A generalized resource-allocation growth law reveals mechanisms of differential responses to intermittent androgen deprivation therapy in prostate cancer
Resource-limited growth frameworks often developed independently. Here, we derived a growth law by assuming a cell balancing internal competing resource demands and generalizing the growth signal through a shape parameter h, the trade-off exponent. In Liebig's single-limiting-resource regime, it recovers Droop's cell quota at h=1 and approaches Prater's growth efficiency as h grows large. h determines how sharply growth declines when allocation departs from optimal, thereby shaping populations evolution. Cyclic environmental stress tests such growth laws. Hence, we built a model from this growth law and applied it to prostate cancer growth in patients undergoing cyclic intermittent androgen deprivation therapy. We showed the model recapitulated longitudinal prostate specific antigen (PSA) and serum androgen measurements from 71 prostate cancer patients using a nonlinear mixed-effects framework (pooled and median individual R^2 {approx} 0.9 for PSA and {approx} 0.8 for serum androgen). h is well-constrained by the data and separates treatment outcomes (success vs. failure, p = 0.003). The results predict that failing cases have cancer that can adapt to a wide range of growth conditions with minimal cost. A simplified model recapitulated PSA dynamics from 32 prostate cancer patients on cyclic adaptive therapy (R^2 {approx} 0.8). Estimates of h for castration resistant cancer in adaptive cohort were similar to failure cases in the intermittent cohort. We found no statistical differences in estimated parameters between the adaptive vs. standard of care arms, suggesting that adaptive schedule may drive the differential outcome. The results support this growth law, providing a foundation for resource-limited growth across biological systems.