TY  - JOUR
AU  - Farias, Matheus dos Santos
PY  - 2026
DA  - 2026/09/20
TI  - Formalism of a Treatment-Modulated Logistic Map for Breast Tumor Growth: Mathematical Formulation, Nonlinear Dynamics, and Clinically Inspired Simulation
JO  - Journal of Nonlinear Dynamics and Applications
T2  - Journal of Nonlinear Dynamics and Applications
JF  - Journal of Nonlinear Dynamics and Applications
VL  - 2
IS  - 3
SP  - 177
EP  - 191
DO  - 10.62762/JNDA.2026.825872
UR  - https://www.icck.org/article/abs/JNDA.2026.825872
KW  - logistic map
KW  - breast cancer
KW  - mathematical oncology
KW  - chemotherapy
KW  - radiotherapy
KW  - Geant4
KW  - nonlinear dynamics
KW  - Lyapunov exponent
KW  - bifurcation
KW  - treatment response
AB  - This study develops a treatment-modulated discrete framework for breast-tumor dynamics, emphasizing dimensional consistency, treatment dependence, and the distinction between scalar tumor burden and spatial tumor-density fields. Starting from the dimensional logistic growth law, treatment is introduced through a cycle-dependent survival factor. Normalization by the continuous carrying capacity $K_c$ yields the exact Euler update $y_{n+1}=S_n[y_n+g\Delta t_n y_n(1-y_n)]$, whereas the canonical non-autonomous map $x_{n+1}=r_nx_n(1-x_n)$, with $r_n=(1+g\Delta t_n)S_n$, is retained as a reduced nonlinear benchmark. This distinction prevents discretization-induced period-doubling and chaos from being interpreted as intrinsic tumor behavior. Invariance and cumulative extinction conditions are derived for non-autonomous sequences $\{r_n\}$, including the periodic product criterion. Radiotherapy is coupled through the linear-quadratic survival model and chemotherapy through an $E_{\max}$ exposure-response formulation. A published breast-cancer case provides the clinical treatment chronology, including chemotherapy and 26~Gy radiotherapy delivered in five fractions. Because longitudinal tumor measurements, voxel-level dose maps, pharmacokinetic data, and tumor-specific radiosensitivity parameters are unavailable, no patient-specific calibration or dose-response validation is claimed. Geant4 outputs are therefore used as clinically inspired spatial visualizations, with dose-to-survival coupling explicitly formulated. The framework also incorporates primitive-parameter uncertainty propagation and state-sensitivity analysis, providing a consistent basis for future patient-specific modeling.
SN  - 3069-6313
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Farias2026Formalism,
  author = {Matheus dos Santos Farias},
  title = {Formalism of a Treatment-Modulated Logistic Map for Breast Tumor Growth: Mathematical Formulation, Nonlinear Dynamics, and Clinically Inspired Simulation},
  journal = {Journal of Nonlinear Dynamics and Applications},
  year = {2026},
  volume = {2},
  number = {3},
  pages = {177-191},
  doi = {10.62762/JNDA.2026.825872},
  url = {https://www.icck.org/article/abs/JNDA.2026.825872},
  abstract = {This study develops a treatment-modulated discrete framework for breast-tumor dynamics, emphasizing dimensional consistency, treatment dependence, and the distinction between scalar tumor burden and spatial tumor-density fields. Starting from the dimensional logistic growth law, treatment is introduced through a cycle-dependent survival factor. Normalization by the continuous carrying capacity \$K\_c\$ yields the exact Euler update \$y\_{n+1}=S\_n[y\_n+g\Delta t\_n y\_n(1-y\_n)]\$, whereas the canonical non-autonomous map \$x\_{n+1}=r\_nx\_n(1-x\_n)\$, with \$r\_n=(1+g\Delta t\_n)S\_n\$, is retained as a reduced nonlinear benchmark. This distinction prevents discretization-induced period-doubling and chaos from being interpreted as intrinsic tumor behavior. Invariance and cumulative extinction conditions are derived for non-autonomous sequences \$\{r\_n\}\$, including the periodic product criterion. Radiotherapy is coupled through the linear-quadratic survival model and chemotherapy through an \$E\_{\max}\$ exposure-response formulation. A published breast-cancer case provides the clinical treatment chronology, including chemotherapy and 26~Gy radiotherapy delivered in five fractions. Because longitudinal tumor measurements, voxel-level dose maps, pharmacokinetic data, and tumor-specific radiosensitivity parameters are unavailable, no patient-specific calibration or dose-response validation is claimed. Geant4 outputs are therefore used as clinically inspired spatial visualizations, with dose-to-survival coupling explicitly formulated. The framework also incorporates primitive-parameter uncertainty propagation and state-sensitivity analysis, providing a consistent basis for future patient-specific modeling.},
  keywords = {logistic map, breast cancer, mathematical oncology, chemotherapy, radiotherapy, Geant4, nonlinear dynamics, Lyapunov exponent, bifurcation, treatment response},
  issn = {3069-6313},
  publisher = {Institute of Central Computation and Knowledge}
}