Research interests
PDE analysis and computations, complex fluids, numerical analysis, mathematical modeling in physics and engineering, mathematical modeling in biology and medicine, bioinformatics, fluid dynamics, finite difference schemes.
PDE analysis and computations, complex fluids, numerical analysis, mathematical modeling in physics and engineering, mathematical modeling in biology and medicine, bioinformatics, fluid dynamics, finite difference schemes.
Colon cancer is the third leading cause of cancer-related deaths in the United States in both men and women. A major clinical challenge is to obtain an effective treatment strategy for each patient or at least identify a subset of patients who could benefit from a particular treatment. Since each colon cancer has its own unique features, it is very important to obtain personalized cancer treatments and find a way to tailor treatment strategies for each patient based on each individual's characteristics, including race, gender, genetic factors, immune response variations. Recently, Quantitative and Systems Pharmacology (QSP) has been commonly used to discover, validate, and test drugs. QSP models are a system of differential equations that model the dynamic interactions between drug(s) and a biological system. These mathematical models provide an integrated “systems level” approach to determining mechanisms of action of drugs and finding new ways to alter complex cellular networks with mono or combination therapy to obtain effective treatments. Since QSP models are a complex system of nonlinear equations with many unknown parameters, estimating the values of the model's parameters is extremely difficult. Existing parameter estimation methods for QSP models often use assembled data from various sources rather than a single curated dataset. These datasets are usually obtained through various biological experiments, in vitro and in vivo animal studies, thus rendering QSP models hard to be practicable for personalized treatments. To the best of our knowledge, no QSP model has been developed for personalized colon cancer treatments. In this project, we propose a unique approach to develop a data-driven QSP software to suggest effective treatment for each patient based on gene expression data from the primary tumor samples. Since signatures of main characteristics of tumors, such as immune response variations, can be found in gene expression profiling of primary tumors, we use gene expression data as input. We develop an innovative framework to systematically employ a combination of data science, mathematical, and statistical methods to obtain personalized colon cancer treatment. We will use these techniques to propose an optimal treatment strategy for each patient and predict the efficacy of the proposed treatment. The model might also suggest alternative therapies in case of low efficacy for some patients.
The main focus of this project is on mathematical modeling and numerical simulation of anisotropic complex fluids whose motion is complicated by the existence of mesoscales or sub-domain structures and interactions. These include multi-component mixtures of immiscible fluids, viscoelastic and polymeric fluids. Such complex fluids are ubiquitous in daily life, e.g., they arise in a wide varieties of mixtures, polymeric solutions, colloidal dispersions, biofluids, electro-rheological fluids, ionic fluids, liquid crystals, and liquid crystalline polymers. Indeed, materials modeled as complex fluids often have great practical utility since the microstructure can be manipulated by external fields or forces in order to produce useful mechanical, optical or thermal properties.
An important application of such complex fluids is the modeling of composites of different materials. The mixing of two (or more) different components can be achieved by deriving various properties from the composite. These properties of a certain composite material can be tuned to suit a particular application, e.g., by varying the composition, concentration and, in many situations, the phase morphology. The modeling of such phenomena is achieved by postulating and analyzing the energy laws of the physical systems and then applying the energetic variational approach for isothermal systems. The advantage of such an approach is that it provides a definitive way to derive a thermodynamically consistent model, which is of critical importance for many physical applications. The next step in such a process is to design efficient numerical simulations approximating the solutions of these advanced models in a way that they preserve the energy laws of the proposed systems. The main goal of my Ph.D. and current research on this topic is to extend the unified energetic variational framework to a wider range of applications, such as mixtures with microstructures and various boundary effects.
To model mixtures of fluids and free boundary motion, I employ the diffuse interface method, which allows seamless integration of the free boundary into the system written on the whole domain. The main focus of my work on this topic is on investigating the effects of different forms of free energy involving phase-field functions on the dynamics of the system (with the Cahn-Hilliard equation governing dynamics of the phase-field). To describe the behavior of mixtures with three components, one has to introduce and follow two phase-field functions, in some models introducing three linearly dependent functions (and expressing one in terms of the others). My main contribution to this area is in showing that in fundamentally different descriptions the free energies are quantitatively similar and the main difference is in the energy dissipation. Also the analysis allowed me to further the understanding of the mixing energy and introduce some additional requirements on the energy coefficients that are useful outside of three-component flow framework. To demonstrate the efficiency of the aforementioned models and further analyze them I developed decoupled unconditionally energy stable numerical discretization, which allows for a better approximation of the models' underlying energetic structure.