Model estimation of cerebral hemodynamics between blood flow and volume changes: a data-based modelling approach

It is well known that there is a dynamic relationship between cerebral blood flow (CBF) and cerebral blood volume (CBV). With increasing applications of functional magnetic resonance imaging (fMRI), where the blood oxygen level dependent (BOLD) signals are recorded, the understanding and accurate modelling of the hemodynamic relationship between CBF and CBV becomes increasingly important. This study presents an empirical and data-based modelling framework for model identification from CBF and CBV experimental data. It is shown that the relationship between the changes in CBF and CBV can be described using a parsimonious autoregressive with exogenous input model (ARX) structure. It is observed that neither the ordinary least squares (LS) method nor the classical total least squares (TLS) method can produce accurate estimates from the original noisy CBF and CBV data, in that the resultant ARX models may be unstable and thus cannot generate stable model predicted outputs. A regularized total least squares (RTLS) method is employed and extended to solve such an error-in-the-variables problem. Quantitative results show that the RTLS method works very well on the noisy CBF and CBV data. Finally, a combination of RTLS with a filtering method can lead to a parsimonious but very effective model that can characterize the relationship between the changes in CBF and CBV.