A Conceptual Stochastic Differential Equation Framework for Quality-of-Life Restoration in Oncology: Simulation Study and Sensitivity Analysis

Despite robust evidence supporting early integration of palliative care, the paradigm remains widely perceived by patients as synonymous with ``end-of-life mitigation'', leading to refusal rates of 30--68\% and delayed referrals \citep{sanders2024,bandieri2023,salins2022,alexander2026}. This paper introduces the \textbf{Restorative Oncology Empowerment Framework (ROEF)}, a conceptual scalar Itô stochastic differential equation (SDE) model (approximating the multidimensional QoL vector) that shifts the ontological focus from symptom mitigation to active restoration of pre-morbid quality-of-life (QoL) trajectories. QoL is modelled as a scalar state \(Q(t) \in [0,100]\) (global health status component of EORTC QLQ-C30) governed by nonlinear drift and multiplicative diffusion terms. The exact analytical solution of the underlying linear ODE is derived, full stochastic Euler-Maruyama Monte Carlo simulations (1000 paths, seed 42), Latin Hypercube sampling (n=1000), bootstrap confidence intervals (10,000 resamples), simple MCMC Bayesian updating (Metropolis-Hastings, 10,000 iterations), and complete global sensitivity analyses (One-at-a-Time, Morris method with 100 paths, Sobol Saltelli n=2000 with pure NumPy implementation, FAST approximation with 1000 samples) are performed. All results are generated from a single self-contained Python 3.12 engine. The model is benchmarked against ASCO 2024 guidelines, generalised to chronic heart failure (CHF), and subjected to Popperian falsifiability criteria. Parameters are calibrated from real longitudinal data \citep{versluis2024,kuan2024,lee2022}. ROEF delivers projected gains consistent with observed clinical ranges. All code, derivations, phase portraits with exact coordinates (derived from executed drift function), Lyapunov stability proofs, detailed Fokker-Planck analysis (including transient and stationary distributions), uncertainty quantification, and 25 recent references (2022--2026) with DOI/PMC/arXiv IDs are provided. \textbf{Keywords:} Restorative Oncology, Stochastic Differential Equations, Sobol Sensitivity Analysis, Quality-of-Life Restoration, Monte Carlo Simulation, Bayesian Inference, Chronic Heart Failure, Palliative Care Stigma, Falsifiability, Fokker-Planck Equation, Uncertainty Analysis