Fig 1.
Three compartment model used for modeling.
The brain was divided into the brain parenchyma, containing the neurons (N) and astrocytes (A); the perivascular space (PVS, blue strip) in the area between the astrocytic foot processes and the brain endothelial cells; and the cerebral blood vessels (B). Interstitial (ISF) flow is unidirectional and laminar in the regions lined by capillaries; its rate is dependent on heart rate (eg. the rate of arterial pulsations) as well as the stiffness of the vessels, which is a function of the presence of atherosclerosis, cerebral amyloid angiopathy (deposition of beta amyloid within the brain vessels) and stiffening secondary to prolonged elevations in systemic blood pressure.
Fig 2.
Flow of brain interstitial fluid in response to pulsations of cerebral arteries and arterioles.
Interstitial fluid is generated at the brain endothelium by an unclear process [30] and surrounds neurons, microglia astrocytes and other cells located within the brain parenchyma, bringing nutrition to the cells and removing wastes. Although its chemical composition is similar to cerebrospinal fluid (CSF), the two fluids serve separate purposes. CSF gives the brain buoyancy and buffers against forces applied to the head; CSF is generated by ependymal cells within the choroid plexus [30]. Brain ISF transports waste from the brain parenchyma via a combination of convection and diffusion towards the perivascular space. At the perivascular space, molecules such as Aβ, are either transported by receptors at the blood-brain barrier, or are transported along the glymphatic pathway. A small percentage of molecules transported in the perivascular space are transported into the cerebrospinal fluid at the arachnoid granulations and are cleared from the brain via CSF drainage pathways. The majority (~60%) of Aβ is transported along the glymphatic system to the cervical lymph nodes [13].
Table 1.
Volume of the human hippocampus as determined by MRI.
Table 2.
Rate constants and initial conditions used in modeling.
Fig 3.
Simulation demonstrating effects of normal aging on Aβ levels within the brain parenchyma and local cerebral vessels.
(A) Deposition of Aβ40 occurred much more rapidly than that of Aβ42 with approximately 6 times the amount of Aβ40 deposited within cerebral vessels. This ratio agrees with what has been observed experimentally. The rate of Aβ increase started to decrease over the simulation as the number of neurons declined with aging and from elevations in the Aβ concentration. (B) The number of endothelial cells (blue) decreased slowly as they entered senescence. Microglia (green) decreased about 80% from their initial values, correlating with conversion to dystrophy. Neurons (red) also decreased secondary to natural loss as well as loss from elevated Aβ levels. (C) The total number of LRP-1 receptors decreased to approximately 60% of the initial value in accordance with the loss of brain endothelial cells.
Fig 4.
Effect of heart rate on Aβ clearance and deposition.
(A, B) Decreasing the heart rate by only 10 beats per minute (60 to 50) led to nearly a 20% increase in the Aβ deposition in the brain parenchyma and a 5% increase in the cerebral vasculature. (C, D) Increasing heart rate by 30 beats per minute (60 to 90) led to the converse, with Aβ deposition levels in brain parenchyma decreasing about 30% and in the cerebral vessels decreasing about 10%.
Fig 5.
Effect of vessel stiffness on Aβ clearance and deposition.
(A) Increasing the stiffness slowly by a factor of 2 during aging led to a significant increase in the amount of Aβ deposited within the brain parenchyma, with a decrease in the amount reaching and thus depositing in cerebral vessels. (B) The high Aβ levels in the brain parenchyma led to accelerated neuronal loss with no change observed in microglia or brain endothelial cell number.
Fig 6.
Effect of ApoE allele on Aβ clearance and deposition.
(A) Levels of Aβ40 deposition in cerebral vessels increased nearly two-fold with only a 67% increase in Aβ42 deposition there. No changes from baseline simulations were noted in Aβ levels in the brain parenchyma. (B) No changes in cell number were observed compared to baseline values for aging.
Fig 7.
Combination effect of bradycardia, ApoE4 allele and increased vessel stiffness.
(A) Having both bradycardia and the ApoE4 allele led to elevated levels of Aβ40 deposition in both the brain parenchyma and the cerebral vessels, with nearly a two-fold increase in Aβ40 in the cerebral vessels. There was a small decrease in the number of neurons at the final simulation endpoint (<10%). (B) In the case where the simulated patient was supposed to have cardiovascular disease (represented by increased vessel stiffness) and carry the ApoE4 allele, Aβ deposition was significantly elevated within the brain parenchyma (100x and 1000x for Aβ40 and Aβ42, respectively). The decreased levels of Aβ within the cerebrovasculature was not quite as significant as with increased vessel stiffness alone, suggesting that the ApoE allele plays an important role in determining deposition and Aβ concentration within the space near cerebral vessels. The significantly elevated Aβ levels led to a decrease in the neuronal cell number secondary to increased neuronal death rate seen at high Aβ levels. (C) The results of increased vessel stiffness in the presence of bradycardia led to results that were similar to those seen with increased vessel stiffness alone.
Table 3.
Summary of simulation results.
Fig 8.
Sensitivity analysis of the model.
(A)The sensitivity of the model to values that were estimated was studied to determine the range over which the model produced stable simulation results. Changing the initial concentration of Aβ from ½ x to 2x led to no changes in simulation result and demonstrating stability of this rate constant. (B) Changing the amount that Aβ generation would increase in response to a stimulus (k4) showed that this parameter had a relatively narrow range of stability.