Department of Mathematics

Purdue University

West Lafayette, IN 47907-1395

Email: chen1447@purdue.edu

Phone: (765)-494-1994

Office: Math 427

- Mathematical modeling and simulations in biology
- Scientific computing and numerical methods, computational
fluid dynamics
- Machine learning and data mining: applying machine learning methods and statistical analysis tools to cancer dataset

- We developed a mathematical model of tumor growth with an elastic membrane, and constructed a stable and efficient numerical implementation to solve the model system. We found that the mechanical forces introduced by the memebrane can enhance the growth of the tumors in duct-like geometries and can retard growth when tumors are encapsulated by a fibuous membrane. The plots below show the evolution of the pressure (backgroud) together with the tumor (red, contours) growing in confined basement membrane (green, contours) with different cell-cell adhesion. The dead cells are located primarily within the magenta curve. Increasing cell-cell adhesion ([a]:small cell-cell adhesion; [b]: large cell-cell adhesion) tends to stabilize the tumor shape. Chen et al., Int J Numer Method. Biomed Eng, 2014.

[b]

- Evolution of tumor cluster in 3D branched ducts showing the tumor (red, isosurfaces), membrane (green, isosurfaces), and necrotic core (magenta, isosurfaces) for different relative strengths of cell-basement membrane (BM) adhesion as labeled via the contact angle. When the cell-BM adhesion is low ([a]), the tumors grow at early times but growth is not sustained and the tumors progress to a steady state. Increasing cell-BM adhesion ([b]) leads to larger, more elongated tumors and to fragmentation. Chen et al., J Theor Biol, 2014.

[b]

- Invasion of the stroma by tumor clusters in 3D simple ducts with different membrane stiffnesses. The tumor (red, isosurfaces), membrane (green, isosurfaces), and necrotic core (magenta, isosurfaces) are shown. A smaller membrane stiffness ([a]) leads to limited invasion and a larger membrane stiffness ([b]) leads to more invasive tumors. Chen et al., J Theor Biol, 2014.

[b]

- We developed a mathematical model of tumor growth and microcalcification in complex, evolving microenvrionments with elastic, deformable membranes. We observed that enhanced membrane deformability promotes tumor growth and tumor calcification. We predicted that correlations between the extents of the mammographic and pathologic tumor change from linear to quadratic as the membrane deformability increases. The plots below show that tumor cluster growth in a 2D simple duct showing the necrotic core (white regions), tumor (regions inside the red curves), microcalcification (regions inside the magenta curves), and membrane (regions inside the green curves) with different relative strengths of cell-basement membrane adhesion as labeled via the contact angle ([a]: small; [b]: large). Increasing cell-membrane adhesion by increasing the contact angle leads to larger, more elongated tumors and microcalcification. Chen et al., J Theor Biol, 2018 (under review).

[b]

[b]

- Schematic diagram of model structure. A short loop, which consists of a descending limb and a contiguous ascending limb and which turns at the outer-inner medullary boundary, is represented. The diagram also dispicts a long loop that turns within the inner medulla (at x_2). Although only one long loop is shown, the model represents one long loop that turns at every spatial point in the inner medulla. Similarly, only two representative descending vasa recta (DVR) are shown, whereas the model represents one DVR that ternimates at every spatial point. A representative collecting duct is shown; the "brnaches" represent the coalescence of the collecting ducts in the inner medulla. The black arrows at the corticomedullary boundary represent boundary flows. The outflow of the ascending limbs determines the inflow of the collecting duct. Collecting duct outflow becomes urine.

- Desending vasa recta (DVR) glucose flow (panel A) and interstitial and DVR fluid glucose concentration (panel B). Medullary depth x=0mm corresponds to the cortico-medullary boundary; x=0.6mm, inner-outer strip boundary; x=2mm, inner-outer medullary boundary; x=7mm, papillary tip. Chen et al., Bullet Math Biol, 2016.

- Descending vasa recta (DVR) lactate flow (panel A) and interstitial and DVR fluid lactate concentration (panel B). Medullary depth x=0mm corresponds to the cortico-medullary boundary; x=0.6mm, inner-outer strip boundary; x=2mm, inner-outer medullary boundary; x=7mm, papillary tip. Chen et al., Math Biosci, 2017.

- Local fluxes and consumption rate of glucose (panel A), as well as local fluxes and production rate of lactate (panel B) within the outer stripe. The long arrows and corresponding values represent glucose/lactate fluxes between regions, in units of pmol/min/nephron. The short arrows and corresponding values represent fluxes vessels and regions, in units of pmol/min/nephron. The number in the regions and vessels not attached to the arrows represent net glucose consumption/lactate production, in units of pmol/min/nephron. In the outer stripe, thick ascending limbs (TALs) are located at regions R3 and R4, which are far away from Oxygen-supplying DVR, and where active Na+ transport is taken place, resulting in higher glucose consumption and lactate production. AVR denotes ascending vasa recta; DL/AL, descending limb/ascending limb; CD, connecting duct; R1/R2/R3/R4 are interstitial regions. Chen et al., Math Biosci, 2017.

- We developed robust, efficient, and stable numerical methods to solve isotropic and strongly anisotropic Cahn-Hilliard systems with a Willmore regularization. The system involves six order derivatives and is highly nonlinear. We used a convex splitting technique to develop energy stable schemes for the isotropic case and extended this approach to account for anisotropy. Chen et al., J Comput Phys, 2018.

[b]

- Simulating incompressible Cahn-Hilliard Navier-Stokes Phase-field Models using decoupled, unconditionally energy stable schemes combined with adaptive mesh, adaptive time and a nonlinear multigrid finite difference method. The new scheme has been proved to be very efficient to solve the phase field models with incompressible flow for both matched and varied densities. A sample shows the evolution of a rising drop with density ratio 1:10. Chen et al., J Comput Phys, 2016.