Multiscale Particle-Based Modeling of Flowing Platelets in Blood Plasma Using Dissipative Particle Dynamics and Coarse Grained Molecular Dynamics

Abstract

We developed a multiscale particle-based model of platelets, to study the transport dynamics of shear stresses between the surrounding fluid and the platelet membrane. This model facilitates a more accurate prediction of the activation potential of platelets by viscous shear stresses—one of the major mechanisms leading to thrombus formation in cardiovascular diseases and in prosthetic cardiovascular devices. The interface of the model couples coarse-grained molecular dynamics (CGMD) with dissipative particle dynamics (DPD). The CGMD handles individual platelets while the DPD models the macroscopic transport of blood plasma in vessels. A hybrid force field is formulated for establishing a functional interface between the platelet membrane and the surrounding fluid, in which the microstructural changes of platelets may respond to the extracellular viscous shear stresses transferred to them. The interaction between the two systems preserves dynamic properties of the flowing platelets, such as the flipping motion. Using this multiscale particle-based approach, we have further studied the effects of the platelet elastic modulus by comparing the action of the flow-induced shear stresses on rigid and deformable platelet models. The results indicate that neglecting the platelet deformability may overestimate the stress on the platelet membrane, which in turn may lead to erroneous predictions of the platelet activation under viscous shear flow conditions. This particle-based fluid–structure interaction multiscale model offers for the first time a computationally feasible approach for simulating deformable platelets interacting with viscous blood flow, aimed at predicting flow induced platelet activation by using a highly resolved mapping of the stress distribution on the platelet membrane under dynamic flow conditions.

Appendix A: Spatiotemporal Analytics at Multiple Scales

Methods

The instantaneous properties of the platelet constituent particles are measured in simulations and they are averaged over a period of time (referred to as the time-averaging technique) and then over a surface volume (referred to as the space-averaging technique). The properties include the velocities and the stress tensors.

The temporal-averaging technique is to average a property ${\mathcal{A}}$ among successive observations over a finite time interval and it is designed for demonstrating a common trend in that interval. Suppose that ${\mathcal{A}}_{i} \left(0 \right)$ is the initial velocity of a particle i and $\left\{{{\mathcal{A}}_{i} \left({j \cdot \delta t} \right)} \right\}_{j = 0}^{\alpha - 1}$ is a collection of α successive points in time. i simply stands for an index of the particle. The time-averaged property ${\mathcal{A}}_{{i\,{\text{time}}}}$ of the particle i in this interval is

$${\mathcal{A}}_{{i\,{\text{time}}}} = \frac{1}{n}\mathop \sum \limits_{j = 0}^{\alpha - 1} {\mathcal{A}}_{i} \left({j \cdot \delta t} \right)$$

(10)

Here, δt is the length of time interval between two successive steps and it is always a multiple of the propagator in simulation. α is the number of instantaneous observations in the succession. Naturally, a larger α implies a more common trend in a longer time interval and the opposite extrema present a very instantaneous behavior.

The spatial-averaging technique is to average a property over a surface volume and it is specially designed for smoothing the spatial relevant fluctuations. Let the total membrane particles form a set of $\left\{{M_{k} \left({\varvec{r}_{k},{\mathcal{A}}_{k}} \right)} \right\}_{k = 1}^{K}$ where r _k and ${\mathcal{A}}_{k}$ are the position and property of a particle k. For convenience, we introduce the set notations: the δ-neighborhood of a point positioned at p is denoted as D(p,δ) where δ is a positive number. A point positioned at q is said within D(p ,δ) if $\parallel\varvec{p} - \varvec{q}\parallel_{2} \le \delta$ and denoted as $\varvec{q} \in D\left({\varvec{p},\delta} \right)$. The intersection D(p, δ) ∩ {M _k }is the set that contains all of the membrane particles that are also within the δ-neighborhood of point p. ϕ is an empty set. |A| is the number of elements in a set A.

The space-averaging technique involves two steps:

• Step 1: Generate a collection of lattices $\left\{{L_{i} \left({p_{i},{\bar{\mathcal{A}}}_{i}} \right)} \right\}_{i = 1}^{\beta}$ where $\varvec{p}_{i}$ and ${\bar{\mathcal{A}}}_{i}$ are the position and unknown property associated with lattice L _i . β = |L _i | is the total number of lattices and the bar stands for the spatial relevant averaging. The collection of lattices forms the coverage of the entire platelet surface if and only if it satisfies: for each $\varvec{r}_{k}$, there is a $D\left({\varvec{p}_{i},\delta} \right)$ such that $\varvec{r}_{k} \in D\left({\varvec{p}_{i},\delta} \right)$ and $D\left({\varvec{p}_{i},\delta} \right) \cap \left\{{r_{k}} \right\} \ne \phi$ for any i.

• Step 2: Calculate the property ${\mathcal{A}}_{i}$ for lattice L _i as

• $${\bar{\mathcal{A}}}_{i} = \frac{1}{{\omega_{i}}}\mathop \sum \limits_{{r_{k} \in D\left({p_{i},\delta} \right)}}^{{}} {\mathcal{A}}_{k}$$

  (11)

Here $\omega_{i} = \left| {D\left({\varvec{p}_{i},\delta} \right) \cap \left\{{\varvec{r}_{k}} \right\}} \right| \ge 1$ and δ depends on the average distance of the nearest lattices.

In total, the temporal and spatial averaging techniques form the multiscale analytic scheme and the attention is paid to the consistency between the observed property ${\mathcal{A}}_{obs}$ and the spatiotemporal-averaged property $\overline{{{\mathcal{A}}_{\text{time}}}}$.

$${\mathcal{A}}_{obs} \cong \overline{{{\mathcal{A}}_{\text{time}}}}$$

(12)

Configurations

We consider four coarsening configurations (Table A1) for the space-averaging technique. The figures are ordered from the finest to the coarsest grained levels.

Table A1 Different collections of lattices for the space-averaging technique

Results

Figure A1 shows the effects of temporal averaging on the platelet-flipping velocity magnitudes. In the legend, “nanosecond” means that we calculate the average magnitude of flipping velocities of the membranous particles every nanosecond (ns). Similarly, “10 ns” and “100 ns” mean the same calculations are performed every 10 and 100 ns, respectively. This figure illustrates the obvious disparity of the fluctuations of numeric values at varied temporal coarse-graining scales. For example, the instantaneous values at the very microscopic level showed the most flickering. Second, the microscopic values of the membrane system at varied temporal scales characterized the same macroscopic quality: the long-term trends. The spatiotemporal averaging technique is a useful method to convert microscopic information to macroscopic qualities for the multiscale modeling and demonstrate a clearer resolution for long-term observables

Figure A1

The effect of temporal averaging on the flipping velocity magnitude

Different combinations of temporal and spatial coarse graining parameters would represent the same metadata with multiple-resolutions. Figures A2 and A3 present the analytic results for ts = 60 and 240 (dimensionless time) respectively. These results re-verify that the instantaneous velocity profile fluctuates very rapidly thanks to the high-frequency oscillations of the massive membrane particles and the profile becomes smoother as the instantaneous velocities are processed by the time-averaging and space-averaging techniques. A rigid platelet model is an extreme case of the coarsening as shown in Fig. A4. We observe that the velocity profile of a deformable model more resembles that of a rigid model, as the temporal coarsening method continues to a larger temporal scale. For instance, both of the models show the consistent phenomena that the high-velocity areas occur merely on the flipping edges while the central stripe has a relatively small flipping rate. Additionally, the spatial coarsening method characterized the “flickering” membrane fluctuations in certain degree, without using the very many particles. For example in Figs. A2 and A3, β = 1350 (bottom right subplot) renders the similar profile as using β = 19,675 (top right subplot), in particular at α = 10,000 (last columns).

Figure A2

The velocity profiles of the case using different coarsening parameters (ts = 60)

Figure A3

The velocity profiles of the case using different coarsening parameters (ts = 240)

Figure A4

The velocity profiles of the rigid platelet model at (ts = 60) and (ts = 240)

Collectively, the results demonstrate: the rigid platelet model never can reflect the molecular-scale fluctuations of natural membranes owing to the complete stiffness. At the opposite extreme, an atomistic-scale simulation provides over detailed access to the spectrum of thermal fluctuations but its application demands the computing power. As a trade-off, the multiscale model does not only reflect the thermal fluctuations of the platelet particles but also maintains the computational feasibility.