A minimal ion–chemistry model for predicting benign paroxysmal positional vertigo risk based on endolymphatic calcium and pH

Benign paroxysmal positional vertigo (BPPV) arises from detachment of otoconia-calcium carbonate (CaCO₃) crystals embedded in a protein matrix-whose stability depends on endolymph ionic composition and <i>pH</i>. Age-related calcium metabolism, acid-base imbalance, and hormonal factors can impair otoconia integrity, yet, to our knowledge, no prior quantitative model integrates these biochemical parameters to predict BPPV risk. Beyond the established mechanical mechanisms of canalithiasis and cupulolithiasis, we introduce a parsimonious biochemical model in which the endolymphatic saturation index (Ω), governed by <i>pH</i> and ionized calcium [<i>Ca</i> ²⁺], delineates an otoconia stability-dissolution boundary (Ω≈1) and complements the mechanical framework. Using carbonate-equilibrium chemistry and the CaCO₃ solubility product (<i>K</i> _<i>sp</i> ), we compute Ω and derive the critical calcium concentration <i>C</i> _<i>crit</i> (<i>pH</i>) at Ω = 1. A logistic mapping of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>Δ</mml:mtext> <mml:mi>C</mml:mi> <mml:mo>=</mml:mo> <mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow> <mml:mrow><mml:mi>c</mml:mi> <mml:mi>r</mml:mi> <mml:mi>i</mml:mi> <mml:mi>t</mml:mi></mml:mrow> </mml:msub> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow><mml:mi>p</mml:mi> <mml:mi>H</mml:mi></mml:mrow> <mml:mo>)</mml:mo></mml:mrow> <mml:mo>-</mml:mo> <mml:mrow><mml:mo>[</mml:mo> <mml:mrow><mml:mi>C</mml:mi> <mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn> <mml:mo>+</mml:mo></mml:mrow> </mml:msup> </mml:mrow> <mml:mo>]</mml:mo></mml:mrow> </mml:math> yields a dimensionless relative-risk score. Systemic and environmental states are represented as shifts in <i>pH</i> and [<i>Ca</i> ²⁺], and a synthetic cohort (<i>N</i> = 10,000) visualizes <i>pH</i>[<i>Ca</i> ²⁺] risk contours and the Ω = 1 boundary. States with Ω>1 (supersaturation) predict otoconia stability, whereas Ω < 1 (undersaturation) predicts dissolution; hyperventilation and thiazide diuretics tend to increase Ω, while metabolic acidosis, hypoventilation, and loop diuretics reduce it; acetazolamide (carbonic-anhydrase inhibition) typically induces metabolic acidosis and therefore lowers Ω. The combination of low <i>pH</i> and reduced [<i>Ca</i> ²⁺] markedly expands the Ω < 1 dissolution-prone domain, with the Ω = 1 contour acting as a dynamic equilibrium sensitive to small biochemical changes. In simulations, the risk distribution was right-skewed (mean <i>R</i>≈0.78; 80% with <i>R</i>>0.68). Because direct endolymph sampling is impractical, we propose serum ionized calcium together with blood-gas-derived <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi> <mml:mi>H</mml:mi> <mml:mo>/</mml:mo> <mml:mi>H</mml:mi> <mml:mi>C</mml:mi> <mml:msubsup><mml:mrow><mml:mi>O</mml:mi></mml:mrow> <mml:mrow><mml:mn>3</mml:mn></mml:mrow> <mml:mrow><mml:mo>-</mml:mo></mml:mrow> </mml:msubsup> <mml:mo>/</mml:mo> <mml:mi>p</mml:mi> <mml:mi>C</mml:mi> <mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:msub> </mml:math> as non-invasive surrogates for relative-risk inference (a blood-based Ω proxy), not one-to-one estimators of absolute vestibular chemistry. This deterministic, two-input minimal framework is hypothesis-generating and complementary to the mechanical model; prospective, surrogate-based calibration and robustness testing (to <i>C</i> _<i>T</i> , ionic strength/activity coefficients, <i>K</i> _<i>sp</i> , and temperature) are required before clinical use.