Mathematical Models for Blood Coagulation

Computational fluid dynamic simulation of two-fluid non-Newtonian nanohemodynamics through a diseased artery with a stenosis and aneurysm

This article presents a two-dimensional theoretical study of hemodynamics through a diseased permeable artery with a mild stenosis and an aneurysm present. The effect of metallic nanoparticles on the blood flow is considered, motivated by drug delivery (pharmacology) applications. Two different models are adopted to mimic non-Newtonian characteristics of the blood flow; the Casson (viscoplastic) fluid model is deployed in the core region and the Sisko (viscoelastic) fluid model employed in the peripheral (porous) region. The revised Buongiorno two-component nanofluid model is utilized for nanoscale effects. The blood is considered to contain a homogenous suspension of nanoparticles. The governing equations are derived by extending the Navier-Stokes equations with linear Boussinesq approximation (which simulates both heat and mass transfer). Natural (free) double-diffusive convection is considered to simulate the dual influence of thermal and solutal buoyancy forces. The conservation equations are normalised by employing appropriate non-dimensional variables. The transformed equations are solved numerically using the finite element method with the variational formulation scheme available in the FreeFEM++ code. A comprehensive mesh-independence study is included. The effect of selected parameters (thermophoresis, Brownian motion, Grashof number, thermo-solutal buoyancy ratio, Sisko parameter ratio, and permeability parameter) on velocity, temperature, nanoparticle concentration, and hemodynamic pressure have been calculated for two clinically important cases of arteries with stenosis and an aneurysm. Skin-friction coefficient, Nusselt number, volumetric flow rate, and resistance impedance of blood flow are also computed. Colour contours and graphs are employed to visualize the simulated blood flow characteristics. It is observed that by increasing the thermal buoyancy parameter, i.e. Grashof number (Gr), the nanoparticle concentration and temperature decrease, whereas velocity increases with an increment in the Brownian motion parameter (Nb). Furthermore, velocity decreases in the peripheral porous region with elevation in the Sisko material ratio (m) and permeability parameter (k'). The simulations are relevant to transport phenomena in pharmacology and nano-drug targeted delivery in haematology.

Automated reduction of blood coagulation models

Mathematical modeling of thrombosis typically involves modeling the coagulation cascade. Models of coagulation generally involve the reaction kinetics for dozens of proteins. The resulting system of equations is difficult to parameterize, and its numerical solution is challenging when coupled to blood flow or other physics important to clotting. Prior research suggests that essential aspects of coagulation may be reproduced by simpler models. This evidence motivates a systematic approach to model reduction. We herein introduce an automated framework to generate reduced-order models of blood coagulation. The framework consists of nested optimizations, where an outer optimization selects the optimal species for the reduced-order model and an inner optimization selects the optimal reaction rates for the new coagulation network. The framework was tested on an established 34-species coagulation model to rigorously consider what level of model fidelity is necessary to capture essential coagulation dynamics. The results indicate that a nine-species reduced-order model is sufficient to reproduce the thrombin dynamics of the benchmark 34-species model for a range of tissue factor concentrations, including those not included in the optimization process. Further model reduction begins to compromise the ability to capture the thrombin generation process. The framework proposed herein enables automated development of reduced-order models of coagulation that maintain essential dynamics used to model thrombosis.

Virtual endovascular treatment of intracranial aneurysms: models and uncertainty

Virtual endovascular treatment models (VETMs) have been developed with the view to aid interventional neuroradiologists and neurosurgeons to pre-operatively analyze the comparative efficacy and safety of endovascular treatments for intracranial aneurysms. Based on the current state of VETMs in aneurysm rupture risk stratification and in patient-specific prediction of treatment outcomes, we argue there is a need to go beyond personalized biomechanical flow modeling assuming deterministic parameters and error-free measurements. The mechanobiological effects associated with blood clot formation are important factors in therapeutic decision making and models of post-treatment intra-aneurysmal biology and biochemistry should be linked to the purely hemodynamic models to improve the predictive power of current VETMs. The influence of model and parameter uncertainties associated to each component of a VETM is, where feasible, quantified via a random-effects meta-analysis of the literature. This allows estimating the pooled effect size of these uncertainties on aneurysmal wall shear stress. From such meta-analyses, two main sources of uncertainty emerge where research efforts have so far been limited: (1) vascular wall distensibility, and (2) intra/intersubject systemic flow variations. In the future, we suggest that current deterministic computational simulations need to be extended with strategies for uncertainty mitigation, uncertainty exploration, and sensitivity reduction techniques. WIREs Syst Biol Med 2017, 9:e1385. doi: 10.1002/wsbm.1385 For further resources related to this article, please visit the WIREs website.