• Geometry
  • Soap Bubbles
  • Medical Imaging
  • Robotics
  • Fluids
  • Semiconductors
  • Wave Propagation
  • Image Denoising
  • Optimal Design
  • Seismic Analysis
  • Tumor Modeling
  • Optimal Control
  • InkJet Plotters
  • Traveling Salesmen
  • ViscoElastic Flow
  • Pinching Droplets
  • Chemical Pathways







    J.A. Sethian
  • Angiogenesis and Tumor Modeling

    Malignant solid tumors generally are described as masses of tissue formed as a result of abnormal and excessive proliferation of mutant (atypical) cells, whose division has escaped the mechanisms that control normal cellular proliferation. This abnormal proliferation of atypical cells in time leads to an uncontrolled growth, extending to the adjacent surrounding tissues, infiltrating and invading them; this invasion is local at first-causing primary tumors, but malignant cells have the ability of migrating through the blood vessels and/or the lymphatic system towards other parts of the body, giving rise to secondary tumors; this process is called metastasis and it is the one responsible for the host death.

    There are different stages of a malignant tumor evolution; described roughly, the main stages are the cellular stage and the macroscopic stage. The cellular stage refers to the early stage of a tumor evolution, when tumor cells are not condensed yet in a macroscopically observable mass. The macroscopic stage corresponds to that phase of a tumor evolution when clusters of atypical (malignant) cells condense together into a quasi-spherical observable mass (nucleus); this stage is sub-divided into two subsequent phases-the avascular phase and the vascular phase. During the avascular phase, the tumor obtains nutrients and "feeds" itself via diffusion processes alone, with nutrients already existing in the environment.

    In the second phase, called the vascular phase, when the tumor grows more rapidly through what is called angiogenesis (i.e., the birth of new blood vessels), malignant tumor cells secrete chemicals that have the ability to diffuse into the surrounding healthy tissues and stimulate the growth of new capillary blood vessels; the newly born blood vessels penetrate into the tumor mass feeding it with nutrients and leading to a rapid growth of the tumor. Tumor growth (spread of malignant cells) occurs basically via two mechanisms: when fed with a sufficient amount of nutrient, malignant cells divide (cellular mitosis); when the density of malignant cells in a specific volume becomes too high, the cells are compressed by their neighbors, so they tend to move to less compressed areas-where they are allowed to continue the division process-and this process is repeated.

    Numerical Simulations

    We built a numerical simulation of a model for angiogenesis and tumor growth first proposed by De Angelis and Preziosi. Their model assumes two regions: an inner region occupied by the tumor mass, which is time-dependent, and a larger fixed domain called the tumor outer environment. The model is described by
    • Living tumor cells, represented by a density
    • Dead tumor cells, represented by a density
    • New capillaries (i.e. endothelial cells)
    • A nutrient concentration
    • A tumor angiogenic factor (TAF) concentration
    These are linked through a set of partial differential equations which describe a continuum mechanics framework. Across the boundary of the time-dependent tumor region, diffusion and transport take place, leading to an intricate interface problem with jump boundary conditions and a delicate behavior.

    Time Evolution of Living Cells Time Evolution of Dead Cells
    (contour density plot) Note development of necrotic region


    The model is a comprehensive continuum model for tumor evolution and development for arbitrary two-dimensional growth. The model represents both avascular and vascular phases of tumor evolution, and is able to simulate when the transition occurs: progressive formations of necrotic cores and rim structures in the tumor during the avascular phase are captured. The computational framework is a Cartesian mesh/narrow band level set method for the coupled advection-diffusion model questions with a moving embedded boundary inside a fixed computational domain.


    • Computational Modeling of Solid Tumor Evolution via a General Cartesian Mesh/level set method ,
            Hogea, C.S., Murray, B.T., and Sethian, J.A., Fluid Dynamics \& Materials Processing, Vol., 1, 2, 2005,
      This paper List of downloadable publications

    • Simulating Complex Tumor Dynamics from Avascular to Vascular Growth using a General Level Set Method, ,
            Hogea, C.S., Murray, B.T., and Sethian, J.A., J. Mathematical Biology, 53, 1, 2005.
      This paper List of downloadable publications