Simulation of Platelets Suspension Flowing Through a Stenosis Model Using a Dissipative Particle Dynamics Approach

Abstract

Stresses on blood cellular constituents induced by blood flow can be represented by a continuum approach down to the μm level; however, the molecular mechanisms of thrombosis and platelet activation and aggregation are on the order of nm. The coupling of the disparate length and time scales between molecular and macroscopic transport phenomena represents a major computational challenge. In order to bridge the gap between macroscopic flow scales and the cellular scales with the goal of depicting and predicting flow induced thrombogenicity, multi-scale approaches based on particle methods are better suited. We present a top-scale model to describe bulk flow of platelet suspensions: we employ dissipative particle dynamics to model viscous flow dynamics and present a novel and general no-slip boundary condition that allows the description of three-dimensional viscous flows through complex geometries. Dissipative phenomena associated with boundary layers and recirculation zones are observed and favorably compared to benchmark viscous flow solutions (Poiseuille and Couette flows). Platelets in suspension, modeled as coarse-grained finite-sized ensembles of bound particles constituting an enclosed deformable membrane with flat ellipsoid shape, show self-orbiting motions in shear flows consistent with Jeffery’s orbits, and are transported with the flow, flipping and colliding with the walls and interacting with other platelets.

Appendix

Dissipative Particle Dynamics Formulation

DPD is a mesoscopic particle method,24,27 where each particle represents a molecular cluster rather than an individual atom, and can be thought of as a soft lump of fluid. The DPD system consists of N point particles of mass m _i , position r _i and velocity v _i . DPD particles interact through three forces: conservative (F ^C _ij ), dissipative (F ^D _ij ) and random (F ^R _ij ) forces given by

$$ {\mathbf{F}}_{ij}^{\text{C}} = F_{ij}^{\text{C}} (r_{ij} ){\mathbf{e}}_{ij} , $$

(1)

$$ {\mathbf{F}}_{ij}^{\text{D}} = - \gamma \omega^{\text{D}} (r_{ij} )({\mathbf{v}}_{ij} \cdot {\mathbf{e}}_{ij} ){\mathbf{e}}_{ij} , $$

(2)

$$ {\mathbf{F}}_{ij}^{\text{R}} = \sigma \omega^{\text{R}} (r_{ij} )\xi_{ij} dt^{ - 1/2} {\mathbf{e}}_{ij} , $$

(3)

where e _ij  = r _ij /r _ij , with r _ij  = r _i  − r _j and r _ij  = (r _ij  · r _ij )^1/2, is a unit vector in the direction of particles i and j, and v _ij  = v _i  − v _j is the relative velocity of particle i with respect to particle j. The coefficients γ and σ define the strength of dissipative and random forces, respectively. In addition, ω ^D and ω ^R are weight functions, and ξ _ij is a normally distributed random variable with zero mean, unit variance and ξ _ij  = ξ _ji . All forces are truncated beyond the cutoff radius r _c, which defines the length scale in the DPD system. The conservative force is given by

$$ F_{ij}^{\text{C}} (r_{ij} ) = \left\{ {\begin{array}{*{20}c} {a_{ij} (1 - r_{ij} /r_{\text{c}} ),} & {{\text{if}}\;r_{ij} \le r_{\text{c}} } \\ {0,} & {{\text{if }}r_{ij} > r_{\text{c}} } \\ \end{array} } \right., $$

(4)

where a _ij is the conservative force coefficient between particles i and j. The random and dissipative forces form a thermostat and must satisfy the fluctuation–dissipation theorem in order for the DPD system to maintain equilibrium temperature T.14 This leads to

$$ \omega^{\text{D}} (r_{ij} ) = [\omega^{\text{R}} (r_{ij} )]^{2} , $$

(5)

$$ \sigma^{2} = 2\gamma k_{B} T, $$

(6)

where k _B is the Boltzmann constant. The choice for the weight functions is

$$ \omega^{\text{R}} (r_{ij} ) = \left\{ {\begin{array}{*{20}c} {(1 - r_{ij} /r_{\text{c}} )^{k} ,} & {{\text{if }}r_{ij} \le r_{\text{c}} } \\ {0,} & {{\text{if }}r_{ij} > r_{\text{c}} } \\ \end{array} } \right., $$

(7)

where k = 1 for the original DPD method. However, other choices (e.g., k = 0.25) have been used in order to increase the viscosity of the DPD fluid.15,18 To mimic the effect of a pressure gradient driving the flow of particles, a body force, given by

$$ {\mathbf{G}} = g_{x} {\mathbf{e}}_{x} + g_{y} {\mathbf{e}}_{y} + g_{z} {\mathbf{e}}_{z} , $$

(8)

acts in particles located within certain regions of space. Particle i can have N ^B _i bonds with other particles. If particle i is bound to particle k, there exists a force in particle i acting in the e _ik direction with magnitude determined by the gradient of the harmonic bond potential

$$ V^{\text{B}} = \frac{{K_{ik}^{\text{B}} }}{2}(r_{ik} - r_{0} )^{2} , $$

(9)

where K ^B _ik is the force constant and r ₀ is the distance when the force is null. Torsional energy is added with the definition of proper dihedrals: particle i can be part of N ^D _i dihedrals. If particles i,j k, and l form a proper dihedral, two planes with normal vectors m and n can be defined with particles i, j and k, and j, k, and l, respectively, i.e.,

$$ {\mathbf{m}} = {\mathbf{r}}_{ij} \times {\mathbf{r}}_{kl} , $$

(10)

$$ {\mathbf{n}} = {\mathbf{r}}_{lk} \times {\mathbf{r}}_{jk} . $$

(11)

The torsional angle is defined as

$$ \phi_{ijkl} = - { \arctan }\left( {\frac{{\sin \phi_{ijkl} }}{{\cos \phi_{ijkl} }}} \right), $$

(12)

with the sine and cosine of the torsional angle being given by

$$ \cos \phi_{ijkl} = \frac{{{\mathbf{m}} \cdot {\mathbf{n}}}}{m \, n}, $$

(13)

$$ \sin \phi_{ijkl} = \frac{{({\mathbf{n}} \cdot {\mathbf{r}}_{ij} )r_{jk} }}{m \, n}, $$

(14)

The harmonic dihedral potential is defined as

$$ V^{\text{D}} = K_{ijkl}^{\text{D}} [1 + \cos (n_{ijkl} \phi_{ijkl} - \phi_{0} )], $$

(15)

where K ^D _ijkl is the torsion constant, $\phi_{0}$ is the angle of minimum potential, and n _ijkl is the multiplicity of minima in a full rotation. Finally, the forces exerted in particle i due to the harmonic bond and dihedral potentials are obtained with the computation of the gradient of the potentials, on which the chain rule is useful, i.e.,

$$ {\mathbf{F}}_{ij}^{\text{B}} = - \frac{{\partial V^{\text{B}} }}{{\partial {\mathbf{r}}_{i} }} = - K_{ij}^{\text{B}} (r_{ij} - r_{0} ){\mathbf{e}}_{ij} , $$

(16)

$$ {\mathbf{F}}_{ijkl}^{\text{D}} = - \frac{{\partial V^{\text{D}} }}{{\partial \phi_{ijkl} }}\frac{{\partial \phi_{ijkl} }}{{\partial {\mathbf{r}}_{i} }}. $$

(17)

The time evolution of velocities and positions of particles is determined by Newton’s second law of motion

$$ \frac{d}{dt}{\mathbf{r}}_{i} = {\mathbf{v}}_{i} , $$

(18)

$$ \frac{d}{dt}(m_{i} {\mathbf{v}}_{i} ) = \sum\limits_{j = 1,j \ne i}^{N} {({\mathbf{F}}_{ij}^{\text{C}} + {\mathbf{F}}_{ij}^{\text{D}} + {\mathbf{F}}_{ij}^{\text{R}} )} + {\mathbf{G}} + \sum\limits_{k = 1}^{{N_{i}^{\text{B}} }} {{\mathbf{F}}_{ik}^{\text{B}} } + \sum\limits_{p = 1}^{{N_{i}^{\text{D}} }} {{\mathbf{F}}_{ip}^{\text{D}} } , $$

(19)

which are integrated using the modified velocity—Verlet algorithm.24 Fluid particles will have no bonds or dihedrals associated to them, i.e., N ^B _i  = N ^D _i  = 0 if particle i is a fluid particle. In our simulations, we consider: (i) all particles have the same mass (i.e., m _i  = m for all i), (ii) share equal conservative force coefficients (i.e., a _ij  = a for all i and j), (iii) a negative pressure gradient is directed along the e _y direction (i.e., g _y  = g and g _x  = g _z  = 0), and (iv) all bonds and dihedrals are equal (i.e., share the same functional form of potential, with $K_{ij}^{\text{B}} = K^{\text{b}} ,\;K_{ijkl}^{\text{D}} = K^{\text{d}} ,\;n_{ijkl} = 1$ and $\phi_{0} = 0$).

No-Slip Boundary Condition Implementation

We have generalized and implemented Willemsen et al. 50 no-slip boundary condition into a three dimensional framework. The boundary condition is composed of three distinct components: (i) inclusion of an additional conservative force to balance void space beyond the wall, (ii) inclusion of additional dissipative and random forces due to interaction with fictitious particles, and (iii) specular reflection of particles crossing a solid wall. Let us consider a particle i located at ${\mathbf{r}}^{i} = (r_{x}^{i} ,r_{y}^{i} ,r_{z}^{i} )$ and let us consider a triangular wall P composed of vertices (P ¹, P ², P ³), each with coordinates ${\mathbf{P}}^{j} = (P_{x}^{j} ,P_{y}^{j} ,P_{z}^{j} )$ for j = 1, 2, 3 (Fig. 12a). With the objective of treating triangles with any orientation, we define a isoparametric transformation from the (x, y, z) coordinate system into a general coordinate system (ξ, η, ζ) and triangle P becomes triangle $\bar{P}$ composed of vertices $({\bar{\mathbf{P}}}^{1} ,{\bar{\mathbf{P}}}^{2} ,{\bar{\mathbf{P}}}^{3} )$ with coordinates ${\bar{\mathbf{P}}}^{j} = (\bar{P}_{\xi }^{j} ,\bar{P}_{\eta }^{j} ,\bar{P}_{\zeta }^{j} )$ for j = 1, 2, 3 given by (0,0,0), (1,0,0), and (0,1,0), respectively, in the general coordinate system (ξ, η, ζ) (Fig. 12b). In order to compute the transformation from P to $\bar{P}$, it is useful to define vectors ${\mathbf{u}} = {\mathbf{P}}^{2} - {\mathbf{P}}^{1}$, ${\mathbf{v}} = {\mathbf{P}}^{3} - {\mathbf{P}}^{1} ,\;{\mathbf{n}} = ({\mathbf{u}} \times {\mathbf{v}})/\left| {{\mathbf{u}} \times {\mathbf{v}}} \right|,\;{\bar{\mathbf{u}}} = {\mathbf{u}}/\left| {\mathbf{u}} \right|$ and ${\bar{\mathbf{v}}} = ({\mathbf{v}} - ({\bar{\mathbf{u}}} \cdot {\mathbf{v}}){\bar{\mathbf{u}}})/\left| {{\mathbf{v}} - ({\bar{\mathbf{u}}} \cdot {\mathbf{v}}){\bar{\mathbf{u}}}} \right|$ such that $({\bar{\mathbf{u}}},{\bar{\mathbf{v}}},{\mathbf{n}})$ form an orthonormal basis. Vectors u and v correspond to two edges of triangle P, ${\bar{\mathbf{u}}}$ is an unit vector aligned with one triangular edge and n is a unit normal defining the interior side of the triangular wall (defined by counter-clockwise numbering of triangular vertices). The Jacobian J of the transformation is given by

Figure 12

The generalization of Willemsen et al. 50 no-slip boundary condition employs isoparametric transformations from all triangular elements defining the complex wall geometry defined in the coordinate system (x, y, z) into coordinate system (ξ, η, ζ) ((a) and (b)). Specular reflection is enforced to particles that cross the wall by reflecting their normal position and normal velocity component (c). Walls enclosing the DPD fluid are concave; hence an algorithm to handle multiple reflections was developed (d)

$$ J = u_{x} (v_{y} n_{z} - v_{z} n_{y} ) - u_{y} (v_{x} n_{z} - v_{z} n_{x} ) + u_{z} (v_{y} n_{x} - v_{y} n_{x} ), $$

(20)

and the transformation matrix M is given by

$$ \left( {\text{\bf M}} \right) = \frac{1}{J}\left( {\begin{array}{*{20}c} {v_{y} n_{z} - v_{z} n_{y} } & {v_{z} n_{x} - v_{x} n_{z} } & {v_{x} n_{y} - v_{y} n_{x} } \\ {u_{z} n_{y} - u_{y} n_{z} } & {u_{x} n_{z} - u_{z} n_{x} } & {u_{y} n_{x} - u_{x} n_{y} } \\ {u_{y} v_{z} - u_{z} v_{y} } & {u_{z} v_{x} - u_{x} v_{z} } & {u_{x} v_{y} - u_{y} v_{x} } \\ \end{array} } \right). $$

(21)

The first step of the boundary condition infers if particle i is within the region of influence of the triangular wall, defined as a layer of thickness r _c above the triangular wall (Fig. 12a)—if outside, the particle does not suffer the influence of the wall, but if inside, it does, and hence, additional interactions in the force computation must be included. Three additional interaction forces will be added to particle i: dissipative and random interactions with fictitious particles beyond the wall and a conservative force that balances particle i with the void space existing beyond the wall.

The position of the particle i with respect to vertex P ¹ is given by vector m ⁱ = r ⁱ − P ^l, which transforms to the general coordinate system simply by ${\bar{\mathbf{m}}}^{i} = {\mathbf{Mm}}^{i}$ and results in coordinates $(\bar{m}_{\xi }^{i} ,\bar{m}_{\eta }^{i} ,\bar{m}_{\zeta }^{i} )$. The necessary conditions for particle i to be inside the region of influence of triangular wall P are $\bar{m}_{\xi }^{i} \ge 0$, $\bar{m}_{\eta }^{i} \ge 0$, $1 - \bar{m}_{\xi }^{i} \ge \bar{m}_{\eta }^{i}$, and $0 \le \bar{m}_{\zeta }^{i} \le 1$ (Fig. 12b).

The additional conservative force introduced to particle i due to the boundary condition is quite straightforward—note that $\bar{m}_{\zeta }^{i}$ is the normal distance of particle i to the wall and n is a unit vector normal to the wall (and directed inwards), thus ${\mathbf{F}}_{i}^{c} = f{\mathbf{n}}$ with $h = \bar{m}_{\zeta }^{i}$ (cf. “Methods”—No-Slip Boundary Conditions in Complex Geometries).

Particle i has M neighbors, i.e., particles k located at r ^k with k = 1, 2,…,M whose distance to particle i is lower than cut-off radius r _c. These neighbor particles k will originate fictitious particles $\tilde{k}$ with which particle i interacts if the fictitious particle lays within one cut-off radius of particle i. Thus, the normal distance of particle i to the wall $\bar{m}_{\zeta }^{i}$ will dictate the thickness of the layer of neighbors that must be reflected beyond the wall, i.e., $h^{\text{layer}} = r_{\text{c}} - \bar{m}_{\zeta }^{i}$ (Fig. 2). The transformation is applied to all neighbor particles k, i.e., ${\mathbf{m}}^{k} = {\mathbf{r}}^{k} - {\mathbf{P}}^{1}$ and ${\bar{\mathbf{m}}}^{k} = {\mathbf{Mm}}^{k}$ such that $\bar{m}_{\zeta }^{k}$ is the normal distance to the wall of neighbor particle k. If $\bar{m}_{\zeta }^{k} < h^{\text{layer}}$, then a reflected fictitious particle $\tilde{k}$ will be considered and dissipative and random interactions between fictitious particle $\tilde{k}$ and particle i included. The location of fictitious particle ${\tilde{\mathbf{r}}}^{k}$ is given by ${\tilde{\mathbf{r}}}^{k} = {\tilde{\mathbf{r}}}^{k} - 2\bar{m}_{\zeta }^{k} {\mathbf{n}} + {\mathbf{h}}^{r}$ where h ^r is the random shift parallel to triangular wall P (Fig. 2). The random shift h ^r is generated with the aid of two random numbers, the random shift distance h ^r and the random shift orientation θ ^r both being generated with independent Gaussian random number generators with zero mean and variance r _c and π respectively, and is given by

$$ {\mathbf{h}}^{\text{r}} = h^{\text{r}} ({\bar{\mathbf{u}}}\cos \theta^{\text{r}} + {\bar{\mathbf{v}}}\sin \theta^{\text{r}} ) . $$

(22)

Random interaction forces are determined simply by distance between fictitious particle $\tilde{k}$ and particle i, i.e., ${\mathbf{r}}_{{i\tilde{k}}} = {\mathbf{r}}_{i} - {\tilde{\mathbf{r}}}_{k}$, whereas dissipative interaction forces take into account the relative velocity between both, i.e., ${\mathbf{v}}_{{i\tilde{k}}} = {\mathbf{v}}_{i} - {\tilde{\mathbf{v}}}_{k}$. The velocity ${\tilde{\mathbf{v}}}_{k}$ of the fictitious particle $\tilde{k}$ is the reflected velocity of neighbor particle k with the tangential component reversed, and is given by ${\tilde{\mathbf{v}}}_{k} = - {\mathbf{v}}_{k} - 2({\mathbf{v}}_{k} \cdot {\mathbf{n}}){\mathbf{n}}$ (Fig. 2). A similar methodology is employed to account for the interaction between particle i and its corresponding fictitious particle $\tilde{i}$ if $\bar{m}_{\zeta }^{i} < r_{\text{c}} /2$. To conclude, the additional conservative, random and dissipative interaction forces added to particle i by the imposition of the no slip boundary condition on triangular wall P are

$$ {\mathbf{f}}_{i}^{{{\mathbf{no - slip}}}} = f \,{\mathbf{n}} + {\mathbf{F}}_{{i\tilde{i}}}^{D} + {\mathbf{F}}_{{i\tilde{i}}}^{R} + \sum\limits_{{\tilde{k} = 1}}^{{\tilde{M}}} {({\mathbf{F}}_{{i\tilde{k}}}^{D} + {\mathbf{F}}_{{i\tilde{k}}}^{R} )} , $$

(23)

where $\tilde{M}$ is the number of fictitious neighbors of particle i introduced.

Lastly, specular reflection of particles that cross the triangular wall P is enforced. We employ the same transformation to evaluate the location of a given particle i with respect to the wall and we define a layer of thickness r _c above the wall (as for the force calculations) but with a clearance of size r _c beyond the triangle edges (Fig. 12b) to account for the possibility of particles that are slightly offset with triangle P but traveling towards it. Particle i is inside the region of possible reflection on triangular wall P if satisfies all of these conditions: $\bar{m}_{\xi }^{i} \ge - r_{\text{c}} /\left| {\mathbf{u}} \right|$, $\bar{m}_{\eta }^{i} \ge - r_{\text{c}} /\left| {\mathbf{v}} \right|$, $(1 - \bar{m}_{\xi }^{i} )/(1 + r_{c} /\left| {\mathbf{u}} \right|) \ge \bar{m}_{\eta }^{i} /(1 + r_{c} /\left| {\mathbf{v}} \right|)$, and $0 \le \bar{m}_{\zeta }^{i} \le 1$ (Fig. 12b). The location of particle i in the current time step ${\mathbf{r}}_{t + \Updelta t}^{i}$ and previous time step r ⁱ _t define the line of its trajectory, i.e., ${\mathbf{r}}_{t}^{i} + \lambda ({\mathbf{r}}_{t + \Updelta t}^{i} - {\mathbf{r}}_{t}^{i} )$ with $\lambda \in ] - \infty ,\infty [$. We identify the location in the trajectory of the intersection point between the line and the triangular wall plane by evaluating λ with

$$ \lambda = - \frac{{{\mathbf{n}} \cdot ({\mathbf{r}}_{t}^{i} - {\mathbf{P}}^{1} )}}{{{\mathbf{n}} \cdot ({\mathbf{r}}_{t + \Updelta t}^{i} - {\mathbf{r}}_{t}^{i} )}}, $$

(24)

which will translate into a valid intersection if 0 < λ < 1. Once λ is determined, the location of the intersection point is given by ${\mathbf{p}}^{i} = {\mathbf{r}}_{t}^{i} + \lambda ({\mathbf{r}}_{t + \Updelta t}^{i} - {\mathbf{r}}_{t}^{i} )$. The intersection point is not guaranteed to be in the triangular region, thus we employ the transformation once again to obtain ${\mathbf{m}}^{\text{p}} = {\mathbf{p}}^{i} - {\mathbf{P}}^{ 1}$, ${\bar{\mathbf{m}}}^{\text{p}} = {\mathbf{Mm}}^{\text{p}}$ and evaluate if the intersection point is indeed in the triangular wall, i.e., if and only if $\bar{m}_{\xi }^{\text{p}} \ge 0$, $\bar{m}_{\eta }^{\text{p}} \ge 0$, and $1 - \bar{m}_{\xi }^{\text{p}} \ge \bar{m}_{\eta }^{\text{p}}$ (note that in this case $\bar{m}_{\zeta }^{p} = 0$).

If so, the particle should be reflected, i.e., its reflected current position ${\tilde{\mathbf{r}}}_{t + \Updelta t}^{i}$ is updated by adding to ${\mathbf{r}}_{t + \Updelta t}^{i}$ twice the normal component of the trajectory beyond the wall (Fig. 12c),

$$ {\tilde{\mathbf{r}}}_{t + \Updelta t}^{i} = {\mathbf{r}}_{t + \Updelta t}^{i} + 2\left( {({\mathbf{r}}_{t + \Updelta t}^{i} - {\mathbf{p}}^{i} ) \cdot {\mathbf{n}}} \right){\mathbf{n}}, $$

(25)

and the velocity is also updated by subtracting twice the normal component (Fig. 12c),

$$ {\tilde{\mathbf{v}}}_{t + \Updelta t}^{i} = {\mathbf{v}}_{t + \Updelta t}^{i} - 2({\mathbf{v}}_{t + \Updelta t}^{i} \cdot {\mathbf{n}}){\mathbf{n}}. $$

(26)

Finally, because we consider walls composed of multiple adjacent triangles defining a concave surface, it is necessary to account for possible multiple reflections occurring in the same time step (Fig. 12d), i.e., a particle is reflected from one triangle in such a way that it should be reflected by adjacent triangles. We implement such multiple reflections for the cases on which the intersection point of the first reflection was located within one cut-off radius inside the triangle (Fig. 12b) determined by conditions $\bar{m}_{\xi }^{\text{p}} \le r_{\text{c}} /\left| {\mathbf{u}} \right|$, or $\bar{m}_{\eta }^{\text{p}} \le r_{\text{c}} /\left| {\mathbf{v}} \right|$, or $(1 - \bar{m}_{\xi }^{\text{p}} )/(1 - r_{\text{c}} /\left| {\mathbf{u}} \right|) \le \bar{m}_{\eta }^{\text{p}} /(1r_{\text{c}} /\left| {\mathbf{v}} \right|)$, each condition corresponding to one edge of the triangular wall. In such case, the previous position is updated with the intersection point, i.e., r ⁱ _t  = p ⁱ, and the same algorithm is performed with respect to the corresponding adjacent triangles.