A density-dependent FEM-FCT algorithm with application to modeling platelet aggregation

Summary

Upon injury to a blood vessel, flowing platelets will aggregate at the injury site, forming an plug to prevent blood loss. Through a complex system of biochemical reactions, a stabilizing mesh forms around the platelet aggregate forming a stable blood clot that further seals the injury. Computational models of clot formation have been developed to a study thrombosis, where a vessel injury does not cause blood leakage outside the blood vessel but blocks blood flow. To model scenarios in which blood leaks from a main vessel out into the extravascular space, new computational tools need to be developed to handle the complex geometries that represent the injury. We have previously modeled intravascular clot formation under flow using a continuum approximation wherein the transport of platelet densities into some spatial location is limited by the platelet fraction that already reside in that location, i.e., the densities satisfy a maximum packing constraint through the use of a hindered transport coefficient. To extend this notion to extravascular injury geometries, we have modified a finite element method flux corrected transport (FEM-FCT) scheme by prelimiting anti-diffusive nodal fluxes. We show that our modified scheme, under a variety of test problems, including mesh refinement, structured vs unstructured meshes and for a range of reaction rates, produces numerical results that satisfy a maximum platelet-density packing constraint.

Graphical Abstract

In this study, we developed a new FEM-based numerical algorithm to solve advection diffusion reaction equations constrained by a physical packing limit. Components of the algorithm include a change in interpolation space of a hindered transport coefficient, which limits the transport of mobile species, and a density dependent prelimiter to a FEM-FCT scheme. Our algorithm, under a variety of test problems and applied to a model of extravascular platelet aggregation, produced numerical results that satisfy the maximum packing constraint.

1 |. INTRODUCTION

When a blood vessel is injured, blood platelets aggregate at the site of injury as part of the normal physiologic response. Briefly, flowing platelets adhere to molecules embedded in the vessel wall at the site of injury, become activated, and cohere with one another to form a massive plug that slows the initial leakage of blood from the vessel. The next stage of the response is the biochemical process of coagulation; the major enzyme produced during coagulation promotes formation of a fibrous mesh to stabilize the platelet aggregates and chemically activates platelets, which further enhances platelet aggregation. The overall blood clotting process is complex and nonlinear and is thus, difficult to predict. For this reason, computational approaches are well-suited to obtain a better understanding of the clotting process as a whole and to make testable hypotheses that can be validated experimentally. For example, our spatial-temporal model of platelet aggregation and coagulation under flow ^1,2 was used to explain differing dynamics of clot formation in hemophilia A due to various treatments³; this was achieved by comparing results from our computational model and those from microfluidic assays. Our model was also extended by others to include fibrin formation and was then compared to microfluidic experiments to understand how the growth and structure of clots that form under flow vary with initial stimulus^4,5.

Most computational models of platelet aggregation, including our previous work, were used to study intravascular clot formation, where there is no leakage of blood from the vessel^{6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}; this scenario models thrombosis, a physiological disorder in which pathological clot formation obstructs normal blood flow. Fewer studies have focused on extravascular clotting where blood leaks out of the vessel from a hole; this scenario could represent the normal hemostatic response where clots form to prevent blood leakage but the blood also retains fluidity in the main blood vessel. Our group has recently begun to investigate extravascular injuries using computational fluid dynamics in conjunction with a microfluidic device²¹; we have modeled the 3D fluid dynamics using the finite element method to estimate pressure drops and wall shear stresses within the complex geometry of the device. Direct application of our previous platelet aggregation model in this geometry has proven to be less straightforward.

In our previous work, we modeled the platelets by using a continuum approximation, i.e., we tracked platelet densities rather than individual platelets. We developed a four-species platelet model that incorporated the adhesion, activation and aggregation of platelets, where mobile species were tracked using advection-diffusion-reaction equations, and the static (bound, aggregated) species were tracked with reaction equations at specified spatial locations. To properly account for the platelet size, advective and diffusive fluxes of platelet densities were scaled by a spatially-varying coefficient that decreased as the platelet fraction increased^1,2. The result of this was a modified, hindered velocity in the advection equation for the mobile platelet species; the advection and diffusion parts of the equations were solved separately with a fractional step method. Using a finite-difference approximation, we solved the advection equation in its conservative form, with Leveque’s high-resolution advection algorithm²² that employs flux-limiters to control oscillations near discontinuities or steep gradients. Within that framework, it was straight-forward to modify the fluxes to incorporate the hindered velocity and control the mobile platelet densities transported from grid cell to grid cell to ensure that platelet densities never exceeded a physical packing limit.

Now that we are interested in complex geometry associated with extravascular injuries, our approach to numerically solving the equations of the platelet model will be to use the finite element method (FEM). Using the FEM requires reexamination of the modified advection-diffusion equation for the mobile platelet species. Flux corrected transport (FCT) methods are another class of algorithms based on the idea of flux-limiters, designed to control oscillations that arise when numerically solving the advection-diffusion equation. Although they were originally developed for use with finite difference approximations^23,24, they have since been extended and further developed for the finite element method (FEM-FCT)^25,26,27,28; we will build on such algorithms in this work. There are other methods to solve advection-diffusion equations using linear finite elements, such as streamline upwind/Petrov-Galerkin (SUPG) methods ²⁹ or spurious oscillations at layers diminishing (SOLD) methods ^30,31. In comparison to these methods, FEM-FCT has been shown to consistently preserve positivity in solutions, a requirement with chemical reacting flow problems.³². A direct application of a FEM-FCT algorithm to our mobile platelet equations results in a violation of a physical packing limit due to platelets being transported into regions that are already filled. The objective of this study is thus to develop a methodology to model platelet transport, subject to a physical packing-limit constraint using the FEM-FCT framework; we do so by modifying the weak form of the transport equations and by extending a current FEM-FCT algorithm.

To develop and test our methodology, we use a simplified version of our previous platelet aggregation model in which there are only two species of platelets, mobile and bound, tracked as number densities; the model is fully described in Section 2. Briefly, mobile platelets are transported through a moving fluid and allowed to bind and unbind to either a wall or a bound platelet aggregate through a reaction term. The fluid velocity is reduced when it coincides with bound platelet aggregates that behave as porous materials. A key component of the mobile platelet transport equation is the hindered transport coefficient, which reduces the total advective and diffusive flux of platelets as a function of the total density of platelets.

In Section 3 we describe the numerical methods used to solve the platelet equations. In Section 3.1, we describe the finite element discretization of the mobile platelet transport equation, used in the FEM-FCT algorithm with piecewise linear $({\mathbb{P}}_1)$ continuous functions as a basis for the solution space. The total platelet density and the hindered transport coefficient are thus also defined using ${\mathbb{P}}_1$ elements. We reason that with this spatial discretization of the mobile platelet transport equation, the total platelet density would fail to satisfy the maximum packing density P_max. To remedy this, we suggest a modification to the interpolation space and to the hindered transport coefficient so that it is a piecewise constant $({\mathbb{P}}_0)$ space defined on the mesh. In Section 3.2, we describe how to represent the platelet-dependent fluid resistance term, α(ϕ^B), in the weak form. In Section 3.3, we give an overview of FEM-FCT methods that are currently available to solve the discretized equations²⁷. In Section 3.4, since there is no guarantee that the current FEM-FCT methods will uphold the platelet packing limit, we propose a modification in which prelimiting fluxes into regions that are maximally packed with platelets become platelet-density dependent. Finally, in Section 3.5 we summarize the complete algorithm for solving the model equations in this framework.

We rigorously tested our methodology with several test problems in Section 4. First, we considered a region of space in which there was a mass of prebound platelets; the test involved flowing mobile platelets through and around the mass and determining if the total platelet fraction, ϕ^T, satisfied the maximum packing limit, ϕ^T ≤ 1. The platelet fraction stayed within the packing limit over a range of densities of mobile platelets flowing continually into the domain and under grid refinement. We quantified any packing-limit overage as a function of the time step and mobile platelet density. In the next problem, mobile platelets were allowed to reversibly bind the prebound mass and here the total fraction was also shown to satisfy ϕ^T ≤ 1. Finally, we showed that ϕ^T ≤ 1 is also satisfied using an unstructured mesh in a complex geometrical domain that models an extravascular injury. Our results suggest that the proposed method allows the maximum packing limit to be satisfied for a variety of modeling situations in addition to platelet aggregation.

2 |. TWO-SPECIES MODEL OF PLATELET AGGREGATION

To describe the details of our density-dependent FEM-FCT algorithm in a straightforward manner, and show how it applies to a model of platelet aggregation, we present a reduced version of our previous four-species platelet model¹. A model for which the algorithm is useful should include at least one mobile species that is carried by a fluid and diffuses within that fluid, in addition to at least one stationary or ‘bound’ species that acts as a porous material through which the mobile species can be transported, and to which the mobile species can bind. The platelet species are coupled to the fluid through the assumption that the bound aggregate is a porous material; fluid motion within an aggregate should decrease with extra resistance that increases due to platelet density. Thus, for simplicity, we consider a minimal, two-species platelet model, where platelets are either mobile (P^m) or bound (P^b). We note, however, that the algorithm applies more generally to cases where there are more than two species. The platelet species are characterized by their the total platelet fraction, ϕ^T, and their the bound platelet fraction, ϕ^B. We also assume there is a physical platelet packing limit, i.e., a maximum number density of platelets at any point in space. We denote this limit as P_max and set its value to be 6.67 × 10⁷ platelets per µL, estimated from a scenario where 20 platelets fit tightly in a 300 µm³ region. The total platelet fraction ϕ^T is the ratio of the sum of P^m and P^b to P_max, while the bound platelet fraction is the ratio of P^b to P^max. The bound platelet fraction affects the fluid velocity, reducing fluid motion within a platelet aggregate, and the total platelet fraction changes how quickly platelets can move due to crowding by one another.

2.1 |. Fluid

The fluid motion is described by the Navier-Stokes-Brinkman equation

$$\rho \left(\frac{\partial \overrightarrow{u}}{\partial t}+\overrightarrow{u}\cdot \mathrm{\Delta }\overrightarrow{u}\right)=-{\nabla }_p-\mu {\nabla }^2\overrightarrow{u}-\mu \alpha ({\varphi }^B)\overrightarrow{u},$$ (1)

$$\nabla \cdot \overrightarrow{u}=0.$$ (2)

where $\overrightarrow{u}$ and p represent fluid velocity and pressure, respectively, ρ is the dynamic viscosity and ρ is the fluid density. The term −µα(ϕ^B) represents a frictional resistance to the fluid motion due to the presence of bound platelets via their platelet fraction, ϕ^B = P^b/P_max. As more platelets become bound at any spatial location, the permeability decreases there and thus the resistance, α(ϕ^B), increases; we assume that α(ϕ^B) satisfies the Carman-Kozeny relation^33,34

$$\alpha ({\varphi }^B)=C\frac{{(0.6{\varphi }^B)}^2}{{\left[1-(0.6{\varphi }^B)\right]}^3},C={10}^8{\text{cm}}^{-2}$$

and a representative plot is shown in Figure 1. Although the Carman-Kozeny model is typically used for spherically-packed granular beds, it has also been shown to have good qualitative agreement with experimental estimates of clot permeability³⁵.

FIGURE 1.

Fluid resistance due to bound platetets: α(ϕ^B)

Unless otherwise noted, all simulations we present in this work consider fluid motion that occurs in a domain Ω, with prescribed parabolic profiles at the inlets of the domain, Γ_in, no-slip boundary conditions $(\overrightarrow{u}=\overrightarrow{0})$ on the walls orthogonal to the direction of flow, Γ_W, and prescribed pressures at the outlets of the domain, Γ_out. We prescribe pressures with open/natural boundary conditions of the form:

$$p\mathbf{n}-\mu (\nabla \overrightarrow{u})\mathbf{n}=\overline{p}\mathbf{n}$$ (3)

where n is the outward normal vector on the boundary Γ_out and $\overline{p}$ is the constant prescribed pressure.

2.2 |. Platelets

The evolution equations for mobile and bound platelets, P^m and P^b, are defined by:

$$\frac{\partial P^m}{\partial t}=-\nabla \cdot \left[W({\varphi }^T)\left(\overrightarrow{u}{P}^m-D_p\nabla P^m\right)\right]-k_{\text{coh}}g(\eta )P_{\text{max}}{P}^m+k_{\text{off}}{P}^b,$$ (4)

$$\frac{\partial P^b}{\partial t}=k_{\text{coh}}g(\eta )P_{\text{max}}{P}^m-k_{\text{off}}{P}^b.$$ (5)

This is a continuum model that tracks number densities of platelets rather than individual discrete platelets and therefore, the model requires an additional procedure to account for the finite size of the platelets. This notion of size is captured by using spatially-varying, phenomenological functions for platelet-platelet cohesion (aggregation) and density-dependent limited transport, as was done in our previous work¹. Briefly, motion of mobile platelets into a region that is already occupied by a platelet aggregate must be limited in some way. This limitation is achieved by modifying the advective and diffusive flux of mobile platelets, ${\mathbf{J}}^m=\overrightarrow{u}{P}^m-D_p\nabla P^m$, which is the first term in equation (4). We multiply this flux by a scalar function W(ϕ^T), which is a monotonically decreasing function of the total platelet fraction, ϕ^T, given by W(ϕ^T) = tanh [π (1 − ϕ^T)]; a schematic of this function is shown in Figure 2A. For increasing values of ϕ^T the shape of W(ϕ^T) represents a gradual hindrance of mobile platelet transport until ϕ^T approaches 0.5, when it begins to more rapidly hinder transport and then drops to zero as ϕ^T approaches 1.

FIGURE 2.

(A) Platelet density-dependent transport limiting function W(ϕ^T), and (B) Binding affinity function.

The second and third terms in equation (4) account for platelet-platelet cohesion and represent the transient binding of mobile platelets on and off of platelet aggregates, respectively. The finite size of platelets must also be considered with cohesion so that the mobile platelets densities are able to build up (bind to aggregates) when they are sufficiently close to aggregates. This is achieved through a virtual substance η, that acts as a proximity function to identify where and how many bound platelets reside in a particular spatial location, and a “binding affinity” function g(η) that, in part, determines the cohesion rate of mobile platelets to an aggregate. The functional form of the binding affinity function is $g(\eta )=g_0\text{max}(0,{(\eta -{\eta }_t)}^3/({\eta }_{\ast }^3+{(\eta -{\eta }_t)}^3))$ and is shown in Figure 2. The parameter η_t is the threshold value before which platelets do not bind, η_* + η_t is the value of η which g(η) changes rapidly, and g₀ is chosen such that g(1) = 1. The evolution equation for η follows the reaction-diffusion dynamics:

$$\frac{\partial \eta }{\partial t}=D_{\eta }{\nabla }^2\eta -{\gamma }_{\eta }\left(\eta -\frac{P^b}{P_{\text{max}}}\right).$$ (6)

This formulation assumes that η decays at a rate γ_η and is produced at that same rate by bound platelets. It also diffuses with diffusion coefficient D_η. As in our previous work, D_η and γ_η are chosen such that the value of η drops substantially at a distance comparable to a platelet diameter d, i.e. $\sqrt{D_{\eta }/{\gamma }_{\eta }}=d$¹.

The cohesion rate, k_coh(s⁻¹ M⁻¹), and the unbinding rate k_off(s⁻¹) are related through a disassociation constant $K_d^{coh}$, defined as the ratio of k_off to k_coh. As a first approximation, we assume that

$$K_d^{coh}=\frac{k_{off}}{k_{coh}}\approx P_0,$$ (7)

where P₀ is the normal platelet count in plasma, P₀ ≈ 2.5 × 10⁵/µL. As presented in the results section, we test the algorithm with various dissociation constants; to do this we use the value k_coh × P_max = 10² s⁻¹ M⁻¹ and vary k_off accordingly.

For the boundary conditions on P^m, we assume constant densities entering through each fluid inlet boundary, Γ_in and on the remainder of the boundaries, Γ_out and Γ_W, P^m satisfies a homogeneous Neumann boundary condition:

$$P^m=P_{up}^m(\overrightarrow{x})\hspace{0.17em}\text{on}\hspace{0.17em}{\Gamma }_{\text{in}}$$ (8)

$$-D_P\frac{\partial P^m}{\partial n}=0\hspace{0.17em}\text{on}\hspace{0.17em}{\Gamma }_{\text{out}}\cup {\Gamma }_W$$ (9)

3 |. NUMERICAL METHODS

In this section, we describe our new numerical methodology that ensures platelet fractions remain within physical limits, comprised of modifications and updates to current algorithms at various stages during the solution process. First, we describe the new spatial and temporal discretizations of the equation for mobile platelets using the finite element method, including the dis-cretization of hindered transport and Brinkman coefficient. Next we give an overview of current FEM-FCT methods in order to highlight the places within its algorithm that require modifications; in particular, we define a new prelimiter that ensures platelet fractions remain within physical limits during the correction step. Finally, we provide an overall summary for the solution algorithm.

3.1 |. Approximating W(ϕ^T) in the Weak Formulation

To understand why a modification to the scalar W(ϕ^T) is needed, we rewrite the partial differential equation of P^m using the finite element method. Let $\mathcal{T}$ be the triangulation of the domain Ω such that $T\in \mathcal{T}$ is a triangular cell on the mesh. For now, we will ignore the reaction terms (i.e. k_cohP_max = k_{of f} = 0) for the PDE of P^m.

To discretize the equations, we first apply a finite-difference approximation in time using a modified Crank-Nicolson time-stepping scheme. Assuming a time step size Δt, equation (4) (without reaction terms) at time step level k + 1 becomes:

$$\frac{P^{m,[k+1]}-P^{m,[k]}}{\mathrm{\Delta }t}=-\frac{1}{2}\nabla \cdot ([W({\varphi }^{T,[k]})({\overrightarrow{u}}^{[k+1]}{P}^{m,[k+1]}-D_p\nabla P^{m,[k+1]})]-[W({\varphi }^{T,[k-1]})(\overrightarrow{u}{P}^{m,[k]}-D_p\nabla P^{m,[k]})]),$$ (10)

where P^m,[k+1] is the mobile platelet density, P^m, at time level k+1. Note that at this stage, we assume that W(ϕ^T) is extrapolated from time level k in both of the transport terms, since ϕ^T depends on P^m.

Next we apply the standard Galerkin method to (10) using ${\mathbb{P}}_1$ elements. Assuming there are N basis functions {ψ₁, … ψ_N} that span $\mathcal{T}$, with P^m being approximated by a linear combination of the basis functions, we have:

$$P^m\approx \sum _{i=1}^N{p}_i^m{\psi }_i.$$ (11)

Multiplying both sides of (10) by a test function ψ_j for j = 1, 2, 5, …, N, integrating over the domain Ω, integrating the diffusive term by parts (where the boundary terms vanish due to equation (9) and for now we ignore the condition from equation (8)), and plugging in (11) for P^m, we obtain the following discrete equation for P^m:

$$\left(M+\frac{\mathrm{\Delta }t}{2}\left[K^{[k+1]}+D^{[k+1]}\right]\right){\mathbf{P}}^{[k+1]}=\left(M-\frac{\mathrm{\Delta }t}{2}\left[K^{[k]}+D^{[k]}\right]\right){\mathbf{P}}^{[k]}.$$ (12)

Here, ${\mathbf{P}}^{[k+1]}={\left(p_1^{m,[k+1]},\hspace{0.17em}\cdots \hspace{0.17em},p_n^{m,[k+1]}\right)}^T$ is a vector of coefficients from the expansion in (11), and the matrices M, K, and D have entries:

$$M_{ij}=\sum _{T\in \mathcal{T}}\underset{T}{\int }{\psi }_i{\psi }_{j}d\mathbf{x},$$ (13)

$$K_{ij}=\sum _{T\in \mathcal{T}}\underset{T}{\int }\nabla \cdot \left(W({\varphi }^T)\overrightarrow{u}{\psi }_i\right){\psi }_{j}d\mathbf{x},$$ (14)

$$D_{ij}=\sum _{T\in \mathcal{T}}\underset{T}{\int }{D}_{P}W({\varphi }^T)\nabla {\psi }_i\cdot \nabla {\psi }_{j}d\mathbf{x}.$$ (15)

By construction, through the use of ${\mathbb{P}}_1$ elements, if i ≠ j, then K_ij and D_ij consist of integrals on at most two cells, and these are the cells that nodes i and j share. Then, K_ij and D_ij can be interpreted as the nodal mass exchange of P^m from node j into node i. We use these matrix properties together to inform our choice in approximating W(ϕ^T), on each cell, as follows.

Since we obtain ϕ^T by summing the nodal values of P^b and P^m, then it is natural to also define W(ϕ^T) on the same nodes. The integrals defined in K_ij and D_ij are computed using Gaussian quadrature on each triangular cell. Since Gaussian quadrature uses interior points of the cell instead of nodal points for the quadrature, then W(ϕ^T), defined as a ${\mathbb{P}}_1$ function, may cause matrices K and D to exchange mass of P^m into nodes where ϕ^T ≈ 1.

Consider the following example where W(ϕ^T), a ${\mathbb{P}}_1$ function, would fail to prevent ϕ^T > 1. Let T be a single triangular cell, and denote the nodal values of ϕ^T at the three vertices of T as ${\varphi }_1^T$, ${\varphi }_2^T$ and ${\varphi }_3^T$, respectively. If we assume that ${\varphi }_1^T=1$, but ${\varphi }_2^T$, ${\varphi }_3^T<1$, then $W({\varphi }_1^T)=0$, while $W({\varphi }_2^T)$, $W({\varphi }_3^T)>0$. Now consider the mass exchange matrix for P^m due to advection from node 2 into node 1 on cell T′, i.e. $K_{12}^{T\prime }$:

$$K_{12}^{T\prime }=\underset{T\prime }{\int }\nabla \cdot \left(W({\varphi }^T)\overrightarrow{u}{\psi }_i\right){\psi }_{j}d\mathbf{x}$$

Since W(ϕ^T) is a ${\mathbb{P}}_1$ function, it is not guaranteed to be zero at the quadrature points evaluated in the integral of $K_{12}^T$. Consequently, mass exchange of P^m from node 2 into node 1 by $K_{12}^T$ will cause ϕ^T > 1 at node 1!

To alleviate this, we modify the interpolant space for W(ϕ^T) by assuming that it is piecewise constant on each $T\in \mathbb{T}$. Let ${\mathbb{P}}_0$ be the space of piecewise constant functions defined on $\mathcal{T}$. Then, using a ${\mathbb{P}}_0$ function for W(ϕ^T) on each cell makes it invariant to the quadrature point choice during the numerical integration. We choose the maximum nodal value of ϕ^T on T to evaluate W(ϕ^T), see Figure 3. This choice was motivated by the worse case scenario in which platelet densities could be transported into a node where ϕ^T = 1.

FIGURE 3.

Schematic of defining W(ϕ^T) on a single mesh cell. Denoting each of the vertices of the cell from 1 to 3, we have the nodal values of ϕ^T on the cell. The maximum between the 3 values is denoted as ϕ^T,*. W(ϕ^T,*) replaces W(ϕ^T) in the weak formulation of P^m.

3.2 |. Defining α(ϕ^B) in a weak formulation

The Brinkman term, α(ϕ^B), is also interpolated using a finite element space. Since ϕ^B is defined as a ${\mathbb{P}}_1$ function on the mesh, the bound platelet fraction ϕ^B should only induce frictional resistance on the fluid in cells occupied by ϕ^B. Given a triangular cell $T\in \mathcal{T}$, if all of the three nodal values of ϕ^B on T are below a threshold value of the bound platelet fraction ϕ^B,* = 0.1, then we set α = 0 on T. If any of the three nodal values exceed this threshold, we compute using the average of ϕ^B over the three nodes in T. The scheme for computing α(ϕ^B) on each cell is summarized in Figure 4. The reason for having a small, nonzero threshold ϕ^B,* is that we assume very small densities of platelets will not significantly contribute to fluid resistance.

FIGURE 4.

Schematic of defining α(ϕ^B) on a single mesh cell. Denoting each of the vertices of the cell from 1 to 3, we have the nodal values of ϕ^B on the cell. The average ϕ^B across 3 values is denoted as ϕ^B,*. α(ϕ^B) is set as α(ϕ^B,*) whenever all nodal values of ϕ^B are greater than a specified threshold value ${\varphi }_{\alpha }^B$.

3.3 |. Overview of the flux-corrected transport (FCT) method

Flux-corrected transport (FCT) methods are designed to solve convection-dominated transport problems where solutions must remain positive for physical reasons, e.g., the solutions represent concentrations or densities²⁶. FCT methods can be interpreted as two-step, predictor-corrector schemes. In the first step, one predicts a low-order solution, obtained by adding artificial diffusion to the discretized model equation (e.g., equation (12)); this smooths the solution to suppress oscillatory overshoots and undershoots. Next, an anti-diffusive flux is constructed by using a residual between the original and low-order equation, and this flux is added to the low-order solution to recover accuracy, while also preserving stability and positivity.

The FCT framework was originally formulated by Boris and Book²³, using finite difference methods to numerically solve the equations of conservation laws. The finite difference methodology has been widely used and extended for multiple dimensions and curvilinear domains; a good review of the history, development and application of FCT methods can be found in^36,24,37,26. FCT algorithms have also been developed for use with the finite element method (FEM), called FEM-FCT algorithms, first developed in 1988 by Löhner²⁵. When using ${\mathbb{P}}_1$ elements to solve linear advection equations and advection-diffusion equations, a good choice is the FEM-FCT algorithm developed by Kuzmin ²⁷, which is what we build on in this work. The approach is to use a linearization of the anti-diffusive flux to avoid an additional iterative solution procedure during the correction step. The algorithm results in a good balance between efficiency and accuracy and is thus potentially the ideal approach to solve the advection-diffusion component of the P^m equation in our model. Unfortunately, a direct application of this method, specifically during correction step, will not guarantee that mobile platelets satisfy ϕ^T ≤ 1. We wish to exploit the efficiency and accuracy of the method so we have extended it by making a modification in the correction step that prevents the anti-diffusive fluxes from moving mobile platelets into regions where ϕ^T = 1. Details of the procedure are as follows.

To begin, rewrite the transport terms as L = K + D in the discrete equation for P^m (12) so that

$$\left(M+\frac{\mathrm{\Delta }t}{2}\left[L^{[k+1]}\right]\right){\mathbf{P}}^{[k+1]}=\left(M-\frac{\mathrm{\Delta }t}{2}\left[L^{[k]}\right]\right){\mathbf{P}}^{[k]}.$$ (16)

To obtain the predicted solution, the consistent mass matrix, M, is replaced by a lumped mass matrix, M_l, and artificial diffusion, $\tilde{D}$, is added to the transport component, L, to obtain the low-order equation:

$$\left(M_l+\frac{\mathrm{\Delta }t}{2}\left[L^{[k+1]}+{\tilde{D}}^{[k+1]}\right]\right){\mathbf{P}}^{[k+1]}=\left(M_l-\frac{\mathrm{\Delta }t}{2}\left[L^{[k]}+{\tilde{D}}^{[k]}\right]\right){\mathbf{P}}^{[k]}.$$ (17)

The lumped mass matrix M_l is a diagonal matrix where each entry M_l,ij is the sum of the entries in the corresponding row of the consistent mass matrix:

$$M_{l,ii}=\sum _{i\ne j}{M}_{ij}.$$ (18)

The artificial diffusion matrix $\tilde{D}$ consists of the entries of L in a symmetric fashion:

$${\tilde{D}}_{ij}=\text{max}\{-L_{ij},-L_{ji},0\}={\tilde{D}}_{ji},\hspace{0.17em}\text{for}\hspace{0.17em}i\ne j,$$ (19)

$${\tilde{D}}_{ii}=-\sum _{i\ne j}{\tilde{D}}_{ij}$$ (20)

With this construction, the artificial diffusion matrix has the zero row (and column) property and thus discrete mass conservation is preserved. We must ensure that the solution to the low-order equation (17) preserves positivity, and thus the time step must be chosen carefully. More specifically, if

$$A^{[k+1]}=M_l+\frac{\mathrm{\Delta }t}{2}{C}^{[k+1]},$$

$$B^{[k]}=M_l-\frac{\mathrm{\Delta }t}{2}{C}^{[k]},$$

are left-hand side of (17) at time level k + 1 and the right-hand side of (17) at time level k, respectively, with $C=L+\tilde{D}$, then the time step must be chosen such that their diagonal entries satisfy

$$A_{ii}^{[k+1]}=M_{l,ii}-\frac{1}{2}\mathrm{\Delta }t{C}_{ii}^{[k+1]}\ge 0,$$ (21)

$$B_{ii}^{[k]}=M_{l,ii}+\frac{1}{2}\mathrm{\Delta }t{C}_{ii}^{[k]}\ge 0,$$ (22)

for all i²⁷.

At this stage, the positivity-preserving solution to equation (17) is overly diffusive due to the mass lumping and added artificial diffusion, thus to recover accuracy we then make an anti-diffusive correction. This is achieved by constructing anti-diffusive fluxes that move into each node from each of its neighboring nodes, throughout the entire computational domain. To estimate how much we have over-diffused, we look at the residual between the low-order equation (17) and the high-order oscillatory Galerkin equation (16). The residual is used to define the nodal components of the anti-diffusive flux ²⁷:

$$\begin{gathered} f_{ij}=\frac{\left[M_{ij}\left(P_i^{m,[k+1]}-P_j^{m,[k+1]}\right)-M_{ij}\left(P_i^{m,[k]}-P_j^{m,[k]}\right)\right]}{\mathrm{\Delta }t} \\ \hspace{1em}\hspace{1em}+\frac{1}{2}\left[{\tilde{D}}_{ij}^{[k+1]}\left(P_i^{m,[k+1]}-P_j^{m,[k+1]}\right)+{\tilde{D}}_{ij}^{[k]}\left(P_i^{m,[k]}-P_j^{m,[k]}\right)\right]. \end{gathered}$$ (23)

Each of these nodal components, f_ij, is such that the flux moves from node j into node i. Using with $C=L+\tilde{D}$ again, the flux-corrected equation includes the sum of all neighboring nodal components as follows:

$$\left(M_l+\frac{\mathrm{\Delta }t}{2}{C}^{[k+1]}\right){\mathbf{P}}^{[k+1]}=\left(M_l-\frac{\mathrm{\Delta }t}{2}{C}^{[k]}\right){\mathbf{P}}^{[k]}+\overline{f}({\mathbf{P}}^{[k+1]},{\mathbf{P}}^{[k]})$$ (24)

where $\overline{f}({\mathbf{P}}^{[k+1]},{\mathbf{P}}^{[k]})={\sum }_{i\ne j}{\beta }_{ij}{f}_{ij}$ is simply called the anti-diffusive flux, and β_ij ∈ [0, 1] are the weighted correction factors. The correction factors allow one to choose how much and at which nodal value, each of the anti-diffusive flux components are incorporated back into the low-order solution. Notice that when β_ij = 1 everywhere, the Galerkin equation (16) is recovered and when β_ij = 0, the low-order equation (17) is recovered.

Choosing the optimal correction weights β_ij is a nontrivial task. Optimizing the weights to minimize the residual error becomes a computationally inefficient task when they must be updated at every time step. Zalesak proposed an alternative flux-limiting algorithm to circumvent this complication³⁷. In this algorithm, the nodal correction factors are chosen so that the solution stays bounded within some local extrema, i.e., the solution at node i does not exceed a maximum value ${\tilde{P}}_i^{\text{max}}$ or dip below a minimum value ${\tilde{P}}_i^{\text{min}}$ but to bound the solution in this way, the positive and negative anti-diffusive fluxes must then be limited separately. Let $\mathcal{N}(i)$ be the set of nodal neighbors that share an edge with node i, then the Zalesak’s flux-limiting algorithm is as follows:

1. Compute ${\mathcal{P}}_i^{\pm }$, the sums of positive and negative nodal anti-diffusive into node i:

   $${\mathcal{P}}_i^{+}=\sum _{j\in {\mathcal{N}}_i}\text{max}\{0,f_{ij}\},{\mathcal{P}}_i^{-}=\sum _{j\in {\mathcal{N}}_i}\text{min}\{0,f_{ij}\}$$ (25)
2. Compute ${\mathcal{Q}}_i^{\pm }$, the distances to a local maxima/minima of the low-order solution ${\tilde{P}}_i^m$ at node i:

   $${\mathcal{Q}}_i^{+}=\text{max}\left\{0,\underset{j\in {\mathcal{N}}_i}{\text{max}}\left({\tilde{P}}_j^m-{\tilde{P}}_i^m\right)\right\},\hspace{0.17em}\hspace{0.17em}{\mathcal{Q}}_i^{-}=\text{min}\left\{0,\underset{j\in {\mathcal{N}}_i}{\text{min}}\left({\tilde{P}}_j^m-{\tilde{P}}_i^m\right)\right\}$$ (26)
3. Compute ${\mathcal{R}}_i^{\pm }$, the nodal correction factors corresponding to the total increment to node i:

   $${\mathcal{R}}_i^{+}=\text{min}\left\{\frac{M_{l,ii}{\mathcal{Q}}_i^{+}}{\mathrm{\Delta }t{\mathcal{P}}_i^{+}}\right\},\hspace{0.17em}{\mathcal{R}}_i^{-}=\text{min}\left\{\frac{M_{l,ii}{\mathcal{Q}}_i^{-}}{\mathrm{\Delta }t{\mathcal{P}}_i^{-}}\right\}$$ (27)
4. Define the nodal correction factors β_ij by limiting the nodal anti-diffusive fluxes f_ij and f_ji symmetrically:

   $${\beta }_{ij}=\{\begin{array}{ll}\text{min}\{{\mathcal{R}}_i^{+},{\mathcal{R}}_i^{-}\},\hfill & \text{if}\hspace{0.17em}{f}_{ij}>0,\hfill \\ \text{min}\{{\mathcal{R}}_i^{-},{\mathcal{R}}_i^{+}\},\hfill & \text{otherwise}\hfill \end{array}$$ (28)

Addition of $\overline{f}$ makes equation (24) nonlinear in P^m. Solution to this nonlinear equation by iterative methods would significantly increase the simulation time since new correction factors are required at every iteration and the iterations are performed at every time step. Following Kuzmin²⁷, to reduce the overall cost of the implicit FCT algorithm, we remove the nonlinearity by using an approximation to the anti-diffusive flux that incorporates the solution to the low-order equation. Denoting P^L as a low-order solution to (17), a new explicit correction becomes

$$M_L{\mathbf{P}}^{[k+1]}=M_L{\mathbf{P}}^L+\mathrm{\Delta }t\overline{f}({\mathbf{P}}^L).$$ (29)

Here, the nodal components in $\overline{f}({\mathbf{P}}^L)$ depend only on the low-order solution in the following way:

$$f_{ij}=M_{ij}\left({\dot{P}}_i^L-{\dot{P}}_j^L\right)+{\tilde{D}}_{ij}^{[k+1]}\left(P_i^L-P_j^L\right)$$ (30)

where ${\dot{P}}^L$ is the time derivative of P^L, approximated by solving:

$${\dot{\mathbf{P}}}^L=M_L^{-1}\left[L^{[k+1]}{\mathbf{P}}^L\right].$$

Hence, the time derivative of ${\dot{P}}^L$ is also approximated by the low-order solution. By using this algorithm, we only have to compute the anti-diffusive flux and its nodal correction factors once per time step. Previous numerical results show equal order error with this method compared to the nonlinear, iterative-solver counterpart, while having significantly smaller CPU time²⁷.

3.4 |. A new density-dependent FCT prelimiter

Since we have constructed W(ϕ^T) to be a piecewise constant function, the low-order solution will surely respect the condition that ϕ^T ≤ 1. However, there is no guarantee that the anti-diffusive fluxes as defined in the previous section will prevent P^m from being transported into a region where ϕ^T = 1. Zalesak’s algorithm does use nodal correction factors constrained by local extrema but there could be a case where f_ij > 0 and ϕ_i = 1 and the nodal correction factor β_ij is positive; this would lead to nodal mass exchange of P^m, from node j into node i, where ${\varphi }_i^T=1$. To ensure that this never occurs in our simulations, we modify the algorithms of Kuzmin and Zalesak. Our approach is to “prelimit” the anti-diffusive fluxes so they are canceled out whenever ϕ^T = 1 at any node.

In the original FCT formulation by Boris and Book²³, flux correction was shown to sometimes reintroduce numerical ripples due to f_ij having the same sign as its corresponding solution gradient, ${\tilde{P}}_j-{\tilde{P}}_i$. Whenever such a ‘defective’ anti-diffusive flux was found, a safe solution was to simply zero-out f_ij whenever it was in the same direction as the gradient of $\tilde{P}$, i.e., f_ij := 0; if $f_{ij}({\tilde{P}}_j-{\tilde{P}}_j)>0$. This technique was coined ‘prelimiting’ since it was performed before the correction factors and flux limiters were computed.

As discussed above, there could be situations with our model where the anti-diffusive fluxes cause P^m to move into regions where ϕ^T = 1. To prevent this, we adopt of the idea of prelimiting, and apply it to ϕ^T in the following way: if a node in the domain has ϕ^T = 1 from the previous time step and the sign of f_ij is such that mass would flow into that node, then we simply set f_ij = 0.

The algorithm is summarized below: When the nodal component is positive, i.e., f_ij > 0, this implies that there is a positive

$$\overline{\underset{\_}{\begin{array}{l}\underset{\_}{\mathbf{A}\mathbf{l}\mathbf{g}\mathbf{o}\mathbf{r}\mathbf{i}\mathbf{t}\mathbf{h}\mathbf{m}\hspace{0.17em}\mathbf{1}\hspace{0.17em}\text{Prelimiting}\hspace{0.17em}{f}_{ij}\hspace{0.17em}\text{as}\hspace{0.17em}\text{a}\hspace{0.17em}\text{function}\hspace{0.17em}\text{of}\hspace{0.17em}{\varphi }^T}\\ \hspace{0.17em}\hspace{0.17em}1:\hspace{0.17em}\hspace{0.17em}\mathbf{p}\mathbf{r}\mathbf{o}\mathbf{c}\mathbf{e}\mathbf{d}\mathbf{u}\mathbf{r}\mathbf{e}\hspace{0.17em}\text{PRELIMIT}(f,{\varphi }^T)\\ \hspace{0.17em}\hspace{0.17em}2:\hspace{1em}\hspace{1em}\mathbf{f}\mathbf{o}\mathbf{r}\hspace{0.17em}\text{all}\hspace{0.17em}\text{nodes}\hspace{0.17em}\hspace{0.17em}i\hspace{0.17em}\hspace{0.17em}\text{from}\hspace{0.17em}\hspace{0.17em}1\hspace{0.17em}\hspace{0.17em}\text{to}\hspace{0.17em}\hspace{0.17em}N\hspace{0.17em}\mathbf{d}\mathbf{o}\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}3:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\mathbf{f}\mathbf{o}\mathbf{r}\hspace{0.17em}\text{all}\hspace{0.17em}\text{neighbors}\hspace{0.17em}j\in {\mathcal{N}}_{\mathbf{i}}\hspace{0.17em}\mathbf{d}\mathbf{o}\\ \hspace{0.17em}\hspace{0.17em}4:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathbf{i}\mathbf{f}\hspace{0.17em}{f}_{ij}>0\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}{\varphi }_i^T\ge 1\hspace{0.17em}\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{n}\\ \hspace{0.17em}\hspace{0.17em}5:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_{ij}:=0\\ \hspace{0.17em}\hspace{0.17em}6:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathbf{e}\mathbf{l}\mathbf{s}\mathbf{e}\hspace{0.17em}\mathbf{i}\mathbf{f}\hspace{0.17em}{f}_{ij}<0\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}{\varphi }_j^T\ge 1\hspace{0.17em}\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{n}\\ \hspace{0.17em}\hspace{0.17em}7:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_{ij}:=0\\ \hspace{0.17em}\hspace{0.17em}8:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathbf{e}\mathbf{l}\mathbf{s}\mathbf{e}\\ \hspace{0.17em}\hspace{0.17em}9:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{continue}\\ 10:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathbf{e}\mathbf{n}\mathbf{d}\hspace{0.17em}\mathbf{i}\mathbf{f}\\ 11:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\mathbf{e}\mathbf{n}\mathbf{d}\hspace{0.17em}\mathbf{f}\mathbf{o}\mathbf{r}\\ 12:\hspace{1em}\hspace{1em}\mathbf{e}\mathbf{n}\mathbf{d}\hspace{0.17em}\mathbf{f}\mathbf{o}\mathbf{r}\\ 13:\hspace{1em}\hspace{1em}\mathbf{r}\mathbf{e}\mathbf{t}\mathbf{u}\mathbf{r}\mathbf{n}\hspace{0.17em}f\\ 14:\hspace{0.17em}\hspace{0.17em}\mathbf{e}\mathbf{n}\mathbf{d}\hspace{0.17em}\mathbf{p}\mathbf{r}\mathbf{o}\mathbf{c}\mathbf{e}\mathbf{d}\mathbf{u}\mathbf{r}\mathbf{e}\end{array}}}$$

flux going from node j into node i; in the first if-statement, this flux is canceled if ${\varphi }_i^T\ge 1$. In the second if-statement, f_ij < 0 means there is a negative flux going from node j into node i. Since f_ij = −f_ji due to mass conservation, this means there is a positive flux going from node i into node j. Thus, if ${\varphi }_j^T\ge 1$ when f_ij < 0, then we also cancel that flux.

3.5 |. Summary of density-dependent FEM-FCT algorithm for modeling platelet aggregation

All partial differential equations are discretized in space using the finite element method (FEM). For the Navier-Stokes-Brinkman equations, we use a modified rotational pressure-correction scheme, originally proposed in³⁸, with Taylor-Hood $({\mathbb{P}}_2-{\mathbb{P}}_1)$ elements for the velocity and pressure terms, respectively. The modification assumes that the Brinkman term is implicit during the momentum step of the projection method. A good review of all steps in the rotational projection method used in our solver can be found in Guermond et. al.³⁹. The advection-diffusion-reaction equation for the mobile platelets are solved using ${\mathbb{P}}_1$ elements with operator-splitting in time, and our modified Flux-Corrected transport (FCT) method with flux linearization for the advection-diffusion operator²⁷. Similarly, η is solved with operating-splitting in time, where the diffusion equation solved using a standard Crank-Nicolson time-stepping scheme with ${\mathbb{P}}_1$ elements. Reactions for P^m, P^b and η are solved nodally using Heun’s Method, a 2nd-order Runge-Kutta scheme. The software, finite element discretization, iterative solvers, and parallelization are handled using the FEniCS software suite⁴⁰. The time-stepping scheme of our two-species platelet model is as follows:

1. Define the initial values for P^m, P^b, $\overrightarrow{u}$, p and η.

2. Starting at t = ∆t:

   1. Compute α(ϕ^B) and solve fluid equations using the rotational pressure-correction scheme.
   2. Update ϕ^T from the previous time step and compute W (ϕ^T,*).
   3. Advect and diffuse P^m using our modified FEM-FCT algoirthm:

      1. Solve for the low-order solution ${\tilde{P}}^m$ using equation (17).
      2. Compute the components of the anti-diffusive flux f_ij using equation (30). While looping through each component, cancel the anti-diffusive flux components according to our density-dependent prelimiter given by Algorithm 1.
      3. Run Zalesak’s flux limiting algorithm using the prelimited components f_ij to obtain the nodal correctional factors β_ij.
      4. Solve for the corrected solution using equation (29).
   4. Diffuse η using a Crank-Nicolson time-stepping scheme.
   5. Compute reactions for P^m, P^b and 4 using Heun’s Method.
   6. Update ϕ^B using the previous timestep value of P ^b.
   7. Advance t = t + ∆t.
   8. Repeat (a)-(g) until desired final time.

4 |. NUMERICAL RESULTS

In this section, we perform a series of numerical tests to show the conditions under which ϕ^T ≤ 1 is satisfied when solving the evolution equation for P^m with our modified FCT scheme. We first consider advection-diffusion and then add the reaction terms as a separate numerical test; both of these tests are performed in a square domain with a circular region that contains nonzero densities of P^b. We test two different models for W (ϕ^T), a piecewise constant W(ϕ^T) and a ϕ^T-dependent prelimiter, and then determine if and how much the total density ϕ^T increases over the value of 1. For the advection-diffusion only case, P^m is transported through and around the domain but P^b remains at a fixed value; when the reactions are reintroduced, P^m is allowed to transiently bind to the prebound aggregate of P^b through k_coh and unbind through k_off. Here we test that the algorithm ensures ϕ^T ≤ 1 for a range of these rates.

Finally, our motivation for developing this algorithm was to use the finite element method to model platelet aggregation within extravascular injuries, i.e., in more complex geometries than a rectangular channel. For our last numerical test, we model platelet aggregation in a domain that represents an extravascular injury where aggregation is initiated by assuming there is a thin layer of prebound platelets lining the injury region. In this test, we show that W as a ${\mathbb{P}}_0$ function together with the use of the density-dependent prelimiter can be used on unstructured meshes in complex geometries. In addition, we show when our method preserves ϕ^T ≤ 1 through a range of binding rates.

4.1 |. Advection-diffusion only

In our first numerical experiment, we test how well our numerical approximation of W (ϕ^T), in combination with a density-dependent FCT prelimiter, prevents mobile platelets from being transported into regions fully occupied by bound platelets. This test is in the absence of reaction terms; setting k_coh and k_off to zero in equations (4) and (5), we are therefore solving

$$\frac{\partial P^m}{\partial t}=-\nabla \cdot \left[W({\varphi }^T)(\overrightarrow{u}{P}^m-D_P\nabla P^m)\right],$$ (31)

$$\frac{\partial P^b}{\partial t}=0.$$ (32)

The computational domain is a 60 by 60 µm box where in the center of the domain, there is a circular patch (9 µm radius) of prebound platelets such that P^b = P^max. There are initially no mobile platelets on the interior of the computational domain, but the boundary conditions are such that platelets flow uniformly from the inlet boundary (x = 0) with zero diffusive flux conditions on the top and bottom walls (y = 0, 60 µm), and at the outlet of the domain (x = 60 µm). For the fluid boundary conditions, a parabolic velocity profile is specified at the inlet boundary corresponding to a 500 s⁻¹ shear rate, a zero pressure is specified at the outlet and no-slip conditions are set on the top and bottom walls. Initially, the fluid starts at rest (i.e. $\overrightarrow{u}=0$) with the Brinkman term α(ϕ^B) defined by the initial concentration of P^b through ϕ^B. The fluid solver is iterated until the velocity field reaches an equilibrium, i.e., we stop solving the fluid equations when

$$\frac{\Vert {\overrightarrow{u}}^k-{\overrightarrow{u}}^{k-1}\Vert }{\Vert {\overrightarrow{u}}^k\Vert }<ϵ$$

at time level k with tolerance ε = 10⁻¹⁰. Initial and boundary conditions are summarized in Figure 5. The resulting steady-state velocity field and the prebound P^b profiles are shown in Figure 6.

FIGURE 5.

Initial and boundary conditions for the prebound platelet flow problem. At the inlet, a parabolic velocity profile for the fluid (not pictured) and a uniform distribution of mobile platelets P^m,up are specified (green), respectively. At the outlet, zero pressure and diffusive flux conditions are specified for the fluid and mobile platelets, respectively. Finally, no slip and no diffusive flux conditions are specified on the horizontal walls for the fluid and mobile platelets, respectively. Initially, the fluid starts at rest and there are no mobile platelets ($\overrightarrow{u}=0$ and P^m = 0, respectively), but a circle of prebound platelets are specified with density P^b,cir at the center of the domain. After the fluid reaches equilibrium, the bulk of the fluid flow moves around the prebound platelet mass (white vectors).

FIGURE 6.

Representative plots of the initial bound platelet fraction ϕ^B (A) and equilibrated velocity field (B). Mesh is constructed to have a cell width of h = 0.25 µm, consisting of 115,200 cells and 60,269 points (nodes), where h refers to the grid-spacing parameter specified near points on the mesh using Gmsh ⁴¹.

To show that both our new numerical approximation of W(ϕ^T) and density-dependent FCT prelimiter are necessary to ensure that ϕ^T ≤ 1, we run the simulation up a set final time for the following three separate cases:

1. W (ϕ^T) is interpolated as a piecewise linear $({\mathbb{P}}_1)$ function using the nodal values of ϕ^T.

2. W (ϕ^T) is a piecewise constant $({\mathbb{P}}_0)$ function as described in Section 3.1.

3. W (ϕ^T) is a piecewise constant $({\mathbb{P}}_0)$ function and P^m is solved using the density-dependent prelimiter described in Algorithm 1.

A normal physiologic platelet count is P₀ = 2.5 × 10⁵ per µL, which implies that P₀ ≈ 0.00375 × P_max. Hence, in order to show the invariance and robustness to magnitude of Pm transporting around prebound platelets, we also consider three uniform upstream concentrations of P^m with concentrations significantly higher than P₀ in each of the 3 scenarios for W(ϕ^T): P^m,up = 0.25, 0.50, and 0.75 × P_max.

To compare all of the scenarios, we compute and report the maximum value of ϕ^T, for each of the three P^m,up, at the final time of t ≈ 1 second. The results are summarized in Table 1.

TABLE 1.

Comparison of the spatial max of ϕ^T at t ≈ 1 second when defining W (ϕ^T) as: A ${\mathbb{P}}_1$ function (first row), A ${\mathbb{P}}_0$ function (second row) and A ${\mathbb{P}}_0$ function with the density-dependent prelimiter (third row). ∆t = 2.5 × 10⁻⁵ sec, h = 0.25µm.

W(ϕT) Space	Prelimiter?		Pm,up = 0.25 × Pmax	Pm,up = 0.5 × Pmax	Pm,up = 0.75 × Pmax
${\mathbb{P}}_1$	No		150.54	1607.75	1274.65
${\mathbb{P}}_0$	No		1.00171	1.00327	1.0038
${\mathbb{P}}_0$	Yes		1	1	1

The results from using an interpolated W (ϕ^T) as a ${\mathbb{P}}_1$ function on the mesh are unsatisfactory, as the max platelet fraction ϕ^T greatly exceeds 1 in all cases. The results for W (ϕ^T) as a ${\mathbb{P}}_0$ function on the mesh are somewhat improved, but there is still overage and it increases as P^m,up increases, reaching approximately 0.38% over 1 in the worse case. The results using the prelimiter are promising; the maximum value of ϕ^T ≤ 1 regardless of the upstream value of P^m.

To ensure that ϕ^T ≤ 1 is preserved under a change in mesh resolution with W (ϕ^T) as a ${\mathbb{P}}_0$ function and the ϕ^T–dependent prelimiter, we performed a number of tests using the same circle problem with P^m,up = 0.5 × P_max, with various grid spacings: h = 0.0625 × n µm, where n is a scalar integer. The results of these tests are shown in Table 2 where we can see that the methodology preserves ϕ^T ≤ 1 under mesh coarsening or refinement. A detailed summary of grid convergence results for this problem is summarized in Appendix C.2.

TABLE 2.

Grid refinement study of the spatial max of ϕ^T in the advection-diffusion circle problem at t ≈ 1 second. All reported spatial data was analyzed using Paraview ⁴².

h (µm)	# of Cells	# of Points	Δt (sec)	max{ϕT}
0.0625	1,843,200	932,600	6.25 ×10−6	1
0.1250	460,800	232,760	1.25 ×10−5	1
0.2500	115,200	60,269	2.5 × 10−5	1
0.5000	28,800	149,12	5.0 × 10−5	1
1.0000	7,200	3,845	1.0 × 10−4	1
2.0000	1,800	1,026	2.0 × 10−4	1
4.0000	450	298	4.0 × 10−4	1

The above results are for the case where the region is completely full, i.e. P^b = P_max. However, when considering the transport of P^m into a region where ϕ^T is nearly 1, the prelimiter may not prevent overage between time steps. Anti-diffusive fluxes can still cause densities of Pm to be transported into nearly packed regions if the hindered fluxes, $W({\varphi }^T)P^m\overrightarrow{u}$, and/or ∆t are sufficiently large. For example, if ${\varphi }_i^T=0.999$, but f_ij has magnitude such that correction term $\mathrm{\Delta }t{M}_L^{-1}\overline{f}$ in equation (17) transports more than P^m = 0.001 × P^max, then this will cause ${\varphi }_i^T>1$; however, this will only occur once at a given spatial point, as the prelimiter prevents any further propagation. The magnitude of this one-time overage is thus affected by time step size and the amount of mobile platelets initially prescribed to the background fluid.

We next tested the robustness of our algorithm to both the prescribed initial platelet fraction (equal to the prescribed boundary value at the inlet) P^m,init and ∆t, by considering a prebound patch of bound platelets as described above. Here we assume that he platelet fraction within the bound patch is nearly 1, i.e., P^b = 0.99 × P_max. For P^m, we set the initial distribution and the upstream boundary condition of mobile platelets to be a fixed density everywhere except within the prebound patch. We varied our time step in multiples of the minimum time step required to satisfy the CFL condition, such that Δt = λ_CFL(h/u_max), where h is the mesh width, u_max is the maximum inlet velocity (defined in Appendix A) and λ_CFL is a multiplier between 0 and 1. Then, we varied λ_CFL and the initial density of mobile platelets and measured any overage of ϕ^T after a final time of t = 1 second.

To gain some insight into what initial values might lead to the highest potential one-time overage, in Figure 7 A, we plotted the hypothetical form of a hindered mobile platelet density, defined as a function of the initial prescribed platelet fraction:$W\left(\frac{P^{m,init}}{P^{max}}\right)\times \frac{P^{m,init}}{P^{max}}$. This function suggests that when the initial prescribed platelet fraction is approximately 0.65 × P^max, we may see the most overage. Figures 7 B,C show the resulting one-time overage for initial platelet fractions between 0.01 and 0.1 times P^max and 0.1 and 0.99 times P^max, respectively, and for various values of the CFL multiplier.

FIGURE 7.

(A) W(P^m/P_max) × (P^m/P_max) vs P^m/P_max shows that the maximum contribution of the advective flux by mobile platelets occurs when P^m ≈ 0.656 × P_max. (B-C) The spatial max of ϕ^T as function of λ_CFL vs P^m/P_max. Simulations were run using a mesh corresponding to a h = 1µm mesh.

When P^m = 0.01 × P_max (Figure 7 B), we observe that the maximum ϕ^T decreases as a function of λ_CFL. When P^m is greater than 0.01 × P_max, the spatial max of ϕ^T always went over 1; the maximum in this case was 1.00000005757, and occurred when λ_CFL = 0.8 and P^m = 0.1 × P_max. In Figure (7C), the spatial max of ϕ^T also went over 1, and this was for all initial values of P^m and λ_CFL. The largest value of ϕ^T was 1.00000035425 and occurred when P^m = 0.6 × P_max, with λ_CFL is 0.99, consistent with the plot in Figure 7 A. Regardless of the initial platelet fraction and the time step choice, assuming the time step satisfies the CFL condition, we see that the worse case one-time overage was on the order of 10⁻⁷.

4.2 |. Advection-diffusion-reaction

Next, we want to know if ϕ^T ≤ 1 is preserved when mobile platelets are allowed to transiently bind to a prebound platelet aggregate. We use the same domain and initial/boundary conditions as in the advection-diffusion case, but now the initial density of bound platelets in the circular region is P^b = 0.5 × P_max. The upstream concentration of P^m is set to P₀ ≈ 0.00375 × P_max to prevent platelet-platelet binding that would take over the entire domain in a short period of time; in addition, P₀ is the typical platelet density observed in plasma. Here we study the sensitivity of the overage to the three different values of the binding rates by setting k_coh × P_max = 10² s⁻¹ and varying the rate of platelet unbinding k_off = 0.1, 0.01, 0. Since ϕ^T ≤ 1 under mesh coarsening (and refinement) from the advection-diffusion test, we use the h = 1 μm mesh for this advection-diffusion reaction problem.

To couple the temporal effects of platelet-platelet binding on the fluid dynamics, the Navier-Stokes-Brinkman equations will also be solved during the time stepping of P^m, P^b and η as described in Section 3.5.

In Figure 8, we show the growth of the total platelet fraction ϕ^T with k_off = 0.1 as a function of time. Mobile platelets transiently bind inside the circular region (Fig. 8A), and gradually bind around the circular region due to the cohesion region designated by the virtual substance η (Fig. 8B,C). The maximum platelet fraction is already close to 1 when t ≈ 15 seconds with max{ϕ^T} = 0.997122. Letting the simulation continue to 60 seconds, we see the the clot grows substantially larger, while still maintaining max{ϕ^T} ≤ 1. In Figure 9, we compare max{ϕ^T} for k_off = 0, 0.01, 0.1 as a function of time. We see that k_off does change how fast the clot has a point where ϕ^T ≈ 1 in the domain, but does not cause ϕ^T > 1.

FIGURE 8.

Spatial distribution of the total platelet fraction ϕ^T at t ≈ 15, 30, 60 seconds (A,B,C respectively) for k_off = 0.1 with h = 1 µm and Δt = 1.0 × 10⁻⁴ seconds. The clot builds gradually near the upstream end of the domain but the platelet fraction ϕ^T never exceeds 1.

FIGURE 9.

max{ϕ^T} as a function of time for k_off = 0, 0.01, 0.1 up to 10 seconds (A) and from 10 to 60 seconds (B). Decreasing k_off delays the time to ϕ^T ≈ 1, but does not cause ϕ^T > 1 in all cases.

4.3 |. Platelet Aggregation in an Extravascular Injury Domain

We have shown that the algorithm using a numerical approximation to W (ϕ^T) to the discrete equations in combination with a density-dependent prelimiter for the FCT method strongly mitigates overage of platelet densities above 1, in a simple domain with a structured mesh. A natural next step is to show how the algorithm behaves with unstructured meshes and in more complicated geometries. Modeling with the finite element method is supposed to make complicated domains such as an extravascular injury domain easy to achieve with the use of unstructured meshes. Furthermore, nothing in our formulation of W (ϕ^T) as a ${\mathbb{P}}_0$ function or the prelimiter is restricted to use with structured meshes. Nevertheless, we will test our method with a model platelet aggregation in a domain with an extravascular injury channel using an unstructured mesh.

The extravascular injury domain represents a model of a microfluidic device described in our previous work where we modeled the 3D fluid dynamics to estimate pressure drops and wall shear stresses within the device 21. Due to the computational complexity of continuum modeling in large domains with long timescales, we formulate the current platelet aggregation model in 2D instead of 3D as we did previously with only the fluid.

The microfluidic device contains two vertical channels, consisting of a “blood” channel and a “wash” channel, connected at the horizontal “injury” channel, forming a “H”-shaped geometry (see Figure 10B). With an imposed flow rate and prescribed pressures, blood flows into the device through the blood channel, and then also through the injury channel (from right to left); blood exits through the outlet of the wash channel and the outlet of the blood channel. The domain is depicted in Figure 10B.

FIGURE 10.

(A) Circuit diagram to setup initial fluid boundary conditions. Quantities in red are inputs, while violet are calculated, and blue are unknown and solved by the circuit analysis equations (see Appendix). (B) A schematic of the domain and how the circuit results are applied to the boundaries. Flow rates are prescribed at the two inlets (∂Ω₁ and ∂Ω₃), while pressures are prescribed at the outlets (∂Ω₂ and ∂Ω₄). A uniform profile of mobile, platelets P^m,up are prescribed at the inlet on the blood channel.

To setup the boundary conditions for the fluid, we impose flow rates Q₁ and Q₃ via a parabolic velocity profile at the blood and wash channels, respectively. We fix the wash channel outlet pressure $P_{C{W}_0}=0$ Pa and use hydraulic circuit analysis (HCA) to set the blood channel outlet pressure $P_{C{B}_0}$ in order to achieve a injury channel flow rate that is approximately equal to Q_M, the prescribed injury channel flow rate (see Figure 10A). A summary of the HCA setup and its parameters can be found in Appendix B.

For the platelet equations, we assume a uniform profile of mobile platelets that enter through the inlet of the blood channel at a concentration of P₀, while no mobile platelets enter through the wash channel. To initiate the reactions for platelet aggregation, we assume there are bound platelets P^b = 0.5 × P_max specified in a thin 4 µm layer on the top and bottom walls of the injury channel (see Figure 10B). A summary of rate constants, and other parameters can be found in the Appendix.

We are mainly interested in the spatial dynamics of platelet aggregate growth inside the injury channel, and thus the finest region of the mesh is taken to be inside the injury channel, while the more coarse mesh is in the rest of the domain. A maximum grid resolution of 2 µm was chosen at the inlet of the blood channel and near the outlet of the wash channel, 1 µm inside the injury channels and the remainder of the domain was chosen to be a 8µm resolution since few platelets reach these regions of the domain. These choices for the mesh yield a good balance of efficiency and accuracy for our current and future modeling objectives. Details of the mesh setup can seen in Figure 11.

FIGURE 11.

(A) Mesh corresponding to the “H” domain. Triangular cells have diameters corresponding to h = 1 µm in the injury channel, 2 µm at the inlet of the blood channel and near the outlet of the wash channel, and 8 µm everywhere else. (B) Zoomed-in view of the injury channel.

In Figure 12, spatial plots of ϕ^T are shown at different times and for each time snapshot, we report the max ϕ^T in the domain. As time evolves, the bound platelet mass grows at the upstream end of the injury channel (Figure 12B), eventually causing the bound platelets from the top and bottom walls of the injury channels to meet (Figure 12C) and ultimately causing occlusion of the injury channel (Figure 12 D), where there is little flow going through the fully bound platelet clot.

FIGURE 12.

Spatial distribution of the total platelet fraction ϕ^T at t ≈ 1, 10, 20, and 30 seconds (A to D, respectively) for k_off = 0.1 for the unstructured “H” mesh with ∆t ≈ 3.21 × 10⁻⁵ seconds. ϕ^T < 1 is observed at all time snapshots (A-D).

Similar to our advection-diffusion-reaction circle test problem, we also measure the max ϕ^T as a function of time for k_off = 0, 0.01, 0.1. In Figure 12, we again observe that the rate of ϕ^T reaching 1 increases as k_off decreases. In all cases, Figure 12 seems to show that ϕ^T ≤ 1 for all time. However, when k = 0, max{ϕ^T} = 1.00000000045 occurs at some time between t = 9 and 10 seconds. After 10 seconds, max{ϕ^T} = 1.00000000045 for the rest of the simulation time.

5 |. DISCUSSION

In our mathematical model of platelet aggregation, we use a continuum approximation to represent platelet densities rather than modeling individual platelets. Within this modeling framework, mobile platelet densities are transported through fluid and into masses of other bound platelet densities, where they can build up to form high-density, stationary aggregates. The build up of platelet densities must obey physical platelet-density packing limits, and this constraint should be satisfied during the numerical solution of the model equations. However, current numerical methods for solving transport equations with FEM do not automatically satisfy the constraint and produce nonphysical results. To our knowledge, there is no general approach to satisfy this constraint for continuum models in complex geometries. In this work, we have developed a density-dependent, numerical algorithm for modeling platelet aggregation using the FEM that produces results within physical packing limits and allow for modeling within complex geometrical domains. Our two contributions to this algorithm include a modification of the W (ϕ^T) function in the weak form of P^m that accounts for the local total platelet density and a density-dependent FEM-FCT prelimiter that prevents anti-diffusive fluxes from further propagating ϕ^T past 1.

We presented a two-species model of platelet aggregation to develop and validate phenomenologically-motivated modifications to the discrete equations and to the FEM-FCT algorithm to ensure that a maximum platelet fraction is preserved numerically. Using our density-dependent FEM-FCT algorithm, we showed that the platelet fraction ϕ^T ≤ 1 is preserved in a variety of test problems that varied the upstream P^m concentration, mesh resolution and element type (structured and unstructured) as well as reaction rates (via k_off) with physiologically-relevant domain sizes and time spans.

There are some limitations to our density-dependent FEM-FCT algorithm. First, in our formulation of W (ϕ^T) as a ${\mathbb{P}}_0$ function, we assume that the function W is evaluated with a constant value, using the maximal nodal value of ϕ^T on each triangular cell, denoted by W (ϕ^T,*). Consequently, if we use the functional form of W and apply it to ϕ^T as a ${\mathbb{P}}_1$ function, we can see that W (ϕ^T,*) ≠ (ϕ^T) at every point in a triangular cell. However, we expect the error between the ${\mathbb{P}}_0$ and ${\mathbb{P}}_1$ function to decrease as the cell size h → 0; to illustrate this, a mesh refinement study of the errors between W as a ${\mathbb{P}}_1$ function versus a ${\mathbb{P}}_0$ function using the initial prebound platelet circle is shown in Figure (C1) in the Appendix. Second, during the prelimiting step, anti-diffusive flux components are set to zero when moving into nodes where ϕ^T = 1 and there is a positive anti-diffusive flux going into that node. Consequently, the solution will possess the overly diffusive qualities given from the predictor step at the neighboring points where ϕ^T = 1. This is the trade off we make in accuracy to achieve the correct physical packing behavior for our application. However, the accuracy-recovering qualities of the correction step remain everywhere in the domain where ϕ^T < 1. Third, the prelimiter may not prevent the transport of P^m into a region where ϕ^T is nearly 1. Our time step sensitivity study in Section 4.1 shows that there will be a negligible one-time overage that depends on the time step and the flux of mobile platelets, as long as the time step is constrained by the CFL condition. We chose the CFL condition as a time step constraint to ensure the numerical speed of platelet transport is within the limits of the maximum fluid velocity. Finally, our modified FEM-FCT algorithm uses an extrapolated value of ϕ^T from the previous time step. However, this issue also addressed in our time step study in Section 4.1.

Addressing dense-packing limits has been previously considered in the FEM-FCT framework, using the group finite element formulation ⁴³ for linear advection only ²⁸. In that study, an additional flux correction step was suggested to limit fluxes of a transported species subject to a maximum packing density. However, densities were only limited at the final correction step, rather than modifying the PDE at the continuous level as we have done here. A combination of our piecewise constant W (ϕ^T) function with this method could obtain higher-order solutions in regions where ϕ^T ≈ 1.

In this study, we examined a two-species platelet model of aggregation, but our methodology can be extended to multiple mobile and bound platelet species that are coupled through reaction terms. With multiple mobile species, the total platelet fraction ϕ^T and the corresponding hindered transport coefficient, W (ϕ^T), would need to be updated between the solution steps for each of the mobile platelet equations, to account for the possibility that if one mobile species fills the spatial region, then the other species should be prevented from moving into that same region. This does suggest that the order in which the equations are solved may be important when the spatial location is nearly full. However, our current work shows that the hindered fluxes into the nearly full regions are an extremely tiny portion of the total platelet fraction in that region and so the order in which the region fills is likely to be biologically insignificant.

Continuum models of cell aggregates are typically set up so that the length scale of the aggregates are much larger than that of the cells. Here, and in our previous work ^1,2, our continuum approach is atypical in that the grid size in the injury channel (≈ 1µm) is smaller than that of one individual cell (≈ 2µm). In our previous works, we describe the methodology that enables he single cell (platelet) length scale to play a role in the model, i.e., through the phenomenological adhesion, cohesion, and hindered transport. There are other approaches to modeling platelet aggregation where each platelet is individually modeled as a discrete object, such as the immersed boundary method, cellular potts methods, lattice boltzmann, or dissipative particle dynamics ^{7,8,9,10,11,12,13,14,15,16,17,18,19,20}; with these methods each platelet is usually modeled as a discrete structure that interacts with the surrounding fluid, and in some instances with fluid on its interior. These modeling techniques have been instrumental in learning about platelet-platelet and platelet-wall interactions and platelet margination. However, when it comes to thrombus formation, there are two major challenges that have to be addressed: (i) clot formation occurs on the timescales of minutes, and (ii) clot formation includes platelet aggregation together with the complex network of the surface-mediated coagulation reactions. The discrete platelet modeling approach works well on the timescale of seconds, but even then it is extremely computationally expensive and thus modeling clot formation on minutes is simply not feasible. Further, it is difficult to model platelet-surface reactions on these structures as they move and deform in time and space, although progress is being made in that direction ⁴⁴. Our modeling approach, albeit atypical in the sense of continuum modeling, can handle both challenges and it simulates growing clots with core-shell characteristics, as seen experimentally ^45,46, and is able to predict clotting time-courses in agreement with the microfluidic device measurements for hemophiliac blood, including its response to treatment ³. Thus, we believe that with our numerical algorithm developed in the current study, this approach is a good one for future modeling of extravascular thrombus formation that occurs in companion microfluidic devices.

Our density-dependent FEM-FCT algorithm has many future applications. As mentioned above, we used FEniCS to implement our FEM equations. However, both the density-dependent prelimiter and the interpolation space for the W (ϕ^T) function could be applied to any existing finite element codes that support nodal and cell-based data structures from the mesh, respectively. Although we are interested in modeling platelet aggregation, this algorithm could also be applied to any chemical or sediment advection-diffusion problem that requires a transport-limitation due to a maximum packing density.

FIGURE 13.

max{ϕ^T} as a function of time for k_off = 0, 0.01, 0.1 up to 10 seconds (A) and from 10 to 32 seconds (B) for the unstructured, “H” mesh. Decreasing k_off delays the time to ϕ^T ≈ 1, but does not cause ϕ^T > 1 in all cases.

APPENDIX. All parameters below are rounded to 5 decimal figures.

A. RATE CONSTANTS AND OTHER PHYSICAL PARAMETERS

The fluid parameters are given below:

Fluid Density (ρ)	1 g/cm3
Fluid Viscosity (µ) (Circle Problem)	4.0 × 10−2 Poise
Fluid Viscosity (µ) (“H” problem)	2.62507 × 10−2 Poise
Re (Circle problem)	1.12500 × 10−3
Re (“H” problem)	1.77773 × 10−2

Characteristic scales for the problem are given below:

xchar	3 µm
tchar (Circle Problem)	2 × 10−3 s
tchar (“H” Problem)	1.92857 × 10−4 s
uchar (Circle Problem)	1.5 × 103 µm/s
uchar (“H” Problem)	1.55556 × 104 µm/s

Finally the rest of the kinetic and other physical parameters are given below:

Mobile Platelet Density Scale (P0)	2.5 ×105 platelets/µL
Max Platelet Density Scale (Pmax)	6.67 ×107 platelets/µL
Platelet Diffusion Coefficent (Dp)	2.5 ×10−7 cm2 s−1
Cohesion rate of platelet-platelet binding (kcoh)	1 ×102/Pmax s−1
Unbinding rate of bound platelets (koff)	0.1, 0.01, 0 s−1
Pe (“Circle” Problem)	1.8 × 102
Pe (“H” Problem)	1.86667 × 103

The Reynolds number, Re, is defined as the ratio of inertial to viscous forces, is given by

$$\text{Re}=\frac{\rho u_{char}{x}_{char}}{\mu }$$

Where u_char and x_char are the characteristic velocity and length scales, respectively.

The Peclet number, Pe, is defined as the ratio of advective and diffusive transport of P^m, is given by:

$$\text{Pe}=\frac{u_{char}{x}_{char}}{D_P}$$

B. FLUID CIRCUIT SETUP AND RESULTS

Using the circuit diagram shown in Figure 10A, we can use a hydraulic analog of linear circuit theory to determine the pressures of the bleeding chip. Between each pressure value on Figure 10A, this requires assuming that flow is fully developed.

Conservation of flow rate tells us that

$$Q_2=Q_1-Q_M,$$ (B1)

$$Q_4=Q_3+Q_M.$$ (B2)

Using our assumption of fully developed full between each pressure segment shown in Figure 10A can be written as ∆P = QR, where ∆P is the difference between two pressure values, Q is the flow rate, and R is the resistance from the channel geometry. For a 2D parallel plate flow, this resistance is given by:

$$R=\frac{12\mu L}{w^3}$$

where µ is the fluid viscosity, L and w is the length and width of the channel, respectively. Hence, the rest of the equations are given by:

$$P_B-P_{CB}=Q_1{R}_{UB},$$ (B3)

$$P_{CB}-P_1=Q_1{R}_1,$$ (B4)

$$P_1-P_{C{B}_0}=Q_2{R}_2,$$ (B5)

$$P_W-P_{CW}=Q_3{R}_{UW},$$ (B6)

$$P_{CW}-P_2=Q_3{R}_3,$$ (B7)

$$P_2-P_{C{W}_0}=Q_4{R}_4,$$ (B8)

$$P_1-P_2=Q_M{R}_M.$$ (B9)

The inlet flow rates Q₁, Q₂ and the injury channel flow rate Q_M are given, while the outlet flow rates Q₂ and Q₄ are unknown. Similarly, we set the outlet pressure $P_{C{W}_0}=0$ Pa and solve for the remainding pressures; this yields a system of equations of 9 equations with 9 unknowns. We can write this as the following matrix system:

$$\left(\begin{array}{ccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& -1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& -1& 0& 0& 0\\ -R_2& 0& 0& 0& -1& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& -1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& -1\\ 0& -R_4& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 0& 0& -1\end{array}\right)\left(\begin{array}{c}{Q}_2\\ Q_4\\ P_B\\ P_{CB}\\ P_{C{B}_0}\\ P_1\\ P_W\\ P_{CW}\\ P_2\end{array}\right)=\left(\begin{array}{c}{Q}_1-Q_M\\ Q_3+Q_M\\ Q_1{R}_{UB}\\ Q_1{R}_1\\ 0\\ Q_3{R}_{UW}\\ Q_3{R}_3\\ P_{C{W}_0}\\ Q_M{R}_M\end{array}\right)$$ (B10)

We summarize the parameters used in the circuit model below:

Dimensions of Circuit

Width of vertical channels	100 µm
Width of injury channel	20 µm
Length from $P_B^{\ast }$ to PB (and $P_W^{\ast }$ to PW)	4915 µm
Length from PCB to P1 (and $P_{CW}^{\ast }$ to P2)	85 µm
Length from P1 to PCB0 (and P2 to $P_{C{W}_0}$)	85 µm
Length of Injury	150 µm
Length of Injury used in HCA	165 µm

HCA Viscosities and Computed Resistances

Blood Channel Viscosity	3.6 × 10−2 Poise
Upstream Wash Channel Viscosity	1 × 10−2 Poise
Injury Channel Viscosity	2.67 × 10−2 Poise
Downstream Wash Channel Viscosity	2.3 × 10−2 Poise
RU B	2.12328 × 108 Pa s/m2
RU W	5.898 × 107 Pa s/m2
R1, R2	3.672 × 106 Pa s/m2
RM	6.60825 × 108 Pa s/m2
R3	1.02 × 106 Pa s/m2
R4	2.346 × 106 Pa s/m2

There are some assumptions with the parameters used in the HCA. First, in the experimental bleeding chip, there are variable viscosities corresponding to the blood, wash and injury channels. Furthermore, the downstream end of the wash channel will be a mix of the blood and wash when blood can flow freely from the blood channel through the injury channel. Therefore, in the HCA we account for the viscosities through the resistance terms as noted in the table above. We assume that the viscosity of the downstream end of the wash channel is an average of the blood and wash viscosities. However, in the fluid simulations, we use a volume-averaged viscosity through entire domain as noted in the table above. Second, we also assume there is an effective length of the injury channel that spans outside its 150 µm length, corresponding to the laminar flow that enters and exits the injury channel. We assume that this extra length is 15 µm, leading to a L = 165µm used in the HCA for the injury channel resistance R_M. This adjustment is required for the fluid simulation to recover an approximate flow rate through the injury channel equal to Q_M.

Finally, we summarize the solving the circuit system in the table below:

Inlet Flow Rate (Blood, Q1)	1.55556×10−6 m2/s
Outlet Flow Rate (Blood, Q2)	1.13056 ×10−6 m2/s
Inlet Flow Rate (Wash, Q3)	2.77778 ×10−6 m2/s
Outlet Flow Rate (Wash, Q4)	3.20278 ×10−6 m2/s
Injury Channel Flow Rate (QM)	4.25 ×10−7 m2/s
Pressure Inlet (Blood, PB)	6.24364 × 102 Pa
Pressure Inlet (Wash, PW)	1.7418 × 102 Pa
Pressure Inlet (Blood, PCB)	2.94076 × 102 Pa
Pressure Outlet (Blood, $P_{C{B}_0}$)	2.84213 × 102
Pressure Inlet (Wash, PCW)	1.03471 × 101 Pa
Pressure Outlet (Wash, $P_{C{W}_0}$)	0 Pa
Pressure Drop Across Injury (P1 − P2)	2.80851 × 102 Pa

C. MESH CONVERGENCE STUDIES

In the following sections, we quantify the error caused by the change in interpolation space for the W(ϕ^T) function from a ${\mathbb{P}}_1$ to a ${\mathbb{P}}_0$ space. We do this in three cases:

1. Compute the error of W(ϕ^T) applied to the prebound clot using a ${\mathbb{P}}_1$ versus a ${\mathbb{P}}_0$ space.

2. Compute the grid error of the advection-diffusion test problem from Section 4.1 at the final time t = 1.0 second. Note that this problem has no analytical solution, so instead we use the solution from the h = 0.0625µm grid to compute the error convergence.

3. Compute the grid error of the advection-diffusion-reaction test problem from Section 4.2. We use the solution from the h = 0.125µm grid to compute the error convergence at different time points.

C.1. W function mesh convergence study

In order for ϕ^T ≤ 1, the equation of P^m uses a hindered transport coefficient W (ϕ^T) which scales the advective and diffusive flux as a function of ϕ^T. When the equation was discretized using the FEM, an approximation of W(ϕ^T) was done in accordance of Figure 3 in place of approximating W(ϕ^T) as a ${\mathbb{P}}_1$ function. We claim that the error between the ${\mathbb{P}}_0$ and ${\mathbb{P}}_1$ forms of W would go to zero as mesh size h → 0. Hence, the objective of this study is show the error difference between the ${\mathbb{P}}_0$ and ${\mathbb{P}}_1$ W functions upon mesh refinement.

To do the comparison between the two functions, we begin by constructing P^b = P_max as described in the advection-diffusion circle problem and shown in Figure 6A. Next, we apply the ${\mathbb{P}}_1$ W function, denoted W(ϕ^T) and the ${\mathbb{P}}_0$ W, denoted W(ϕ^T) using the prebound circle of platelets. Next, the error is between the two functions is defined by the L² norm:

$$\text{Error}=\frac{{\Vert W({\varphi }^T)-W(\varphi T,\ast )\Vert }_{L^2(\Omega )}}{{\Vert W({\varphi }^T)\Vert }_{L^2(\Omega )}}$$ (C11)

where Ω is the domain and the L2-norm applied to a function f on Ω is defined by:

$${\Vert f\Vert }_{L^2(\Omega )}=\sqrt{\underset{\Omega }{\int }{[f(x)]}^{2}d\hspace{0.17em}x}$$ (C12)

Integrals are computed directly on the mesh using the mesh sizes specified in Table 1. Results in Figure C1 show that the error decreases as h decreases.

FIGURE C1.

A mesh convergence study between the ${\mathbb{P}}_0$ vs ${\mathbb{P}}_1$ W function, denoted as W(ϕ^T,*) and W(ϕ^T), respectively. The relative L² error is computed using relative to W(ϕ^T).

C.2. Advection-Diffusion Only

In this section, we compute the error of the advection-diffusion test problem by comparing the solution between a fine and coarse grid. For this study, we use P^m,up = 0.5 × P_max as the inlet boundary condition and P^b = P_max in the 9 µm radius circle as decribed in Section 4.1. We denote the the finest grid with as h_fine, where u_fine is the solution on the finest grid. In order to compare the solutions on different grid resolutions, we interpolate the solutions onto a mesh grid using Scipy’s griddata() function using the same number (3000²) of points for each of the grids with a piecewise linear interpolant. We also set the time step as a function of h that satisfies the CFL condition: ∆t = (3/4)h/(u_max), where u_max is defined in Appendix A.

We define the grid norm applied to a solution vector $P_{h_k}$ with grid width h_k as

$$E(P_{h_k})=\Vert \frac{P_{\text{fine}}-P_{h_k}}{u_{\text{fine}}}\Vert$$ (C13)

We use 3 different norms to look at the error of the solution applied to a n-tuple vector solution ν_h = (ν₁, ν₂, … ν_n):

1. ${\ell }_2$-norm:

   $${\Vert {\nu }_h\Vert }_{{\ell }_2}=\sqrt{h\sum _{i=1}^n{\left|{\nu }_i\right|}^2}$$ (C14)
2. ${\ell }_1$-norm:

   $${\Vert {\nu }_h\Vert }_{{\ell }_1}=h\sum _{i=1}^n\left|{\nu }_i\right|$$ (C15)
3. ${\ell }_{\text{max}}$-norm:

   $${\Vert {\nu }_h\Vert }_{{\ell }_{\text{max}}}=h\underset{i=1\cdots n}{\text{max}}\left|{\nu }_i\right|$$ (C16)

We summarize the error convergence in Figure (C2). Using Numpy’s polyfit() function to fit the error vectors to a line on a log-log scale, we find an observed order of convergence of approximately 1.61, 1.55 and 0.97 for the ${\ell }_2$, ${\ell }_1$, and ${\ell }_{\text{max}}$ norms, respectively.

C.3. Advection-Diffusion-Reaction

In this section, we compute the error of the advection-diffusion-reacction test problem by comparing the solution between the finest grid and a coarser one. For this study, we use P^m,up = P₀ as the inlet boundary condition, and initially set P^b = 0.99 × P_max in the 9 µm radius circle as decribed in Section 4.1. We also use the same k_coh value from Section 4.2 and set k_off = 0.1 s⁻¹. We use h = 0.125 µm for h_fine for these studies and compare the solution for h = 4, 2, 1, 0.5, 0.25 µm. We use the same grid norms as defined in Appendix C.2. Finally, we compute the error at three different time points: t = 0.1, 0.5, 0.7, 1.0 seconds. The grid convergence is summarized in the figure below.

FIGURE C2.

A mesh convergence study on the advection-diffusion test problem at t ≈ 1 second. The fine grid solution P^m (h = 0.0625µm) is used as a replacement for an analytical solution to compare to the P^m solution on a grid with h = 4, 2, 1, 0.5, 0.25, 0.125 µm.

FIGURE C3.

A mesh convergence study on the advection-diffusion-reaction test problem at t ≈ 0.1 seconds (left) and ≈ 0.5 seconds (right). The fine grid solution P^m (h = 0.125µm) is used as a replacement for an analytical solution to compare to the P^m solution on a grid with h = 4, 2, 1, 0.5, 0.25 µm.

For t ≈ 0.1 seconds, the observed order of convergence of approximately 1.95, 1.67 and 1.48 for the ${\ell }_2$, ${\ell }_1$, and ${\ell }_{\text{max}}$ norms, respectively. At the final time of t ≈ 2.0 seconds,the observed order of convergence of approximately 1.87, 1.71, and 1.17 for the ${\ell }_2$, ${\ell }_1$, and ${\ell }_{\text{max}}$ norms, respectively.

FIGURE C4.

A mesh convergence study on the advection-diffusion-reaction test problem at t ≈ 1.0 seconds (left) and ≈ 2.0 seconds (right). The fine grid solution P^m (h = 0.125µm) is used as a replacement for an analytical solution to compare to the P^m solution on a grid with h = 4, 2, 1, 0.5, 0.25 µm.