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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...
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Consecutive reactions involve a sequence where the product of a preceding reaction becomes the reactant for the subsequent one. In a simple scheme, A transforms into B, which further reacts to form C, with rate constants k1 and k2, respectively. This concept is evident in the radioactive decay series. Assuming an initial state with only A present, the conservation of matter leads to three coupled differential equations, determining the concentrations of A, B, and C over time.The rate of change...
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The linear concentration–effect model, underpinned by the principle that pharmacological effect (E) is directly proportional to plasma drug concentration (C), emerges as a pivotal simplification of the Emax model for conditions where C is significantly less than EC50. This model portrays a linear trajectory of the concentration–effect relationship when drug levels are markedly below the EC50 threshold.Despite its inherent assumption of continuous effect augmentation with increasing...
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The Emax drug-concentration effect model is central to pharmacodynamics in drug discovery and development. This model is predicated on the receptor occupancy theory, which posits that the effect of a drug is directly related to the number of receptors occupied by the drug and the resultant complex formation.The model describes the reversible interaction between a drug (C) and a receptor (R) to form a drug-receptor complex (RC). The kinetics of this interaction are quantified by an equation that...
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Modelling the radiotherapy effect in the reaction-diffusion equation.

Giovanni Borasi1, Alan Nahum2

  • 1University of Milano Bicocca, Milano, Italy.

Physica Medica : PM : an International Journal Devoted to the Applications of Physics to Medicine and Biology : Official Journal of the Italian Association of Biomedical Physics (AIFB)
|September 4, 2016
PubMed
Summary
This summary is machine-generated.

This study corrects the external radiotherapy term in the Fisher-Kolmogorov equation, crucial for modeling glioblastoma multiforme invasion. The new derivation ensures accurate simulation of tumor behavior under radiation therapy.

Keywords:
Fisher-Kolmogorov equationGlioblastomaLinear-Quadratic ModelOncologyRadiotherapyReaction-diffusion equation

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Area of Science:

  • Oncology
  • Mathematical Biology
  • Radiotherapy Physics

Background:

  • The Fisher-Kolmogorov equation models glioblastoma multiforme (GBM) invasion.
  • Accurate modeling of GBM resistance to therapy is critical.
  • Previous models used an incorrect term for external radiotherapy effects.

Purpose of the Study:

  • To derive the correct analytical expression for the external radiotherapy (R) term in the reaction-diffusion equation.
  • To address inaccuracies in existing models of GBM invasion under radiotherapy.
  • To improve the fidelity of mathematical models used in oncology research.

Main Methods:

  • The external radiotherapy (R) term was derived using the Linear-Quadratic (LQ) model of cell killing.
  • Fundamental principles of differential calculus were applied to the derivation.
  • The derived R term was integrated into the reaction-diffusion equation.

Main Results:

  • A correct analytical expression for the external radiotherapy term (R) was successfully derived.
  • The compatibility of the new R term with the reaction-diffusion equation was confirmed.
  • Simulations using the corrected R term showed differences compared to models with the incorrect term, particularly for glioblastoma tumors.

Conclusions:

  • The derived R term provides a more accurate representation of external radiotherapy effects in GBM models.
  • This correction is essential for reliable simulations of GBM invasion and treatment response.
  • The study highlights the importance of precise mathematical formulation in cancer modeling.