Modeling of Blood Flow Dynamics in Rat Somatosensory Cortex

Abstract

Background: The cerebral microvasculature forms a dense network of interconnected blood vessels where flow is modulated partly by astrocytes. Increased neuronal activity stimulates astrocytes to release vasoactive substances at the endfeet, altering the diameters of connected vessels. Methods: Our study simulated the coupling between blood flow variations and vessel diameter changes driven by astrocytic activity in the rat somatosensory cortex. We developed a framework with three key components: coupling between the vasculature and synthesized astrocytic morphologies, a fluid dynamics model to compute flow in each vascular segment, and a stochastic process replicating the effect of astrocytic endfeet on vessel radii. Results: The model was validated against experimental flow values from the literature across cortical depths. We found that local vasodilation from astrocyte activity increased blood flow, especially in capillaries, exhibiting a layer-specific response in deeper cortical layers. Additionally, the highest blood flow variability occurred in capillaries, emphasizing their role in cerebral perfusion regulation. We discovered that astrocytic activity impacted blood flow dynamics in a localized, clustered manner, with most vascular segments influenced by two to three neighboring endfeet. Conclusions: These insights enhance our understanding of neurovascular coupling and guide future research on blood flow-related diseases.

Keywords: astrocytic endfoot; blood flow; neuro-glia-vasculature; simulation; vasodilation.

Figure A1

Effective resistance. A scatter plot depicting the effective resistance (ER) as a function of the logarithm of the exit node area. This analysis involves the calculation of effective resistance between each of the three selected entry nodes and all exit nodes in the network.

Figure A2

The calibration of the Ornstein–Uhlenbeck (OU) process. (A) The sample path of a reflected OU (ROU) process with $T=1$, $\kappa =5$, and $\sigma =4$. The solid black line represents the reflecting model, while the dashed black line denotes a realistic maximum value for the ROU process. (B) A comparison of the asymptotic OU and ROU distributions. We simulated 5000 ROU paths and constructed a histogram using the values of $X_T$ at $T=1$. The solid red line represents the ROU asymptotic distribution, the dashed red line represents the OU asymptotic distribution, and the dashed black line denotes a realistic maximum value for the ROU process.

Figure A3

Fit of a sinusoidal blood flow signal. Comparison of blood velocity signal (gray dashed line) obtained from [103], with fitted sine function $v\left(t\right)=Asin\left(2\pi ft\right)+C$ (orange continuous line) with $A=6119\mathsf{\mu }\mathrm{m}·{\mathrm{s}}^{-1}$, $f=8{\mathrm{s}}^{-1}$, and $C=35,000\mathsf{\mu }\mathrm{m}·{\mathrm{s}}^{-1}$.

Figure A4

Validation of simulation results against literature data. Distribution of blood flow in capillaries categorized by diameter plotted on logarithmic scale, with dashed lines from previous studies (see Appendix B). (A) 1–2 $\mathsf{\mu }\mathrm{m}$, (B) 2–3 $\mathsf{\mu }\mathrm{m}$, (C) 3–4 $\mathsf{\mu }\mathrm{m}$, (D) 4–5 $\mathsf{\mu }\mathrm{m}$, (E) 5–6 $\mathsf{\mu }\mathrm{m}$, and (F) 6–7 $\mathsf{\mu }\mathrm{m}$.

Figure A5

Volume discrepancy: the misalignment between the vasculature and microcircuit in the rat SSCx. The pink rectangle represents the vasculature volume, while the green one shows the cortical column volume. The (B) striped volume represents the volume of the vasculature above L1, while the (A) hatched volume represents the missing vasculature from three-fourths of L6 down to the white matter (WM).

Figure A6

The dynamical evolution of the average radii. A comparison of the average radii over time (gray dashed line), with a fitted exponential function $f\left(t\right)=a{e}^{-bt}+c$ (orange continuous line) with the fitted parameters $a=-0.23$, $b=21.38$, and $c=3.03$.

Figure 1

Model overview. (A) The left side shows a complete circuit with synthesized astrocytic morphologies developed by Zisis et al. [8]. Astrocytes are shown in blue and blood vessels in red. On the right, there is a schematic depiction of a realistic microvascular network sample, comprised of nodes and edges. A dot represents a node, two dots connected by a line represent an Section (an edge between every bifurcation), a black line framed by vertical orange lines represents a Segment (an edge between every Section), and a star represents an endfoot. (B) NGV unit. Neurons are depicted in gray, astrocytes in blue, and capillaries in red. Astrocytes contact synapses, wrap around them, and extend their perivascular projections to the surface of blood vessels, where they form endfeet. r stands for radius.

Figure 2

Model presentation. (A) The vasculature segmented into six cortical layers in the rat SSCx separated by gray planes. (B) The endfeet distribution along the cortical depth in the vasculature [8]. (C) The spatial distribution of capillaries and large vessels along the cortical depth [8]. (D) A time series illustrating the flow variations averaged across all large vessels (top) and capillaries (bottom) in each cortical layer (color-coded) in response to a three-second astrocytic stimulation (gray rectangle). Since there are no large vessels in L6, a corresponding time series is missing from the legend. (E) HA heatmap of the mean flow in the vasculature at a specific time ($t=3\mathrm{s}$) within the simulation, calculated as an average over a surface area of 17 $\mathsf{\mu }\mathrm{m}$× 21 $\mathsf{\mu }\mathrm{m}$ within large vessels (diameter $d\ge 14\mathsf{\mu }\mathrm{m}$). (F) A heatmap of the mean flow in the vasculature at a specific time ($t=3\mathrm{s}$) within the simulation, calculated as an average over a surface area of 17 $\mathsf{\mu }\mathrm{m}$× 21 $\mathsf{\mu }\mathrm{m}$ within capillaries (diameter $d\approx 4–6\mathsf{\mu }\mathrm{m}$). The vertical axis in (B,C,E,F) represents the cortical depth.

Figure 3

Dynamic analysis of blood vessel radii and flow in response to astrocytic activity. (A) Histograms showing the distribution of $RS{r}_{i,k}^e$ and (B) $RS{Q}_{i,k}^e$ for each edge connected to only one endfoot and for each time point (see Section 2.7). (C) A time series plot showing the radius dynamics for a single segment. The solid black line represents the resting state radius, the dashed black line represents the average radius over time, and the dotted black line indicates the theoretical maximal extension of the radius. (D) The time evolution of the mean radius across all blood vessels when the stimulation was halted after three seconds. (E,F) A presentation of the equivalent time series for the flow. (D–F) The solid gray line represents the stimulation period, while the dashed orange line illustrates the passive phase, after the stimulus ceased.

Figure 4

The validation of our simulation results against data in the existing literature. Blood flows (A) and velocities (B) were evaluated for both capillaries and large vessels. The mean values of the simulated flow/velocity were produced by averaging over all segments throughout the active astrocytic phase. Their distribution is shown in the boxplots. Gray dots show outliers, and orange lines indicate the median flow in blood vessels. Dot markers represent values from previous studies [73,74], color-coded by species. Error bars in turquoise–green depict the flow and velocity minimum and maximum observed for rats in [72]). The distribution of the blood flow (C) and velocity (D) in the capillaries (diameter d $\approx 4--6\mathsf{\mu }\mathrm{m}$) plotted on a logarithmic scale. The dashed lines indicate values in the literature for rats and mice. The distribution of the blood flow (E) and velocity (F) in the large vessels ($d\ge 14\mathsf{\mu }\mathrm{m}$) plotted on a logarithmic scale. Error bars in turquoise–green depict the flow and velocity ranges observed for rats in [72].

Figure 5

Layer-specific and localized astrocytic activity. (A) The average resting state flow ratio, $AR{Q}_i$ from Equation (7), during a five-second period, with the astrocytic stimulation occurring in the first three seconds. The average taken over large vessels is illustrated in orange, that for capillaries is in purple, and that for other vessels is in gray. (B) The percentage contribution of the average resting state flow ratio, $ARQ$ from Equation (8), attributed to each vessel type averaged over the initial three seconds of the stimulation. Each bar corresponds to a cortical layer, with orange representing variation due to large vessels, purple due to capillaries, and gray due to other vessels. (C,D) Violin plots presenting the distribution of the mean flow over time across the six cortical layers in capillaries (C) and in large vessels (D). Each violin corresponds to a specific cortical layer, and the height of the plot reflects the range of blood flow values within that layer. The width represents the number of segments in that range.

Figure 6

The effect of astrocytic activity on the local vasculature. (A) The order flow ratio, $OR{Q}_m$ from Equation (9), of the change in the flow between edges linked to an endfoot and their adjacent edges (see Section 2.7). The x-axis shows the order, m, defined as the number of segments starting from the one linked to an endfoot. Data are shown in gray (dotted line), the 0.1 threshold is indicated in pink (dashed line), and the neighbor threshold (order of m = 20) is depicted in blue–green (dotted line). The data can be fitted accurately by an exponential function (orange line): $f\left(m\right)=a{e}^{-bm}+c$. The final fitting parameters are $a=0.39$, $b=0.07$, and $c=0.01$. (B) The distribution of the number of edges based on their proximity to the endfeet. The x-axis indicates the number of endfeet neighbors (close to each edge), while the y-axis represents the number of edges. (C) The variation in the flow concerning the nearest endfeet. The x-axis represents the number of closest endfeet, while the y-axis illustrates the average resting state flow ratio, $ARQ$, defined in Equation (8). (D) The distribution of the absolute flow variation with respect to the nearest endfeet. The x-axis denotes the number of nearest endfeet related to each edge above a $0.1$ threshold (see (A)), while the percentage of edges is depicted on the y-axis. The left panel illustrates the distribution of flow across L2/L3, while the right panel displays the distribution of flow across L5.