This study examines the effects of ultra-low-dose hormone therapy (Ultra-LD HT) on bone turnover in postmenopausal women through the lens of a mathematical shock model. In this framework, intermittent physiological stressors such as fluctuations in bone remodeling dynamics or estrogen depletion are conceptualized as mathematical shocks occurring in a Poisson pattern. These shocks model sudden, discrete changes in bone resorption risk. When the interval between shocks is too short (below a defined threshold) the cumulative effect of estrogen deficiency may trigger osteoporosis. To reflect biological adaptation under increasing physiological stress, we propose that these threshold intervals grow geometrically, representing a system that becomes more resilient (requires longer intervals between insults to trigger damage). Conversely, during recovery following estrogen supplementation, threshold intervals are assumed to decrease geometrically, modeling a return to homeostasis. Using clinical data on bone turnover markers (BTM) including PINP, BSAP, CTX-I, and NTX from postmenopausal women treated with E2 0.5/NETA 0.1 or placebo, we fit this shock-based model to assess the optimal timing for dose adjustment. The switch point N^* minimizes average estrogen exposure while improving BTM outcomes over 24 weeks.
