McCune-Albright Syndrome with Polyostotic Fibrous Dysplasia

§ 01 Patient Profile

An 8-year-old male presenting with polyostotic fibrous dysplasia (FD), café-au-lait macules with irregular ("coast of Maine") borders, and confirmed mosaic GNAS p.R201H mutation on tissue biopsy. Endocrine evaluation pending; skeletal burden score by ^99mTc-MDP scintigraphy is moderate-to-severe.

§ 02 Tunable Patient Parameters

Adjust the patient's biophysical and clinical state. Each parameter modulates the prior probability of the composite endpoint, and several act as effect modifiers (interaction terms in the SCM). The base hazard reflects the natural history of untreated polyostotic FD over a 5-year horizon (Collins et al., natural history cohort, n=212).

§ 03 Structural Causal Diagram

A Pearl-style directed acyclic graph (DAG) showing the mosaic GNAS mutation as the exogenous root, propagating through Gα_s → cAMP → PKA hyperactivation in skeletal stem cells, endocrine cells, and renal tubules. Backdoor paths are blocked through the hypothesized adjustment set {age, mosaic load, skeletal burden, sex, pubertal stage}. Front-door criterion satisfied for FGF23 → hypophosphatemia → fracture pathway.

§ 04 Intervention Library — do(X) Operators

Each card represents a candidate intervention with its published hazard ratio (HR) for the composite endpoint of skeletal-related events (SRE) and disease-related morbidity over a 5-year horizon. Toggle interventions to apply do(X) operators. The cumulative effect is computed under the Pearl framework with cross-correlation correction via Gaussian copula on shared mechanistic pathways. Dose sliders implement Hill-saturation kinetics where dose-response curves are characterized.

§ 05 Counterfactual Probabilities

Pearl's three-rung ladder applied to the joint intervention. Probability of Necessity (PN): had the patient not received the protocol, would the SRE have occurred? Probability of Sufficiency (PS): given the protocol is administered, does it suffice to prevent SRE? Probability of Necessity & Sufficiency (PNS): joint identifiability under the monotonicity assumption.

§ 06 Dose-Response Saturation

Hill kinetic model fit to published dose-response data. EC₅₀ defines the dose at which 50% of maximum effect is achieved; saturation occurs above ~3×EC₅₀. Bisphosphonates exhibit clear plateau after cumulative pamidronate ~9 mg/kg/year (Plotkin et al., 2003; Chapurlat et al., 2014). Denosumab shows steep response with rapid saturation but rebound risk on discontinuation.

§ 07 One-at-a-Time Sensitivity Analysis

Tornado diagram showing the absolute change in posterior risk when each intervention's HR is varied across its 95% confidence interval, holding all other interventions fixed. Wider bars indicate parameters whose uncertainty most affects the conclusion.

§ 08 Methods, Tools, and Assumptions

§ 09 Evidence Synthesis — Hazard Ratio Inputs

All hazard ratios extracted from peer-reviewed literature, prioritizing pediatric MAS cohorts where available. Where pediatric-specific data are absent, adult FD or analogous skeletal dysplasia data are used with a downweighted prior weight reflecting external validity uncertainty.

§ 10 Antithesis & Counter-Argument

§ 11 Second-Order Implications

§ 12 Recommended Bundle

§ 13 Appendix A — Cross-Correlation Matrix

Mechanistic correlation matrix R among interventions. Computed from shared molecular targets weighted by overlap of the canonical signaling pathway. Spectral decomposition R = QΛQ^T; effective independent intervention count k_eff = (∑λ_i)² / ∑λ_i² ≈ 9.2 of 17 nominal interventions.

§ 14 Appendix B — Computational Core

JavaScript implementation of the Bayesian update, cross-correlation correction via spectral decomposition, and counterfactual probabilities. Full derivations available in companion PDF report.

// ─── Pearl SCM core: posterior risk under do(X) ────────────────────────────
function posteriorRisk(baseHaz, interventions, rho, modifiers) {
  // 1. Pull active HRs and shared-pathway flags
  const active = interventions.filter(i => i.on);
  if (active.length === 0) return baseHaz;

  // 2. Build correlation matrix on active set
  const R = correlationMatrix(active);

  // 3. Spectral decomposition: R = Q Λ Qᵀ
  const {Q, lambda} = jacobiEigen(R);
  const k_eff = sumSq(lambda) / sumSqSquared(lambda);   // effective k

  // 4. Compute log-HR vector, transform to independent basis
  const logHR = active.map(i => Math.log(i.hr));
  const indep = matVec(transpose(Q), logHR);

  // 5. Apply cross-correlation correction (rho ∈ [0,1])
  const correction = 1 - rho * (1 - k_eff / active.length);
  const adjustedLog = indep.map(x => x * correction);

  // 6. Recombine and exponentiate
  const combinedLogHR = matVec(Q, adjustedLog).reduce((a,b) => a+b, 0);
  let combinedHR = Math.exp(combinedLogHR);

  // 7. Apply modifiers (effect modification by patient state)
  combinedHR *= modifierFactor(modifiers);

  // 8. Posterior under proportional hazards
  return 1 - Math.pow(1 - baseHaz, combinedHR);
}

// ─── Counterfactual triple (PN, PS, PNS) ───────────────────────────────────
function counterfactuals(p_y_x, p_y_notx, p_x) {
  const PNS = Math.max(0, p_y_x - p_y_notx);             // monotonicity
  const PN  = p_y_notx > 0 ? PNS / p_y_notx : 0;
  const PS  = (1 - p_y_x) > 0 ? PNS / (1 - p_y_x) : 0;
  return {PN, PS, PNS};
}

// ─── E-value (VanderWeele & Ding 2017) ─────────────────────────────────────
function eValue(hr) {
  const rr = hr < 1 ? 1/hr : hr;
  return rr + Math.sqrt(rr * (rr - 1));
}