About the Marty model of blood-brain barrier closure after its disruption using focused ultrasound

Many studies have demonstrated that pulsed ultrasound combined with circulating microbubbles can permeate the blood-brain barrier in a reversible manner. In 2012, our group demonstrated that the BBB remains permeable to small MRI contrast agents up to 24 h after ultrasound application and also that this duration was dependent on nanoparticle size. We derived a simple theoretical model explaining these observations (Marty et al 2012 J. Cereb. Blood Flow Metab. 32 1948-58). However, in this original paper the expression of the BBB closure time (t _1/2) as a function of the size of delivered contrast agents (d _H) could not be mathematically derived from the model but rather from a guessed function that is fit to the numerical solution of the model. In this context, the two numeric parameters of this fitting function could not be related to the other physical parameters of the model. Here, we present a formal solution, finding the same expression of t _1/2 already published and linking t _1/2 to relevant physical variables such as the molecular hydrodynamic diameter d _H, the BBB closure rate k and the standard deviation of the initial BBB gap sizes distribution σ ₀.