Multilevel Mathematical Model for Manual Therapy: Integration of Viscoelastic Mechanics, Neurophysiology of Pain and Clinical Dose-response

Abstract

Aim: To propose a multilevel mathematical model that quantitatively formalizes the mechanisms of action of Manual Therapy, integrating the viscoelastic mechanics of the tissue, the neurophysiology of pain control, and the clinical dose-response relationship in a system of coupled equations suitable as a calculation engine for clinical software.
Methodology: A three-level coupled mechanistic-theoretical model was developed: (1) a standard linear solid model with differential mechanotransduction (fast/slow adapting populations) for the tissue level; (2) a system of ordinary differential equations (ODEs) based on the mathematical formalization of Melzack and Wall's Gate Control Theory, with slow-time-scale central sensitization dynamics, for the neurophysiological level; and (3) a sigmoidal Emax dose-response curve with time-normalized effective dose, for the clinical level. The model was implemented numerically (Python, Runge-Kutta RK45 integrator) and subjected to a physiological coherence validation protocol with parametric sensitivity analysis.
Main results: The EDO system produced physiologically consistent behavior: dynamic closure of the spinal gate during the manual technique (T reduction to 85% of the baseline value), central sensitization with a time constant ε << α consistent with the slowness observed in chronic pain, and a sigmoid clinical response curve that gradually distinguishes between techniques of different intensity/duration. Sensitivity analysis confirmed that the clinical-level parameters ED50, Emax, and h are independently identifiable, a prerequisite for their calibration with real-world data. The incorporation of two mechanoreceptor populations (rapid/slow adaptation, differentiated Hill type) allowed the model to predict substantial responses for both oscillatory (Maitland) and sustained gliding techniques.
Conclusion: The model offers the first multilevel quantitative architecture that connects, in a computable way and with clinically interpretable parameters, the force applied by the therapist with the clinical change in pain. It constitutes a formal framework for: (a) generating testable hypotheses about the optimal dosage of manual technique, (b) informing the design of clinical decision support tools, and (c) guiding the design of calibration studies. Its direct clinical application requires parametric calibration with data from real patients.