Directional flow in perivascular networks: Mixed finite elements for reduced-dimensional models on graphs

TL;DR

These findings reveal that even long wavelength arterial pulsations can induce directional flow in asymmetric, branching perivascular networks, and establish fundamental mathematical and numerical properties of these Stokes–Brinkman network models, with particular attention to increasing graph order and complexity.

Abstract

The flow of cerebrospinal fluid through the perivascular spaces of the brain is believed to play a crucial role in eliminating toxic waste proteins. While the driving forces of this flow have been enigmatic, experiments have shown that arterial wall motion is central. In this work, we present a network model for simulating pulsatile fluid flow in perivascular networks. We establish the well-posedness of this model in the primal and dual mixed variational settings, and show how it can be discretized using mixed finite elements. Further, we utilize this model to investigate fundamental questions concerning the physical mechanisms governing perivascular fluid flow. Notably, our findings reveal that arterial pulsations can induce directional flow in branching perivascular networks.

Figures (7)

• Figure 1: The perivascular space consists of a network of annular flow channels surrounding the vasculature. We organize the channels as a graph $\mathcal{G}=(\mathcal{V}, \mathcal{E})$, with internal vertices $\mathcal{I}$ and boundary vertices $\partial \mathcal{V}$. Each branch $i$ of the network consists of annular generalized cylinders with centerline $\Lambda_i$. The channel cross-sections $C_i(s,t)$ are characterized by a given inner radius $R^1(s,\theta,t)$ and outer radius $R^2(s,\theta,t)$.
• Figure 2: Velocity profile $v^{vp}$ associated with idealized and image-based pial PVS cross-section shapes. The top and bottom rows show shapes associated with arterial and venous PVSs, respectively. We see that the asymmetry of the image-based pial artery PVS yields an increase in the velocity profile magnitude. We therefore expect a considerably lower resistance offered by this domain.
• Figure 3: Pressure $p$ and cross-section flow $q$ due to a fixed pressure drop through an arterial tree, computed using resistance due to idealized (left) and image-based (right) arterial PVS cross-sections. Due to the lower resistance in the image-based cross-sections, the pressure drop in this tree yields a larger cross-section flux.
• Figure 4: Examples of functions that are in $H^1(\Lambda)$ (left) and $H(\mathrm{div}; \mathcal{G})$ (right).
• Figure 5: The (a) arterial tree and (b) honeycomb networks used for numerical experiments. From left to right, the networks grow by the addition of more edges. The arterial tree networks are grown by adding more generations; while the honeycomb networks grow by increasing the number of cycles.

Theorems & Definitions (14)

• Remark 1: Both Stokes flow and Stokes--Brinkman flow yield Stokes--Brinkman type network models
• Lemma 3.1: Integration by parts
• proof
• Remark 2: Relation to quantum graphs
• Theorem 1
• proof
• Theorem 2: Higher regularity
• proof
• Remark 3: Connection to finite volume schemes
• Theorem 3.1