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Growth Models with Integration: Problem Solving01:27

Growth Models with Integration: Problem Solving

In population modeling, integration provides a systematic way to determine accumulated quantities from known rates of change. One such application arises in ecology, where the total weight of a fish population in a body of water is referred to as its biomass. When the rate of growth of this biomass is known as a function of time, calculus can be used to determine the total biomass at a future date.Growth Rate and Biomass FunctionLet the growth rate of the fish population be represented by a...

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Related Experiment Video

Updated: Jul 5, 2026

Quantification of Breast Cancer Cell Invasiveness Using a Three-dimensional (3D) Model
08:08

Quantification of Breast Cancer Cell Invasiveness Using a Three-dimensional (3D) Model

Published on: June 11, 2014

Three-dimensional multispecies nonlinear tumor growth--I Model and numerical method.

S M Wise1, J S Lowengrub, H B Frieboes

  • 1Mathematics Department, University of Tennessee, Knoxville, TN 37996-1300, USA.

Journal of Theoretical Biology
|May 20, 2008
PubMed
Summary

This study introduces a diffuse interface model for multispecies tumor growth, enabling detailed simulations of complex tumor morphologies. The model accurately captures avascular tumor growth and is computationally efficient for large-scale 3D analyses.

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Last Updated: Jul 5, 2026

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Published on: June 11, 2014

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Longitudinal Morphological and Physiological Monitoring of Three-dimensional Tumor Spheroids Using Optical Coherence Tomography
08:50

Longitudinal Morphological and Physiological Monitoring of Three-dimensional Tumor Spheroids Using Optical Coherence Tomography

Published on: February 9, 2019

Area of Science:

  • Computational biology
  • Mathematical modeling
  • Biophysics

Background:

  • Tumor growth and angiogenesis are complex processes.
  • Existing models often lack detailed descriptions of cell-cell interactions and morphology.
  • Previous simulations of tumor growth did not fully detail the underlying diffuse interface model.

Purpose of the Study:

  • To develop and analyze a diffuse interface continuum model for multispecies tumor growth and angiogenesis.
  • To describe the model's thermodynamic consistency and relation to mixture models.
  • To present a new numerical method for efficient simulation of tumor growth.

Main Methods:

  • Developed a diffuse interface continuum model replacing sharp interfaces with transition layers based on differential adhesion.
  • Introduced a continuum model of adhesion, thermodynamically consistent with mixture models.
  • Employed a fully adaptive, nonlinear multigrid/finite difference method for efficient simulations.

Main Results:

  • Demonstrated that the diffuse interface model reduces to a sharp interface model as interface thickness approaches zero.
  • Presented simulations of unstable avascular tumor growth in 2D and 3D.
  • Showcased the computational feasibility of large-scale 3D simulations for complex tumor morphologies.

Conclusions:

  • The diffuse interface model provides a thermodynamically consistent and detailed description of tumor progression.
  • The developed numerical techniques enable efficient, large-scale 3D simulations of tumor growth.
  • This work lays the foundation for investigating tumor invasion and angiogenesis in subsequent studies.