PINN-ing the Balloon: A Physically Informed Neural Network Modelling the Nonlinear Haemodynamic Response Function in MRI

Accurate characterisation of the haemodynamic response function (HRF) is central to interpreting blood-oxygen-level-dependent (BOLD) signals in functional magnetic resonance imaging, yet standard estimation approaches remain centred around phenomenological formulations lacking biophysical grounding. We present a physics-informed neural network (PINN) framework that bridges these paradigms by embedding the Balloon-Windkessel model directly into the training objective of a multi-headed Neural Network. Our aproach simultaneously estimates probable latent neurovascular state variables such as cerebral blood inflow, metabolic rate of oxygen consumption, blood volume, and deoxyhaemoglobin content, through an indirect optimisation scheme in which the predicted BOLD signal is obtained via convolution of the estimated HRF with experimental stimuli. Training is governed by a composite loss, balancing differential-equation residuals, physiological initial conditions and data fidelity. In simulations with temporal signal-to-noise ratios representative of clinical acquisitions, the framework recovered ground-truth state variables with coefficients of determination exceeding 0.99 and mean squared errors below 10-3, at a physics-to-data weighting of 0.40:0.60. Application to 1.5 T block-design fMRI data from an ischaemic stroke patient yielded physiologically plausible, subject-specific HRF estimates, establishing feasibility of single-subject, physics-constrained HRF inference without reliance on fixed gamma basis assumptions.To our knowledge, this constitutes the first deployment of a single PINN incorporating the full Balloon-Windkessel model within an indirect training objective, reconstructing full BOLD observations, positioning PINN-based haemodynamic modelling as a principled and personalised route towards more interpretable and patient-specific fMRI biomarkers.