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 (Sr²⁺) 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 SiO₂–P₂O₅–CaO–SrO–Ag₂O glasses (Ag₂O: 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⁺, Ca²⁺, Sr²⁺, P⁵⁺, Si⁴⁺); 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% (Ca²⁺), 11% (P⁵⁺), 0.72% (Sr²⁺), 1.84% (Si⁴⁺), 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⁺, Ca²⁺, Sr²⁺, P⁵⁺, and Si⁴⁺, and enhanced osteoblast proliferation and differentiation, across 0–10 mol% Ag₂O and 1–21 days of immersion. The analysis identifies 5 mol% Ag₂O (BG-5A) as the optimal formulation, striking an ideal balance between robust antibacterial efficacy and maximal osteogenic stimulation without inducing cytotoxicity.