Sr and Ag Co‐Doped Bioactive Glasses: Modeling and Monte Carlo Simulation of pH, Ion Release, and Osteoblast Response

ABSTRACT

Multifunctional bioactive glasses co‐doped with silver (Ag ⁺ ) and strontium (Sr ^{2 +} ) represent a compelling approach for engineering biomaterials capable of concurrently supporting bone regeneration and suppressing microbial infection. Leveraging experimental data from our prior work on sol–gel‐derived SiO ₂ –P ₂ O ₅ –CaO–SrO–Ag ₂ O glasses (Ag ₂ O: 1–10 mol%), a novel multiscale computational framework is presented that predicts the coupled evolution of physicochemical and biological responses during in vitro degradation. The framework unifies three physically informed sub‐models: (i) a modified Jones‐type kinetics model for time‐ and composition‐dependent pH dynamics in simulated body fluid (SBF); (ii) a pH‐coupled diffusion–reaction formalism capturing the release profiles of five ionic species (Ag ⁺ , Ca ²⁺ , Sr ²⁺ , P ⁵⁺ , Si ⁴⁺ ); and (iii) dual hormesis‐based equations describing the biphasic effects of Ag ⁺ on osteoblast viability (MTT) and differentiation (ALP). All model coefficients were calibrated using experimental datasets and rigorously validated against a held‐out test subset of the experimental dataset, yielding exceptionally low coefficients of variation: 0.30% (pH), 0.92% (Ag ⁺ ), 0.60% (Ca ²⁺ ), 11% (P ⁵⁺ ), 0.72% (Sr ²⁺ ), 1.84% (Si ⁴⁺ ), and 1.85% (both MTT and ALP). Monte Carlo simulations were applied to quantify the probability of satisfying key design criteria, pH > 8.3, controlled release of Ag ⁺ , Ca ²⁺ , Sr ²⁺ , P ⁵⁺ , and Si ⁴⁺ , and enhanced osteoblast proliferation and differentiation, across 0–10 mol% Ag ₂ O and 1–21 days of immersion. The analysis identifies 5 mol% Ag ₂ O (BG‐5A) as the optimal formulation, striking an ideal balance between robust antibacterial efficacy and maximal osteogenic stimulation without inducing cytotoxicity.