Reduced-order modeling of peristaltic pumping of cerebrospinal fluid in the glymphatic system of mice

Abstract

In peripheral organs, byproducts of cellular activities are discharged into the interstitium, transported to the lymphatic system, and drained into venous blood. However, the central nervous system (CNS) lacks conventional lymphatic vessels. Growing experimental evidence demonstrates that cerebrospinal fluid (CSF), in addition to providing buoyancy to the brain and protecting it inside the skull, also serves an important pseudolymphatic function. CSF from the subarachnoid space (SAS) enters the CNS via perivascular pathways surrounding blood vessels, forming a pseudolymphatic network. A leading hypothesis calls this network the "glymphatic system". According to this hypothesis, the brain's waste removal system is formed by a network of perivascular spaces (PVSs), which are annular channels surrounding blood vessels. The inner walls of penetrating PVSs are formed by smooth muscle cells of arteries, while the outer walls are circumscribed by astrocyte endfeet (AE), constructing annular PVS pathways. PVSs surrounding pial arteries in SAS bifurcate into smaller arterioles with PVSs that penetrate the parenchyma. As penetrating arteries pierce through the parenchyma, the cross-sectional area and thickness of their surrounding PVSs decrease, vanishing eventually into capillary beds. These capillaries then merge to form venules and veins, which also have PVSs that serve as a primary outflow route. It is posited that wastes are removed from the brain by interstitial fluid (ISF) moving through the parenchyma, mixing with CSF flowing along PVSs, and ultimately draining into venous PVSs or along spinal and cranial nerves. Similar to arterial PVSs, venous PVSs are annular spaces surrounding veins within the brain parenchyma that facilitate the drainage of CSF and ISF from the brain. Recent evidence shows that disruption to the glymphatic system can lead to the accumulation of proteins such as amyloid-β (Aβ) plaques in the parenchyma, which play a key role in the development of neurodegenerative diseases such as Alzheimer's. Some aspects of the glymphatic system, particularly the driving mechanism of fluid transport and the outflow pathways, are highly debated. The experiments required to investigate the glymphatic system are quite challenging and often are unable to quantitatively characterize the flow, failing to assess the viability of different hypotheses. Mathematical and computational approaches are excellent tools to verify the feasibility of different theories and can be informative for future experiments. The diversity of modeling methods and their varying levels of physical realism lead to a wide range of results that are sometimes contradictory. In this work, we utilize reduced-order models to investigate fluid transport in the glymphatic system. Reduced-order modeling allows us to simulate fluid flow in complex geometries that are computationally intractable using higher dimensional methods while producing results much faster and without losing much accuracy. In this thesis, we developed two numerical models: (I) A one-dimensional (1D), second-order accurate finite difference model, and (II) A reduced-order model based on finite volume method (FVM). Both methods enable us to investigate fluid transport in the glymphatic system and whether different arterial pulsation waves such as spreading depolarizations (SD), cardiac pulsations (CP), and neurovascular coupling (NVC) are capable of driving CSF flow in the glymphatic system. The PVSs in the brain are represented as idealized concentric annuli surrounding blood vessels. Depending on the brain region and vessel type, PVSs may be classified as either open (non-porous) or porous. There is also the possibility of fluid exchange between the PVS and the surrounding parenchyma. We assume the inner wall of PVSs oscillates as arterial pulsation waves propagate along the vessel. We use both idealized and realistic waveforms obtained from in vivo experiments to examine flow dynamics in PVSs. The proposed methods are validated using available analytical solutions and experimental measurements. We characterize volume flow rates and pressure gradients generated by various arterial pulsation waves in pial and penetrating PVSs. We are the first to characterize pressure gradient distributions in penetrating PVSs, which play a pivotal role in regulating the movement of CSF, influencing brain tissue deformability, and maintaining brain homeostasis. Based on our findings, arterial pulsations are capable of driving CSF flow in PVSs. We find that as a solitary arterial pulsation wave propagates along the vessel, the CSF volume flow rate and pressure gradients depend intricately on different domain and wave parameters such as PVS length and width, vessel radius, waveform, wave frequency, amplitude, and wavelength. We obtain analytical expressions for pressure and volume flow rates of CSF through the PVS for a given SD wave and quantify CSF flow variations when two SD waves collide. Our model predicts that both CP and NVC waves generate comparable CSF volume flow rates in pial PVSs. Additionally, NVC waves have a more prominent role in driving CSF flow in penetrating PVSs compared to CP waves. Our approach is very general and could be extended in the future to obtain novel, quantitative insights into how CSF flow in the brain couples with seizures or externally applied neural stimulations. This approach will enable new insights into CSF flow behavior in the brain's waste removal system and pave the way for a new understanding of the mechanisms leading to neurological diseases.