A Comprehensive Numerical Model of Thrombus Embolization: Fluid-Thrombus Interactions Through a Coupled Computational Fluid Dynamics - Peridynamics Framework

Abstract

Thromboembolic diseases, which account for approximately one-third of global deaths, are characterized by thrombus embolization—the process whereby blood clots undergo cohesive/adhesive fracture under fluid-induced forces. Comprehensive numerical modeling of this multiphysics phenomenon has remained challenging, and robust computational frameworks are scarce. This work substantially advances the state-of-the-art of numerical models of thrombus embolization by presenting a framework that couples Lagrangian non-ordinary state-based peridynamics with Eulerian computational fluid dynamics. The formulation can (i) incorporate complex macroscale constitutive laws for a thrombus, (ii) capture delamination physics between a thrombus and a contacting surface, and (iii) couple thrombus structural dynamics with fluid flow discretized on arbitrary unstructured meshes. The latter capability is achieved through the development of a robust interpolation scheme for fast bi-directional data transfer between polyhedral meshes and peridynamic particle clouds. This also advances the literature of peridynamic coupled fluid numerical models, which are mostly limited to bond-based peridynamics and Cartesian meshes. The framework undergoes rigorous validation through five computationally unique benchmarks. The validated framework is then applied to reproduce reported experimental results of embolization of preformed thrombus within a polycarbonate cylindrical tube under varying flow rates. To demonstrate that the numerical framework can support spatially heterogeneous material properties (a capability absent in other approaches), a synthetic test case is reported. Unlike homogeneous assumptions that always predict leading-edge detachment, heterogeneous material properties enable trailing-edge detachment, thus elucidating this previously unexplained experimental observation.

1. Introduction

Thrombus embolization - fragmentation of blood clots (thrombus) under blood flow - is a precursor to several life-threatening cardiovascular conditions, such as pulmonary embolism [1], myocardial infarction [2], and ischemic stroke [3]. Such conditions account for about one-third of the global deaths [4]. Despite such grim statistics, relatively little attention has been paid to understanding and modeling this crucial phenomenon. This research gap stems from the complexity involved in coupling the non-linear mechanics of fluid, soft-tissue fracture, and dynamic clot growth. The few studies in the literature that investigate fluid-induced thrombus deformations are described below.

Fogelson and Guy [5] proposed a 2D immersed boundary method on Cartesian meshes that captured the micro-scale adhesive and cohesive interaction of platelets and vessel walls. Platelet-platelet interactions were modeled through springs, which are inadequate in capturing the complex constitutive response that characterizes a blood clot [6]. Xu et al. [7] presented a 2D phase field model for thrombus formation and embolization, wherein the thrombus was modeled as a Kelvin–Voigt material. Tobin et al. formulated a 3D numerical model of thrombus embolization based on the multiphase framework, where the thrombus was modeled as a Phien- Thanner viscoelastic material [8]. The latter two continuum-based approaches are limited by their assumption of representing thrombi as Eulerian fields. Using an Eulerian framework to solve thrombus structural dynamics equations is problematic because (i) mass conservation is compromised, (ii) it cannot incorporate the inherent (often point-wise) heterogeneous material properties of a thrombus, and (iii) error analysis of the transport of tensorial quantities, like deformation gradient and stresses, is non-trivial. Mitigating these shortcomings necessitates highly refined meshes to minimize numerical diffusion, which severely constrains the applicability of Eulerian-based numerical frameworks used for solving large-deformation structural dynamics to only trivial cases. Therefore, a Lagrangian formulation is required to model the structural response of a thrombus as it will make assignment of spatially heterogeneous material properties straightforward, mass conservation is obtained for free, and numerical diffusion of quantities associated with the solution of structural equations will be completely eliminated.

The numerical framework presented herein constitutes a comprehensive model for thrombus embolization, offering four critical capabilities: (i) accommodation of arbitrarily complex macroscale hyperelastic constitutive models to capture the nonlinear mechanical response of a thrombus, (ii) representation of arbitrary spatial material heterogeneity of a thrombus, (iii) explicit modeling of thrombus-wall delamination physics, and (iv) a numerical formulation that is agnostic to fluid mesh discretization. We achieve these capabilities by solving the governing equations of solid mechanics using peridynamics (PD) and coupling with the fluid flow equations solved using the finite volume method on unstructured polyhedral meshes. We justify our choice of PD as the modeling framework for capturing the structural response of a blood clot in the next three paragraphs.

Classical finite element methods (FEM), whose governing equations are formulated as partial differential equations, require special numerical treatment at displacement discontinuities. Nodal splitting [9] and the cohesive zone method (CZM) [10] extend the FEM framework to handle such discontinuities. In thrombus embolization, crack initiation sites and propagation paths are governed entirely by the evolving hemodynamic loading and cannot be known before the simulation begins — making mesh-independent crack representation fundamentally unachievable with nodal splitting [11], and rendering the computational overhead of CZM very high for widespread fragmentation native to thrombus embolization. The extended finite element method (XFEM) [12] removes the requirement for crack-conforming meshes but requires explicit crack-front tracking, most commonly achieved via level-set methods [13, 14]. The number of level sets scales linearly with the number of cracks to be resolved, making its adoption difficult in complex 3D fragmentation scenarios characteristic of problems involving thrombus fracture. Phase-field methods (PFMs) [15, 16] circumvent crack tracking by modeling fracture as a damage-accumulation process, and consequently, cracks are naturally represented as diffuse damage fields. However, PFMs require highly refined meshes in potential fracture zones to resolve the regularization length scale, necessitating a priori knowledge of where fracture might occur [17]. More broadly, the requirement for mesh refinement around crack tips is a limitation shared by all mesh-based fracture methods. This limitation is further exacerbated when thrombus growth is considered, as the continuously evolving clot geometry necessitates repeated mesh regeneration at every time step, rendering such approaches computationally intractable. These considerations motivate the adoption of a meshless framework for modeling the structural physics of thrombus embolization.

Among meshless methods, the material point method (MPM) [18] and smoothed particle hydrodynamics (SPH) [19] are prominent methods for modeling large deformation and fracture problems. In MPM, fracture is modeled either as a strong discontinuity by allowing multiple velocity fields at nodes cut by crack surfaces (similar to the nodal splitting approach in FEM) or through continuous approaches in which crack surfaces are not explicitly represented [20]. The former requires tracking of evolving crack surfaces, with implementation becoming particularly intricate for complex crack patterns. The latter MPM approach has been shown to produce mesh-dependent results in which accurate solutions are achieved only when crack orientation coincides with the background grid cell orientation [21]. SPH suffers from tensile instability [22] and difficulties enforcing essential boundary conditions [23], limiting its accuracy for solid fracture mechanics. To this end, we use PD.

PD is a non-local reformulation of classical continuum mechanics (CCM) formulated in terms of integral equations rather than partial differential equations [24], which allows the governing equations to be well defined around displacement discontinuities. Fracture and damage are modeled naturally through inter-particle bond failure and, more importantly, it does not require crack front tracking, explicit crack branching and coalescence criteria, or mesh topology updates. Furthermore, the inbuilt non-locality of PD through a radius of influence of parameter (δ, also called the horizon), implicitly regularizes the fracture process zone, thereby avoiding mesh-dependent damage localization of classical damage models [25]. Thereby, making it suitable for capturing the fracture of thrombi under dynamic hemodynamic loading, where fracture locations are not known a priori. Since its introduction in 2000, PD has been used to model the fracture of many kinds of materials [26–28], including biological materials [29, 30]. Multiple variants of PD have been proposed: (i) Bond-based (BB-PD), (ii) Ordinary state-based (OSB-PD), and (iii) Non-ordinary state-based (NOSB-PD) [24, 31, 32]. Among these formulations, BB-PD constrains the Poisson ratio to ν = 0.25 in 3D, which is incompatible with the near-incompressibility of blood clots (ν > 0.4). OSB-PD partially relaxes this constraint but restricts the force state to act along bond directions, precluding the direct incorporation of arbitrary CCM constitutive laws - an uncompromising requirement for a numerical framework that needs to model the complex materials response of blood clots. NOSB-PD overcomes both limitations through the correspondence principle [31, 33], which provides a systematic framework for converting arbitrary CCM constitutive laws into the PD framework without reformulation of the governing equations of PD. Consequently, NOSB-PD is our choice of numerical framework to solve the structural dynamics of a thrombus. In our previous work [30], we developed the first NOSB-PD framework specifically designed for blood clot constitutive modeling, rigorously validated against standard benchmarks, and subsequently calibrated against published experimental data. This validated framework is adopted directly in the present work.

The numerical framework presented herein couples an NOSB-PD particle cloud with fluid flow solved on arbitrary unstructured polyhedral meshes. This is a significant advance over existing PD-fluid coupling frameworks in which the fluid governing equations are discretized and solved on Cartesian meshes [34, 35], restricting their applicability to simple geometries. This limitation is not unique to PD-fluid frameworks as general particle-fluid coupling strategies based on direct forcing immersed boundary methods [36–40], have similarly been demonstrated predominantly on Cartesian meshes. Attempts to generalize these frameworks to unstructured meshes, such as through modifications to the standard delta function spreader and accumulation operators [41] or through cut-cell methods [42], introduce significant computational overhead. The former requires the solution of large matrix systems, while the latter involves complex interface reconstruction algorithms that become particularly intricate and computationally heavy for dynamically evolving solid geometries such as fracturing thrombi. Furthermore, the use of Cartesian meshes handicaps the deployment of classical coupling techniques to most clinically relevant geometries, such as ventricular assist devices (VADs) or in patient-specific vasculature, where unstructured polyhedral meshes are required for accurate representation. Since fluid-solid force exchange must be performed at every time step, the computational efficiency of the interpolation procedure is critical. To address these limitations, we present a matrix-free bi-directional interpolation technique that is computationally efficient and applicable for coupling particle clouds with arbitrary fluid meshes. Although demonstrated here for NOSB-PD particle clouds, the proposed interpolation scheme is generic and applicable to any Lagrangian particle system required to be coupled to a finite volume solver on arbitrary polyhedral meshes. Furthermore, under a Cartesian mesh assumption, the scheme recovers classical coupling methods as a special case, confirming consistency with established approaches. Furthermore, using the proposed interpolation operators, an explicit derivation establishes that the coupling scheme is predominantly dissipative, a property critical for numerical stability in fluid-structure-interaction (FSI) problems involving severe fracture.

The structure of the remaining text is as follows. Sections 2.1–2.2 present the basics of PD for fluid flow, followed by the modifications required to achieve coupling between the uncoupled fluid-structure equations. Sections 2.4–2.5 detail the interpolation and the numerical algorithm procedure. The framework is validated rigorously through a series of benchmarks, examining its ability to: (i) preserve spatial flow structures when interpolating between CFD meshes and particle clouds, (ii) confine flows within PD particle clouds, (iii) strongly enforce particle cloud velocity and effectively reproduce the response of the surrounding fluid, (iv) predict drag forces of flow around bluff bodies, and (v) handle large-deformation of a body immersed in flow in a rectangular channel. The validated framework is subsequently applied to reproduce the experimental observations of Tobin et al. [8], namely, embolization of pre-formed clots within a polycarbonate cylindrical tube under varying flow rates. Finally, the implications of the assumptions underlying this framework are discussed, and the capability of the framework to represent and preserve spatially heterogeneous material properties is presented through a synthetic test case.

2. Methods

2.1. Non-ordinary State-Based Peridynamics (NOSB-PD)

NOSB-PD is chosen as the modeling framework for solving solid mechanics equations in this study. The mathematical framework of PD represents a body ${ℬ}_0$ as a set of finite-sized particles of diameter $d_p$, where each particle interacts with all other particles within its neighborhood (See Fig. 1(a)). For a particle P, with a reference position ${\mathit{x}}_P$, the neighborhood ${\mathscr{H}}_P$ is defined as

$${\mathscr{H}}_P=\left\{{\mathit{x}}_N\in {ℬ}_0:\left|{\mathit{x}}_N-{\mathit{x}}_P\right|<\delta \text{and}{\mathit{x}}_N\ne {\mathit{x}}_P\right\},$$ (1)

where δ is a scalar quantity called the horizon. The collective set of particles that are within ${\mathscr{H}}_P$ is referred to as the family of particle P, and each family member is said to be bonded to P. The bond between a particle pair ${\mathit{x}}_P$ and ${\mathit{x}}_N$ is denoted as ${\mathit{\xi }}_{PN}$ where

$${\mathit{\xi }}_{PN}=\left({\mathit{x}}_N-{\mathit{x}}_P\right).$$ (2)

Figure 1:

Shows (a) PD discretization of an arbitrary body. For a particle colored in light green, the horizon $\delta$ shows all the neighboring particles with which the light green particle interacts. (b) Kinematical quantities for PD. Given two particle pairs P and N, the reference configuration is shown on the left, where the bond vector is ${\mathit{\xi }}_{PN}$, and the current configuration on the right, where the current bond vector is $\underset{\_}{\mathbf{Y}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩$.

In NOSB-PD, special mathematical objects called states are introduced to capture the collective response of a family. Formally, a state is a map of all bonds to a list of scalars, vectors, or tensors [31]. It is useful to understand the notation of a generic state □ for the family of particle P as stated below

$$\underset{\_}{\square }\left[{\mathit{x}}_P\right]=\text{list of}\underset{\_}{\square }\text{associated with all bonds}$$ (3)

$$\underset{\_}{\square }\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩=\text{quantity}\underset{\_}{\square }\text{associated with a specific bond}{\mathit{\xi }}_{PN}.$$ (4)

For example, the deformation state Y[x_P] contains the deformations of all family members of particle P, and Y[x_P]⟨ξ_PN⟩ stores the current deformation of the bond ξ_PN. With y_P and y_N denoting the current positions of particles P and N, Y[x_P]⟨ξ_PN⟩ is defined

$$\underset{\_}{\mathbf{Y}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩=\left({\mathit{y}}_N-{\mathit{y}}_P\right).$$ (5)

The governing equation of motion for a particle P with density (${\rho }_s$) and external body force density (${\mathit{b}}_P$) is the following

$$\left({\int }_{{\mathscr{H}}_P}\left\{\underset{\_}{{\mathbf{T}}_{\mathbf{c}}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩-\underset{\_}{{\mathbf{T}}_{\mathbf{c}}}\left[{\mathit{x}}_N\right]⟨{\mathit{\xi }}_{NP}⟩\right\}d{V}_{\xi }\right)+{\mathit{b}}_P={\rho }_s{\dot{\mathit{v}}}_{sP},$$ (6)

where ${\mathit{v}}_{sP}$ is the velocity of particle $\mathrm{P},\psi =\psi (\underset{\_}{\mathbf{Y}})$ is the strain-energy function (SEF), and the force vector state $\underset{\_}{\mathbf{T}}\left[{\mathit{x}}_P\right]$ is the work-conjugate of $\underset{\_}{\mathbf{Y}}\left[{\mathit{x}}_P\right]$ defined

$$\underset{\_}{{\mathbf{T}}_{\mathbf{c}}}\left[{\mathit{x}}_P\right]={\left.\frac{\partial \psi }{\partial \underset{\_}{\mathbf{Y}}}\right|}_{\underset{\_}{\mathbf{Y}}\left[{\mathit{x}}_P\right]}.$$ (7)

$\underset{\_}{{\mathbf{T}}_{\mathbf{c}}}\left[{\mathit{x}}_P\right]$ is the central mathematical quantity used to account for different constitutive behaviors. Using PD correspondence law [33], an explicit expression for the bond level force vector state can be found by matching internal stress power expressed using CCM and PD. The obtained force state expression is

$$\underset{\_}{{\mathbf{T}}_{\mathbf{c}}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩={\omega }_{PN}{\mathbb{P}}_P{𝕂}_P^{-1}{\mathit{\xi }}_{PN}.$$ (8)

where $\mathbb{P}$ is the first Piola–Kirchhoff tensor, ${\omega }_{PN}=\omega \left(\left|{\mathit{\xi }}_{PN}\right|\right)$ is an influence function that weights each bond according to some function of its length, and 𝕂_P is the shape tensor associated with the point x_P and is defined

$${𝕂}_P=\left({\int }_{{\mathscr{H}}_P}{\omega }_{PN}{\mathit{\xi }}_{PN}\otimes {\mathit{\xi }}_{PN}d{V}_{\xi }\right).$$ (9)

Furthermore, calculating stress fields in a body requires the deformation gradient ℱ. Using purely PD quantities, a first-order approximation of the classical definition of the deformation gradient is

$${ℱ}_P=\left({\int }_{{\mathscr{H}}_P}{\omega }_{PN}\underset{\_}{\mathbf{Y}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩\otimes {\mathit{\xi }}_{PN}d{V}_{\xi }\right){𝕂}_P^{-1}.$$ (10)

2.1.1. Stabilized peridynamics

In this paper, zero-energy modes in PD solution due to the weak definition of ℱ are avoided by the addition of a correction term to (8) as shown

$$\underset{\_}{\mathbf{T}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩=\underset{\_}{{\mathbf{T}}_{\mathbf{c}}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩+\underset{\_}{{\mathbf{T}}_{\mathbf{s}}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩$$ (11)

The correction term $\underset{\_}{{\mathbf{T}}_{\mathrm{s}}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩$ used in this paper takes the form as shown below

$$\underset{\_}{{\mathbf{T}}_{\mathbf{s}}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩={\omega }_{PN}\left(\frac{9{\kappa }_c}{\pi {\delta }^4}\right)\left(\frac{{\mathit{\xi }}_{PN}\cdot {\mathbf{z}}_{PN}}{{\left|{\mathit{\xi }}_{PN}\right|}^3}\right){\mathit{\xi }}_{PN},$$ (12)

$$\text{where}{\mathit{z}}_{PN}={\mathit{y}}_{PN}-{𝔽}_P{\mathit{\xi }}_{PN},$$ (13)

where ${\kappa }_c$ is the bulk modulus of the material. The force vector correction was originally proposed by Li et al. [] and has been used previously by us without any problems. The stabilized version of the force vector state $\underset{\_}{\mathbf{T}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩$ is used in place of $\underset{\_}{{\mathbf{T}}_{\mathbf{c}}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩$ in (8) as the stabilized PD governing equation for running all simulations presented in this paper.

2.1.2. Fracture modeling in PD

Fracture within a material is captured through a damage accumulation approach. For modeling damage, interaction between particle pairs P and N is systematically modified by multiplying ${\omega }_{PN}$ with a scale factor $\mu \left({\mathit{\xi }}_{PN}\right)$ defined as

$$\mu \left({\mathit{\xi }}_{PN}\right)=\left\{\begin{array}{ll}1& \text{if}0\le s<s_1\\ \left(\frac{s_2-s}{s_2-s_1}\right)& \text{if}{s}_1\le s<s_2\\ 0& \text{if}s\ge s_2,\end{array}\right.$$ (14)

where s is the ratio of the current bond length $\left(\left|\underset{\_}{\mathbf{Y}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩\right|\right)$ to the reference bond length $\left(\left|{\mathit{\xi }}_{PN}\right|\right),s_1$ and $s_2$ are calibrated material specific constants that govern crack growth. The bilinear indicator function is chosen as as our damage evolution function as it is simple and has provided great results in calibrating fracture properties of thrombi in our previous work [30]. To satisfy the second law of thermodynamics, the quantity $\mu \left({\mathit{\xi }}_{PN}\right)$ is evolved as

$${\mu }^n\left({\mathit{\xi }}_{PN}\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left({\mu }^{n-1}\left({\mathit{\xi }}_{PN}\right),\mu \left({\mathit{\xi }}_{PN}\right)\right)\text{with}{\mu }^0\left({\mathit{\xi }}_{PN}\right)=1,$$ (15)

where $n$ is a time-step index. The above equation ensures that bonds between particle pairs are broken irreversibly beyond critical stretch thresholds. Damage at a point is computed according to

$$d\left({\mathbf{x}}_P\right)=1-\left(\frac{{\int }_{{\mathscr{H}}_P}\mu \left({\mathit{\xi }}_{PN}\right)dV}{{\int }_{{\mathscr{H}}_P}dV}\right).$$ (16)

For particles with non-zero damage, the first Piola–Kirchhoff stress tensor is modified as

$$\mathbb{P}\left({\mathbf{x}}_P,d\left({\mathbf{x}}_P\right)\right)=\left(1-\alpha \left(d\left({\mathbf{x}}_P\right)\right)\right)\mathbb{P}\left({\mathbf{x}}_P\right),$$ (17)

wherein the postulated function $\alpha \left(d\left({\mathbf{x}}_P\right)\right)$ must be non-decreasing. This general stress modifier equation is consistent with previously proposed continuum damage models [43]. For our usage to thrombus fracture the specific form is

$$\alpha (d)=\left(\frac{1-e^{-a{d}^m}}{1-e^{-a}}\right),$$ (18)

where $a$ and $m$ were set to 1.7 and 0.5, respectively. The specific form has been validated for soft-tissue fracture applications in our prior work [30].

2.1.3. Time Integration

Similar to our previous work [30], the governing ODE of PD is solved using the explicit Verlet time integration with artificial damping ($\vartheta$)

$${\ddot{\mathbf{y}}}_P+\vartheta {\dot{\mathbf{y}}}_P={\mathbf{a}}_P,$$ (19)

where

$${\mathit{a}}_P=\frac{1}{{\rho }_s}\left[\left({\int }_{{\mathscr{H}}_P}\left\{\underset{\_}{\mathbf{T}}\left[{\mathit{x}}_P\right]⟨{\mathit{\xi }}_{PN}⟩-\underset{\_}{\mathbf{T}}\left[{\mathit{x}}_N\right]⟨{\mathit{\xi }}_{NP}⟩\right\}d{V}_{\xi }\right)+{\mathit{b}}_P\right].$$

Using a similar analysis like the one used to derive the Verlet time advancement procedure, the position and velocities of a particle P with damping ϑ are updated as (n is the current time step)

$${\mathbf{y}}_P^{n+1}={\mathbf{y}}_P^n+{\dot{\mathbf{y}}}_P^n(1-\gamma )\mathrm{\Delta }t+{\mathbf{a}}_P^n\frac{(\mathrm{\Delta }t{)}^2}{2},$$ (20)

$${\mathbf{v}}_P^{n+1}=(1-\gamma {)}^2{\mathbf{v}}_P^n+\frac{\mathrm{\Delta }t}{2}\left((1-\gamma ){\mathbf{a}}_P^n+{\mathbf{a}}_P^{n+1}\right),$$ (21)

$$\gamma =\frac{\vartheta \mathrm{\Delta }t}{2}.$$ (22)

2.2. Fluid flow equations

The mechanics of the fluid is modeled through the incompressible Navier-Stokes equations

$$\nabla \cdot {\mathit{u}}_f=0$$ (23)

$$\frac{\partial {\rho }_f{\mathit{u}}_f}{\partial t}+\nabla \cdot \left({\rho }_f{\mathit{u}}_f\otimes {\mathit{u}}_f\right)=\nabla \cdot {\mathit{\sigma }}_f+{\rho }_f\mathit{f}.$$ (24)

Following the approach of Peskin [44], the influence of the moving solid is incorporated via a source term $\mathit{f}$. The exact form of the source term is given in later sections of the manuscript. In the present work, the fluid stress tensor is described by the linear viscous constitutive relation

$${\mathit{\sigma }}_f=-p\mathbf{I}+\mu \left(\nabla {\mathit{u}}_f+\nabla {\mathit{u}}_f^T\right),$$ (25)

where I is the identity tensor.

2.3. Coupling between CFD and PD

Modeling the fracture of a PD particle cloud under fluid flow requires a two-way coupling scheme: (i) transfer fluid forces onto PD particles to update their deformation, and (ii) transfer particle positions and velocities onto the CFD mesh so that the fluid flow can adjust to the motion and configuration of the PD particles.

2.3.1. Fluid-to-Solid Coupling

Given a particle P with volume V_P, the particle-centered hydrodynamic force ${\mathit{b}}_{h,P}\left(\mathrm{N}/{\mathrm{m}}^3\right)$ is

$${\mathit{b}}_{h,P}=\frac{1}{V_P}{\text{∮}}_{\partial V_P}{\mathit{\sigma }}_f\cdot \mathit{n}dS,$$ (26)

where n is the outward-pointing unit normal on the particle surface. Direct evaluation of Eq. (26) presents two challenges: (i) interpolation of all six fluid stress tensor components at the particle center, and more importantly, (ii) specification of an appropriate surface area metric for the particles. To alleviate this, the RHS of Eq. (26) is rewritten using the integral form of the Navier-Stokes Eq. (24) as shown below [45]:

$${\text{∮}}_{\partial V_P}{\mathit{\sigma }}_f\cdot \mathit{n}dS={\rho }_f\left(\frac{d}{dt}{\int }_{V_P}{\mathit{u}}_{f}dV-{\int }_{V_P}\mathit{f}dV\right)$$ (27)

Plugging Eq. (27) into Eq. (26), a manageable expression in terms of fluid velocity and the forcing term is obtained

$${\mathit{b}}_{h,P}=\frac{{\rho }_f}{V_P}\left(\frac{d}{dt}{\int }_{V_P}{\mathit{u}}_{f}dV-{\int }_{V_P}\mathit{f}dV\right).$$ (28)

For a time step n, the time-discrete form of the hydrodynamic force can then be obtained by using the explicit finite difference approximation of the individual terms on the RHS of (28) as

$${\left[\frac{d}{dt}{\int }_{V_P}{\mathit{u}}_{f}dV\right]}^n=V_P\left(\frac{{ℐ}_P^{f\to s}\left[{\mathit{u}}_f^{n-1}\right]-{ℐ}_P^{f\to s}\left[{\mathit{u}}_f^{n-2}\right]}{\mathrm{\Delta }t}\right)$$ (29)

$${\left[{\int }_{V_P}\mathit{f}dV\right]}^n=V_P\left(\frac{{\mathit{u}}_{sP}^n-{ℐ}_P^{f\to s}\left[{\mathit{u}}_f^{n-1}\right]}{\mathrm{\Delta }t}\right),$$ (30)

where ${ℐ}_P^{f\to s}[◻]$ is an operator that interpolates a quantity □ defined on a CFD mesh to the center of particle P. Details of the interpolation operator are provided in section 2.4. Discrete approximation of the forcing term is obtained by assuming explicit forcing Eq. (35). Plugging in Eqs. (29) – (30) into Eq. (28) yields a closed-form expression

$${\mathit{b}}_{h,P}^n={\rho }_f\left(\frac{2{ℐ}_P^{f\to s}\left[{\mathit{u}}_f^{n-1}\right]-{ℐ}_P^{f\to s}\left[{\mathit{u}}_f^{n-2}\right]-{\mathit{u}}_{sP}^{n-1}}{\mathrm{\Delta }t}\right).$$ (31)

The above equation requires storing interpolated fluid velocities at two time levels per particle. In most practical cases, a near-steady state approximation, i.e., ${ℐ}_P^{f\to s}\left[{\mathit{u}}_f^{n-1}\right]\approx {ℐ}_P^{f\to s}\left[{\mathit{u}}_f^{n-2}\right]$, suffices. Therefore, we approximate the discrete particle-centered hydrodynamic force as

$${\mathit{b}}_{h,P}^n\approx {\rho }_f\left(\frac{{ℐ}_P^{f\to s}\left[{\mathit{u}}_f^{n-1}\right]-{\mathit{u}}_{sP}^{n-1}}{\mathrm{\Delta }t}\right).$$ (32)

2.3.2. Solid-to-Fluid Coupling

To establish solid-to-fluid coupling, the forcing term f in (24) is specified such that the fluid velocity matches the PD particle cloud velocity. Following the analysis of Uhlmann [36], a procedure for implicitly calculating the forcing term from the discrete Navier-Stokes equation is shown below. If the discretized fluid equation at CFD cell P is

$$a_P{\mathit{u}}_{fP}=\mathit{H}\left({\mathit{u}}_f\right)-\nabla p+{\mathit{f}}_P,$$ (33)

where u_fP is the cell-centered fluid velocity, H(u_f) contains the discretized convective and diffusive terms, a_P (units 1/s) are diagonal entries in the discretized fluid velocity equation, and f_P is the cell-centered discrete volumetric forcing term to be determined. To impose the no-slip condition at the fluid-solid interface, the forcing is

$${\mathit{f}}_P=g\left({\alpha }_s\right)\left(a_P{ℐ}_P^{s\to f}\left[{\mathit{u}}_s\right]-\mathit{H}\left({\mathit{u}}_f\right)+\nabla p\right),$$ (34)

where ${ℐ}_P^{s\to f}[\underset{\_}{\square }]$ is an operator that interpolates a quantity □ defined on a PD cloud to CFD cell P, and g(α_s) is a decreasing function with respect to the volume fraction of the PD cloud (α_s) with constraints: g(1) = 1 and g(0) = 0. The immersed boundary forcing is computed iteratively because of its implicit dependence on the pressure gradient.

The above procedure exactly enforces the particle cloud velocity in the Navier-Stokes solution. While rigorous, this approach demands extensive pressure iterations in practice, particularly in two-way coupled flows. If the pressure gradient from the previous time-step is used in (34), we get the simplified version

$${\mathit{f}}_P=g\left({\alpha }_s\right)a_P\left({ℐ}_P^{s\to f}\left[{\mathit{u}}_s\right]-{\mathit{u}}_{fP}^{\mathrm{*}}\right),$$ (35)

where u_fP* is the fluid velocity obtained by solving Eq. (33) without the forcing term. This explicit forcing formulation is mathematically equivalent to Eq. (34). The coefficient a_P incorporates the term 1/Δt alongside contributions from the diffusion and convection terms when implicit discretization schemes are employed. In the limit of small time steps, the coefficient a_P asymptotically approaches 1/Δt. Substituting this limiting value into Eq.(35) recovers the forcing formulation presented in prior studies [38, 46].

2.4. Interpolation

Coupling between the fluid and the solid requires bidirectional information transfer, and this is achieved through interpolation. Since these operations must be performed at every time step, computational efficiency is critical. Data (scalar, vector, or tensor) field transfer between an arbitrary particle cloud and unstructured CFD meshes is achieved via an intermediate pMesh structure. A brief description of the utility of pMesh in our numerical framework is provided below.

2.4.1. The pMesh

A pMesh is a voxelized CFD mesh (See Fig. 2). This entity is primarily used to generate and track particle clouds and to transfer data bidirectionally between the CFD mesh and the particle clouds. In our framework, PD particles are sized to match individual pMesh cells, both of which are represented as cuboids. The precise particle shape is inconsequential because particle-level rotational effects are not included in the current numerical formulation.

Figure 2:

(a) Supplied CFD mesh of a half-torus arbitrarily oriented in space. (b) pMesh obtained from the given half-torus CFD mesh. We show this geometry in this figure because a 3D half-torus provides a simple geometry with non-trivial topological features for creating a boundary-adhering pMesh.

Associating PD particle clouds with the pMesh circumvents many challenges that arise when particles directly interface with an unstructured CFD mesh: (i) particle tracking on unstructured meshes is computationally expensive, requiring hierarchical testing against cell faces → face edges to transfer particle ownership/interpolation of flow quantitites and, (ii) constructing dynamic neighbor lists —a fundamental operation in PD simulations— becomes computationally prohibitive on dynamically adapting CFD meshes. In contrast, we exploit the computational efficiency of geometric operations on structured meshes (pMesh), enabling negligible-cost particle tracking, quick neighbor cell query of arbitrary radii, and efficient runtime construction of the dynamic neighbor list, regardless of whether the CFD mesh is adapting dynamically. The computational implementation used for generating results in this paper required near-zero runtime neighbor list generation cost, as horizon cells for all pMesh cells are pre-computed during initialization.

Bidirectional interpolation between particle data and unstructured meshes is non-trivial due to the difficulty of defining regularized function support for data spreading and accumulation operations. Usage of the pMesh as an intermediate data object helps simplify the implementation of interpolation operations. The interpolation procedure comprises two distinct pathways: (i) particle cloud to CFD mesh interpolation (defining operator $\left.{ℐ}^{s\to f}[\cdot ]\right)$, accomplished via transferring data from the CFD mesh to the pMesh, and then subsequently, transferring the interpolated data from the pMesh to the particle cloud, and (ii) the reverse transfer from CFD mesh to particle cloud (defining operator $\left.{ℐ}^{f\to s}[\cdot ]\right)$, achieved by reversing the process detailed in (i). The subsequent sections detail the atomic operations that constitute these two cases of interpolation.

2.4.2. Interpolation between pMesh ↔ CFD mesh

Bidirectional pMesh ↔ CFD mesh interpolation is achieved through two operators: (i) $ℱ\left(\mathrm{\Phi },{𝒫}_j\right)$, which maps quantity $\mathrm{\Phi }$ from the CFD mesh to pMesh cell ${𝒫}_j$, and (ii) $ℬ\left(\mathrm{\Psi },{𝒞}_i\right)$ maps $\mathrm{\Psi }$ from pMesh to CFD mesh cell ${𝒞}_i$. For bounded, volume-conservative interpolation, the idealized definition of both these operators are

$$ℱ\left(\mathrm{\Phi },{𝒫}_j\right)=\frac{\sum _i\mathrm{\Phi }\left({𝒞}_i\right)𝒱\left({𝒞}_i\cap {𝒫}_j\right)}{\sum _i𝒱\left({𝒞}_i\cap {𝒫}_j\right)},$$ (36)

$$ℬ\left(\mathrm{\Psi },{𝒞}_i\right)=\frac{\sum _j\mathrm{\Psi }\left({𝒫}_j\right)𝒱\left({𝒫}_j\cap {𝒞}_i\right)}{\sum _j𝒱\left({𝒫}_j\cap {𝒞}_i\right)},$$ (37)

where operator 𝒱 returns the volume of its argument. Equations (36) – (37) are straightforward; however, it requires computing the intersection volume between an arbitrary polyhedron and a cuboid, which is non-trivial. In the present work, the volume of intersection between a CFD mesh cell 𝒞_i and a pMesh cell 𝒫_j is approximated as:

$$𝒱\left({𝒞}_i\cap {𝒫}_j\right)\approx 𝒱\left(BB\left({𝒞}_i\right)\cap BB\left({𝒫}_j\right)\right)$$ (38)

where the operator BB returns the axis-aligned bounding box of its argument. See illustration of interpolation weight calculation in Fig. 3.

Figure 3:

((a) Shows a pMesh cell (colored green) which overlaps with 3 CFD mesh cells (colored in green). (b)-(d) shows the approximated volume of intersection (colored in hatched blue). These volume approximations are used in Eq. (36) and (37).

2.4.3. Interpolation between particle cloud ↔ pMesh

In our numerical framework, a PD particle is essentially a floating pMesh cell. This allows for exact, volume-weighted, conservative bidirectional interpolation between an arbitrary particle cloud and a pMesh. For interpolating a quantity χ defined on the pMesh to a particle P, the interpolation operator ℱ′(χ, P) is defined as

$${ℱ}^{\text{'}}(\chi ,P)=\frac{1}{𝒱(P)}\sum _j\chi \left({𝒫}_j\right)𝒱\left({𝒫}_j\cap P\right).$$ (39)

Similarly, for transferring data of a quantity Ω defined on the particle cloud to pMesh cell 𝒫_j, the operator used is ℬ′(Ω, 𝒫_j) and is defined as

$${ℬ}^{\text{'}}\left(\mathrm{\Omega },{𝒫}_j\right)=\frac{1}{𝒱\left({𝒫}_j\right)}\sum _I {\mathrm{\Omega }}_I𝒱\left(P_j\cap I\right).$$ (40)

Consequently, the interpolation operators ${ℐ}^{f\to s}[\cdot ]$ and ${ℐ}^{s\to f}[\cdot ]$ can be defined as a composition of operators defined in Eqs. (36), (37), (39), and (40) as

$${ℐ}^{f\to s}[\cdot ]={ℱ}^{\text{'}}\circ ℱ\text{and}{ℐ}^{s\to f}[\cdot ]=ℬ\circ {ℬ}^{\text{'}}.$$ (41)

2.5. Numerical algorithm

An in-house parallel C++ code was developed to: (i) generate the pMesh for arbitrary unstructured polyhedral mesh configurations, (ii) solve the PD equations, and (iii) perform data transfer operations across the three computational domains: the particle cloud, pMesh, and CFD mesh. This code was interfaced with OpenFoam [47] to solve the fluid equations. The solution procedure for our two-way coupled PD-CFD framework is shown below. Assuming we are progressing from time-step n − 1 to n (where n = 1, 2, 3, …):

1. Interpolate CFD mesh → pMesh → particle cloud: The fluid velocity field (${\mathit{u}}_f^{n-1}$) is transferred from the CFD mesh onto the pMesh. This step entails the calculation of the quantity $ℱ({\mathit{u}}_f^{n-1})$ using the procedure outlined in Section 2.4.2. Subsequently, the fluid velocity residing on the pMesh is interpolated on the particle cloud using equation (39).

2. Update the configuration of the particle cloud: Assemble hydrodynamic forces ${\mathit{b}}_h^n$ on the particle cloud using (32), and solve the PD equation of motion (6) with stabilization (11). Particle equations are subcycled if required. In the current paper, time sub-cycling was performed to maintain an upper limit for the particle Courant number at 0.01, and damping parameter $\gamma$ was set to 0.005.

3. Interpolate particle cloud → pMesh → CFD mesh: The particle cloud volume and velocity are interpolated onto the pMesh according to equation (40). The interpolation of particle cloud volume onto the pMesh yields a volume fraction field (α_s), which is obtained by setting Ω_I = 1 in equation (40). Subsequently, these interpolated quantities residing on the pMesh are transferred to the CFD mesh through the interpolation operator ℬ, defined in equation (37).

4. Update CFD fields: The obtained particle cloud volume fraction and velocity fields from step 3 are then used to solve the Navier-Stokes equations using either implicit (34) or explicit (35) treatment of the body force f.

The above steps are repeated throughout the simulation. For spatial discretization of the non-linear convective terms of (24), a second-order scheme using upwind interpolation with explicit correction based on local gradients was used.

3. Results

3.1. Benchmarking the interpolation procedure

We validate our interpolation procedure against a numerical test case of flow through a 3D half-torus (a bent pipe). The CFD mesh consisted of hexahedral elements and contained both boundary layers and adaptive elements. The geometry and dimension are shown in Fig. 4(a). For getting non-trivial asymmetric flow structures in the full domain, the inflow boundary condition was set to (0 0 0) m/s in the hatched blue section (Fig. 4(a)), and for the red hatched section, the flow velocity was set to (0 5 0) m/s. Kinematic viscosity of the fluid was set to 0.01 m²/s. The flow simulation was run until t = 10s with a time step of Δt = 0.01s using the SIMPLE algorithm, and at each time step, the flow velocity calculated on the CFD mesh was transferred to the pMesh. The average spacing of the pMesh and particle diameter were set to 0.032 m. Total pMesh cell count was 911,284, and volume was 29.4656 m³ against the original CFD domain volume of 29.4477 m³ . Average wall-clock time for transferring data between meshes was about 17 milliseconds when run in parallel with meshes decomposed across 6 processors.

Figure 4:

Benchmarking the interpolation procedure (a) Geometry of the 3D half torus. R_i = 2 m and R_o = 4 m. The diameter of the circular cross-section is r = 1 m. The red-hatched section of the inflow patch consisted of all points that satisfy x > −2.5 m and x < −3.5 m. (b) Snapshot of the velocity magnitude field on the CFD mesh at t = 10 s. (c) Interpolated velocity field on the pMesh at t = 10 s.

Fig. 4 shows snapshots of the flow field obtained at t = 10 s on the CFD mesh (Fig. 4(b)) and the interpolated values on the pMesh (Fig. 4(c)). Both sub-figures show excellent qualitative agreement between the source and interpolated velocity fields. Fig. 5 shows a quantitative comparison of the source and interpolated velocity fields by extracting planar and line data from the field values. Fig. 5 (e) and (f) show the excellent agreement between the velocity fields obtained at the line that passes through the origin and makes angles of 45° and 135° with the negative x-axis. The Pearson correlation coefficient for both curves is greater than 0.99. Sub-figure pairs 5(a–b) and 5(c–d) provide a qualitative comparison of the velocity fields on the CFD mesh and pMesh. Each pair shows a 2D slice of the 3D velocity field obtained using a plane that passes through the origin and makes an angle of 45° or 135° with the negative x-axis, respectively.

Figure 5:

Quantitative comparison of source and interpolated velocity fields for flow in a 3D half-torus. Sub-figure pairs (a-b) and (c-d) show 2D slices of the velocity field obtained by slicing the full field by planes that pass through the origin, which make angles of 45° and 135° with the negative x-axis, respectively. (e-f) Shows the quantitative comparison of line plots of the CFD and the interpolated pMesh velocity field through lines that pass through the points (R, −R, 0) and (R, R, 0), respectively. Here, R is an arbitrary variable that takes values between 0 ≤ R ≤ 4. Coordinates are defined according to the system shown in Fig. 4.

3.2. Validation Study 1: Lid-Driven Cavity

Our coupled PD-CFD framework is first validated using the classic lid-driven cavity case. This is a simple test case that evaluates the effectiveness of an immersed PD body with prescribed zero velocity to contain a flow. The geometry and boundary conditions are shown in Fig. 6(a). The immersed body is modeled as a cloud of static particles within the hatched region. The diameter of PD particles was set to 0.02 m. The number of particles across the wall thickness was set such that there was at least one layer of CFD cells in which the particle-cloud volume fraction (α_s) is 1. Results for three Reynolds numbers 100, 1000, and 5000 are generated by varying the kinematic viscosity of the fluid. Simulations were run until a steady state was reached with Δt = 0.01s. Figs. 7 (a), (c), and (e) show the streamlines of the calculated velocity field inside the cavity for all Reynolds numbers listed previously. Figs. 7(b), (d), and (f) present quantitative comparisons of the velocity components (U_x, U_y) along the x- and y-axes, respectively, against numerical results of Ghia et al. [48]. The fluid velocity flux measured at the CFD domain outer boundaries walls (magenta boundaries in Fig. 6(a)) was negligible, with magnitudes less than 10⁻¹⁵ indicating almost no numerical fluid leakage.

Figure 6:

Geometry and dimensions for validation cases 1 and 2. Regions where the coupled equations are solved are colored in light blue. (a) The 2D lid-driven cavity case is a square with L = 1 m with an actuated top wall with velocity U₀ = 1 m/s. The wall of the cavity is replaced with a finite-sized body with prescribed zero velocity indicated by the hatched cross-section. An additional fluid domain is created outside the cavity and the immersed body to ensure well-posedness of the fluid flow solution. The magnet-colored line indicates dummy walls where zero-gradient boundary conditions for both the pressure and velocity fields are imposed. (b) Domain definition of the oscillating cylinder case. Diameter D of the cylinder is 1 m.

Figure 7:

(a), (c), and (e) show the streamlines of the classic driven cavity case solved using our proposed numerical framework at Re 100, 1000, and 5000. (b), (d), (e) show the quantitative validation of velocity profiles against numerical results published by Ghia et al. [48]. In each sub-figure, the red axes show the plot of the x-component of the fluid velocity at x/L = 0.5, and the blue axes show the plot of the y-component of the fluid velocity at y/L = 0.5.

3.3. Validation Study 2: Oscillating cylinder in a quiescent flow

Our second validation study evaluates and compares the flow field generated by an oscillating cylinder in quiescent flow against published experimental data. This represents an important validation case for future cardiovascular applications where the governing scenario involves prescribed motion of solid boundaries (such as ventricular or atrial walls) to which the blood flow must conform. The geometry, dimensions, and coordinate system are shown in Fig. 6(b). PD particles are generated inside the cylindrical body, and their velocities are prescribed as a cosine function in accordance with the reference [49].

$$u_x(t)=-u_m\mathrm{c}\mathrm{o}\mathrm{s}(2\pi ft)$$ (42)

where u_m = 0.1 m/s and f = 0.02 Hz. The kinematic viscosity was set to 10⁻³ m²/s, yielding Reynolds and Keulegan–Carpenter numbers of 100 and 5, respectively. Simulations were run for four cycles (i.e., 200s) with Δt = 0.1 s. Results in Fig. 8 are shown for the last cycle. The sub-figures 8 (a) and (d) show the snapshots of the streamlines of the fluid velocity field at times t = 175 s and t = 179 s, respectively, that correspond to phase angles of 180° and 210°.

Figure 8:

Data shown in (a-c) and (d-f) are obtained when the accumulated phase angles of the translational motion of the cylinder (comprised of PD particles) were 180° and 210°, respectively. (a) and (d) show snapshots of the fluid velocity streamlines around the cylindrical particle cloud. Figure pairs (b) and (e), and (c) and (f) show the quantitative comparison of the fluid velocity components x and y around the cylinder against the numerical and experimental data published in [49].

These specific values were selected to enable quantitative comparison of velocity profiles at x/D = −0.6 between our results and the experimental data of Dütsch et al. [49]. Subfigures 8(b–c) and (e–f) show excellent correspondence of our results with reference experimental and numerical data.

3.4. Validation Study 3: Evaluation of drag forces around bluff bodies

A two-way coupled fluid structure interaction (FSI) framework requires the accurate evaluation of fluid forces. In the third validation study, the force prediction capability of the solver is established by benchmarking drag coefficients of flow past a cylinder and sphere against published data. Computational domains of dimensions (30D × 15D) and (10D × 10D × 10D) were used for calculating the drag coefficients for sphere and cylinder, respectively. Here, D is the symbol used for both the diameter of a sphere and a cylinder. A collection of static PD particles, each with an effective diameter of D/25, was used to represent the geometry of the cylinder and sphere in the center of the flow domain. Like in the previous validation study, volumes of each of these particles were interpolated on the CFD mesh to calculate the field α_r to be used for equation (34). At the domain walls, no-slip boundary conditions were enforced for the cylinder case, whereas slip boundary conditions were employed for the sphere to minimize computational expense in the 3D case. The drag coefficient was calculated as

$$C_d=-\frac{\int {\rho }_f\mathbf{f}\cdot {\stackrel{\mathit{ˆ}}{\mathit{u}}}_{f}dV}{\frac{1}{2}{\rho }_f{U}_0^{2}A},$$ (43)

where f is the immersed body force defined in (24), ${\tilde{\sum }}_f$ is the specified unit inflow fluid velocity vector, U₀ is the magnitude of the inlet velocity used to define Re, and A is the reference area. Mean drag coefficients for both the cylinder and sphere cases are reported under Table 1. We report values for Re 100 and 200 for a cylinder and Re 100 and 250 for a sphere, which match with published data very well. Re for both cases is defined using the hydraulic diameter of the entity, which is the diameter (D) for both cases.

Table 1:

Mean drag coefficient calculation.

	Re	Reference	$\overline{C_d}$		Re	Reference	$\overline{C_d}$
		Mittal et al. [50]	1.35			Schiller and Naumann [51]	1.09
		Uhlmann [36]	1.45			Flemmer and Banks [52]	1.17
Cylinder	100	Silva et al. [53]	1.39	Sphere	100	Hausmann et al. [37]	1.15
		Azis et al. [54]	1.37			Fadlun et al. [55]	1.07
		Present	1.39			Present	1.11
		Yang and Balaras [39]	1.36
		Pinelli et al. [41]	1.43			Schiller and Naumann [51]	0.74
Cylinder	200	Yi et al. [56]	1.36	Sphere	250	Flemmer and Banks [52]	0.74
		Azis et al. [54]	1.36			Present	0.75
		Present	1.37

3.5. Validation Study 4: Thick Elastic Beam under Flow

While the previous validation studies focused on benchmarking one-way coupling, modeling thrombus embolization necessitates two-way coupling. This validation study benchmarks the two-way coupling capabilities of the current numerical framework by comparing the displacement of a probed point of a 3D beam deforming under flow. The dimensions of the flow domain are (1.25 m × 0.40 m × 0.80 m) (shown in Fig. 9(a)). The beam with dimensions (0.1 m × 0.2 m × 0.4 m) is placed inside the channel such that the front face is located at 0.4 m from the inlet of the domain. This case was originally proposed by Richter [57] with a stiff material model with the Young’s modulus and Poisson’s ratio of 10⁶ Pa and 0.4, respectively. However, due to high stiffness values, the displacement of the beam is very small, so to get greater displacement values, the Young’s modulus of the beam was reduced to 10⁴ Pa by Gillebaart et al. [58]. We reproduce the setup of Gillebaart.

Figure 9:

(a) Geometry and domain definition for the fourth validation study. (b) Displacement evolution of the probed point (0.45, 0.15, 0.15) on the beam for the Gillebaart case [] (Young’s modulus = 10⁴ Pa and Poisson’s ratio = 0.4). (c) Spatial distribution of the beam displacement and the fluid velocity at steady state.

A parabolic velocity profile was set such that the peak inlet velocity was 0.3 m/s. The velocity profile at the inlet (x = 0) was (U₀(y, z) 0 0) m/s where U₀(y, z) is evaluated as

$$U_0(y,z)=\frac{0.3}{{0.2}^2{0.4}^2}[y(0.4-y)]\left[{0.4}^2-z^2\right].$$ (44)

PD particle clouds were generated inside the beam geometry, where particle diameter was set to 0.01 m, resulting in a total of 8000 particles. Both the fluid and PD equations of motion were solved with Δt = 10⁻³ s, and the simulation was run until steady state (i.e., average PD particle velocity less than 10⁻⁵ m/s). Fluid forces were computed on the PD particle cloud and were scaled according to

$$f_s=\left\{\begin{array}{ll}0& \text{if}0\le t<t_s\\ \left(\frac{1}{2}\left[1-\mathrm{c}\mathrm{o}\mathrm{s}2\pi \left(t-t_s\right)\right]\right)& \text{if}{t}_s\le t<\left(t_s+1\right)\\ 1& \text{if}t\ge \left(t_s+1\right),\end{array}\right.$$

where t_s is the solution start time for solving the PD equation of motion. This scaling was multiplied by the assembled fluid force on a single PD particle (32) to avoid destabilization of the solver. Fig. 9(b) shows that the steady state displacement of the beam for the Gillebaart case at the probed point located at (0.45, 0.15, 0.15) is independent of the parameter t_s. Steady state displacement of the probed point is calculated to be (14.51, 4.74) mm, which compares very well with the displacements reported by Tuković et al. [59] who reported values of (14.63, 5.0) mm.

3.6. Embolization of a pre-formed blood clot under fluid flow

In this section, we reproduce in-silico the experimental results reported in Tobin et al. [8]. The setup consisted of a tube of diameter D = 12.7 mm. To form an attached clot, bovine blood of volume 160 μL was deposited on the curved surfaces of the tube and incubated for about an hour at 37 °C. After incubation, phosphate-buffered saline (PBS) was flushed through the tube at flow rates ranging from 2 to 10 liters per minute (L/min) until the clot delaminated from its anchoring surface. It was reported that embolization occurred at an average flow rate of 5 LPM. The simulation setup consisted of a cylindrical domain of diameter D and a length of 4D (See Fig. 10(a) where a = 1 and b = 3). Density and kinematic viscosity of PBS were set to 10³ kg/m³ and 10⁻⁶ m²/s, respectively. A neutrally buoyant in-silico blood clot with dimensions described in Fig. 10(b) was placed such that its leading edge was positioned at x = 0. The constitutive property of the clot was modeled using a first-order Ogden model

$$W\left(ℂ,{\mu }_c,{\beta }_c,{\kappa }_c\right)=\frac{{\mu }_c}{{\beta }_c}\left({\bar{\lambda }}_1^{{\beta }_c}+{\bar{\lambda }}_2^{{\beta }_c}+{\bar{\lambda }}_3^{{\beta }_c}-3\right)+\frac{{\kappa }_c}{2}\left(\frac{J^2-1}{2}-\mathrm{l}\mathrm{n}J\right).$$ (45)

where J is the Jacobian (square root of ℂ), ${\overline{\mathrm{\lambda }}}_i$ are the square roots of the eigenvalues of J^−2/3ℂ, and (μ_c, β_c, κ_c) are material specific parameters. The values of these parameters were set to μ_c = 360 Pa, β_c = 7.00, and κ_c = 10 kPa, to match the material parameters reported in the reference. Choice of using an Ogden constitutive law was motivated due to its wide-spread use in modeling soft-tissue material behavior [60]. Parameter value used in this paper are only a small perturbation of previously reported clot constitutive parameters in [30]. The geometry of the clot Fig. 10(b) was discretized using particles of effective diameter 0.2 mm, resulting in a total of 20,780 particles. Interfacial force modeling was achieved via the procedure detailed by Karmakar et al. [30], which is briefly described here. Adhesive force on a particle P (${\mathit{b}}_P^a$) is modeled as a source terms in Eq. (6), and it takes the form

$${\mathit{b}}_P^a=\sum _{{𝒩}_P^a}\left\{\omega \left(\left|{\mathit{\xi }}_{PN}^a\right|\right){\mu }^a\left({\mathit{\xi }}_{PN}^a\right)\left(s^a\left({\mathit{x}}_P,{\mathit{x}}_N^a\right)-1\right)\left(\frac{18{\kappa }_a}{\pi {\delta }^4}\right)\frac{\left({\mathit{y}}_P-{\mathit{x}}_N^a\right)}{\left|{\mathit{y}}_P-{\mathit{x}}_N^a\right|}{V}_a\right\}.$$ (46)

$${\mathit{\xi }}_{PN}^a={\mathit{x}}_N^a-{\mathit{x}}_{P,},$$ (47)

$$s^a\left({\mathit{x}}_P,{\mathit{x}}_N^a\right)=\frac{\left|{\mathit{y}}_P-{\mathit{x}}_N^a\right|}{\left|{\mathit{x}}_P-{\mathit{x}}_N^a\right|},$$ (48)

$${\mu }^a\left({\mathit{\xi }}_{PN}^a\right)=\left\{\begin{array}{ll}1& \text{if}0\le s^a<s_1^a\\ \left(\frac{s_2^a-s^a}{s_2^a-s_1^a}\right)& \text{if}{s}_1^a\le s^a<s_2^a\\ 0& \text{if}{s}^a\ge s_2^a,\end{array}\right.$$ (49)

where ${\mathit{x}}_{\mathit{Ni}}^a$ is the reference position vector of an anchor particle generated immediately outside the CFD boundary (See Fig. 10(a)); ${𝒩}_P^a$ is the set of all particles ${\mathit{x}}_{\mathit{Ni}}^a$ such that $|{\mathit{x}}_{\mathit{Ni}}^a-{\mathbf{x}}_P|<\delta$; k_a is a surface specific adhesive constant that governs the “stickiness” of a surface; $V_{{\mathit{N}}_i^a}$ is the volume of an anchor particle; $s^a({\mathit{x}}_P,{\mathit{x}}_{\mathit{Ni}}^a$ is bond stretch defined in equation (48); and μ^a is a bond indicator function analogous to equation (14) defined in (49). The light-green particles generated outside the curved walls (Fig. 10(a)) serve a dual purpose: first, they anchor adjacent bodies (through adhesive forces, described above), and secondly, they keep a body from escaping the fluid domain through collision forces. Collision force (${\mathit{b}}_P^c$) is modeled as

$${\mathit{b}}_P^c=\sum _{{𝒩}_P^c} \left\{\mathrm{m}\mathrm{a}\mathrm{x}\left[\left(1-\frac{\left|{\mathit{y}}_P-{\mathit{x}}_{Ni}^a\right|}{2r_c}\right),0\right]\left(\frac{18{\kappa }_b}{\pi {\delta }^4}\right)\frac{\left({\mathit{y}}_P-{\mathit{x}}_{Ni}^a\right)}{\left|{\mathit{y}}_P-{\mathit{x}}_{Ni}^a\right|}{V}_{N_i^a}\right\},$$ (50)

where ${𝒩}_P^c$ is the set of all particles ${\mathit{x}}_{\mathit{Ni}}^a$ such that $|{\mathit{y}}_P-{\mathit{x}}_{\mathit{Ni}}^a|\le r_c$ and no-adhesive bond exists between the considered particles; r_c is the collision radius; and k_b is the boundary collision constant. Values for collision parameters r_c and k_b were set to 0.55d_p and 100k_c, respectively. Three flow cases are considered (see Fig. 10(c)): two constant flow-rates of 4 and 5 LPM, and a step-wise increment of the flow-rate ϕ(t) defined as

$$\varphi \left(t\right)=\left\{\begin{array}{ll}4& \text{if}0\le t<0.2\\ 4+0.25⌈\frac{t-0.2}{0.1}⌉& \text{if}0.2\le t<0.8\\ 5.5& \text{if}t\ge 0.8\end{array}\right..$$ (51)

Figure 10:

(a) Shows half of the PD-CFD simulation setup. The red particles represent the clot geometry, and the light-green colored particles represent static anchor particles, which serve the dual purpose of both anchoring and containing the clot within the fluid domain. These anchor particles are generated outside the walls (light-gray surface) of the fluid domain. (b) Geometry of the clot obtained through the intersection of two orthogonal cylinders, such that L = 14.2 mm and H = 2.5 mm. (c) and (d) show the flow rate (ϕ) specified at the inlet of the domain and the corresponding evolution of surface damage with time.

Delamination of the clot from the anchoring surface is quantified through the surface damage parameter

$$d_{\mathrm{\Gamma }}\left({\mathit{x}}_P\right)=\left(1-\left(\frac{{\int }_{{\mathscr{H}}_P\cup {𝒩}_P^a}{\mu }^{a}dV}{{\int }_{{\mathscr{H}}_P\cup {𝒩}_P^a}dV}\right)\right)\text{and}{d}_{\mathrm{\Gamma }}=\frac{2}{N_{bc}}\sum _{P\in {ℬ}_c^b} d_{\mathrm{\Gamma }}\left({\mathit{x}}_P\right),$$ (52)

where ${ℬ}_c^b$ is the set of all clot particles (red) located next to the curved cylindrical wall at $t=0$, and $N_{bc}$ is the number of particles which are in the set ${ℬ}_c^b$. The term $d_{\mathrm{\Gamma }}\left({\mathit{x}}_P\right)$ calculates the ratio of broken anchoring bonds to all the bonds of a particle P . The factor of two is introduced to scale the surface damage to one. We define a clot to be completely detached from a surface when $d_{\mathrm{\Gamma }}\ge 0.9$. The case of step-wise flow-rate increment was used to calibrate the adhesive force constant $k_a$ under the constraint that $s_1^a=1.03$ and $s_2^a=1.05$. These adhesive critical stretch values were previously used in our work [30] to model the detachment of clots from biomaterial surfaces. Both the fluid and PD equations of motion were solved with $\mathrm{\Delta }t=5\times {10}^{-5}\mathrm{s}$, and the simulation was run for a total of one second. We found that for $k_a=60\mathrm{k}\mathrm{P}\mathrm{a}$, clot embolization occurs at a flow-rate threshold of 5 LPM (See Fig. 10(d)). Figure 10 shows the internal stresses of the clot at various time snapshots of the two constant flow-rate cases. This simulation consisted of solving 250,000 timesteps, which took about 8 hrs of wall clock time for a parallel computer with 12 cores. In addition, the per-time step cost ratio of PD to CFD is less than 0.5. The reader is referred to the Supplementary Materials of this paper to see an animation of the embolization behavior of the in-silico blood clot.

4. Discussion

This paper presents for the first time a numerical framework that couples CFD performed on arbitrary unstructured polyhedral meshes with PD to resolve and predict the physics of blood clot fragmentation and delamination under fluid dynamic forces. In this section, we discuss the fidelity of the various components that make up the proposed numerical framework.

4.1. Comments on the Interpolation Procedure

Bi-directional interpolation is an important aspect of the proposed numerical framework, as it facilitates the transfer of fluid velocity onto the particle cloud to compute fluid forces as well as the transfer of particle cloud velocity and volume to the CFD mesh. To demonstrate the robustness and accuracy of the interpolation method, a simple yet challenging test case is presented in Section 3.1. This particular problem was chosen as a benchmark because the geometry is inherently curved, and the asymmetry of the fluid velocity field helps evaluate the ability of the interpolation technique to capture non-trivial flow structures. Qualitative and quantitative comparisons in Figs. 4–5 show the excellent pointwise agreement between the source and interpolated data. An animation included in the Supplementary materials shows a time sequence of the source and interpolated flow fields for the entire simulation time.

4.1.1. Favorable global conservation properties

By the definitions of operators ℐ^f→s and ℐ^s→f in Eq. (41), conservation errors arising from interpolation will only be due to operators ℱ and ℬ, since operators ℱ′ and ℬ′ conserve quantities exactly. We demonstrate this by evaluating the cumulative error metric ℰ_c after one complete cycle of interpolation.

$${ℰ}_c(\mathrm{\%})=\left(1-\frac{{\int }_{V_{CFD}}ℬ(ℱ(\mathrm{\Lambda }))dV}{{\int }_{V_{CFD}}\mathrm{\Lambda }dV}\right)\times 100$$ (53)

where Λ is an arbitrary smooth field with spatial variation takes the mathematical form

$$\mathrm{\Lambda }(x,y,z)=1+\mathrm{s}\mathrm{i}\mathrm{n}(2\pi x)\mathrm{s}\mathrm{i}\mathrm{n}(2\pi y)\mathrm{s}\mathrm{i}\mathrm{n}(2\pi z).$$ (54)

This quantity is defined on a unit cube CFD mesh centered at (0 0 0) and has been used to evaluate interpolation errors in [61]. ℬ(ℱ(Λ)) represents the interpolation of Λ from the CFD mesh onto the pMesh using the operator defined in Eq. (36), and then transferring the field ℱ(Λ) from the pMesh back onto the CFD mesh using the operator defined in Eq. (37). The error metric Eq. (53) provides a cumulative measure of the relative error in interpolation with regards to the global conservation of Λ.

Two CFD meshes were generated for the unit cube domain: (i) a pure tetrahedral mesh containing 573497 cells and (ii) a pure polyhedral mesh containing 53145 cells (dual mesh of the tetrahedral mesh). Face count per cell for the polyhedral mesh ranged from 7 to 24. Figure 12(a) shows the rate of convergence of the error metric as a function of pMesh spacing for the two CFD meshes. As expected, the interpolation error reduces as the pMesh spacing is decreased. The order of error convergence for the tetrahedral and polyhedral meshes is evaluated to be 0.8 and 1.7, respectively. The convergence rate reported herein applies to the practically usable range of pMesh spacing for this particular case. At impractically fine resolutions on the order of 10⁻⁴ in this test case, the interpolation error saturates to a small asymptotic value that is independent of further mesh refinement, though this asymptotic error remains smaller than the values reported here. The first-order convergence of interpolation errors for pure tetrahedral meshes is due to the assumption made in Eq. (38), namely, the intersection volume between an arbitrary CFD mesh cell and a pMesh cell is approximated through their bounding box intersections. For tetrahedra, Eq. (38) over-estimates the volume intersection, which leads to data diffusion. This necessitates the establishment of a practical criterion for selecting pMesh spacing such that diffusive errors are minimized. To obtain such a criterion, define an average CFD mesh cell size as

$$\mathrm{\Delta }{x}_{\mathrm{C}\mathrm{F}\mathrm{D}}=\frac{1}{N_{\mathrm{C}\mathrm{F}\mathrm{D}}}\sum _{i=1}^{N_{\mathrm{C}\mathrm{F}\mathrm{D}}}{\left[𝒱\left({𝒞}_i\right)\right]}^{1/3},$$ (55)

where N_CFD is the number of CFD mesh cells, and 𝒞_i is the ith CFD mesh cell. Through many numerical tests, we found that setting pMesh spacing (Δx_pMesh) equal to Δx_CFD yields good quality interpolation results. This is also evidenced in Fig. 12(b), which shows that when (Δx_pMesh/Δx_CFD) ≈ 1, the value of the error metric falls below 0.001. This heuristic spacing criterion is sufficient for most practical simulation runs. To provide context, for a reference value of unity (analytical volume integral of Λ), a value of 0.001 for the error metric (53) corresponds to a volume-integrated Λ field value of 0.99999.

Figure 12:

(a) Relative interpolation error convergence as a function of pMesh spacing (Δx_pMesh) for different types of CFD meshes (pure tetrahedral and polyhedral). (b) Interpolative fidelity of data shown in (a) as a function of the quantity (Δx_pMesh/Δx_CFD). Mean order of convergence is 1.3. (c) Field data representation of the quantity Λ Eq. (54) on a pure polyhedral mesh, interpolated onto the pMesh via operator ℱ(·), and then interpolated back on the CFD mesh using operator ℬ(·).

4.1.2. Efficiency of interpolation

Not only is the interpolation procedure accurate, but it is also fast. Table 2 shows profiling results for different types of meshes. The largest pure polyhedral mesh interpolation case examined in this study involved mapping 53145 cells to a pMesh containing 1 million hexhedral cells and required 4.53 seconds of wall clock time (single core). In comparison, the study reported in [61], which presents a state-of-the-art mesh-to-mesh interpolation procedure, required 103.73 seconds for mapping a mesh containing 43731 polyhedral cells to 243422 tetrahedral elements (single core). While direct comparison is complicated by differences in hardware specifications (specifically, 16 GB RAM and an Intel(R) Xeon(R) CPU at 3.20 GHz in the present study versus 4 GB RAM and an Intel Core2 Quad processor at 2.83 GHz in the reference), our significantly lower wall clock time is noteworthy. Furthermore, in our interpolation-weight generation step, the cell search currently scales logarithmically but can be reduced to constant time using hash tables—a future implementation that would further improve overall performance by at least an order of magnitude. In addition, the parallel scaling of our interpolation procedure is quite good, as evidenced in Table 2.

Table 2:

Interpolation time profiling results. The parameters N_CFD and N_pMesh denote the number of computational cells in the CFD mesh and pMesh, respectively. The variable n_p denotes the number of processors utilized; N_Faces is the total number of faces on a CFD mesh; t_w denotes the computational time required for constructing interpolation weights between the CFD mesh and pMesh; and t_f and t_b denote the computational times required for executing the operations ℱ and ℬ, respectively, after the calculation of interpolation weights. It is to be noted that the t_f and t_b are the run-time costs, and t_w is a one-time only cost.

CFD cell-type	N Faces	N CFD	N pMesh	Execution Time np = 1			Execution Time np = 4
				tw (s)	tf (s)	tb (s)	tw (s)	tf (s)	tb (s)
tetrahedral	1129196	573497	1000000	26.969	0.338	0.180	6.033	0.099	0.05
hexahedral	120050	122480	1000000	2.070	0.068	0.025	0.854	0.027	0.012
polyhedral	373999	53145	1000000	4.528	0.099	0.024	1.053	0.026	0.008

4.2. Fidelity of coupling strategy and benchmark results

Typical fluid-solid coupling strategies, such as [36, 38, 44, 45], solve FSI problems on Cartesian meshes and utilize an isotropic discrete Dirac-delta function to distribute interface quantities, which guarantees local force conservation and hence momentum conservation. However, constructing an analogous discrete Dirac-delta function for arbitrary unstructured polyhedral meshes is considerably more involved. The proposed scheme makes no assumption about the mesh structure. It is demonstrated in Appendix A.2 that the scheme recovers the exactness of classical approaches on fixed- and adaptive- Cartesian meshes, where the geometric interpolation errors in our scheme vanish exactly.

For applications involving thrombus fracture under flow, numerical stability is of paramount importance, and we have prioritized it over strict local force conservation. For the polyhedral meshes considered in this work, the local conservation error of an extensive quantity under our interpolation scheme is observed to be approximately 1%, meaning interpolated values are subject to a 1% over- or under-estimation (See Appendix A.3). We further demonstrate in the Appendix A.1 that employing a generic conservative force interpolation scheme does not, in general, guarantee that the net energy transfer across the fluid-solid interface remains dissipative. Using our interpolation strategy, we show in Appendix A.4 that the net energy transfer across the interface is predominantly dissipative in practice. Given the severity of the numerical challenges inherent to ill-conditioned problems such as thrombectomy (a future application of our numerical method), where thrombus fracture occurs under near-choked flow conditions, prioritizing numerical stability over strict local force conservation is a justified compromise.

Additionally, we also show in Appendix A.5 that the forces defined in Eqs. (35) and (32) can be corrected to yield an unconditionally dissipative scheme regardless of the approximation errors. The corrections take the form

$${\mathit{f}}^m\left({𝒞}_i\right)=\left(1+\frac{ℰ\left({𝒞}_i\right)}{𝒱\left({𝒞}_i\right)}\right)\mathit{f}\left({𝒞}_i\right)\text{and}{\mathit{b}}_h^m\left({𝒫}_j\right)=\left(1+\frac{ℰ\left({𝒫}_j\right)}{𝒱\left({𝒫}_j\right)}\right){\mathit{b}}_h\left({𝒫}_j\right).$$ (56)

where ℰ(𝒞_i) and ℰ(𝒫_j) are the error in approximating the volume of the CFD mesh cell (𝒞_i) using pMesh cells, and vice-versa. Formal definitions of these errors can be written down as

$$ℰ\left({𝒞}_i\right)=\sum _j 𝒱\left({𝒞}_i\cap {𝒫}_j\right)-𝒱\left({𝒞}_i\right)\text{and}ℰ\left({𝒫}_j\right)=\sum _i𝒱\left({𝒞}_i\cap {𝒫}_j\right)-𝒱\left({𝒫}_j\right).$$ (57)

Nevertheless, the uncorrected forces are employed throughout this work, as the net energy transfer across the interface is observed to be predominantly dissipative in practice, and it yielded excellent results across all benchmark cases without any spurious numerical instabilities, as discussed below.

The solver is validated through a multiple-step process where constraints on the coupled physics are systematically relaxed. The first three validation cases demonstrate that our FSI framework exhibits the following properties: (i) flow confinement within immersed external bodies prescribed to have zero velocity boundary conditions (Section 3.2), (ii) accurate prediction of velocity fields in the fluid domain surrounding immersed bodies with prescribed non-zero motion (Section 3.3), and (iii) precise evaluation of fluid-induced forces (Section 3.4). Finally, the fourth validation case (Section 3.5) benchmarked two-way coupling by predicting the large deformations of a thick-elastic beam immersed in flow. Results of all these benchmarks yield excellent correlation with published experimental/numerical data.

In the current framework, the PD solid is represented as a volume fraction field α_s calculated by interpolating the particle cloud volume onto the CFD mesh, resulting in a diffused solid-fluid interface. Such representation of the interface can lead to over-prediction of drag forces imparted to the fluid or to delayed flow separation from bluff bodies (e.g., flow past a cylinder at Re = 100). This is a well-known issue in diffused interface immersed body methods [62]. The two most common fixes proposed in the literature are: (i) refining the mesh around the fluid-solid interface, and (ii) inwardly shrinking the outer boundary of an immersed body [63]. Refining the mesh around the solid-fluid interface is not always feasible, particularly when the body undergoes large deformation. While inward-boundary shrinkage has been successfully used in previous studies, the amount of retraction required to generate correct results is often arbitrary and body-shape specific. This is non-ideal for our framework that seeks to handle dynamically evolving arbitrarily shaped bodies, namely, clot growth, deformation, and embolization. In our framework, the immersed boundary interface is sharpened by applying a sharpening function g(α_s) which is used in Eq. (34). Common forms used in this work are

$$g\left({\alpha }_s\right)={\alpha }_s^m;\text{or}g\left({\alpha }_s\right)=\left[\frac{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{{\alpha }_s-{\alpha }_s^t}{l_w}\right)+\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{{\alpha }_s^t}{l_w}\right)}{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{1-{\alpha }_s^t}{l_w}\right)+\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{{\alpha }_s^t}{l_w}\right)}\right],$$ (58)

where α_s, ${\alpha }_s^t\in [0,1]$, m ≥ 1 is the polynomial exponent, and l_w is a characteristic length of sharpening. The hyperbolic tangent function helps achieve arbitrarily sharp interfaces, but it can cause pressure oscillations near the interface in under-resolved CFD meshes, a known problem with immersed body methods [64]. Therefore, for general-purpose use and eliminating pressure oscillations, a sufficiently resolved CFD mesh combined with a polynomial sharpening function is recommended. This strategy with m = 1.2 was employed for running all our two-way coupled cases.

It was interesting to note that although the nonlocal formulation of NOSB-PD naturally increases computational and parallelization requirements, it did not bottleneck our framework. This was because PD particles occupy only a small fraction of the overall domain, leaving the simulation runtime primarily governed by the CFD pressure Poisson solver.

4.3. Modeling thrombus embolization

The physical phenomenon of thrombus embolization occurs through two main (competing) pathways: (i) fragmentation of blood clots by blood flow forces and (ii) delamination of blood clots anchored to surfaces. Simulating clot embolization is complex. The complexity stems from the need to simulate multiple coupled physical phenomena simultaneously that include: (i) heterogeneous non-linear constitutive law of a blood clot, (ii) fracture mechanics of a blood clot, (iii) adhesive mechanics of a blood clot anchored on a biomaterial surface, (iv) coupling with blood flow, and (v) dynamic growth of a thrombus through thrombogenesis pathways. The presented numerical framework can incorporate all the above-mentioned physics, thereby making it the first comprehensive numerical framework able to simulate all key phenomena associated with thrombus embolization. We justify this bold assertion in the following paragraphs to convince the reader that our numerical framework is indeed what we claim it to be.

In section 3.6 we replicated in silico the experimental results of Tobin et al. [8] who observed the flow-induced dynamics of a pre-formed blood clot deposited inside a poly-carbonate tube and its embolization at a mean flow-rate of ≈ 5 LPM. In our simulation of the experimental conditions, the in silico blood clot was modeled as a homogeneous material whose mechanical response was described by a first-order Ogden material model with parameters corresponding to an elastic modulus of ≈ 3.5 kPa, and a peak force reaction of 0.08 N at 10 % compressive strain for a cylindrical clot (diameter 15 mm, height 10 mm). The in silico blood clot was anchored to the tube walls where uniform adhesive strength was assumed. Fig. 13(a) shows a plot of surface adhesive damage for different values of adhesive strength k_a (Eq. 50) when subjected to flow rates specified by Eq. (51). k_a was calibrated to be 60 kPa such that the clot embolized at ≈ 5 LPM in the simulation case. This value can be contextualized using our previously published work [30], where we calibrated adhesive strength required for pulled-force detachment of blood clots (incubated for 3 hrs) from biomaterial surfaces, specifically, k_a = 122 kPa for polyurethane and k_a = 146 kPa for polytetrafluoroethylene. Blood clots used in the experiment of Tobin [8] were incubated for one hour. Kannojia et al. [65] show that adhesive strength reduces by a factor of 0.6 and 0.4 when clot incubation time is reduced from three to one and 0.5 hours, respectively. Based on this observation, a linear scaling applied to our previous calibration in [30] yields an expected adhesive strength within the range 50 – 80 kPa for the Tobin experiment. These expected values are inline with our in silico determined value of 60 kPa for the Tobin experiment, which validates the physical consistency of our PD model of clot-surface adhesion via two experimental studies focused on two different types of release mechanisms.

Figure 13:

Additional plots for the simulation study reported in Section 3.6. (a) Temporal evolution of surface adhesive damage for different values of surface adhesive constant k_a. (b) Detachment times of particles initially located at (x, 0, 0) at the start of the simulation. For instance, the particle positioned at (0.2, 0, 0) at t = 0 detached from the anchoring surface at t ≈ 0.25 s. The axial direction of the domain is aligned along the x-axis. A particle is said to be detached from an anchoring surface when the quantity d_Γ(x_P) exceeds 0.9.

A truly comprehensive numerical model of thrombus embolization should be able to predict not only the flow rate at which a thrombus embolizes, but the mechanism of embolization, i.e., the time-course evolution of the thrombus deformation and fracture. For the 32 experiments conducted in the reference [8], it is documented that embolization occurred in 23 cases due to trailing edge detachment, one case due to leading edge detachment, and the remaining 8 cases showed ambiguous detachment initiation patterns. In our simulations, under the assumptions of homogeneous material properties and uniform adhesive strength, the clot always detached from the leading edge. This is intuitive given the flow diverting nature of the leading edge and is confirmed from the vonMises stress distribution shown in Figs. 11(d–f). Due to the re-circulation zone at the wake of the clot, there are lift forces acting on the trailing edge. These forces are insufficient to initiate detachment, but as the clot starts to detach, the flow becomes very chaotic, leading to the amplification of lift forces and causing the trailing edge to lift in a very pronounced manner (Fig. 11(e)). These observations suggest that embolization initiated by the trailing edge is probably caused due to non-uniform adhesive strength distribution and strong flow buffeting that induces strong aperiodic recirculation zones. In addition, it is well documented that deposited blood droplets dry nonuniformly, making its material properties inhomogeneous [66]. We hypothesize that for experimental embolizations initiated through trailing edge detachment, those clot specimens had inhomogeneous material stiffness and reduced nonuniform adhesive strength. This is evidenced by the video of an experiment in reference [8] that shows that the leading edge firmly anchored through 80% of the video duration, while the trailing edge begins to flap immediately upon flow initiation.

Figure 11:

Time-evolution of Von-Mises stress distribution as the clot interacts with the fluid under a constant flow rate of 4 LPM (a-c) and 5 LPM (d-f). Results for half the computational domain are shown here for clarity. Streamlines colored orange-red are shown to depict the chaotic wake. Part of the curved cylindrical wall of the tube is shown as a shaded grey colored structure.

We put our hypothesis to the test by subjecting a synthetic inhomogeneous clot to a constant flow rate of 5 LPM (Fig. 14(a)). Both regions of the clot were modeled using a first-order Ogden model, assigning the red region stiff material properties (μ_c = 360 Pa, β_c = 7.00, κ_c = 10 kPa) and the blue region to be less rigid (μ_c = 250 Pa, β_c = 2.00, κ_c = 2 kPa). Additionally, adhesive strength k_a in the red and blue regions was set to 200 kPa and 0 kPa, respectively, such that the area averaged k_a ≈ 60 kPa. Figs. 15(a)–(c) show the time-course evolution of deformation for the heterogeneous clot. It can be clearly seen that the trailing edge of the clot lifts before the leading edge, and the embolus is transported downstream by the flow after complete wall detachment. Although the variation of parameters chosen for this synthetic test case is large, they effectively demonstrate that trailing edge clot embolization can be predicted by our comprehensive numerical framework, provided the constitutive parameters of the clot and its adhesive strength distribution are known. Mesh convergence studies are reported in Appendix B.

Figure 14:

(a) Clot divided into regions based on material stiffness. The blue region is softer than the red region. L = 14.2 mm and H = 2.5 mm. (b) Comparison of the temporal evolution of surface adhesive damage for the homogeneous and inhomogeneous cases. Detachment times of the entire clot were ≈ 0.1 and ≈ 0.3 s, respectively. (c) Comparison of detachment times for particles located at (x, 0, 0) for the homogeneous and inhomogeneous cases. In the heterogeneous case, detachment from the anchoring surface initiates at the trailing edge, in contrast to the homogeneous case, where detachment initiates near the leading edge.

Figure 15:

Evolution of Von-Mises stress (a-c) and particle-wise surface adhesive damage (d-f) for a heterogeneous clot (g-i) as it interacts with fluid flow. For the heterogeneous case (softer material properties and weaker adhesive strength at the trailing edge of the clot), the trailing edge detaches before the leading edge. (g-i) Preservation of clot region volumes as the clot moves and deforms in the fluid domain. Region-wise volume conservation is naturally achieved in a Lagrangian framework, whereas it poses non-trivial numerical challenges in an Eulerian framework.

In summary, through the results reported in Section 3.6 and the synthetic case presented above, we have demonstrated that the PD-CFD coupled numerical framework can incorporate four of the five physical phenomena identified in the opening paragraph of this section. The fifth requirement - the ability to include dynamic thrombus growth physics - is also possible with this framework. Section 4.4 provides a brief description of how this can be achieved. In addition, the applicability of the current framework goes beyond thrombus embolization, as the core physics governing thrombus embolization - fracture of soft-tissue under fluid forces - also governs phenomena like (i) aspiration thrombectomy where successful clot retrieval through catheters depends critically on maintaining thrombus integrity and preventing fracture [67], (ii) atherosclerotic plaque rupture where fluid shear stresses induce fibrous cap failure [68], and (iii) thrombolysis where clots are piecemeal dissolved and transported away as anticoagulants slowly degrade its internal fibrin networks. This study has demonstrated that the model can capture fundamental physics of clot detachment and embolization, albeit for an idealized, controlled in-vitro experiment and based on a published constitutive model. Therefore, future efforts are required to implement the model with patient-specific (vascular) geometry. Also, it behooves us to validate the versatility of the constitutive model (stiffness and fracture) over a range of real-life blood clots. The framework is designed to accommodate more sophisticated material models if required.

4.4. Limitations and Future Work

Primary future work includes the coupling of the presented framework with Eulerian models of thrombogenesis, such as those presented in [69–71]. The main output of these models is a time-evolving volume fraction field representing thrombus growth. To incorporate thrombus growth into the PD framework, two fields must be maintained: ${\alpha }_t^o$, representing the previous thrombus configuration, and α_t, representing the current. Assuming the existing particle cloud corresponds to ${\alpha }_t^o$, new particle generation proceeds by identifying locations in the pMesh where $ℱ({\alpha }_t-{\alpha }_t^o)$ exceeds a certain threshold. Following this particle update, the PD solution proceeds normally as described herein. Implementation and error analysis of such a procedure will be addressed in future work.

While the model of clot detachment and embolization has been demonstrated for a specific, well-controlled experiment, its future practical application needs to include patient-specific geometry, boundary conditions, as well as the composition of the clot. Our vision is to inform a physician contemplating a thrombectomy procedure in a patient presenting with a clot-occluded blood vessel. A simulation could be conducted with boundary conditions specified by clinical imaging and measured hemodynamics to guide the optimal positioning of the thrombectomy (suction) catheter and flow rate to retrieve the blood clot in one piece. Alternatively, the simulation would help provide guidance on whether or not an intervention is even necessary. Therefore, future steps will be conducted to implement the model for several patient-specific boundary value problems to calibrate and validate it as a clinical tool.

In this paper, thrombus material parameters were initialized from the calibrated values reported in our previous work [30], in which the Ogden and damage parameters were determined using a manual iterative approach. These parameters were subsequently perturbed by a small amount to match the specific clot mechanical properties reported in Tobin et al. [8]. The sole independently calibrated parameter is the adhesive strength of a thrombus with a polycarbonate substrate, which is reported to be 60 kPa. The uncertainty associated with this quantity is minimal as embolization flow rate scales linearly with adhesive strength (See Fig. 13), making the parameter search trivial. Blood clot mechanical properties exhibit significant patient-to-patient variability depending on clot age, fibrin network density, and red blood cell content [72], and is a critical avenue to explore in future work. Specifically, the implementation of a Bayesian framework following [73] for calibrating thrombus mechanical properties would be ideal. Such a framework would allow for systematic quantification of how experimental uncertainties translate into model predictions, ultimately providing a more robust characterization of biological variability on embolization dynamics.

Accurate constitutive modeling of thrombi is critical for predicting accurate embolization dynamics. Consequently, future work will also incorporate viscoelastic constitutive models, such as those presented in [74], as well as composition-informed material property variation [75]. In particular, it will be very interesting to evaluate the influence of viscoelasticity on the embolic behaviors and dynamics of a thrombus through parametric studies.

5. Conclusion

This paper presents a comprehensive numerical model of thrombus embolization through the coupling of non-ordinary state-based peridynamics with fluid flow equations. The framework can resolve the two competing physics of thrombus embolization: (a) cohesive fracture of blood clots and (b) adhesive fracture of blood clots from anchored surfaces. The coupling methodology is independent of fluid mesh topology, which is enabled by a robust interpolation algorithm that facilitates efficient bidirectional transfer of field quantities between arbitrary polyhedral meshes and peridynamic particle clouds. The framework was rigorously validated through a series of benchmarks, where it is established that: (i) the interpolation procedure preserves both the magnitude and spatial variation of quantities when transferring data between CFD meshes and particle clouds, (ii) stationary particle clouds can successfully contain flow within their boundaries, as demonstrated by the lid-driven cavity case, (iii) particle cloud velocities are strongly enforced, thereby allowing correct prediction of flow around an oscillating cylinder in quiescent flow, (iv) the solver can correctly predict drag forces exerted by stationary bluff bodies imparted to fluid, as verified by mean drag coefficient evaluation for flow past cylinders and spheres, and (v) large deformations of bodies immersed in flow are correctly predicted, as shown by the thick elastic beam in flow case. The validated framework successfully reproduced experimental observations of the embolization of pre-formed thrombus in polycarbonate cylindrical tubes under varying flow rates. Finally, a synthetic test case was run to demonstrate the capability of the framework to incorporate spatially heterogeneous material properties. The deformation dynamics of a heterogeneous clot explained the experimentally observed trailing edge detachment, in contrast to the leading edge detachment predicted when clot homogeneity is assumed.

Highlights.

• A robust interpolation procedure for fast bi-directional data transfer between arbitrary polyhedral meshes and peridynamic particle clouds.

• Rigorously validated numerical framework that couples Non-Ordinary State-Based peri-dynamics with the finite-volume method.

• A comprehensive model of thrombus embolization capable of capturing the complex coupled non-linear mechanics of fluid flow, thrombus deformation, and cohesive and adhesive fracture.

A. Interpolation error analysis

Let ℰ (𝒞_i) and ℰ (𝒫_j) be the errors in approximating the exact volumes of CFD mesh cell 𝒞_i and pMesh cell 𝒫_j, respectively. The mathematical form of this approximation can be written as

$$\sum _i𝒱\left({𝒞}_i\cap {𝒫}_j\right)=𝒱\left({𝒫}_j\right)+ℰ\left({𝒫}_j\right),$$ (A1)

$$\sum _j𝒱\left({𝒞}_i\cap {𝒫}_j\right)=𝒱\left({𝒞}_i\right)+ℰ\left({𝒞}_i\right).$$ (A2)

A.1. Quantification of local conservation errors

Let $w_{ji}^f$ be interpolation weights of an exactly conservative interpolation scheme that interpolates an extensive quantity $\varphi \left({𝒞}_i\right)𝒱\left({𝒞}_i\right)$ directly from the CFD mesh onto the pMesh. The relation between the equivalent quantity on the $\mathrm{p}\mathrm{M}\mathrm{e}\mathrm{s}\mathrm{h}{ℱ}^{\mathrm{*}}\left(\varphi ,{𝒫}_j\right)𝒱\left({𝒫}_j\right)$ and $\varphi \left({𝒞}_i\right)𝒱\left({𝒞}_i\right)$ on the CFD mesh can be written as

$${ℱ}^{\mathrm{*}}\left(\varphi ,{𝒫}_j\right)𝒱\left({𝒫}_j\right)=\sum _i{w}_{ji}^f\varphi \left({𝒞}_i\right)𝒱\left({𝒞}_i\right).$$ (A3)

For exact local conservation, the following condition needs to be satisfied

$$\sum _j{w}_{ji}^f=1.$$ (A4)

According to our operator ℱ(·), the weight $w_{ji}^f$ is

$$ℱ\left(\varphi ,{𝒫}_j\right)=\frac{\sum _i𝒱\left({𝒞}_i\cap {𝒫}_j\right)\varphi \left({𝒞}_i\right)}{𝒱\left({𝒫}_j\right)+ℰ\left({𝒫}_j\right)}⟹w_{ji}^f=\left(\frac{𝒱\left({𝒞}_i\cap {𝒫}_j\right)}{𝒱\left({𝒞}_i\right)}\right)\left(\frac{𝒱\left({𝒫}_j\right)}{𝒱\left({𝒫}_j\right)+ℰ\left({𝒫}_j\right)}\right).$$ (A5)

Computing $\sum _j{w}_{ji}^f$ for our operator, the error in local conservation is Starting from Eq. (A5), simplifying and using the definition of $ℬ(·)$, we have

$$\begin{gathered} \sum _j{w}_{ji}^f=\frac{1}{𝒱({𝒞}_i)}\sum _j\frac{𝒱(P_j)}{𝒱(P_j)+ℰ(P_j)}𝒱({𝒞}_i\cap P_j)=\frac{1}{𝒱({𝒞}_i)}\sum _j𝒱({𝒞}_i\cap P_j)\left(1-\frac{ℰ(P_j)}{𝒱(P_j)+ℰ(P_j)}\right) \\ =1+\frac{1}{𝒱({𝒞}_i)}\left(ℰ({𝒞}_i)-\sum _j\frac{𝒱({𝒞}_i\cap P_j)ℰ(P_j)}{𝒱(P_j)+ℰ(P_j)}\right)=1+\frac{1}{𝒱({𝒞}_i)}(ℰ({𝒞}_i)-ℬ(ℰ(P_j),{𝒞}_i)) \end{gathered}$$

Therefore, the local error in extensive interpolation from CFD mesh to pMesh is

$$\sum _j{w}_{ji}^f-1=\frac{1}{𝒱\left({𝒞}_i\right)}\left(ℰ\left({𝒞}_i\right)-ℬ\left(ℰ\left({𝒫}_j\right),{𝒞}_i\right)\right)$$ (A6)

Fig. A1 shows a plot of a cell-wise local conservation error and the max errors are about 1%. Note that these errors is independent of the quantity being interpolated and is a function of the error in geometric approximation which become better as the pMesh/CFD mesh is refined. Cells with highest errors are concentrated at the boundary, which is expected as the bounding boxes of the boundary cells are incompletely covered by pMesh cells. A similar calculation can be done for interpolation in the other direction i.e., pMesh to CFD mesh, and the error are of the same order as the geometric approximation errors are coupled.

Figure A1:

Absolute value of the local conservation error calculated using Eq. (A6) is shown for an arbitrary polyhedral mesh (a). Max local conservation errors are seen to be about 0.01 (1%).

A.2. Exact conservation in Cartesian meshes

It is easy to see that local conservation is exactly guaranteed in Cartesian meshes as both the errors $ℰ\left({𝒞}_i\right)$ and $ℰ\left({𝒫}_j\right)$ are exactly zero, and thereby making the right hand side of Eq. (A6) exactly zero.

A.3. Net power transfer under exact conservative force interpolation

Using symbols ${\mathit{F}}_f$ and ${\mathit{F}}_s$ as interface forces acting on fluid and the solid phases, exact local force conservation requires

$${\mathit{F}}_f\left({𝒞}_i\right)𝒱\left({𝒞}_i\right)=-\sum _j{b}_{ij}{\mathit{F}}_s\left({𝒫}_j\right)𝒱\left({𝒫}_j\right)\text{with}\sum _i{b}_{ij}=1.$$ (A7)

For energy conservation to hold exactly, ${ℒ}_e=0$, where

$${ℒ}_e=\sum _i{\mathit{u}}_f\left({𝒞}_i\right)\cdot {\mathit{F}}_f\left({𝒞}_i\right)𝒱\left({𝒞}_i\right)+\sum _j{\mathit{u}}_s\left({𝒫}_j\right)\cdot {\mathit{F}}_s\left({𝒫}_j\right)𝒱\left({𝒫}_j\right).$$ (A8)

Assuming $f_{ji}$ are interpolation weights used for interpolating ${\mathit{u}}_f$ to calculate the force on the solid, such that

$${\mathit{F}}_s\left({𝒫}_j\right)=\frac{{\rho }_f}{\mathrm{\Delta }t}\left(\sum _i{f}_{ji}{\mathit{u}}_f\left({𝒞}_i\right)-{\mathit{u}}_s\left({𝒫}_j\right)\right)$$

Proceeding by simplifying $\sum _i{\mathit{u}}_f\left({𝒞}_i\right)\cdot {\mathit{F}}_f\left({𝒞}_i\right)𝒱\left({𝒞}_i\right)$,

$$\sum _i{\mathit{u}}_f\left({𝒞}_i\right)\cdot \left(-\sum _j{b}_{ij}{\mathit{F}}_s\left({𝒫}_j\right)𝒱\left({𝒫}_j\right)\right)=\sum _j{\mathit{F}}_s\left({𝒫}_j\right)𝒱\left({𝒫}_j\right)\cdot \left(-\sum _i{b}_{ij}{\mathit{u}}_f\left({𝒞}_i\right)\right)$$

Implies,

$$\begin{gathered} {ℒ}_e=\sum _j{\mathit{F}}_s({𝒫}_j)𝒱({𝒫}_j)\cdot \left(u_s({𝒫}_j)-\sum _i{b}_{ij}{\mathit{u}}_f({𝒞}_i)\right) \\ =-\frac{{\rho }_f}{\text{Δ}t}\sum _j\left(u_s({𝒫}_j)-\sum _i{f}_{ji}{\mathit{u}}_f({𝒞}_i)\right)\cdot \left({\mathit{u}}_s({𝒫}_j)-\sum _i{b}_{ij}{\mathit{u}}_f({𝒞}_i)\right)𝒱({𝒫}_j) \end{gathered}$$

It is evident from the final expression that ℒ_e ≤ 0 if and only if f_ji = b_ij. Since this condition is not guaranteed, the net power transfer across the fluid-solid interface is not guaranteed to remain dissipative under a generic conservative force interpolation scheme. Classical IBMs satisfy ℒ_e ≤ 0 due to the discrete Dirac delta function which cannot be trivially extended to unstructured polyhedral meshes.

A.4. Net power transfer using our interpolation scheme

Power transfer associated with the fluid and solid phases:

$$\sum _i{\mathit{u}}_f\cdot {\mathit{F}}_f𝒱\left({𝒞}_i\right)=\frac{{\rho }_f}{\mathrm{\Delta }t}\sum _i{\mathit{u}}_f\cdot \left(ℬ\left({\mathit{u}}_s\right)-{\mathit{u}}_f\left({𝒞}_i\right)\right)𝒱\left({𝒞}_i\right)$$ (A9)

$$\sum _j{\mathit{u}}_s\cdot {\mathit{F}}_s𝒱\left({𝒫}_j\right)=\frac{{\rho }_f}{\mathrm{\Delta }t}\sum _j{\mathit{u}}_s\cdot \left(ℱ\left({\mathit{u}}_f\right)-{\mathit{u}}_s\left({𝒫}_j\right)\right)𝒱\left({𝒫}_i\right)$$ (A10)

Define

$${ℒ}_{ef}=\sum _i\left({\left|ℬ\left({\mathit{u}}_s\right)\right|}^2-{\left|{\mathit{u}}_f\right|}^2\right)𝒱\left({𝒞}_i\right)\text{and}{ℒ}_{es}=\sum _j\left({\left|ℱ\left({\mathit{u}}_f\right)\right|}^2-{\left|{\mathit{u}}_s\right|}^2\right)𝒱\left({𝒫}_j\right)$$ (A11)

Writing out expressions for power for both phases

$$\begin{gathered} \frac{2\mathrm{\Delta }t}{{\rho }_f}\sum _i{\mathit{u}}_f\cdot {\mathit{F}}_f𝒱\left({𝒞}_i\right)=-\left(\sum _i{\left|ℬ\left({\mathit{u}}_s\right)-{\mathit{u}}_f\right|}^2𝒱\left({𝒞}_i\right)\right)+{ℒ}_{ef} \\ \frac{2\mathrm{\Delta }t}{{\rho }_f}\sum _j{\mathit{u}}_s\cdot {\mathit{F}}_s𝒱\left({𝒫}_j\right)=-\left(\sum _j{\left|ℱ\left({\mathit{u}}_f\right)-{\mathit{u}}_s\right|}^2𝒱\left({𝒫}_j\right)\right)+{ℒ}_{es} \\ \frac{2\mathrm{\Delta }t}{{\rho }_f}{ℒ}_e=-\left(\sum _i{\left|ℬ\left({\mathit{u}}_s\right)-{\mathit{u}}_f\right|}^2𝒱\left({𝒞}_i\right)\right)-\left(\sum _j{\left|ℱ\left({\mathit{u}}_f\right)-{\mathit{u}}_s\right|}^2𝒱\left({𝒫}_j\right)\right)+{ℒ}_{ef}+{ℒ}_{es} \end{gathered}$$ (A12)

In the above expression, it is trivial to see that

$$\left(\sum _i{\left|ℬ\left({\mathit{u}}_s\right)-{\mathit{u}}_f\right|}^2𝒱\left({𝒞}_i\right)\right)\ge 0\text{and}\left(\sum _j{\left|ℱ\left({\mathit{u}}_f\right)-{\mathit{u}}_s\right|}^2𝒱\left({𝒫}_j\right)\right).\ge 0$$

Indicating that the first two terms are always less than zero. We now need to simplify ℒ_ef and ℒ_es. Using Jenson’s inequality, we can state

$${\left|ℬ\left({\mathit{u}}_s\right)\right|}^2\le ℬ\left({\left|{\mathit{u}}_s\right|}^2\right),\text{and}{\left|ℱ\left({\mathit{u}}_f\right)\right|}^2\le ℱ\left({\left|{\mathit{u}}_f\right|}^2\right).$$ (A13)

Proceeding further with the analysis

$$\begin{gathered} {ℒ}_{ef}+{ℒ}_{es}\le \sum _i\left(ℬ\left({\left|{\mathit{u}}_s\right|}^2\right)-{\left|{\mathit{u}}_f\right|}^2\right)𝒱\left({𝒞}_i\right)+\sum _j\left(ℱ\left({\left|{\mathit{u}}_f\right|}^2\right)-{\left|{\mathit{u}}_s\right|}^2\right)𝒱\left({𝒫}_j\right) \\ =-\left(\sum _i{\left|{\mathit{u}}_f\right|}^2𝒱\left({𝒞}_i\right)-\sum _jℱ\left({\left|{\mathit{u}}_f\right|}^2\right)𝒱\left({𝒫}_j\right)\right)-\left(\sum _j{\left|{\mathit{u}}_s\right|}^2𝒱\left({𝒫}_j\right)-\sum _iℬ\left({\left|{\mathit{u}}_s\right|}^2\right)𝒱\left({𝒞}_i\right)\right) \end{gathered}$$

The above expression is deliberately not simplified further, as it shows the difference in volume integration of ${\left|{\mathit{u}}_s\right|}^2$ and ${\left|{\mathit{u}}_f\right|}^2$ on source and target meshes. By construction of the interpolation operators and validated through our numerical tests, the source volume integral is always greater than or equal to the target interpolated volume integral, making ${ℒ}_{ef}+{ℒ}_{es}\le 0$. Indicating that the net power transfer across the interface is dissipative, guaranteeing a stable numerical solution. Furthermore, the quantities ${ℒ}_{ef}$ and ${ℒ}_{es}$ are dependent on the geometric approximation errors, which again decrease with refinement.

A.5. Net power transfer using modified forces

To keep the analysis clean, introduce

$$\overline{𝒱\left({𝒞}_i\right)}=𝒱\left({𝒞}_i\right)+ℰ\left({𝒞}_i\right)\text{and}\overline{𝒱\left({𝒫}_j\right)}=𝒱\left({𝒫}_j\right)+ℰ\left({𝒫}_j\right)$$ (A14)

Power transfer associated with the fluid phase using the modified force (56):

$$\sum _i{\mathit{u}}_f\cdot {\mathit{F}}_f^m𝒱\left({𝒞}_i\right)=\frac{{\rho }_f}{\mathrm{\Delta }t}\sum _i{\mathit{u}}_f\cdot \left(ℬ\left({\mathit{u}}_s\right)-{\mathit{u}}_f\left({𝒞}_i\right)\right)\overline{𝒱\left({𝒞}_i\right)}$$ (A15)

Power transfer associated with the solid phase using the modified force (56):

$$\sum _j{\mathit{u}}_s\cdot {\mathit{F}}_s^m𝒱\left({𝒫}_j\right)=\frac{{\rho }_f}{\mathrm{\Delta }t}\sum _j{\mathit{u}}_s\cdot \left(ℱ\left({\mathit{u}}_f\right)-{\mathit{u}}_s\left({𝒫}_j\right)\right)\overline{𝒱\left({𝒫}_j\right)}$$ (A16)

Following the same algebra as before:

$$\frac{2\mathrm{\Delta }t}{{\rho }_f}{ℒ}_e=-\left(\sum _i{\left|ℬ\left({\mathit{u}}_s\right)-{\mathit{u}}_f\right|}^2\right)\overline{𝒱\left({𝒞}_i\right)}-\left(\sum _j{\left|ℱ\left({\mathit{u}}_f\right)-{\mathit{u}}_s\right|}^2\overline{𝒱\left({𝒫}_j\right)}\right)+{ℒ}_{ef}^m+{ℒ}_{es}^m$$ (A17)

Applying Jenson’s inequality as before

$${ℒ}_{ef}^m+{ℒ}_{es}^m\le \sum _i\left(ℬ\left({\left|{\mathit{u}}_s\right|}^2\right)-{\left|{\mathit{u}}_f\right|}^2\right)\overline{𝒱\left({𝒞}_i\right)}+\sum _j\left(ℱ\left({\left|{\mathit{u}}_f\right|}^2\right)-{\left|{\mathit{u}}_s\right|}^2\right)\overline{𝒱\left({𝒫}_j\right)}$$

It can be shown that

$$\begin{gathered} \sum _iℬ\left({\left|{\mathit{u}}_s\right|}^2\right)\overline{𝒱\left({𝒞}_i\right)}=\sum _j{\left|{\mathit{u}}_s\right|}^2\overline{𝒱\left({𝒫}_j\right)} \\ \sum _jℱ\left({\left|{\mathit{u}}_f\right|}^2\right)\overline{𝒱\left({𝒫}_j\right)}=\sum _i{\left|{\mathit{u}}_f\right|}^2\overline{𝒱\left({𝒞}_i\right)} \end{gathered}$$

Adding the above two relations up, we exactly get

$${ℒ}_{ef}^m+{ℒ}_{es}^m\le 0$$ (A18)

Thus, using modified forces and our interpolation method, the net power is unconditionally dissipative with force modification.

B. Mesh convergence studies

Mesh convergence of the coupled system is demonstrated through four studies. The first two studies assess the convergence of quantities directly relevant to the fluid-solid coupling. Fig. B2(a)–(b) shows the x-component of the fluid velocity and normalized particle cloud volume fraction along a line through the domain, computed on three successive CFD mesh

Figure B2:

Mesh convergence of a coupled FSI-PD system. (a) Fluid velocity u_fx and (b) normalised particle cloud volume fraction α_r/α_rm along a line perpendicular to the anchoring surface, shown for three CFD mesh levels (Nm = 114,564, 229,080, 458,186) with the peridynamic cloud held fixed at N_p = 20,780. (c) x-displacement u_sx and (d) von Mises stress σ_vm through the full length of the clot, shown for three peridynamic cloud counts (N_p = 10,390, 20,780, 41,560) with the CFD mesh held fixed. All quantities are reported at t =0.3 s.

refinement levels with the PD cloud count held fixed at 20,780. Data is sampled along a line passing perpendicular to the anchoring surface and through the full length of the clot, and all quantities are reported at t = 0.3. Both quantities are virtually indistinguishable with small variations seen near the discontinuous diffused interface that improves with refinement. Pearson coefficient between the medium and fine refinement levels is greater than 0.98, consequently N_m = 229, 080 is chosen as the CFD mesh discretization level used in simulation of embolization of a preformed clot. Fig. B2(c)–(d) shows the vonMises stress distribution and the x-component of displacement through the thrombus as the PD cloud is refined at constant horizon-to-particle-spacing ratio δ/d_p = 3, with the CFD mesh held fixed at 229,080. For both quantities, the coarsest cloud slightly underestimates the peak values near x ≈ 2 mm, while the two finer clouds are in close agreement throughout the full clot length. Based on these results, the intermediate refinement levels N_m = 229, 080 and N_p = 20, 780 are used for our simulations.

Fig. B3 shows the adhesive surface damage as a function of flow rate for the two remaining convergence studies, which directly assess independence of the primary quantity of interest — the embolization onset flow rate. Simulations for this convergence study was run is a slightly

Figure B3:

Convergence of the embolization onset flow rate with respect to discretisation. (a) Adhesive surface damage as a function of flow rate for two CFD mesh levels (N_m = 229,080 and 458,186) with the peridynamic cloud held fixed at N_p = 20,780. (b) Adhesive surface damage as a function of flow rate for two peridynamic cloud counts (N_p = 20,780 and 41,560) with the CFD mesh held fixed at N_m = 229,080. The embolization onset flow rate is independent of discretisation in both cases.

different way where the flow-rate was incremented much slowly compared to the what was done in the main manuscript. This was done to resolve the embolization flow-rate with sufficient precision for use as a convergence metric as a coarser flow rate increment would introduce ambiguity in identifying the threshold at which adhesive surface damage first becomes non-zero. The acceptable uncertainty was set to 2% of the exact value, so, the flow-rate variation was set to

$$\varphi (t)=\left\{\begin{array}{ll}4.7& \text{if}0\le t<0.2,\\ 4.7+0.1⌈\frac{t-0.2}{0.1}⌉& \text{if}0.2\le t.\end{array}\right.$$ (B1)

Since full 3D simulations take time, we show two levels of refinement. In panel (a), the CFD mesh is refined from $N_m=\mathrm{229,080}$ to $N_m=\mathrm{458,186}$ with the PD cloud held fixed; the damage curves are almost identical across both levels. The damage slightly differs before 5 LPM but majority of the non-zero damage happens at when a flow-rate of 5 LPM is reached. In panel (b), the PD cloud is refined from $N_p=\mathrm{20,780}$ to $N_p=\mathrm{41,560}$ with the CFD mesh held fixed; again, the embolization onset flow rate is unchanged, with only minor differences in damage magnitude visible at intermediate flow rates of 4.8 and 4.9 LPM. Together, these results confirm that the reported embolization onset flow rate is independent of both CFD mesh size and PD cloud count at the refinement levels adopted in this work.

Data availability

Data will be made available on reasonable request by the corresponding author.