A Four-Compartment Mass-Action Model of Toxin-Driven Secondary Injury: Analytical Solutions and Global Sensitivity Analysis

Intracerebral hemorrhage (ICH), malignant gliomas, subarachnoid hemorrhage (SAH), traumatic brain injury (TBI), and several neuroinflammatory and autoimmune disorders of the central nervous system share a common downstream feature: a self-amplifying extracellular milieu of bioactive toxins (heme-iron, thrombin, lactate, kynurenine, pro-inflammatory cytokines) that impairs endogenous clearance, drives effector-cell dysfunction, and suppresses recovery. This paper develops, purely as a theoretical and computational modeling exercise, a four-compartment mass-action ordinary differential equation (ODE) system that couples residual lesion volume V(t), toxin concentration T(t), effector dysfunction E(t), and a recovery/immune-response index R(t), with a Hill-type saturable nonlinearity governing toxin-driven effector exhaustion and a switch-controlled term representing a hypothetical, idealized localized clearance-enhancing intervention.We derive the exact closed-form solution of the linearized toxin subsystem via the integrating-factor method, cross-validate it against the full numerical integration to machine precision (maximum deviation between numerical and analytical solutions of approximately 2.6 times 10 to the power of negative 11), and establish, via the Routh Hurwitz criterion, the asymptotic stability of the (V, T, E) subsystem's trivial equilibrium. We identify and explicitly correct a structural property of the recovery compartment R(t): it lacks an intrinsic decay term and therefore does not admit a finite equilibrium, growing asymptotically at a constant rate once (V, T, E) approach (0, 0, 0), and we discuss the implications of this property for model interpretation.Latin Hypercube Monte Carlo uncertainty propagation (N equals 5,000, plus or minus 15 percent parameter uncertainty) and a from first principles Saltelli (2002, 2010) variance based global sensitivity analysis (first order and total order Sobol indices, N equals 1,024 base samples, 7,168 total model evaluations) are implemented without reliance on third party sensitivity analysis libraries. Under an illustrative, biologically motivated parameter regime, the model exhibits an 80.7 percent reduction in the toxin load area under curve (AUC) metric under the idealized intervention arm relative to the reference arm (mean 80.6 percent, 95 percent confidence interval 77.7 to 83.3 percent across the Monte Carlo ensemble), with the immune boost parameter identified as the dominant driver of model variance (first order and total order Sobol index of 0.836).We emphasize throughout that all quantitative outputs describe the behavior of the mathematical model under stated assumptions and parameter choices, not observed, expected, or predicted clinical outcomes; no patient data, animal data, or empirical calibration underlie this work, and the model's translational relevance is contingent on future wet laboratory and in vivo parameter estimation that has not yet been performed. We conclude with an explicit statement of the model's structural assumptions, a set of falsifiable, model internal predictions suitable for future empirical testing, a scientific and technical risk assessment, and a staged roadmap for the empirical grounding that would be required before any translational or clinical interpretation becomes appropriate.