Journal of Cancer and Tumor International

Colorectal cancer (CRC) is the most frequently diagnosed cancer in Romania, according to the most recent GLOBOCAN 2022 estimates. We developed a homogeneous Markov-chain model describing the treatment-stage transitions of a cohort of 100 CRC patients undergoing cytostatic (chemotherapy) treatment at a hospital in Sibiu, Romania, distributed across four states of decreasing severity (S₁–S₄), with S₄ representing patients no longer requiring cytostatic treatment. Transition probabilities, estimated from the observed data conditioned on Body Mass Index via Bayes' formula, define an absorbing Markov chain: numerical evolution of the state distribution over 100 treatment doses shows the cohort converging steadily toward S₄, with over 99% of patients reaching this state by the 100^th dose. We derive semi-empirical exponential approximations that closely reproduce this recursion in closed form for two of the three transient states across the full dose range, and for the third once early transient effects have decayed (n≳30), providing readers with an accessible pedagogical bridge from matrix recursion to closed-form approximation. Building on these empirically grounded results, we additionally propose, as methodological frameworks rather than empirically validated findings, two forward-looking extensions: a quality-of-life-weighted formulation compatible with the existing transition matrix, and a constrained-optimization framework identifying an optimal dose count n₀ that balances treatment benefit against accelerating patient toxicity and financial cost, illustrated numerically with hypothetical weights. As a single-center study with a modest sample size (N=100) and no external validation of the estimated transition probabilities, the results should be read as an approximate, illustrative picture of Romanian clinical practice rather than a nationally representative or externally validated estimate. The present work provides a transparent, reproducible modeling framework - including a didactic Appendix - intended both as a practical tool and as a teaching resource for graduate students in mathematical epidemiology.