V}}_{2t}$, such that for all $\left\{{\mathit{w}}_1,q\right\}\in {\mathcal{V}}_1\times \mathcal{P},{\mathit{w}}_2\in {\mathcal{V}}_2$, the following holds.

$$\begin{array}{l}{B}_1\left(\left\{{\mathit{w}}_1,q\right\},\left\{{\mathit{v}}_1,p\right\};v_m\right)\hfill \\ -F_1\left({\mathit{w}}_1\right)+B_2\left({\mathit{w}}_2,{\mathit{v}}_2\right)-F_2\left({\mathit{w}}_2\right)\hfill \\ -{(〚\mathit{w}〛,\{\mathit{t}\})}_{{\text{Γ}}_I}-{(\{\mathit{\delta }\mathit{t}\},〚\mathit{v}〛)}_{{\text{Γ}}_I}+{\left(〚\mathit{w}〛,{\mathit{\tau }}_S〚\mathit{v}〛\right)}_{{\text{Γ}}_I}\hfill \\ -{\left(\mathit{\delta }{\mathit{t}}_1+\mathit{\delta }{t}_2,{\mathit{\delta }}_S\left({\mathit{t}}_1+t_2\right)\right)}_{{\text{Γ}}_I}=0\hfill \end{array}$$ (20)

The four interface terms in the fourth line enforce the two interface conditions in (16)-(17). The discrete form (20) is produced by employing the appropriate discrete version of the admissible spaces of functions. In (20) $B_1\left(\left\{w_1,q\right\},\left\{v_1,p\right\};v_m\right),F_1\left(w_1\right)$ is the VMS stabilized weak form for the fluid, and $B_2\left(w_2,v_2\right),F_2\left(w_2\right)$ is the standard Galerkin weak form for the nonlinear solid which is written in the updated-Lagrangian form.

$$\begin{gathered} B_1\left(\left\{{\mathit{w}}_1,q\right\},\left\{{\mathit{v}}_1,p\right\};{\mathit{v}}_m\right)={\left({\mathit{w}}_1,{\rho }_1\frac{\partial {\mathit{v}}_1}{\partial t}\right)}_{{\text{Ω}}_1} \\ +{\left({\mathit{w}}_1,{\rho }_1\left({\mathit{v}}_1-{\mathit{v}}_m\right)\cdot \nabla \mathit{v}\right)}_{{\text{Ω}}_1}+{\left(\nabla {\mathit{w}}_1,{\sigma }_{\mathit{v}}\left({\mathit{v}}_1\right)\right)}_{{\text{Ω}}_1} \\ -{\left(\nabla \cdot {\mathit{w}}_1,p\right)}_{{\text{Ω}}_1}+{\left(q,\nabla \cdot {\mathit{v}}_1\right)}_{{\text{Ω}}_1}+{\left(\mathit{\chi },\mathit{\tau }{\mathit{r}}_1\right)}_{{\text{Ω}}_1} \end{gathered}$$ (21)

$$F_1\left({\mathit{w}}_1\right)={\left({\mathit{w}}_1,{\rho }_1{\mathit{f}}_1\right)}_{{\text{Ω}}_1}+{\left({\mathit{w}}_1,{\mathit{h}}_1\right)}_{{\text{Γ}}_1^{\mathit{h}}}$$ (22)

$$B_2\left({\mathit{w}}_2,{\mathit{v}}_2\right)={\left({\mathit{w}}_2,{\rho }_2\frac{\partial {\mathit{v}}_2}{\partial t}\right)}_{{\text{Ω}}_2}+{\left(\nabla {\mathit{w}}_2,{\mathit{\sigma }}_2\right)}_{{\text{Ω}}_2}$$ (23)

$$F_2\left({\mathit{w}}_2\right)={\left({\mathit{w}}_2,{\rho }_2{\mathit{f}}_2\right)}_{{\text{Ω}}_2}+{\left(w_2,{\mathit{h}}_2\right)}_{{\text{Γ}}_2^{\mathit{h}}}$$ (24)

and

$$\begin{gathered} \delta t_1={\rho }_1\left(\left({\mathit{v}}_1-{\mathit{v}}_m\right)\cdot {\mathit{n}}_1\right){\mathit{w}}_1+2\eta (\dot{\gamma })\mathit{\epsilon }\left({\mathit{w}}_1\right){\mathit{n}}_1+q{\mathit{n}}_1 \\ +4{\eta }^{\prime }(\dot{\gamma }){\dot{\gamma }}^{-1}\left(\nabla {\mathit{w}}_1:\epsilon \left({\mathit{v}}_1\right)\right){\mathit{n}}_1\cdot \epsilon \left({\mathit{v}}_1\right) \end{gathered}$$ (25)

$$\delta {\mathit{t}}_2=\text{Δ}t\left[\left({\mathit{\sigma }}_2{\mathit{n}}_2\right)\cdot \nabla {\mathit{w}}_2+\left(\nabla {\mathit{w}}_2:\mathit{c}\right)\cdot {\mathit{n}}_2\right]$$ (26)

$$\left\{\mathit{t}\right\}={\mathit{\delta }}_1{\mathit{t}}_1-{\mathit{\delta }}_2{\mathit{t}}_2$$ (27)

where ${\mathit{t}}_{\alpha }={\mathit{\sigma }}_{\alpha }{\mathit{n}}_{\alpha }$ is the Cauchy traction, $\mathit{\delta }{\mathit{t}}_{\alpha }$ is the variation of the traction on either side of ${\text{Γ}}_I$ . The weighted average operator for the numerical flux at the interface is defined in (28). The traction stability tensor ${\mathit{\tau }}_{\alpha }$ that implicitly accounts for the effects of element size, material properties, and partial differential operators has the following expression.

$${\mathit{\tau }}_{\alpha }={\left[\text{meas}\left({\text{Γ}}_{\alpha }^e\right)\right]}^{-1}{\left({\displaystyle {\int }_{{\text{Γ}}_{\alpha }^e}{b}_{\alpha }d\text{Γ}}\right)}^2{\tilde{\mathit{\tau }}}_{\alpha }$$ (28)

where

$${\tilde{\tau }}_1={\left[\begin{array}{l}\underset{{\text{Ω}}_1^e}{{\displaystyle \int }}{\rho }_1{\left(b_1\right)}^2\nabla v_{1}d\text{Ω}+\underset{{\text{Ω}}_1^e}{{\displaystyle \int }}{\rho }_1{b}_1\left(v_1-v_m\right)\cdot \nabla b_{1}d\text{Ω}1\hfill \\ +\underset{{\text{Ω}}_1^e}{{\displaystyle \int }}\eta (\dot{\gamma })\left(\nabla b_1\otimes \nabla b_1\right)d\text{Ω}+\underset{{\text{Ω}}_1^e}{{\displaystyle \int }}\eta (\dot{\gamma }){\left|\nabla b_1\right|}^{2}d\text{Ω}1\hfill \\ +\underset{{\text{Ω}}_1^e}{{\displaystyle \int }}4{\eta }^{\prime }(\dot{\gamma }){\dot{\gamma }}^{-1}\left(\mathit{\epsilon }\left(v_1\right)\cdot \nabla b_1\right)\otimes \left(\nabla b_1\cdot \mathit{\epsilon }\left(v_1\right)\right)d\text{Ω}\hfill \end{array}\right]}^{-1}$$ (29)

$${\tilde{\mathit{\tau }}}_2={\left[\begin{array}{l}\underset{{\text{Ω}}_2^e}{{\displaystyle \int }}\text{Δ}t{\mathit{\sigma }}_2:\left(\nabla b_2\otimes \nabla b_2\right)d\text{Ω}\hfill \\ +\underset{{\text{Ω}}_2^e}{{\displaystyle \int }}\text{Δ}t\left(\nabla b_2\cdot c\cdot \nabla b_2\right)d\text{Ω}\hfill \end{array}\right]}^{-1}$$ (30)

The stability tensor ${\mathit{\tau }}_S$ and the flux weighting tensor ${\mathit{\delta }}_{\alpha }$ are defined in terms of ${\mathit{\tau }}_{\alpha }$ as

$${\mathit{\tau }}_S={\left({\mathit{\tau }}_1+{\mathit{\tau }}_2\right)}^{-1}$$ (31)

$${\mathit{\delta }}_{\alpha }={\mathit{\tau }}_s\cdot {\mathit{\tau }}_{\alpha }$$ (32)

Note that the summation of the flux weighting tensors is equal to the identity tensor, ${\mathit{\delta }}_1+{\mathit{\delta }}_2=1$. The additional interfacial stability tensor ${\mathit{\delta }}_S$ for the traction continuity is defined by

$${\mathit{\delta }}_S={\mathit{\tau }}_1\cdot {\mathit{\delta }}_2={\mathit{\tau }}_2\cdot {\mathit{\delta }}_1={\left({\mathit{\tau }}_1^{-1}+{\mathit{\tau }}_2^{-1}\right)}^{-1}$$ (33)

4. Numerical Results

We have implemented the stabilized FSI method using linear tetrahedral elements. The generalized-α method is used for time integration. The nonlinear problem is solved using the Newton-Raphson method and the linear system of equations is solved via the GMRES solver with additive Schwarz preconditioner.

Accurate prediction of stress and deformation in arterial wall in patient specific applications requires physiologically relevant constitutive models for the arteries that are comprised of soft tissue with embedded collagen fibers. In this work the multiple layers of artery wall, intima, media and adventitia, are modeled via hyperelastic energy functionals that accounts for finite stretching of the soft tissue as well as anisotropy induced by the directionally oriented collagen fibers. The reinforcing fibers are laid helically around the artery wall with alternate layers placed to form a network of directionally oriented layers.

An aneurysm in the artery leads to larger local deformation of the artery, resulting in local thinning of the wall. Blood-artery simulations provide local stress distribution ${\tilde{f}}_{loc}(x,t)$ in the artery, and the norm of maximum stress can be compared with the norm of critical stress that can trigger dissection. Based on this information a limit-state function g(x,t) can be defined for the probability of local rupture (i.e., local failure of the arterial system).

$$g(x,t)=1-\frac{\max \left({\tilde{f}}_{\text{loc }}(x,t)\right)}{{\tilde{F}}_{\max }(x,t)}$$ (34)

where ${\tilde{F}}_{\max }(x,t)$ is the norm of maximum allowable stress in the tissue. The distribution of g(x,t) can help identify the zones in the arteries that are most vulnerable to stress overloading, and therefore help with determining the percutaneous access route in TVAR patients by avoiding the procedures that may overload these regions.

4.1. Progressive growth of aortic aneurysm and blood-artery Interaction and

In the first test case we employ a Neo-Hookean model with embedded anisotropy. The energy functional is given as:

$$\begin{gathered} W(C)=\frac{{\mu }_1}{2}\left(I_1-3\right)+\frac{{\mu }_2}{2}{\left(J_4-1\right)}^2 \\ -{\mu }_1(J-1)+\frac{\kappa +{\mu }_1}{2}{(J-1)}^2 \end{gathered}$$ (35)

where $I_1=\text{tr}\text{C},J=\det F,J_4=\text{tr}[CM]$ are invariants of the deformation gradient tensor and the right Cauchy-Green deformation tensor. $M=\mathbf{a}\otimes \mathbf{a}$ is a second-order tensor that models anisotropy of the fibrous material with a direction vector a . μ₁ , $\kappa$ are the Lame parameter and the bulk modulus of Neo-Hookean solid, and μ₂ is the material parameter for reinforcing fibers. The corresponding stress tensors are provided in Appendix A.

The blood is modeled as shear-thinning fluid using the Carreau-Yasuda model given in Equation (7). The material parameters for Carreau-Yasuda fluids are set equal to μ₀ = 0.056 dyne s/cm² , λ=1.902 , n = 0.22 , a =1.25. The density of blood is ρ₁ =1.05 g/cm³ . We represent the diseased artery, i.e., aneurysm, by varying the value of d wherein Young’s modulus for the healthy and diseased arteries are $E_{healthy}=2.98\times {10}^6\text{ dyne }/{\text{cm}}^2$ and $E_{diseased}=9.95\times {10}^6\text{\hspace{0.17em}dyne }/{\text{cm}}^2$ respectively. Poisson’s ratio ν= 0.45 , and the density of artery is ρ₂ =1.16 g/cm³ . The stiffness of the fibers is set equal to μ₂ = 0.25 μ₁.

The mesh is comprised of 4-node tetrahedral elements wherein the number of fluid and solid elements is 106,888 and 92,863, respectively. The outflow boundary conditions are weakly applied [10,17,18,28]. The parabolic profile of the velocity is prescribed at the inflow and the RCR boundary conditions with the Windkessel model are applied at the outflow [18]. The proximal resistance is R_p =1014 dyn s/cm⁵ , the distal resistance is R_d =11055 dyn s/cm⁵ , and the capacitance is C = 9.5×10⁻⁵ cm / dyn.⁵ The support of the external tissue is modeled via elastic spring analogy [29].

We simulated seven cardiac cycles with a time increment of Δt = 0.005 sec , which corresponds to 220 time steps per typical cardiac cycle. The first few cycles model the evolution of the aneurysm, showing the progression of the disease which in fact occurs over months or years. This is shown in Fig. 1.

Figure 1.

Growth of the abdominal aortic aneurysm (colored by the deformation of artery wall).

The last 3 cycles correspond to flow through developed aneurysm and the last image in Fig. 1 presents an instantaneous snapshot of the deformed geometry at the peak systole. This test case shows that numerical simulations can be employed for accelerated modeling of disease progression as well as to decipher the flow features at the advanced stage of the arterial disease. The computed results in Fig. 2(a) show the snapshot of the velocity field of blood in the instantaneous deformed shape the blood vessel. Also shown is the iso-surface of blood viscosity, that enables us to identify the regions with high-viscosity where blood coagulation can potentially take place. We have also projected the Wall Shear Stress (WSS) on the arterial wall as shown in Fig. 2(a) and 2(b) for the first and the fifth cycles, respectively. It is important to note that WSS is one of the most significant factors for the progression of arterial disease. It is important to note that obtaining WSS data is quite difficult to obtain via vivo experiments. Fig. 3 presents spatial distribution of the limit-state function given in Equation (34) at the instant of maximum systole. The local dilation of artery vessel leads to higher local stretch and therefore higher tensile stress which results in lower value for the index g(x) that identifies the regions of high risk for the rupture of the aneurysm.

Figure 2.

(a). Cross-sectional view of the flow physics in blood artery interaction: velocity streamlines and viscosity contour in blood projected onto instantaneous snapshot of deformation of the artery wall.

(b). Wall shear stress (WSS) on the interface between blood and artery at the systole and diastole of the seventh cardiac cycle.

Figure 3.

Spatial distribution of the limit-state function g(x) on the arteries and aneurysm.

4.2. Blood-artery Interaction in Calcified Aorta

Calcification of the arterial walls is a natural biological process associated with arterial disease. It leads to local stiffening of the arteries, thereby increasing the risk of rupture.

The anisotropic component of the material model used in this section is derived from [30]:

$$\begin{gathered} W(C)=\frac{1}{2}\lambda {(\ln J)}^2+\frac{1}{2}\mu \left(I_1-3-2\ln J\right) \\ +\sum _{a=1}^2\frac{{\alpha }_1}{2}{\left(I_1{J}_4^{(a)}-J_5^{(a)}-2+\left|I_1{J}_4^{(a)}-J_5^{(a)}-2\right|\right)}^{{\alpha }_2} \end{gathered}$$ (36)

where $I_1=trC,J^2=\det C,J_4=tr[CM],J_5=tr\left[C^{2}M\right]$ are invariants of the right Cauchy-Green deformation tensor. $M=a\otimes a$ is a second-order tensor that models anisotropy of the fibrous material with a direction vector a . λ and μ are the material parameters for Neo-Hookean type solid, and α₁ and α₂ are material parameters for reinforcing fibers. The corresponding stress tensors are provided in Appendix B. Fig. 4 shows the case of severe calcification in abdominal aorta where four zones of calcified plaques can be seen in Fig. 4(a). Fig. 4(b) shows the cross sections of the blood-artery models with matching a non-matching meshes at the interface. The computed flowrate and pressure drop along the length of the artery are shown in Fig. 4(c) and good comparison is attained for both matching and non-matching meshes, and for the two types of outflow boundary conditions imposed, that model the flow resistance from the arterial system distal to the domain of computation.

Figure 4.

(a) Calcified abdominal aortic model, (b) Matching and non-matching meshes, (c) Computed flowrate and pressure drop.

Fig. 5(a) shows flow through calcified abdominal aorta. A portion of the longitudinal section of the wall is removed to help visualize the streamlines in the interior that are colored with fluid velocity. Artery wall is color with the magnitude of local deformation in the artery. Fig. 5(b) shows wall shear stress (WSS) at the interface and fluid vorticity inside the fluid domain at peak systole. An objective of blood flow modeling through the aorta along with the compliant motion of the arterial walls is to estimate the risk of artery wall rupture in the sections of the weakened and tortuous arterial walls that are most vulnerable to failure because of the local stiffening of the walls due to calcifications, or due to local aneurysms as shown in Fig. 5(c).

Figure 5.

(a) Wall deformation and streamlines colored with fluid velocity, (b) Wall shear stress (WSS) at the interface and fluid vorticity inside the fluid domain at peak systole, and (c) Spatial distribution of the limit-state function g(x).

Fig. 6(a) shows wall deformation and streamlines colored with fluid velocity at peak systole and mid diastole, while Fig. 6(b) presents WSS at the interface and fluid vorticity inside the fluid domain at peak systole and mid diastole. Once again, the local changes in the flow features are due to tortuosity and increased stiffness of the artery in a highly nonlinear, coupled, blood-artery interaction problem.

Figure 6.

(a) Wall deformation and streamlines colored with fluid velocity at peak systole and mid diastole.

(b) Wall shear stress (WSS) at the interface and fluid vorticity inside the fluid domain at peak systole and mid diastole. (Unit: dyne s/cm²)

5. Conclusion

We have presented a stabilized method to couple non-Newtonian shear-rate model for blood with anisotropic hyperelastic model for finitely deforming arteries. Employing velocity as the primary unknown field in the fluid and solid subdomains results in a first-order system of coupled equations. Variationally derived interface coupling terms stabilize the interface that is exploited in the monolithically coupled method for concurrent solution of the problem. A significant advantage of the method in variationally imposing the interface conditions is the flexibility it provides for not requiring nodally matched discretizations at blood-artery interfaces. This relaxation helps in developing discretizations for the blood and artery subregions independently, and then glued together with the proposed formulation. The method is applied to a patient-specific geometries with progressive abdominal aortic aneurysm as well as to blood-artery interaction in calcified and narrowed aorta. Enhanced stability of the method results in high fidelity simulations that provide insights into local pressure and velocity fluctuations in blood, and the deformation and stress in the artery walls. Although aneurysms take a long time to grow, in a virtual environment the growth process can be accelerated as shown in the simulations presented. Such simulations can enhance our understanding of the underlying biomechanical processes involved in the progression of the disease and also help with vascular surgical planning.

Table 1.

Mechanical material properties for the aorta and calcification [31]

	Young’s Modulus (dyne/cm2 )	Poisson’s Ratio
Aorta	6 10× 6	0.3
Calcification	6 10× 8	0.3

Appendix A.

Energy Functional

$$\begin{gathered} W(C)=\frac{{\mu }_1}{2}\left(I_1-3\right)+\frac{{\mu }_2}{2}{\left(J_4-1\right)}^2 \\ -{\mu }_1(J-1)+\frac{\kappa +{\mu }_1}{2}{(J-1)}^2 \end{gathered}$$ (37)

Derivation of the Stress Tensors:

$$\begin{gathered} P=\frac{\partial W(F)}{\partial F}=\frac{{\mu }_1}{2}\frac{\partial I_1}{\partial F}+{\mu }_2\left(J_4-1\right)\frac{\partial I_4}{\partial F} \\ -{\mu }_1\frac{\partial J}{\partial F}+\left(\kappa +{\mu }_1\right)(J-1)\frac{\partial J}{\partial F} \end{gathered}$$ (38)

Noting that $\partial I_4/\partial \mathit{F}=2\mathit{F}\mathit{N}\otimes \mathit{N}$ , and using the standard relations $\partial J/\partial F=J{\mathit{F}}^{-T}$ , and $\partial I_1/\partial \mathit{F}=2\mathit{F}$ we arrive at the following relation.

$$\begin{gathered} P={\mu }_{1}F+{\mu }_1(J-2)J{F}^{-T} \\ +2{\mu }_2\left(J_4-1\right)FN\otimes N+\kappa (J-1)J{F}^{-T} \end{gathered}$$ (39)

Second PK Stress Tensor

$$\begin{gathered} \mathit{S}={\mathit{F}}^{-1}\mathit{P} \\ ={\mu }_1\mathit{I}+{\mu }_1(J-2)J{\mathit{C}}^{-1} \\ +2{\mu }_2\left(J_4-1\right)N\otimes N+\kappa (J-1)J{C}^{-1} \end{gathered}$$ (40)

where $\mathit{C}={\mathit{F}}^T\mathit{F}$.

Cauchy Stress:

$$\begin{gathered} \sigma =J^{-1}P{F}^T \\ ={\mu }_1{J}^{-1}b+\left[{\mu }_1(J-2)+\kappa (J-1)\right]I \\ +2{\mu }_2\left(I_4-1\right)J^{-1}(FN)\otimes (FN) \end{gathered}$$ (41)

Tensor of Material Moduli

$$C_{IJKL}=\left\{\begin{array}{l}\left[2{\mu }_{1}J(J-1)+\kappa J(2J-1)\right]C_{IJ}^{-1}{C}_{KL}^{-1}\hfill \\ +4{\mu }_2{N}_I{N}_J{N}_K{N}_L\hfill \\ -\left[\left({\mu }_1+\kappa \right)(J-1)-{\mu }_1\right]J\left(\begin{array}{c}{C}_{IK}^{-1}{C}_{JL}^{-1}\\ +C_{IL}^{-1}{C}_{JK}^{-1}\end{array}\right)\hfill \end{array}\right\}$$ (42)

Tensor of material moduli in the current (spatial) configuration:

$$\begin{gathered} c_{ijkl}=J^{-1}{F}_{iI}{F}_{jJ}{F}_{kK}{F}_{lL}{C}_{IJKL} \\ =\left\{\begin{array}{c}\left[2{\mu }_1(J-1)+\kappa (2J-1)\right]{\delta }_{ij}{\delta }_{kl}\\ +4{\mu }_2{J}^{-1}{F}_{iI}{N}_I{F}_{jJ}{N}_J{F}_{kK}{N}_K{F}_{lL}{N}_L\\ -\left[\left({\mu }_1+\kappa \right)(J-1)-{\mu }_1\right]\left({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}\right)\end{array}\right\} \end{gathered}$$ (43)

Appendix B.

Energy Functional

$$\begin{gathered} W(C)=\frac{1}{2}\lambda {(\ln J)}^2+\frac{1}{2}\mu \left(I_1-3-2\ln J\right) \\ +\sum _{a=1}^2{\alpha }_1{\langle K_3\rangle }^{{\alpha }_2} \end{gathered}$$ (44)

where $K_3=I_1{J}_4^{(a)}-J_5^{(a)}-2$, and $\langle ·\rangle =1/2(·+|·|)$

Derivation of the Stress Tensors:

$$\begin{gathered} \mathit{P}=\frac{\partial W(\mathit{F})}{\partial \mathit{F}}=(\lambda \ln J-\mu )J^{-1}\frac{\partial J}{\partial \mathit{F}}+\frac{\mu }{2}\frac{\partial I_1}{\partial \mathit{F}} \\ +\sum _{a=1}^2{\alpha }_1{\alpha }_2{\langle K_3\rangle }^{{\alpha }_2-1}\left(J_4^{(a)}\frac{\partial I_1}{\partial \mathit{F}}+I_1\frac{\partial J_4^{(a)}}{\partial \mathit{F}}-\frac{\partial J_5^{(a)}}{\partial \mathit{F}}\right) \end{gathered}$$ (45)

Note that $\partial J_4/\partial \mathit{F}=2\mathit{F}\mathit{M},\partial J_5/\partial F=2\mathit{F}{F}^T\mathit{F}\mathit{M}+2\mathit{F}\mathit{M}{\mathit{F}}^T\mathit{F}$.

Using the standard relations $\partial J/\partial \mathit{F}=J{\mathit{F}}^{-T}$ and $\partial I_1/\partial \mathit{F}=2\mathit{F}$ we arrive at the following:

$$\begin{gathered} \mathit{P}=(\lambda \ln J-\mu ){\mathit{F}}^{-T}+\mu \mathit{F} \\ +\sum _{a=1}^{2}2{\alpha }_1{\alpha }_2{\langle K_3\rangle }^{{\alpha }_2-1}\left(\begin{array}{l}{J}_4^{(a)}\mathit{F}+I_1\mathit{F}{\mathit{M}}^{(a)}\\ -\mathit{F}{\mathit{F}}^T\mathit{F}{M}^{(a)}-\mathit{F}{\mathit{M}}^{(a)}{\mathit{F}}^T\mathit{F}\end{array}\right) \end{gathered}$$ (46)

Second PK Stress Tensor

$$\begin{gathered} \mathit{S}={\mathit{F}}^{-1}\mathit{P} \\ =(\lambda \ln J-\mu )C^{-1}+\mu \mathit{I} \\ +\sum _{a=1}^{2}2{\alpha }_1{\alpha }_2{\langle K_3\rangle }^{{\alpha }_2-1}\left(\begin{array}{l}{J}_4^{(a)}I+I_1{\mathit{M}}^{(a)}\\ -C{\mathit{M}}^{(a)}-{\mathit{M}}^{(a)}C\end{array}\right) \end{gathered}$$ (47)

Where $\mathit{C}={\mathit{F}}^T\mathit{F}$

Cauchy Stress:

$$\begin{gathered} \sigma =J^{-1}P{F}^T \\ =(\lambda \ln J-\mu )J^{-1}I+\mu J^{-1}b \\ +\sum _{a=1}^{2}2{\alpha }_1{\alpha }_2{J}^{-1}{\langle K_3\rangle }^{{\alpha }_2-1}\left(\begin{array}{l}{J}_4^{(a)}C+I_{1}F{M}^{(a)}{F}^T\\ -bF{M}^{(a)}{F}^T-F{M}^{(a)}{F}^{T}b\end{array}\right) \end{gathered}$$ (48)

Tensor of Material Moduli

$$C_{IJKL}=\lambda C_{IJ}^{-1}{C}_{KL}^{-1}-(\lambda \ln J-\mu )\left(\begin{array}{l}{C}_{IK}^{-1}{C}_{JL}^{-1}\hfill \\ +C_{IL}^{-1}{C}_{JK}^{-1}\hfill \end{array}\right)$$ (49)

$$\begin{gathered} +\sum _{a=1}^{2}4{\alpha }_1{\alpha }_2\left({\alpha }_2-1\right){\langle K_3^{(a)}\rangle }^{{\alpha }_2-2}\left(\begin{array}{l}{\delta }_{KL}{J}_4^{(a)}\hfill \\ +I_1{M}_{KL}^{(a)}\hfill \\ -2M_{KR}{C}_{RL}\hfill \end{array}\right)\left(\begin{array}{l}{J}_4^{(a)}{\delta }_{IJ}\hfill \\ +I_1{M}_{IJ}^{(a)}\hfill \\ -C_{IK}{M}_{KJ}^{(a)}\hfill \\ -M_{IK}^{(a)}{C}_{KJ}\hfill \end{array}\right) \\ +\sum _{a=1}^{2}4{\alpha }_1{\alpha }_2\mid {\langle K_3^{(a)}\rangle }^{{\alpha }_2-1}\left(\begin{array}{l}{\delta }_{IJ}{M}_{KL}^{(a)}+{\delta }_{KL}{M}_{IJ}^{(a)}\hfill \\ -\frac{1}{2}\left(\begin{array}{c}{\delta }_{IK}{M}_{JL}^{(a)}+{\delta }_{IL}{M}_{JK}^{(a)}\\ +M_{IK}^{(a)}{\delta }_{JL}+M_{IL}^{(a)}{\delta }_{JK}\end{array}\right)\hfill \end{array}\right) \end{gathered}$$ (50)

Tensor of material moduli in the current (spatial) configuration:

$$\begin{gathered} c_{ijkl}=J^{-1}{F}_{iI}{F}_{jJ}{F}_{kK}{F}_{lL}{C}_{IJKL} \\ {c}_{ijkl}=\lambda J^{-1}{\delta }_{ij}{\delta }_{kl}-J^{-1}(\lambda \ln J-\mu )\left({\delta }_{ik}{\delta }_{kl}+{\delta }_{il}{\delta }_{jk}\right) \end{gathered}$$ (51)

$$\begin{gathered} +\sum _{a=1}^{2}4{\alpha }_1{\alpha }_2{J}^{-1}{\langle K_3\rangle }^{{\alpha }_2-2} \\ \left[\begin{array}{l}\left({\alpha }_2-1\right)\left(\begin{array}{l}{b}_{kl}{J}_4^{(a)}\hfill \\ +I_1{F}_{kK}{M}_{KL}^{(a)}{F}_{lL}\hfill \\ -2F_{kK}{M}_{KR}{C}_{RL}{F}_{lL}\hfill \end{array}\right)\left(\begin{array}{l}{J}_4^{(a)}{b}_{ij}\hfill \\ +I_1{F}_{iI}{M}_{IJ}^{(a)}{F}_{jJ}\hfill \\ -F_{iI}{C}_{IK}{M}_{KJ}^{(a)}{F}_{jJ}\hfill \\ -F_{iI}{M}_{IK}^{(a)}{C}_{KJ}{F}_{jJ}\hfill \end{array}\right)\\ +\langle K_3\rangle \left(\begin{array}{l}{b}_{ij}{F}_{kK}{M}_{KL}^{(a)}{F}_{lL}+F_{iI}{M}_{IJ}^{(a)}{F}_{jJ}{b}_{kl}\\ -\frac{1}{2}\left(\begin{array}{c}{b}_{ik}{F}_{jJ}{M}_{JL}^{(a)}{F}_{lL}+b_{il}{F}_{jJ}{M}_{JK}^{(a)}{F}_{kK}\\ +F_{iI}{M}_{IK}^{(a)}{F}_{kK}{b}_{jl}+F_{ik}{M}_{IL}^{(a)}{F}_{lL}{b}_{jl}\end{array}\right)\end{array}\right)\end{array}\right] \end{gathered}$$ (52)