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Related Concept Videos

Modeling with Differential Equations01:25

Modeling with Differential Equations

Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...
Population Growth00:57

Population Growth

Population size is dynamic, increasing with birth rates and immigration, and decreasing with death rates and emigration. In ideal conditions with unlimited resources, populations can increase exponentially, which plots as a J-shaped growth rate curve of population size against time. This type of curve is characteristic of newly-introduced invasive species, or populations that have suffered catastrophic declines and are rebounding.However, realistic environmental conditions limit the number of...
Exponential Equations for Modeling Growth01:26

Exponential Equations for Modeling Growth

Exponential models are essential for describing rapid, multiplicative changes in natural systems, such as population growth. When a population doubles at regular intervals, the process can be modeled using a suitable base. For instance, a bacterial culture that doubles every three hours follows the model n(t)=n0⋅2t/3, where n(t) is the population at the time t.A more general model uses the natural base e, especially for continuous growth. This takes the form n(t)=n0⋅ert, where r is the relative...
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...
Tumor Progression02:07

Tumor Progression

Tumor progression is a phenomenon where the pre-formed tumor acquires successive mutations to become clinically more aggressive and malignant. In the 1950s, Foulds first described the stepwise progression of cancer cells through successive stages.
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Tumor Progression

Tumor progression is a phenomenon where the pre-formed tumor acquires successive mutations to become clinically more aggressive and malignant. In the 1950s, Foulds first described the stepwise progression of cancer cells through successive stages.
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Related Experiment Video

Updated: Jul 11, 2026

A Computational Modeling Approach to Investigate the Influence of Hyperthermia on the Tumor Microenvironment
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Published on: December 1, 2023

[Population dynamics in tumor growth modeling].

V A Slepkov, V G Sukhovol'skiĭ, R G Khlebopros

    Biofizika
    |October 3, 2007
    PubMed
    Summary

    This study models solid tumor growth using local cell interactions, revealing how cancer cell reproduction influences tumor shape and spread. The findings explain complex tumor structures and metastasis formation.

    Area of Science:

    • * Mathematical modeling
    • * Cancer biology
    • * Tissue engineering

    Context:

    • * Solid tumor formation and growth dynamics
    • * Interplay between cancer cells and host tissues
    • * Tumor progression across tissue boundaries

    Purpose:

    • * To model the onset and growth of solid tumors
    • * To investigate tumor behavior in homogeneous and heterogeneous tissues
    • * To explore the development of complex tumor structures and metastasis

    Summary:

    • * A novel approach models tumor growth based on local cancer-tissue cell interactions.
    • * The model assumes cancer cell reproduction rate is a nonmonotone function of local concentration and surface curvature.
    • * This approach explains spherical tumor growth, transitions to complex structures (rough interfaces, outgrowths, dendrites), and metastasis.

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    Published on: September 20, 2016

    Impact:

    • * Provides insights into the mechanisms driving tumor morphology and invasion.
    • * Offers a framework for understanding how tissue boundaries affect tumor progression.
    • * Contributes to the understanding of metastasis and complex tumor development.