# Development of an SPH-FEM coupling platform for post-TAVI hemodynamic assessment

Jolan Lopez, Zhe Li, David Le Touzé, Julien Michel, Guillaume Oger

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Jolan Lopez, Zhe Li, David Le Touzé, Julien Michel, Guillaume Oger. Development of an SPH-FEM coupling platform for post-TAVI hemodynamic assessment. 18th International SPHERIC Workshop, SPHERIC, Jun 2024, Berlin, Germany. [hal-05103733](https://hal.science/hal-05103733v1)

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# Development of an SPH-FEM coupling platform for post-TAVI hemodynamic assessment

Jolan Lopez¹, Zhe Li¹, David Le Touzé¹, Julien Michel¹, Guillaume Oger¹

¹ Nantes Université, Ecole Centrale Nantes, CNRS, LHEEA, UMR 6598, F-44000 Nantes, France

***Abstract***—The Trans-Aortic Valve Implantation (TAVI) procedure has become a standard treatment for aortic valve failure where a bio-prosthetic valve replaces the dysfunctional native one. However, early degradation of the bio-prosthetic valve is often observed due to local hemodynamic effects. This paper deals with the first developments of an FSI numerical plateform based on LBM-FEM and SPH-FEM coupling, able to predict the changes in hemodynamics before TAVI.

## I. INTRODUCTION

As demography in developed nations is shifting towards an older population, cardio-vascular diseases are becoming an important public health concern. In particular, aortic valve failures can lead to many cardiac symptoms such as elevated blood pressure, coronary stroke, heart failure. A standard treatment for aortic valve failure consists in replacing the dysfunctional valve by a rigid mechanical one implanted in open surgery, or by a bio-prosthetic valve which aims at reproducing the behavior of the native one implanted during a Trans-Aortic Valve Implantation (TAVI) procedure. However, after the valve replacement, complications are observed for some patients. Therefore, a better understanding of patient-specific modified hemodynamics due to valve implantation is important to understand such complications. To this end, researchers have conducted experimental campaigns (see e.g. [13], [16]) and have developed numerical platforms, for both mechanical (see e.g. [4]) and bio-prosthetic valves (see e.g. [14]–[16]). It is important to highlight that while complex fluid flows are involved in both mechanical and bio-prosthetic valves, the valves motion differs since the former can be considered as rigid whereas the latter presents large deformations. In the present work, we focus on developing a numerical platform dedicated to simulations of trans-aortic flows through a bio-prosthetic valve, to anticipate early valve degradation and prevent post-TAVI complications.

The physics of the problem is characterized by strong Fluid-Structure Interactions (FSI) where complex fluid flows involve large deformation of thin structures, with highly coupled responses. Consequently, studying numerically this phenomenon represents a challenging topic and some works have been initiated in this sense in the literature. In Siguenza et al. [13], a Finite Volume (FV) method for the fluid has been coupled with a Finite Element Method (FEM) for the deformable structure via an Immersed Thick Boundary Method (ITBM), and the results obtained are compared to those of the experimental campaign performed in the same article, where a homemade

valve is subjected to pulsatile flow. Furthermore, Li et al. [1] developed an in-house FSI numerical platform where the fluid flow solution is approximated by the Lattice Boltzmann Method (LBM) and the structure by a Finite Element method, the results obtained being compared to those given in [13].

Considering that an FSI framework based on the coupling of the Lattice Boltzmann Method (LBM) with an in-house FEM solver has already been proposed in [1], development of Smoothed Particle hydrodynamics (SPH) method coupled with the same FEM solver aims to compare both approaches in terms of accuracy and efficiency.

The paper is organized as follows. In Sec. II a description of the SPH model is provided. The numerical validation is performed in Sec. III where we start with the validation of the SPH model considering a simplified 2D case involving a rigid leaflet. Sec. IV is dedicated to the description of both the FEM model and the SPH-FEM coupling strategy. Finally, Sec. V shows initial results for the development of a complex 3D case with a FSI simulation modeling trans-aortic flow through a tricuspid aortic valve model inside a simplified aorta geometry.

## II. SPH FORMULATION

On the one hand, for the targeted application, the capture of the vortical structures is crucial. With a conservative pressure gradient formulation, Tensile Instability and consequently noisy vorticity fields have been observed in the SPH literature (see e.g. [8]). By contrast, precise vorticity fields are obtained with non-conservative but accurate pressure gradient formulation [8]. Therefore, the latter pressure gradient approximation is retained in the present work.

On the other hand, the targeted application is characterized by complex geometries. Since Boundary Integral Methods (BIM) are suitable to model solid boundary conditions in the presence of such geometries, the BIM method derived by Chiron et al. [7] is adopted in the present work. In [7] a conservative pressure gradient approximation was used, and, in the present work, this BIM is extended to deal with non-conservative but accurate pressure gradient approximation.

Regular spatial particle distributions are maintained through Particle Shifting Technique (PST), with the PST velocity $\delta u$ derived in [10]. As initially proposed by Sun et al. [6], the quasi-Lagrangian derivative $\frac{d(\bullet)}{dt} := \frac{\partial(\bullet)}{\partial t} + \nabla(\bullet) \cdot (u + \delta u)$, is used. The resulting spatial differential operators are extended for particles close to solid boundaries modeled by the BIM selected in the present paper. The numerical diffusion is
obtained by means of Riemann solver (see [9] for more details). With these choices the SPH formulation used in the present paper finally reads:

$$ egin{cases} 
\frac{d\rho_i}{dt} = -\rho_i \operatorname{div}(\mathbf{u}_i) - \rho_i \operatorname{div}(\boldsymbol{\delta}\mathbf{u}_i) + \operatorname{div}(\rho_i \boldsymbol{\delta}\mathbf{u}_i) + \Theta_{i,Rie}^{\rho} \\ 
\rho_i \frac{d\mathbf{u}_i}{dt} = -\nabla P_i + \boldsymbol{F}_i^{\mu} + \operatorname{div}(\rho_i \mathbf{u}_i \otimes \boldsymbol{\delta}\mathbf{u}_i) + \Theta_{i,Rie}^{\mu} \\ 
\frac{d\mathbf{x}_i}{dt} = \mathbf{u}_i + \boldsymbol{\delta}\mathbf{u}_i, \quad V_i(t) = m_i / \rho_i(t), \quad P_i = c_0^2(\rho_i - \rho_0) 
\end{cases} 
\quad (1) $$

where $\rho$, $V$, $m$, $P$, $\mathbf{u}$, $c_0$, $\mathbf{x}$ represent the density, volume, mass, pressure, velocity, speed of sound and position. The subscript $i$ refers to the generic $i$-th particle while $-\nabla P_i$ and $\boldsymbol{F}_i^{\mu}$ are respectively the pressure and viscous forces acting on the particles. $\Theta_{i,Rie}^{\rho}$ and $\Theta_{i,Rie}^{\mu}$ denote the numerical diffusive terms obtained by means of Riemann solver.

The particle masses $m_i$ remain constant in time and they are set using the initial particle volumes, $V_i(0)$, and the initial particle densities $\rho_i(0)$, i.e. $m_i = \rho_i(0)V_i(0)$. The velocity divergence operator $\operatorname{div}(\mathbf{u}_i)$ is discretized following [7] as

$$ \operatorname{div}(\mathbf{u}_i) = \frac{1}{\gamma_i} \left[ \sum_j (\mathbf{u}_j - \mathbf{u}_i) \cdot \nabla_i W_{ij} V_j + \sum_s (\mathbf{u}_s - \mathbf{u}_i) \cdot \mathbf{n}_s W_{is} s_s \right] \quad (2) $$

where $W$ is the kernel function ($\mathcal{C}^2$-Wendland [18] in the present work), $\gamma_i$ is the renormalization factor, $\mathbf{u}_s$ and $\mathbf{s}_s$ are the velocity of the barycenter and the area of the panel $s$ respectively and $\mathbf{n}_s$ is the vector normal to the panel oriented outwards the fluid domain. Regarding $\nabla P_i$, a formulation ensuring the exact approximation of the gradient of a linear pressure field, valid both for particles within the fluid bulk or close to a solid boundary is retained:

$$ \nabla P_i = \mathbb{L}_i^T \left[ \sum_j (P_j - P_i) \nabla_i W_{ij} V_j + \sum_s (P_s - P_i) \mathbf{n}_s W_{is} s_s \right] \quad (3) $$

with $\mathbb{L}_i$ the renormalization matrix:

$$ \mathbb{L}_i = \left[ \sum_j (\mathbf{x}_j - \mathbf{x}_i) \otimes \nabla_i W_{ij} V_j + \sum_s (\mathbf{x}_s - \mathbf{x}_i) \otimes \mathbf{n}_s W_{is} s_s \right]^{-1} \quad (4) $$

and $P_s$ the pressure at the barycenter of the panel, computed following [12] as $P_s = \tilde{P}_s + \tilde{\rho}_s c_0 (\tilde{\mathbf{u}}_s - \mathbf{u}_s) \cdot \mathbf{n}_s$, with $\tilde{\rho}_s$ and $\tilde{\mathbf{u}}_s$ the extrapolated density and velocity of the fluid at the panel barycenter location computed as $\tilde{f}_s = f_i + \nabla f_i \cdot (\mathbf{x}_s - \mathbf{x}_i)$, $\tilde{P}_s$ being obtained through the equation of state with $\tilde{\rho}_s$. Note that the renormalization factor $\gamma_i$ is not taken into account in Eq. (3) thanks to the renormalization matrix. Furthermore, since no free-surface is considered in the present work, Eq. (3) remains valid in the whole fluid domain.

The spatial differential operators relative to the PST velocity

$\boldsymbol{\delta}\mathbf{u}$ are discretized as:

$$ egin{cases} 
\operatorname{div}(\boldsymbol{\delta}\mathbf{u}_i) = \frac{1}{\gamma_i} \left[ \sum_j (\boldsymbol{\delta}\mathbf{u}_j - \boldsymbol{\delta}\mathbf{u}_i) \cdot \nabla_i W_{ij} V_j + \sum_s (\boldsymbol{\delta}\mathbf{u}_s - \boldsymbol{\delta}\mathbf{u}_i) \cdot \mathbf{n}_s W_{is} s_s \right] \\ 
\operatorname{div}(\rho_i \boldsymbol{\delta}\mathbf{u}_i) = \frac{1}{\gamma_i} \left[ \sum_j (\rho_i \boldsymbol{\delta}\mathbf{u}_i + \rho_j \boldsymbol{\delta}\mathbf{u}_j) \cdot \nabla_i W_{ij} V_j + \sum_s (\rho_i \boldsymbol{\delta}\mathbf{u}_i + \tilde{\rho}_s \boldsymbol{\delta}\mathbf{u}_s) \cdot \mathbf{n}_s W_{is} s_s \right] \\ 
\operatorname{div}(\rho_i \mathbf{u}_i \otimes \boldsymbol{\delta}\mathbf{u}_i) = \frac{1}{\gamma_i} \left[ \sum_j (\rho_i \mathbf{u}_i \otimes \boldsymbol{\delta}\mathbf{u}_i + \rho_j \mathbf{u}_j \otimes \boldsymbol{\delta}\mathbf{u}_j) \nabla_i W_{ij} V_j + \sum_s (\rho_i \mathbf{u}_i \otimes \boldsymbol{\delta}\mathbf{u}_i + \tilde{\rho}_s \tilde{\mathbf{u}}_s \otimes \boldsymbol{\delta}\mathbf{u}_s) \mathbf{n}_s W_{is} s_s \right] 
\end{cases} 
\quad (5) $$

with $\boldsymbol{\delta}\mathbf{u}_s$ the PST velocity at the solid boundary.

The viscous forces $\boldsymbol{F}_i^{\mu}$ are modelled using the Morris formulation [5] further adapted to BIM in [7] which reads:

$$ \boldsymbol{F}_i^{\mu} = \mu \frac{2}{\gamma_i} \left[ \sum_j (\mathbf{u}_j - \mathbf{u}_i) \frac{\mathbf{x}_j - \mathbf{x}_i}{\|\mathbf{x}_j - \mathbf{x}_i\|^2} \cdot \nabla_i W_{ij} V_j + \sum_s \frac{\mathbf{u}_i - \mathbf{u}_s}{d_n} W_{is} s_s \right] \quad (6) $$

with $\mu$ the dynamic viscosity and $d_n$ the normal distance between the particle and the solid panel $s$.

The scheme (1) is integrated in time by using a 4th-order Runge-Kutta scheme, with the following Courant-Friedrichs-Lewy condition:

$$ \Delta t = \min \left( \alpha_h \frac{R_w}{c_0}, \alpha_v \frac{R_w^2}{v} \right); \quad \alpha_h = 0.375; \quad \alpha_v = 0.025 $$

A ratio $R_w/\Delta x = 4$ is used in 2D whereas $R_w/\Delta x = 3$ is chosen in 3D, where $R_w$ is the radius of the kernel support and $\Delta x$ is the initial particle distance.

Note that inside the fluid bulk, the scheme (1) is strictly equivalent to the one in [8].

Finally, due to thin geometries involved in the targeted application, a visibility criterion is employed for particles close to thin solid boundaries. Precisely, when the kernel support of a particle $i$ interacts with both sides of a thin boundary, the faces located on the wrong side are deleted from the neighbourhood of $i$. Additionally, the particles located further the wrong side are not considered as neighbours particles of $i$.

### III. VALIDATION: RIGID LEAFLET UNDER A PULSATILE FLOW

In order to validate development of Sec. II, the test-case proposed by Stijnen et al. in [3] is studied. It aims to model a rigid mechanical valve prosthesis behavior in interaction with a pulsatile flow. The 3D experimental test-case is reduced to a 2D case where only one rigid leaflet is considered. Initial setup is presented in Fig.1. The inlet velocity signal is imposed similarly to the one in [1] except for a minor difference in the first period of the signal, so as to apply a temporal ramp. As visible in Fig.2, maximum amplitude of inlet velocity of this
period is reduced before gradually up to the maximum regime. This inlet signal has a period $T_p$ of 2.45 s with an end-time of 25 s, which has been chosen so that the pulsatile behavior is well established. Note that spatial resolution in Li et al. [1] is expressed as $\Delta x_f$ and is proportional to the fluid inlet height $H$ via $\frac{H}{\Delta x_f} \in [80; 100; 160]$. A convergence study in [1] has shown that smaller discretization parameter $\Delta x_f$ has little to no impact onto the leaflet motion. Outlet boundary conditions are imposed via an outlet pressure set here as $P_{outlet} = 0$ Pa. The initial rigid leaflet angle is 22°.

In Fig.3, a comparison of the temporal evolution of the leaflet angle between present work results, experimental results from Stijnen et al. [3] and numerical results from Li et al. [1] is presented. A representative period, chosen far enough from $t=0$, is retained so that periodic regime is well established. A good agreement of amplitude and frequency has been found between present work results and those in [3] and [1] and are both well maintained throughout the simulation as observed in Fig.4.

In addition, convergence study is proposed using three spatial resolutions: $\frac{H}{\Delta x} = 50$, $\frac{H}{\Delta x} = 66.6$ and $\frac{H}{\Delta x} = 100$. In Fig.5, it is shown that the influence of discretization is limited to the maximum angle of the rigid leaflet. As $\Delta x$ decreases, so does the maximum angle of the rigid leaflet. Furthermore, the maximum amplitude of the rigid leaflet angle gets closer to the results of Li et al. as $\Delta x$ decreases. A possible explanation lies in a better approximation of local velocity within the central part of the domain when the rigid leaflet reaches its maximum amplitude. As visible in Fig.6, some similarities can be observed in velocity fields obtained in the present work and Stijnen et al. [3] and Li et al. [1].

As pointed out in Sec. I and Sec. II, accurate capture of flow vortices is important. As one can see in Figs. 7 and 8 vortical structures are well captured and both velocity fields and pressure fields are particularly smooth.

## IV. FEM FORMULATION AND COUPLING SCHEME

Regarding Finite-Element Method (FEM) formulation, a nonlinear approach within a total lagrangian framework is adapted from [11] into an in-house FEM solver as it fits the targeted application requirements. Considering this framework, the momentum equation gives:

$$ \nabla_0 \cdot \mathbf{P} + \rho_0 \mathbf{b} = \rho_0 \ddot{\mathbf{u}} \quad (7) $$

<figure><img src="image_2.png" /><figcaption>Fig. 1. Rigid leaflet test case: initial setup.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. Rigid leaflet test case: Inlet velocity signal</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Rigid leaflet test case: Comparison of leaflet angular motion to literature results ([3], [1])</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. Rigid leaflet test case: Comparison of leaflet angular motion throughout simulation between present work and Li et al. [1]</figcaption></figure>

where $\rho$ is the solid density, $\ddot{u}$ its acceleration, $\mathbf{P}$ being the nominal stress tensor, $\mathbf{b}$ is the external force, $\nabla \cdot$ is the divergence operator and the subscript 0 stands for the initial configuration. In such framework, the deformation gradient tensor $\mathbf{F}$, the jacobian $\mathbf{J}$ that characterizes volume variation, the Green-Strain Tensor $\mathbf{E}$ and the right Cauchy-Green strain tensor $\mathbf{C}$ are given as:

$$ \mathbf{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} ; \quad \mathbf{J} = \det(\mathbf{F}) \quad (8) $$

$$ \mathbf{E} = \frac{1}{2} (\mathbf{F}^T \cdot \mathbf{F} - \mathbb{I}_d) \quad (9) $$

$$ \mathbf{C} = \mathbf{F}^T \mathbf{F} \quad (10) $$
where **x** and **X** are material point positions, in the present and initial configurations respectively. For the targeted application, a Neo-Hookean compressible and isotropic hyperelastic material model as defined in [11] is considered. Stress **S** is defined

<figure><img src="image_6.png" /><figcaption>Fig. 5. Rigid leaflet test case: Leaflet angular motion for various Δx</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Rigid leaflet test case: Velocity fields, from top to bottom: Stijnen et al [3], Li et al. [1] and present work (Δx = H/100)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Rigid leaflet test case: Vorticity field for Δx = H/100 at t = 9.5 s</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. Rigid leaflet test case: Pressure field for Δx = H/100 at t = 9.5 s</figcaption></figure>

by a stored energy function Ψ that acts as a potential, giving:

$$S = \frac{\partial \Psi(C)}{\partial C} \qquad (11)$$

$$S = \lambda_0 \ln(J) C^{-1} + \mu_0 (\Pi_d - C^{-1}) \qquad (12)$$

$$\lambda_0 = \frac{E v_s}{(1+v_s)(1-2v_s)}; \quad \mu_0 = \frac{E}{2(1+v_s)}$$

with λ₀ and μ₀ Lamé coefficients in the initial configuration, E Young's modulus and v_s Poisson's ratio. As for the discrete equations, considering that discrete and continuous motions are linked via:

$$\mathbf{x}(\mathbf{X}, t) = \mathbf{x}_I(t)\mathbf{N}_I(\mathbf{X}) \qquad (13)$$

with N_I the nodal shape function, indexed by I and is expressed in terms of the initial configuration in the total lagrangian framework. Note that in the present section all repeated indices are summed.

Hence, (8) gives at a discrete level:

$$F = \frac{\partial N_I}{\partial \mathbf{X}} \mathbf{x}_I \qquad (14)$$

Discrete equations for (9) and (10) are directly derived from (14). Time integration is performed via a Newark Implicit scheme as in [11] that reads:

$$\mathbf{u}_s^{n+1} = \mathbf{u}_s^n + \Delta t \mathbf{v}_s^n + \Delta t^2 \left( \frac{1-2\beta}{2} \mathbf{a}_s^n + \beta \mathbf{a}_s^{n+1} \right) \qquad (15)$$

$$\mathbf{v}_s^{n+1} = \mathbf{v}_s^n + \Delta t \mathbf{v}_s^n + \Delta t [(1-\gamma)\mathbf{a}_s^n + \gamma \mathbf{a}_s^{n+1}] \qquad (16)$$

where u_s is the displacement vector, v_s is the velocity vector and a_s is the acceleration vector; Δt is the time-step ; β
and $\gamma$ are scheme coefficients. As for the coupling, a similar algorithm to the one presented in [17] is adopted (see Fig. 9).

A Conventional Serial Staggered (CSS) scheme is used in the present, where the SPH solver explicitly resolves Equations (1) considering a solid state at the current instant to compute the external force $F_{ext}^{n+1}$ sent to the FEM solver, which implicitly updates the solid state the next instant as represented in Fig. 10.

## V. 3D TEST-CASE: PULSATILE FLOW THROUGH A DEFORMABLE TRICUSPID AORTIC VALVE

This 3D test-case is inspired by the work of Li *et al.* [1]. In [1], the aorta is modeled as a straight cylinder with symmetric sinuses, and the aortic valve is similar to the one in [1]. It is composed of three symmetrical leaflets, shaped similarly to a bio-prosthesis valve. Valve tightness is obtained using an additional rigid wall that models the aortic valve root. Initial setup is shown in Fig. 11 and 12.

All solid boundaries are considered as no-slip boundary conditions. Regarding solid material parameters, the deformable tricuspid aortic valve material model is an hyperelastic Neo-Hookean one (see Sec. IV Equation (12)), with a Young's modulus $E = 4.8$ MPa and a Poisson's ratio $\nu_s = 0.4$ chosen to approach a nearly-incompressible material. Its density is $1000 \text{ kg.m}^{-3}$ and the initial configuration of the valve is considered stress-free. The fluid density is set at $1100 \text{ kg.m}^{-3}$ and its dynamic viscosity as $\mu = 0.0036 \text{ Pa.s}$. Spatial discretization is $\Delta x = D/42$ with D the inner upstream diameter and the sound speed is $c_0 = 6 \text{ m.s}^{-1}$.

Preliminary simulation results of a 3D FSI between a prosthetic valve and a trans-aortic flow are encouraging. As one can see in Figs. 13 and 14, the aortic valve is opening under the increasing positive upstream velocity. Validation of those results are still ongoing work and needs to be pursued.

<figure><img src="image_10.png" /><figcaption>Fig. 9. Coupling approach between SPH solver and FEM Solver</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10. Conventional Serial Staggered (CSS) scheme</figcaption></figure>

## VI. CONCLUSION

In the present work, the SPH formulation derived in [8] has been adapted to deal with solid boundaries modeled using the Boundary Integral Method from [7]. The validation of this formulation has then been performed on a 2D rigid leaflet test-case, showing a good agreement with numerical and experimental results, both regarding the fluid flow solutions and the solid boundary displacement obtained. Particularly, smooth pressure and vorticity fields have been noticed thanks to the absence of Tensile Instability occurrences, showing the capability of this formulation to deal with such physical problems.

<figure><img src="image_12.png" /><figcaption>Fig. 11. Trans-aortic flow FSI: valve geometry</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12. Trans-aortic flow FSI: initial setup (red: aortic valve root, blue: aortic valve)</figcaption></figure>
First attempt of FSI simulations on a full 3D case with complex geometries and no-slip boundary conditions have shown that some numerical problems still need to be tackled. Sudden variations in the fluid velocity during the valve opening have been observed at early stages of the simulation, leading to miscalculation of external forces. In a future work, modification of the coupling scheme to allow sub-iteration via CSS subcycling [19] [20], will have to be developed so as to reduce velocity variation and boundary forces instability.

## ACKNOWLEDGMENT

This work was supported by Agence Nationale de la Recherche (grant ANR-21-CE45-0008) and Région Pays de la Loire.

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<figure><img src="image_14.png" /><figcaption>Fig. 13. Trans-aortic flow FSI: Velocity at $t = 0.056$ s (XY plane slice)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 14. Trans-aortic flow FSI: Velocity at $t = 0.056$ s (XZ plane slice)</figcaption></figure>

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