Research

The research conducted in our group is focused on using mathematics to gain insight into the complex biological world around us. It involves developing mathematical models of a biological systems, solving the equations of those models, analyzing the results, and translating the results into the appropriate scientific context. The two main biological systems/areas that we focus on are blood coagulation and the fluid-structure interaction between microorganisms and their fluid environment.

Mathematical modeling of blood clotting and bleeding

Our group develops spatial-temporal mathematical models to study the influence of biochemical and biophysical mechanisms on blood coagulation, clot formation, and bleeding. Our mathematical models have suggested i) clot porosity strongly impacts clot growth and structure, ii) hindered transport of proteins within the clot reduces thrombin production and thus slows clot growth, and iii) clots have a dense core and a less dense periphery similar to those described in animal models. We work closely with experimental collaborators that develop microfluidic models that are physical analogs of our our mathematical models. We work on models of thrombosis, the formation of an occlusive intravascular clot , and hemostasis, the arrest of extravascular bleeding where blood through the main vessel retains its fluidity. The cellular and biochemical processes in hemostasis and thrombosis are largely the same, but the fluid dynamics and vascular components differ greatly. To understand the mechanisms of recovery of hemostasis from bleeding, we have developed preliminary ODE and PDE models of extravascular clotting. This work is in close collaboration with Prof. Keith Neeves at CU Denver and Dr. Aaron Fogelson and the U. of Utah.

Phenotypic heterogeneity in hemophilia A

Hemophilia is a genetic disorder that manifests as a deficiency in clotting proteins that are necessary to produce stable blood clots. Bleeding frequency and severity within clinical categories of hemophilia A are highly variable and the origin of this variation is unknown. Solving this mystery in coagulation requires the generation and analysis of large data sets comprised of experimental outputs and/or patient samples, both of which are subject to limited availability. In this project we use a computationally driven approach to bypasses such limitations by generating large synthetic patient data sets. We create these data sets with a mechanistic mathematical model of flow-mediated coagulation, and by varying the model inputs such as clotting factor and inhibitor concentrations within their normal physiologic ranges. We choose specific mathematical metrics from the model output as surrogate measures for bleeding severity, and statistically analyze for further exploration and hypothesis generation. An exciting result from our recent study identified FV as a key modifier of thrombin generation in hemophilia A, which was confirmed with complementary experimental assays. The mathematical model was used further to propose a potential biochemical mechanism for these observations, which is being tested with biochemical assays. This is a collaboration with Prof. Keith Neeves at CU Denver, Dr. Dougald Monroe at UNC Chapel Hill, Dr. Aaron Fogelson and the U. of Utah, Dr. Dr. Suzanne Sindi at UC Merced, Dr. Michael Stob at Coe College, and Dr. Kathryn Link at UC Davis.

Uncertainty quantification in blood coagulation

Many biochemical details needed in our mathematical models come from purely biochemical studies. Due to the surface-dependence of the coagulation network, some biochemical assays for coagulation require a cellular surface component (lipid vesicles or platelets) to be carried out. The most common is the thrombin generation assay where blood plasma is combined with tissue-factor-bearing lipid vesicles, and thrombin generation is measured by indirect monitoring of it cleaving a synthetic substrate (chromogenic of fluorogenic). Thus, if one wants to probe such a system with a mathematical model, the model must correctly account for lipids as a model variable and give the correct response to lipid variations. Additionally, experimental uncertainty in pipetting, plate readers, and synthetic substrates, for example, should necessarily inform and couple to the mathematical models. We are building new mathematical models of lipid-mediated enzyme reactions in which the association rates between lipid-bound reactants are modified by a surface interaction probability. We are further employing modern parameter estimation and uncertainty quantification techniques along with experimental data to better understand the intrinsic kinetic rates of lipid-dependent enzymes. This is joint work with biochemist Dr. Dougald Monroe (UNC Chapel Hill), mathematician Dr. Suzanne Sindi (UC Merced).

Computational modeling of cilia and flagella in a Stokes or Brinkman fluid

The scientific theme of this work is to study the interaction of cilia and flagella with their complex fluid environments. For example, our group would like to know how the fluid environment influences the emergent beat patterns of cilia and waveforms of flagella. Answering these questions has important implications, some of which include improvement of aerosol drug delivery and a better understanding of the fluid mechanics of reproduction. Our work has mostly focused on the development of numerical methods to solve the underlying PDEs of the fluid-structure interaction problems. This is joint work with Dr. Sarah Olson at WPI.

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