A ballooning family: A review of models of the hemodynamic response in blood oxygen level dependent (BOLD) images.

Mechanistic modelling of the blood oxygenation level-dependent (BOLD) response is essential for interpreting functional magnetic resonance imaging (fMRI) in physiological terms. A central framework in this literature is the balloon model, which links changes in cerebral blood flow, venous blood volume, and deoxyhemoglobin content to the observed BOLD signal through a compliant venous compartment. This review examines the development of models that extend or embed this balloon/Windkessel lineage. We define the balloon framework as hemodynamic models that retain a compliant venous or Windkessel-like compartment, represent blood volume and deoxyhemoglobin as latent dynamical states, and generate BOLD contrast through balloon-type vascular behaviour. The definition presented intentionally distinguishes the balloon family from a broader set of compliant-compartment hemodynamic models. Within this scope, we review classical balloon variants, extensions involving viscoelasticity, autoregulation, oxygen-extraction assumptions, and multicompartment physiology, anatomically structured adaptations for laminar fMRI and white-matter BOLD modelling, and broader generative formulations including dynamic causal modelling, stochastic state-space models, and multimodal electroencephalogram-fMRI (EEG-fMRI) frameworks. We also consider studies of validation, sensitivity, and parameter identifiability. Across this literature, the balloon model emerges not as a single fixed formulation but as a flexible family of mass-balance-based hemodynamic models. Its continued value lies in its physiological interpretability, modularity, and compatibility with inference frameworks, while its main limitations remain limited anatomical specificity, uncertainty in neurovascular coupling, and limited identifiability from BOLD data alone.