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Gompertz model with delays and treatment: mathematical analysis
Marek Bodnar1, Monika Joanna Piotrowska, Urszula Foryś
1Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland. mbodnar@mimuw.edu.pl
Mathematical Biosciences and Engineering : MBE
|August 3, 2013
Summary
This study analyzes delayed tumor growth models, exploring how treatment timing impacts tumor suppression and stability. Mathematical models predict how delays in chemotherapy affect tumor growth dynamics and treatment effectiveness.
Area of Science:
- Mathematical Biology
- Tumor Growth Dynamics
- Pharmacokinetics
Background:
- The Gompertz model is a standard for simulating tumor growth.
- External interference, like chemotherapy, significantly impacts tumor progression.
- Time delays in biological systems can alter model stability and outcomes.
Purpose of the Study:
- To investigate the stability of delayed Gompertz models for tumor growth under external interference (treatment).
- To analyze the impact of different delay types (single vs. double) on tumor growth dynamics.
- To examine Hopf bifurcations and the stability of periodic solutions in response to constant and time-varying treatments.
Main Methods:
- Mathematical modeling of delayed Gompertz equations.
- Stability analysis of steady states with respect to delay parameters.
- Bifurcation analysis to identify transitions in system behavior.
- Numerical simulations to validate analytical findings using pharmacokinetic models.
Main Results:
- Identified conditions for stability switches in tumor growth models as delay increases.
- Characterized Hopf bifurcations, leading to periodic tumor growth patterns.
- Demonstrated the influence of delay in treatment application on overall tumor suppression effectiveness.
Conclusions:
- Delayed models provide crucial insights into tumor growth and treatment response.
- The timing of therapeutic intervention significantly affects treatment efficacy.
- Mathematical modeling, including delay dynamics, is essential for optimizing cancer treatment strategies.
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