Continuous Modeling of Arterial Platelet Thrombus Formation Using a Spatial Adsorption Equation

In this study, we considered a continuous model of platelet thrombus growth in an arteriole. A special model describing the adhesion of platelets in terms of their concentration was derived. The applications of the derived model are not restricted to only describing arterial platelet thrombus formation; the model can also be applied to other similar adhesion processes. The model reproduces an auto-wave solution in the one-dimensional case; in the two-dimensional case, in which the surrounding flow is taken into account, the typical torch- like thrombus is reproduced. The thrombus shape and the growth velocity are determined by the model parameters. We demonstrate that the model captures the main properties of the thrombus growth behavior and provides us a better understanding of which mechanisms are important in the mechanical nature of the arterial thrombus growth

Overcoming conventional modeling limitations using image- driven lattice-boltzmann method simulations for biophysical applications

The challenges involved in modeling biological systems are significant and push the boundaries of conventional modeling. This is because biological systems are distinctly complex, and their emergent properties are results of the interplay of numerous components/processes. Unfortunately, conventional modeling approaches are often limited by their inability to capture all these complexities. By using in vivo data derived from biomedical imaging, image-based modeling is able to overcome this limitation. In this work, a combination of imaging data with the Lattice-Boltzmann Method for computational fluid dynamics (CFD) is applied to tissue engineering and thrombogenesis. Using this approach, some of the unanswered questions in both application areas are resolved. In the first application, numerical differences between two types of boundary conditions: “wall boundary condition” (WBC) and “periodic boundary condition” (PBC), which are commonly utilized for approximating shear stresses in tissue engineering scaffold simulations is investigated. Surface stresses in 3D scaffold reconstructions, obtained from high resolution microcomputed tomography images are calculated for both boundary condition types and compared with the actual whole scaffold values via image-based CFD simulations. It is found that, both boundary conditions follow the same spatial surface stress patterns as the whole scaffold simulations. However, they under-predict the absolute stress values approximately by a factor of two. Moreover, it is found that the error grows with higher scaffold porosity. Additionally, it is found that the PBC always resulted in a lower error than the WBC. In a second tissue engineering study, the dependence of culture time on the distribution and magnitude of fluid shear in tissue scaffolds cultured under flow perfusion is investigated. In the study, constructs are destructively evaluated with assays for cellularity and calcium deposition, imaged using µCT and reconstructed for CFD simulations. It is found that both the shear stress distributions within scaffolds consistently increase with culture time and correlate with increasing levels of mineralized tissues within the scaffold constructs as seen in calcium deposition data and µCT reconstructions. In the thrombogenesis application, detailed analysis of time lapse microscopy images showing yielding of thrombi in live mouse microvasculature is performed. Using these images, image-based CFD modeling is performed to calculate the fluid-induced shear stresses imposed on the thrombi’s surfaces by the surrounding blood flow. From the results, estimates of the yield stress (A critical parameter for quantifying the extent to which thrombi material can resist deformation and breakage) are obtained for different blood vessels. Further, it is shown that the yielding observed in thrombi occurs mostly in the outer shell region while the inner core remains intact. This suggests that the core material is different from the shell. To that end, we propose an alternative mechanism of thrombogenesis which could help explain this difference. Overall, the findings from this work reveal that image-based modeling is a versatile approach which can be applied to different biomedical application areas while overcoming the difficulties associated with conventional modeling

Application of smooth particle hydrodynamics method for modelling blood flow with thrombus formation

Thrombosis plays a crucial role in atherosclerosis or in haemostasis when a blood vessel is injured. This article focuses on using a meshless particle-based Lagrangian numerical technique, the smoothed particles hydrodynamic (SPH) method, to study the flow behaviour of blood and to explore the flow parameters that induce formation of a thrombus in a blood vessel. Due to its simplicity and effectiveness, the SPH method is employed here to simulate the process of thrombogenesis and to study the effect of various blood flow parameters. In the present SPH simulation, blood is modelled by two sets of particles that have the characteristics of plasma and of platelet, respectively. To simulate coagulation of platelets which leads to a thrombus, the so-called adhesion and aggregation mechanisms of the platelets during this process are modelled by an inter-particle force model. The transport of platelets in the flowing blood, platelet adhesion and aggregation processes are coupled with viscous blood flow for various low Reynolds number scenarios. The numerical results are compared with the experimental observations and a good agreement is found between the simulated and experimental results

Smoothed Particle Hydrodynamics for Computational Fluid Dynamics

Smoothed particle hydrodynamics (SPH) is a simple and effective numerical method that can be used to solve a variety of challenging problems in computational mechanics. It is a Lagrangian mesh-free method ideal for solving deformation problems. In the SPH method, the state of a system is represented by a set of particles, which possesses individual material properties and interact with each other within a specific range defined as a support domain by a weight function or smoothing function. SPH features flexibility in handling complex flow fields and in including physical effects. In theory, the basic concept of the SPH method is introduced in this paper. Some detailed numerical aspects are discussed including the kernel approximation in continuous form and particle approximation in discrete form, the properties for the smoothing functions and some of the most frequently used ones in the SPH literature, the concept of support and interface domain, SPH formulations for Navier-Stokes equation, time integration, boundary treatment, particle interaction, artificial viscosity, laminar viscosity, shifting algorithm, and so on. In applications, this paper presents an improved SPH method for modeling the diffusion process of a microneedle and using smoothed particle hydrodynamics (SPH) method to simulate the 25% cross-section stenosis blood vessel model and the 75% crosssection stenosis blood vessel model. The obtained numerical results are in close agreement with available theoretical and experimental results in the literature. As an emerging transdermal drug delivery device, microneedles demonstrate some superior potential and advantages over traditional metallic needles-on-syringes in skin injection and vaccine [1]. However, very few research papers are available. This project uses a high order continuous method, the spectral element method (SEM), and a low order discrete method, the Smoothed Particle Hydrodynamics (SPH), to investigate this new drug delivery system. The incompressible Navier-Stokes equations were solved with SEM under appropriate initial and slip boundary conditions for the transport of medicine inside microneedles of rectangular and circular cross-sections. In addition, Darcy-Brinkman equations and a concentration equation were solved with SEM under appropriate initial and boundary conditions for the infiltration of medicine solution through porous media of the dermis tissue once a microneedle enters the skin. Meanwhile, the Lagrangian form of the Navier-Stokes equations were solved with the weighted interpolation approach via numerical integrations without inverting any matrices. Results from the mesh-based SEM and the mesh-free SPH simulations revealed technical details about the processes of delivery of medicine particles through microneedles and diffusion in the skin tissue, and the medicine concentration changes with space and time. The overall effect of medicine delivery under initial concentration and conditions were simulated and the effect of drug delivery were assessed. The formation of thrombus is a complicated process. The existing literature rarely has a model for high-fidelity simulation of the effects and hazards of blood clots on blood flow. In this model, high-fidelity simulations are performed for complex human internal environments. The result of this simulation indicates high pressure area in blood vessel wall which matches the real condition of the vessel experiment

The role of platelets in blood coagulation during thrombus formation in flow

Hemostatic plug covering the injury site (or a thrombus in the pathological case) is formed due to the complex interaction of aggregating platelets with biochemical reactions in plasma that participate in blood coagulation. The mechanisms that control clot growth and which lead to growth arrest are not yet completely understood. We model them with numerical simulations based on a hybrid DPD-PDE model. Dissipative particle dynamics (DPD) is used to model plasma flow with platelets while fibrin concentration is described by a simplified reaction-diffusion-convection equation. The model takes into account consecutive stages of clot growth. First, a platelet is weakly connected to the clot and after some time this connection becomes stronger due to other surface receptors involved in platelet adhesion. At the same time, the fibrin network is formed inside the clot. This becomes possible because flow does not penetrate the clot and cannot wash out the reactants participating in blood coagulation. Platelets covered by the fibrin network cannot attach new platelets. Modelling shows that the growth of a hemostatic plug can stop as a result of its exterior part being removed by the flow thus exposing its non-adhesive core to the flow

Hypoxic Cell Waves around Necrotic Cores in Glioblastoma: A Biomathematical Model and its Therapeutic Implications

Glioblastoma is a rapidly evolving high-grade astrocytoma that is distinguished pathologically from lower grade gliomas by the presence of necrosis and microvascular hiperplasia. Necrotic areas are typically surrounded by hypercellular regions known as "pseudopalisades" originated by local tumor vessel occlusions that induce collective cellular migration events. This leads to the formation of waves of tumor cells actively migrating away from central hypoxia. We present a mathematical model that incorporates the interplay among two tumor cell phenotypes, a necrotic core and the oxygen distribution. Our simulations reveal the formation of a traveling wave of tumor cells that reproduces the observed histologic patterns of pseudopalisades. Additional simulations of the model equations show that preventing the collapse of tumor microvessels leads to slower glioma invasion, a fact that might be exploited for therapeutic purposes.Comment: 29 pages, 9 figure

von Willebrand Factor unfolding mediates platelet deposition in a model of high-shear thrombosis

Thrombosis under high-shear conditions is mediated by the mechanosensitive blood glycoprotein von Willebrand Factor (vWF). vWF unfolds in response to strong flow gradients and facilitates rapid recruitment of platelets in flowing blood. While the thrombogenic effect of vWF is well recognized, its conformational response in complex flows has largely been omitted from numerical models of thrombosis. We recently presented a continuum model for the unfolding of vWF, where we represented vWF transport and its flow-induced conformational change using convection-diffusion-reaction equations. Here, we incorporate the vWF component into our multi-constituent model of thrombosis, where the local concentration of stretched vWF amplifies the deposition rate of free-flowing platelets and reduces the shear cleaning of deposited platelets. We validate the model using three benchmarks: in vitro model of atherothrombosis, a stagnation point flow, and the PFA-100, a clinical blood test commonly used for screening for von Willebrand Disease (vWD). The simulations reproduced the key aspects of vWF-mediated thrombosis observed in these experiments, such as the thrombus location, thrombus growth dynamics, and the effect of blocking platelet-vWF interactions. The PFA-100 simulations closely matched the reported occlusion times for normal blood and several hemostatic deficiencies, namely, thrombocytopenia, vWD Type 1, and vWD Type 3. Overall, the multi-constituent model of thrombosis presented in this work enables macro-scale 3-D simulations of thrombus formation in complex geometries over a wide range of shear rates and accounts for qualitative and quantitative hemostatic deficiencies in patient blood. The results also demonstrate the utility of the continuum model of vWF unfolding that could be adapted to other numerical models of thrombosis