Mathematical frameworks for left ventricular assist device therapy: Ventricular mechanics, blood rheology, haemodynamics, control, and nonlinear dynamics.

Ventricular assist devices (VADs) integrate multiple branches of applied mechanics within a single implanted system, spanning rotor-scale haemodynamics, nonlinear ventricular wall mechanics, blood trauma, and closed-loop control under changing physiological loads. This review aims to unify five mathematical frameworks central to VAD modelling: ventricular mechanics, blood rheology and damage, partial differential equation (PDE)-based device haemodynamics, pump engineering, and nonlinear heart-device dynamics. By bringing these domains together, the review clarifies their interactions and highlights unresolved mathematical challenges that limit progress in design, control, and prediction. An expository narrative review was conducted in accordance with the Scale for the Assessment of Narrative Review Articles (SANRA); a completed SANRA checklist is provided as Supplementary Material. Relevant literature was identified through targeted searches of PubMed, Scopus, and Web of Science, supplemented by citation tracking. Studies were selected for mathematical relevance, with emphasis on formulations that recur across VAD research, reveal model limitations, or connect analytical structure to clinically important complications. Major LVAD complications, including pump thrombosis, haemolysis, suction instability, and acquired von Willebrand syndrome, map onto distinct but interacting mathematical domains. Important cross-disciplinary links emerge between statistical mechanics and continuum damage models, between bifurcation theory and proportional-integral controller design, and between reduced-order cardiovascular models and full fluid-structure interaction simulations. Several formulations currently used in clinical, or engineering practice appear to extend beyond their original validation range. The mathematical problems underlying VAD therapy are strongly coupled and, in several areas, remain open. Advances in fluid-structure interaction theory, first-principles haemolysis modelling, and bifurcation analysis of the heart-pump oscillator could substantially improve device design, controller safety, and clinical outcome prediction.