A mathematical perspective on the paradox that chemotherapy sometimes works backwards

Luis A. Fernández, Isabel Lasheras, Cecilia Pola

wrongmedium confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s constant-effects classification for Ec>1 asserts that, for any initial condition other than the equilibrium (V̂c,V̂c), solutions satisfy exactly one of lim V(t)=0 or lim V(t)=+∞ (using Theorem 2 and then mapping constants), but it omits the nonempty 1D stable manifold of the hyperbolic saddle at (V̂c,V̂c) when b/(1+Ea)>λ2. The system’s Jacobian at (V̂c,V̂c) has negative determinant, so there must exist initial data (inside the positive quadrant) that converge to (V̂c,V̂c) along this stable manifold. Thus the paper’s statement is false as written; the model’s solution correctly includes the standard “stable-manifold exception” and otherwise matches the Ec<1 and Ec>1 classifications and equilibrium formulae presented in the paper (systems (9)–(10); Theorem 1; constant-effects mapping) .

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper offers a useful synthesis of the Hahnfeldt-type model with constant and time-dependent therapies and presents informative simulations, but its analytical classification for the Ec>1 regime is overstated. When b/(1+Ea)>λ2, the equilibrium is a hyperbolic saddle with a nontrivial stable manifold inside the positive quadrant; the manuscript’s theorem excludes this set. This is a substantive correctness issue. Adding the standard stable-manifold exception, with a brief linearization and phase-portrait illustration, will resolve the flaw and improve precision.