Focus on bridging the gap between theoretical mathematics and biological phenomena to understand how malignant cells disseminate, biological impulses propagates and tissues develop.
Migration of malignant cells in epithelial tissue
– Focus on the collective motion and antagonistic interactions between healthy and cancerous cell populations plos20
– Active Vertex Models to simulate competitions for space where fluid-like Ras-oncogene (malignant) cells invade solid-like wild-type (healthy) cells
– Demonstrate that the success of malignant invasion often depends on junction tensions between different cell types, which determines whether the populations mix or segregate
– Topological Data Analysis (TDA) and persistent homology to track and classify the advancing interfaces of cell aggregates, helping to automate the study of how malignant cells overcome healthy tissue barriers
Propagation of biological impulses in nerves and muscles
– Focus on how these impulses move through complex, discrete biological structures rather than continuous media.
– Utilizes discrete reaction-diffusion equations to model «depinning transitions,» which explain how biological signals overcome resistance to move through a cellular lattice.
siap03 prl01
– Developed an asymptotic construction of pulses for the discrete Hodgkin–Huxley model. This shows that nerve impulses in myelinated axons act as traveling waves with two distinct components: they behave as discrete pulses at the Nodes of Ranvier, which then drive wave motion through the internodal regions. pre05 jns11
– Analyzed muscle contraction as a core area of biomedical modeling, exploring how electrical impulses translate into mechanical force within tissues. pd05
– Identified critical parameter ranges (such as coupling strength and time scales) that lead to propagation failure in discrete systems. This is vital for understanding neurological conditions where signal transmission is interrupted or when muscle fibers loose their ability to relax to their normal state.
– Predicted the speed and shape of wave trains and to study synchronization phenomena.
– Work that extends to networks of coupled cells that can behave as either excitable or self-oscillatory media siap03 pd05
– Work that extends to explain folding/unfolding experiments with biomolecules and DNA
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Angiogenesis (formation of new blood vessels)
– Analysis of integrodifferential kinetic equations of Fokker-Planck type to describe the density and velocity of growing vessel tips, establishing well posedness results.
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– Developed positivity-preserving schemes, which prevent the model from producing physically impossible results like «negative» cell densities, to ensure that simulations are biologically accurate. ijnsns21 jcp18
Research on inverse problems for biological structure reconstruction, including electrical impedance tomography and holography is discussed in the imaging section.
