Keywords: Adaptive time stepping, Artificial intelligence, Multiscale modeling, Platelet dynamics

3.1. Overview

The current study of fluid-platelet interactions follows our prior MSM framework [3–5,8–10] by incorporating the DPD method for the macroscopic transport of blood plasma in the vessel, governed by

where${\mathbf{F}}_{ij}^C$,${\mathbf{F}}_{ij}^D$,${\mathbf{F}}_{ij}^R$ are the conservative, dissipative, random forces acting on particle$i$ by particle$j$ respectively, and${\mathit{F}}^E$ are external forces exerted on each particle [27].$r_{ij}=\left|{\mathbf{r}}_{ij}\right|=\left|{\mathbf{r}}_i-{\mathbf{r}}_j\right|$ is the inter-particle distance,${\mathbf{v}}_{ij}={\mathbf{v}}_i-{\mathbf{v}}_j$ is the relative velocity,${\mathbf{e}}_{ij}$ is the unit vector in the direction${\mathit{r}}_{ij}$,${\zeta }_{ij}$ is a random variable following normal distribution, and parameters$\alpha$,$\gamma$,$\sigma$ control conservative, dissipative and random forces strength [27]. Español and Warren [28] introduced${\omega }^D\left(r_{ij}\right)={\left[{\omega }^R\left(r_{ij}\right)\right]}^2$ and${\sigma }^2=2\gamma k_{B}T$, where$k_B$ is the Boltzmann constant and$T$ is the temperature of the system.

The classical DPD scheme, favored for its simple formulation, suffers from limitations in the thermodynamics behavior, too simplistic friction forces, lack of scaling, and sustaining temperature gradient [29]. The many-body DPD (MDPD) method [30] relaxes some of these limitations in allowing general equations of state by modulating the thermodynamics behavior at the interaction level between particles. The energy-conserving DPD (EDPD) model [31] extends the DPD model to non-isothermal situations for sustaining temperature gradients. The fluid particle model (FPM) [32] addresses the simplistic friction issue by incorporating shear forces between dissipative particles. By employing discretization of the Navier-Strokes equations in a Lagrangian moving Voronoi gird and thermal fluctuations, the smoothed DPD (SDPD) model [33] encapsulates all benefits of the three models discussed above.

The CGMD method for the dynamics of individual platelet in response to extracellular hydrodynamic stresses governed by

The last two terms are bonded interactions in which$k_b$ and$k_{\theta }$ are the force constants and$r_0$ and${\theta }_0$ are the equilibrium distance and angle, existing in the membrane and cytoskeleton. The Lennard-Jones (LJ) potential describes nonbonded interactions between any particle pairs within a cutoff range defined as

where$\epsilon$ is the depth of the potential well and$\sigma$ indicates the distance at which the inter-particle potential is zero. The disparate spatial scales between DPD and CGMD are handled by a hybrid force field [5] which establishes a functional spatial interface between the platelet membrane and the surrounding viscous fluid, governed by

The platelet moves continuously according to the fluid-induced instantaneous stresses and, conversely, the flow’s state varies by the platelet’s motion.

In this MSM system, the flow-driven dynamics of individual platelet whose motion can be modeled as a rigid body in circumstances like flipping platelets in the free flow [5] and marginated platelets before adhesion to blood vessel walls [8], described as

where$M$ is the mass,${\mathbf{V}}_{\text{com}}$ is the center-of-mass (COM) velocity,$\mathbf{\omega }=\left[{\omega }_x, {\omega }_y, {\omega }_z\right]$ is the angular velocity,$\mathbf{I}$ is the moment of inertia, of the platelet. Among them, we choose$\mathbf{\omega }$ and rotational and translational kinetic energies,$E_{ro}$ and$E_{tr}$, to demonstrate the platelet’s rotation and translation. The fluid state influencing the platelet is measured by the surrounding flow rate${\mathbf{v}}_f$, the flow speed at the platelet’s COM. As the platelets flip slow (fast), the system evolves at large (small)$\text{Δ}t$, adaptively. Algorithmically, the$\text{Δ}t$ should change along with$\omega$ to facilitate capture of adequate dynamics at a low cost.

3.2. Data-Driven Time Stepping

Time stepping algorithms use time discretization to integrate the governing equations with a timestep size$\text{Δ}t$ such as the scheme of the velocity Verlet algorithm [6]

With initial conditions$\left\{\mathbf{x}\left(0\right),\mathbf{v}\left(0\right)\right\}$, the simulator iterates to time$t$ by computing the velocity at a half step and the position at full step, and then completing the calculation of the second half step velocity. In STS, all particles share a constant$\text{Δ}t$ throughout the entire simulation, a conservative approach allowing capture of the fastest motion. In MTS, the system is decomposed into sub-systems according to spatial scales of system constituents or interaction force types. Then, small (large)$\text{Δ}t$’s are prescribed for sub-systems at bottom (top) scales. We consider finding$\text{Δ}t$ as a function that maps system states to$\text{Δ}t$’s as$f: \text{Φ}\left(\text{t}\right)\to \mathrm{\Delta }t$ subject to particles’ physical states. STS maps everything to a fixed conservative$\text{Δ}{t}_{\text{min}}$ as$f_{\text{STS}}: \text{Φ}\left(t\right)\to \mathrm{\Delta }{t}_{\text{min}}$ regardless of the discrepancy between the system time scales. Comparing to STS, MTS considers multiple spatial scales and the mapping function follows$f_{\text{MTS}}: \text{Φ}\left(t\right)\to \mathrm{\Delta }\mathbf{t}$, where$\text{Δ}\mathit{t}$ is a vector of characteristic$\text{Δ}t$’s for component sub-systems [6,7]. To further improve simulation efficiency, a regulation on$\text{Δ}t$ for the time integrator is derived to accommodate the constantly varying system dynamics.

The system state evolves with particles’ dynamics and the$\text{Δ}t$’s should synchronize itself with the underlying dynamics by mapping states to a variable$\text{Δ}t$ as$f_{\text{ATS}}: \text{Φ}\left(t\right)\to \mathrm{\Delta }t\left(t\right)$, where$\mathrm{\Delta }t\left(t\right)$ is an adaption at time$t$ bounded by the$\text{Δ}{t}_{\text{min}}$ at bottom scales and$\text{Δ}{t}_{\text{max}}$ at top scales. Intuitively, the adaptive$\text{Δ}t$ can be computed by following the particle motion within a safe range. A data-driven ATS detects system dynamics and adjusts$\text{Δ}t$ automatically. At the end of each integration step, ATS reassesses the most recent system states and alters$\text{Δ}t$ to match the states when necessary.

We design the workflow for ATS to handle large streaming data generated during the simulation. ATS maintains the velocity Verlet algorithm as demonstrated inEq. (7). Next, instantaneous platelet and fluid properties are calculated and collected as raw data frequently from the simulator. Parallel to the simulator, ATS analyzes the raw data to predict an optimal$\text{Δ}t$ for the near-future integrations. As a feedback, the ATS outputs are transferred back to the simulator to update the internal$\text{Δ}t$, and the simulation moves on with the new$\text{Δ}t$. Then, ATS waits for the time trigger for the next inference by repeating the same procedure above. Apparently, the frequency of ATS arbitration adapts to the state changes for an optimal examination of the system. For this optimization, a variable, step jump$n_r$, is introduced to indicate the number of consecutive integration steps that can be skipped without computing$\text{Δ}t$. As depicted inFig. 1, ATS produces a pair of$\left(\text{Δ}t,n_r\right)$, where$n_r$ is a feedback to updating the dynamic trigger of invoking ATS.

Because of the high-frequency oscillation, the dynamics requires pre-processing to better monitor the core motion. A denoising technique compresses the high-frequency signals along the time in each measurement separately as the first step in ATS. A long-term analysis is ideal to capture the present state and predict the trend. Therefore, we formulate denoised data points in a certain period in time series format to maintain the continuity of the data, and one sequence of each measurement is treated as an entry. To study the pattern in the large data sets, we integrate AI techniques with the data-driven ATS to predict$\text{Δ}t$ and$n_r$.

3.3. Learning Architecture

AI-ATS aims to predict the next$\text{Δ}t$, from the historical system states, by a deep learning framework that integrates RNN-based AEs and FCNs. The framework (Fig. 1) is composed of three components: 2-stage denoising by moving average (MA) and wavelet transform (WT) filters to clean high-frequency noise in raw data, two RNN-based AEs to compress and extract latent features, and two FCNs to predict$\text{Δ}t$ and$n_r$.

The platelet, modeled from the finest molecular scales, allows dynamic interactions with the surrounding flow and, for the first time, this study depicts a high-fidelity flow-induced stress mapping as the basis for accurate prediction of platelet activation and aggregation [8,10,34]. On the other hand, this made it challenging to extract the long-term patterns of platelet’s motion mixed by intense high-frequency oscillations on the membranes. Algorithmically in MSM, a platelet object is described as a collection of in-homogenous particles (membrane, receptors, cytoplasm, and cytoskeleton) interacted by a variety of governing functions. Among these massive particles’ interactions, our special interest is the large-scale motion and energy of the platelet subsystem under external forces. As an example,Fig. 1 depicts a snapshot of the raw data for the platelet rotational energy$E_{ro}$ at which the features of interest are buried with the noisy raw data. To capture physics-informed features of a platelet dynamics in MSM (Sec. 3.1), we develop a 2-stage data denoising approach. Initially, the raw data is sampled directly from the MSM simulator with the same frequency as$\text{Δ}t$ and stored as a time series. Stage-1: a MA filter is used for regulating time series of sampled raw data. It takes the samples of a time window$\left[t-w_{\text{ma}},t\right]$ of a fixed size$w_{\text{ma}}$ and outputs the time-averaged values at the time point$t$. The stride of MA is fixed as$t_{\text{stride}}$. After being processed by the MA filter, the unevenly sampled raw data are converted to an evenly spaced and filtered time series. Stage-2: a WT filter, when applied to the Stage-1 outputs for the noise removal, decomposes the input time series as the low and high-frequency components by [35]

where

The$s\left(t\right)$ is the input data,$A_{J,k}$ and$D_{j,k}$ are the approximate and detail coefficients representing the orthogonal relationship between wavelet at resolution level$j$ and$s\left(t\right)$, in which$0<j\le J$ and$0\le k<2^j$. The smoothed approximation is described by a function${\varphi }_{J,k}$, translating and dilating of a scaling function$\varphi$, and the fine-scale details are generated by an orthonormal basis function${\psi }_{j,k}$, scaling of a mother wavelet$\psi$ [35]. By cleaning coefficients with thresholds at different resolutions, a smooth estimated signal is reconstructed as inverting the transform. The hard and soft thresholds are adopted, governed by operators [36]

where the function$\text{sgn}$ extracts the signum,${\lambda }_j$ is the threshold at resolution level$j$, for which we used a universal threshold$\lambda =\eta \sqrt{2\cdot \ln N_s}$ [37] where$N_s$ is the length of the signal, and$\mathbb{I}$ is an indicator function. The noise level is estimated from the finest scale wavelet coefficient defined as$\eta =\text{median}\left(\left|D_1\right|\right)/0.6745$. By examining the snapshots showing$E_{ro}$ passes through MA and WT filters inFig. 1, we find that the high-frequency oscillation movements are sufficiently suppressed and meanwhile informative dynamics are attained for being processed by the Autoencoders (AEs) to extract physics-informed learnable features.

The AEs transform these large amounts of filtered high-dimensional inputs further into a coded representation, a smaller amount of lower-dimensional latent features which, by design, preserve sufficient properties of original inputs. To achieve this, we employed RNN cells in AEs to process those time series inputs as RNNs “memorize” the relevant long-term information inside their hidden states throughout the processing of the time series. In this study, we used two state-of-the-art RNNs: Long Short-Term Memory (LSTM) [38] and Gated Recurrent Unit (GRU) [39] (Fig. 1). LSTM is developed to regulate memories by mechanisms called gates to better analyze long sequences obeying

where the subscript$t$ indicates the time step,$\sigma$ is the sigmoid function, and$\mathcal{F}$ is the customized activation. The cell state$C_t$, acting as the internal memory of the network, and the hidden state$h_t$ outputs the current state of the cell. GRU is a variant of LSTM and governed by

where the hidden state$h_t$ acts as the internal memory. Update gate$z_t$ and reset gate$r_t$ are similar to$i_t$ and$f_t$ in the LSTM cell. We construct two AEs by LSTM and GRU respectively, and the training processes are discussed inSec. 3.4. The resultant latent features obtained from AEs are inputs of following FCNs.

In the last step, two FCNs using perceptron neurons are constructed for predicting$\mathrm{\Delta }t$ and$n_r$ independently (Fig. 1). The input vector for both FCNs consists of latent features and most recently used$\mathrm{\Delta }t$’s. In$\mathrm{\Delta }t$ prediction, a discrete value is outputted as a candidate of adaptive$\mathrm{\Delta }t$. In$n_r$ prediction, the integer used to measure the duration before the next inference, we select among a group of predefined categorical values by their predicted probability and this predication is a classification problem. At a time, a pair of$\mathrm{\Delta }t$ and$n_r$ are provided as feedbacks to the simulator and the AI-ATS program: the$\mathrm{\Delta }t$ is used in the integrator of the simulator and the$n_r$ is used by the AI-ATS scheme as a time trigger for the next inference.

3.4. Training Method

Training samples cover the duration of the platelet rotation of π in each of four typical motions, to be discussed inSec. 4.1, of platelets rotating and translating under shear blood flow. In the training and inferencing, we follow the same dimensionless unit system as in MSM. The conversion between the dimensionless unit and the SI unit can be found inTable 2 of [8]. During the data production process, six physics measurements (Sec. 3.1) are recorded as time series. In the MA filter, the sampling step size follows$\mathrm{\Delta }t$, and the moving window size$w_{ma}$ and the moving stride$t_{\text{stride}}$ are both 0.02. The$w_{\text{ma}}$ in each time series covers 100–170 consecutive data points. One training sample is produced by averaging the time series within$w_{\text{ma}}$ for all six measurements. In this study, we collected 400 samples for each of these four motions, resulting in a total of 1,600 samples prepared in the training database. In the training processes, 90% of these samples are used for training while the remaining 10% for validation.

The labeling process of finding the optimal$\mathrm{\Delta }t$ and$n_r$ is based on a trial-and-error approach to determine the most aggressive$\mathrm{\Delta }t$ and$n_r$ values that allow platelet-fluid dynamic interactions in our MSM. The search space for$\mathrm{\Delta }t$ and$n_r$ are based on our prior knowledge as in [5,6]. At the trial, possible values are examined for performance tests; at the error, particle-based dynamics breaks the constraints discussed inSec. 3.2, determining the$\mathrm{\Delta }t$ and$n_r$.

In the training processes, AEs and FCNs are trained iteratively: the training of AEs is performed first for clearer physics-informed latent features, followed by the training of FCNs for the better approximation to labels. In the implementation, an AE is constructed as a 2-layer encoder of 16 and 4 units respectively and a 1-layer decoder of 16 units. Two types of AEs are implemented using LSTM and GRU cells (Fig. 1). In the LSTM-AE, mish and tanh are selected as the activation functions$\mathcal{F}$ in the encoder and mish in the decoder, respectively. In the GRU-AE, the$\mathcal{F}$’s in the first and last layers are replaced by swish. Activation functions used in the model are illustrated inAppendix A. Each training sample is formatted as a 2-D matrix of dimensions 50×6 (steps×features). The mean square error (MSE) is employed as the training loss function.

Fig. 2 illustrates the performance of feature engineering processes by showing the linear correlation among the raw data, filtered data, latent features, and the target$\mathrm{\Delta }t$’s. When correlation is less than 0.3, most of the raw data cease to impact$\mathrm{\Delta }t$ because particles’ high-frequency vibrations are mistreated in the presence of the generation motions. The data processed by the MA and WT filters exploit the hidden behaviors of measurements to amplify correlation. For example, those rotation-relevant measurements achieved more than 0.5. RNN-based AEs map data into a latent space where each projection combines partial original measurements and forms the most representative new features, boosting up the correlation to above 0.8. The correlation enhancement, aided by the feature engineering, boosts the ultimate determination of$\mathrm{\Delta }t$. The scatter plot (b) inFig. 3 uses the raw data$E_{ro}$ (and the filtered${\tilde{E}}_{ro}$ and the latent feature$x_1$) as an example to show a gradual improvement of the feature engineering performance. It is intuitively evident that our data denoising and feature extracting methods can effectively extract physics-informed features out of the noisy raw data with high-frequency motions.

Regarding FCNs, the input vector is composed of eight latent features obtained from AEs and five historical$\mathrm{\Delta }t$’s evenly selected from the most recent time window of 1 with interval 0.2. In the regression FCN for$\mathrm{\Delta }t$ prediction, we construct a 4-layer NN of {32, 16, 8, 1} neurons in each layer respectively. Rectified linear units (ReLU) (All activation functions used in the NN model are depicted inFig. A.1.) are used as activation functions for all neurons, and the MSE is the loss function. We introduce a threshold$\kappa$ to restrict the regulation strength on$\mathrm{\Delta }t$ to prevent sharp changes by

where$\delta t$ controls the maximum deviation in one step, and the indicator function$\mathbb{I}=1$ as the deviation between currently used$\mathrm{\Delta }t$ and predicted$\text{Δ}{t}^{\left(\text{pred}\right)}$ reaches a threshold$\kappa \cdot \delta t$. In the classification FCN for$n_r$ prediction, the same NN architecture is applied in which the activation functions are swish and the last layer is replaced by a softmax layer. The target categories of$n_r$ are prescribed as {1,2,4} where each category represents a time duration of$0.04n_r$. For training purposes, target categories are transformed by one-hot encoding, using a binary value to indicate the presence of a sample at each category. The categorical cross-entropy is used as the loss function, defined as$L=-{\sum }_{i=1}^C{y}_i\log \left(p_i\right)$, where$C$ is the number of categories,$y_i$ is the ground truth (0 or 1), and$p_i$ is the predicted probability falls into the category$i$.

The inference model is trained before deploying to test cases. In performance testing, the pre-trained model is used to collaborate with the simulator for$\mathrm{\Delta }t$ determination. Relative to the underlying MSM time, the inferencing time is insignificantly small. The performance gain of AI-ATS highly depends on dynamics being modeled. Our current efforts focus on proving feasibility while showing the performance for a moderately steady motion

3.5. Inferencing Workflow

The inferencing method integrates the AI-ATS with the MSM simulator. The MSM simulator carries out time integration and passes simulation data to the AI-ATS program who analyzes data and predicts$\mathrm{\Delta }t$ and$n_r$ for the simulator. The inferencing workflow is depicted inFig. 3. In the MSM simulator, per-particle properties$\mathbf{x}$,$\mathbf{v}$, and$\mathbf{f}$ are updated in each time step following the velocity Verlet algorithm asEq. (7). Then, platelet$E_{ro}$ and$E_{tr}$ at the current time step are calculated based on in-memory per-particle data. Together with$\mathbf{\omega }$ and${\mathbf{v}}_f$, totally six physics measurements showing platelet and flow instantaneous states are exported from the simulator to the external datastore. Once the AI-ATS program is invoked by the trigger emitted from the simulator, it begins to import MSM raw data from the datastore and processes them through the pipeline consisting of 2-stage denoising filters, AEs, and FCNs. The inference outputs,$\text{Δ}{t}^{\left(\text{new}\right)}$ and$n_r^{\left(\text{new}\right)}$, are transferred back to the MSM simulator through the other external datastore. Meanwhile, the simulator waits for the feedback from the AI-ATS program to update its internal$\mathrm{\Delta }t$ and$n_r$. After synchronizing parameters by the end of step, all simulator processors continue to the next step.

Algorithm 1 illustrates the pseudocode of the MSM simulator. The velocity Verlet algorithm is implemented in Lines 2–5, followed by the computation and exportation of six physics measurements in Lines 6–7. The timer$l$ records simulated time since last inference on$\mathrm{\Delta }t$ and$n_r$. Each time$l$ reaches$n_r$, the simulator triggers the AI-ATS program. As illustrated in Algorithm 2, the AI-ATS program starts with importing raw MSM data into$X$ in time series format. The Stage-1 denoising, the MA filter, processes the data with window size$w_{\text{ma}}$ and moving stride$t_{\text{stride}}$ and assembles evenly spaced results in$\overline{X}$; the Stage-2 denoising, the WT filter, outputs denoised time series in$\tilde{X}$. The following AEs compress$\tilde{X}$ and extract eight physics-informed latent features$X_L$. Then, both regression and classification FCNs take$X_L$ and historical$\text{Δ}{\mathit{t}}_{\text{hist}}$ as inputs and give the inference on$\text{Δ}{t}^{\left(\text{new}\right)}$ and$n_r^{\left(\text{new}\right)}$ simultaneously. A physics constraint as inEq. (13) is applied to the final examination on$\text{Δ}{t}^{\left(\text{new}\right)}$. The AI-ATS program is concluded with exporting outputs to the other external datastore. The MSM simulator imports$\text{Δ}{t}^{\left(\text{new}\right)}$ and$n_r^{\left(\text{new}\right)}$ and updates its internal parameters accordingly. At the end of each time step, parameters are broadcasted to all processors to synchronize the status. We set the MA window size$w_{\text{ma}}=0.02$, the moving stride$t_{\text{stride}}=0.02$, and the maximum allowance of$\mathrm{\Delta }t$ variation$\delta t=2\times {10}^{-5}$. The threshold for$\mathrm{\Delta }t$ variation is$\kappa =0.6$.

The inference model is trained before deploying to test cases. In performance testing, the pre-trained model is used to collaborate with the simulator for$\mathrm{\Delta }t$ determination. Relative to the underlying MSM time, the inferencing time is insignificantly small. The performance gain of AI-ATS highly depends on dynamics being modeled. Our current efforts focus on proving feasibility while showing the performance for moderately steady motion.

4.1. Simulations

In the MSM, we study platelet dynamics in a variety of shear blood flow. A microchannel is simulated by a rectangular region with dimensions of 16×16×8$\left(x\times y\times z\right)$ in$\text{μm}$ as displayed inFig. 4. The periodic boundary condition is applied along$x$- and$z$-axes. The blood vessel walls are modeled at top and bottom boundaries in$y$-axis and no-slip boundary condition [40] is applied. The entire system incorporated 1,091,360 fluid particles and 140,303 platelet particles. The shear blood flow is modeled by counter Couette flow by driving top and bottom walls in opposite directions along$x$-axis. We consider two cases of platelet initial positions, near the central line–rotation motion (R) is dominant–and near the blood vessel–mixture of rotation and translation (RT). With different moving velocities of walls, flow conditions of shear stresses {30, 40, 50} in dyne/cm² are applied to the system. A total of 18 simulations (Table 1) were conducted to cover three different algorithms, two platelet motion types, and three shear stresses. The same force field parameters in [8] are used in this work. Simulations under shear stresses 30 and 50 in dyne/cm² are used for training purposes. Adaptive$\mathrm{\Delta }t$’s are selected from four empirical values of {416, 624, 832, 936} in ps. The instantaneous physics measurements are collected at the 40-ns basis from the simulator. With the trained model, AI-ATS is benchmarked in simulations under shear stresses 40 dyne/cm² against a reference system of using STS at the smallest$\mathrm{\Delta }t=416$ ps. All numerical simulations are implemented by modified LAMMPS package [41] on the IBM’s WSC Cluster consisting of IBM AC-922 nodes.

4.2. Analysis

We performed simulations of platelet rotation in R.40 and RT.40 by STS, ATS, and AI-ATS. Performance metrics are defined to quantize measurement precisions. Accuracy is investigated in terms of platelet dynamics and mechanics. Speed is analyzed in terms of the variation of$\mathrm{\Delta }t$’s and simulation steps.

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