The Abel differential equation of the first kind has no general closed-form solution, and most recent results on its rational solutions give upper bounds on their number or describe qualitative properties. This paper describes a direct algebraic procedure for finding rational solutions when the coefficients are polynomials. Writing the unknown as $y=P/Q$ with coprime polynomials $P$ and $Q$, then applying the quotient rule and clearing denominators, one obtains the polynomial identity $(P'Q-PQ')Q=f_0Q^3+f_1PQ^2+f_2P^2Q+f_3P^3$. For prescribed degrees of $P$ and $Q$, comparing coefficients gives a finite system of polynomial equations in the unknown coefficients, and rational solutions of tho... More >

Compartmental and tracer-kinetic models are widely used to describe radiopharmaceutical uptake in brain PET and SPECT, but in simple analytical models the fraction of transferred activity that reaches the brain is seldom isolated as a parameter of its own. This study defines a dimensionless Brain Uptake Constant, $\psi$, as the uptake fraction in a two-stage first-order model of cerebral radiopharmaceutical activity and examines its behaviour in a purely synthetic setting. Activity in the systemic pool, $A(t)$, is lost at a first-order rate $k$; cerebral activity, $A_c(t)$, receives a source term $\psi k A(t)$ and is lost at an effective decay rate $\lambda$. The closed-form solution was eva... More >

This study formulates a mathematical and computational model of tumor-cell-density regression in pancreatic cancer treated with carbon-ion hadron therapy. A reaction-diffusion equation with logistic proliferation and a local radiation-induced loss term is derived term by term, discretized with a two-dimensional finite-difference scheme and forward Euler time stepping, and implemented in Python. The treatment input follows a published carbon-ion schedule of 48~Gy(RBE) in 12 fractions of 4~Gy(RBE). The simulation produces sequential two- and three-dimensional maps of the modeled tumor-cell density as a function of accumulated dose; in the three-dimensional reconstruction the relative tumor vo... More >

Glioblastoma (GBM) is a highly infiltrative primary brain tumor, and reaction-diffusion models are widely used to describe how its cells spread and proliferate. We derive the classical Fisher-KPP equation from mass conservation, Fickian diffusion, and logistic growth, and extend it to fractionated radiotherapy through a linear-quadratic survival update applied to the tumor-density field at each fraction. The Fisher-KPP equation is solved by finite differences on a three-dimensional voxel grid with 0.5-mm spacing; Geant4 is used only for Monte Carlo photon transport and dose calculation. The schedule of 60~Gy in 30 fractions from the clinical case reported by Roda et al. is reproduced, and th... More >

Mechanistic tumor-response models that are transparent and updatable with patient data are important for model-informed precision radiotherapy. This study presents a reproducible computational framework integrating an explicit mathematical derivation, numerical solver, clinical protocol-derived parameters, and three-dimensional visualization to simulate simplified lung tumor regression during radiotherapy for non-small cell lung cancer (NSCLC). A first-order Taylor approximation, equivalent to the forward Euler method, is derived from Taylor's formula with integral remainder, with local truncation error of order $h^2$. The resulting update rule is applied to a first-order regression model wi... More >