Fig 1.
Schematic diagram of a cross-section of a cerebral artery demonstrating the geometry and location of the IPAD pathway.
The artery lumen is surrounded by an endothelium (yellow), which forms a BM (dark green). Several layers of SMC, which express their own type of BM (light green) surround the artery. A pial or leptomeningeal sheath (pink), which is derived from the pia mater, surrounds the outside of the artery. Arteries are covered tightly by astrocyte end-feet (grey), forming the pial-glial BM (glia limitans) [11, 14, 15].
Fig 2.
Effect of the release of K+ and Glu into the synaptic space of a neuron and an astrocyte process.
Glu here refers to the ratio of bound/unbound Glu receptors (dimensionless). The arterial radius is modelled using a system of ODE [18] with corrections to the equations listed in [24]. The model shows dilatation of the artery of 20%. This figure only shows the input (a) and result (b) of the functional hyperaemia model. We refer the reader to the original and reimplementation publications for the full details of the chemical cascade implemented in the model and the source code.
Fig 3.
IPAD inside a cerebral arteriole.
(a) Displacement and stress of the arteriole wall due to U(t) at t = 20 s of a single astrocyte end-foot. Because displacement is fixed to u = 0 at the ends, stresses at the ends are high. Thus, all following results are presented for 10 μm ≤ z ≤ 190 μm. (b) IPAD velocity at various time points over the length of the arteriole wall using K0/K1 = 0.1. The average velocity over time and space is −8.37 μm s−1. (c) IPAD flow rate at various time points over the length of the arteriole wall using K0/K1 = 0.1. The average flow rate over time and space is −7.24 × 10−8 μl/min for a single arteriole. Extrapolated over 6.5 billion arterioles estimated for the human brain it would take 9.92 h to process the total amount of ISF in the brain (280 ml). (d) IPAD velocity for k = 1 × 10−3 (blue), 1 × 10−4 (orange), 1 × 10−5 μm2 (green) over the strength of the valve mechanisms K0/K1. Values are always negative, except at K0/K1 = 1.0. The effect of the valve mechanism is decreased with decreasing k.
Fig 4.
Comparison of IPAD velocities in an arteriole for varying numbers of astrocytes acting on the wall and strength of the valve-like mechanism.
An astrocyte end-foot is modelled with a width of 10 μm and gap between astrocytes 1 μm. Length has a negative effect on IPAD whilst the number of astrocytes acting on the arteriole simultaneously has a positive effect. k = 1 × 10−3 μm2, arteriole length l = 309 μm, number of astrocytes: 10 (black), 5 (white), 2 (gray), 1 (dotted).