0967-3334/42/5/054003
Abstract
The brain's inability to store nutrients for more than a few seconds makes it one of the most tightly regulated systems in the body. Driven by metabolic demand, cerebral autoregulation (CA) ensures a constant cerebral blood flow (CBF) over a ±50% change in arterial blood pressure (ABP) from baseline. Recent evidence suggests that pericytes, contractile cells in the capillary bed, play a previously-ignored regulatory role. To elucidate the CA phenomenon, the role of oxygen metabolism, pericyte activity and neural signaling in CBF modulation were quantified. Driven by nutrient metabolism in the tissue and pressure sensitivity in the vasculature, the model introduced here successfully replicates CA. To highlight the role of different vessel sizes, vessels with a diameter above 1 mm were represented using a lumped parameter model while the microvasculature was illustrated as a branching tree network model. This novel approach elucidated the relationship between the microvasculature's nutrient supply and arterial regulation. Capillary responses to local increases in neuronal activity were experimentally determined, showing that pericytes can increase the diameter of the adjacent vessel by 2.5% in approximately 1 s. Their response was quantified and included in the computational model as an active component of the capillary bed. To compare the efficacy model presented here to existing ones, four feedback mechanisms were tested. To simulate dynamic CBF regulation a 10% increase in ABP was imposed. This resulted in a 23.79%–34.33% peak increase in CBF, depending on the nature of the feedback mechanism of the model. The four feedback mechanisms that were studied significantly differ in the response time, ultimately highlighting that capillaries play a fundamental role in the rapid regulation of CBF. Conclusively, this study indicates that while pericytes do not greatly alter the peak CBF change, they play a fundamental role in the speed of regulation.
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To ensure an adequate tissue perfusion, the brain relies on a tightly regulated system of vessels. Both large and small, each vessel category has a specific and indispensable function. Arteries operate at the highest blood pressure, regulating flow through a smooth muscle cell (SMC) driven compliance (Cipolla 2016). Capillaries penetrate the white matter and supply the tissue with nutrients through a network of predominantly bifurcating vessels (Cassot and Lauwers 2006). Ultimately, nutrient-poor blood flows back to the heart through a series of veins and sinuses.
Being highly sensitive to perfusion changes, the brain relies on an autoregulation mechanism, driven by an active response of the arteries and arterioles. The regulatory drivers differ in nature, being either global (i.e. set by the systemic circulation) or local (i.e. set by changes in metabolic demand) (Payne 2006, 2016). Global and local regulating mechanisms are interlinked and changes in one therefore result in an alteration in behavior of the other. Sometimes these mechanisms compensate for each other's deficiency, yet the magnitude of their importance of governing agents is yet to be defined.
Nutrient supply takes place in the microvasculature of the brain, targeting the capillary network. Pericytes, contractile cells found on capillary vessels, have been recently proved to actively regulate capillary vessel diameter (Burdyga and Borysova 2014). A dysfunction in this further regulating mechanism is linked to several pathological conditions (Hall et al 2014, Attwell et al 2016).
Numerous studies (Hall et al 2014, Sweeney et al 2019) support the notion that sufficient tissue perfusion is essential for a healthy brain function. Indeed, low cerebral blood flow (CBF) has been associated with dementia onset and tissue atrophy. Alzheimer's disease is also linked to a lack of tissue perfusion, underlining the importance of adequate oxygenation levels (Sweeney et al 2019). Neurons are particularly susceptible to drops in oxygen supply, as almost 75% of the brain's energy supply is consumed by the neural network (Bélanger et al 2011). Low levels of partial tissue pressure are characteristic of several ischemic conditions, such as stroke and neurogenesis (Hall et al 2014).
Imaging alone has proven to be an insufficient tool when investigating the mechanisms of CBF regulation within the microvasculature, as human imaging data are restricted to a resolution of order 1 mm (Wintermark et al 2005, Cheng et al 2017). Mathematical models therefore represent a suitable approach for investigating the coupling between flow and metabolism which occurs at the microvasculature length scale. Here, a metabolic demand-driven model on the arteriole scale was coupled with a pericyte activity-driven one on the capillary scale.
Metabolism is considered one of the key drivers of vasodilation (Reglin et al 2009, Bulte et al 2012) with oxygen partial tissue pressure acting as a modulator of arterial compliance. Furthermore, the activity of pericyte cells was represented as a dilation of the capillary vessels, further contributing to an active regulation of CBF. Ultimately, neural activity was introduced to emulate a fast vasoreactivity, resulting in a brain vasculature model whose time response for dynamic changes of CBF faithfully represents reality.
To achieve this, a microvasculature network of arterioles, capillaries and venules was coupled with a lumped parameter model of large arteries and veins. This resulted in a model of the complete brain vasculature whose local and global responses were analyzed. The activity of pericytes, defined by a variable capillary diameter in the model, was derived from experimental data on capillary diameter variation in mice. Finally, the time response of the model to changes in pressure was analyzed, aiming to disentangle the individual and cumulative role of the proposed CBF regulators.
With the aim of designing a physiologically representative CBF regulation mechanism, the computational model presented here comprises two parts: the first part consists of large arteries and veins whose response is modeled through an electrical equivalent model while the second part extrapolates from experimental data on pericyte-induced dilation, which are included in a microvasculature network model. Figure 1 is a representation of the multi-scale model proposed here.
Dynamic changes in flow were calculated as described in Payne and El-Bouri (2017). This formulation of flow calculation was designed particularly for dynamic changes in flow in vessels whose diameter is smaller than 1 mm. Therefore, inlet and outlet flow were calculated for each generation using the following equations:
Figure 5. Model fit of capillary dilation for (a) large (6–10 μm) and (b) small (2–6 μm) vessels.
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Standard image High-resolution image3.1.2. Response time increases with depth
The collected data are from a wide range of imaging depths 0–350 μm. To investigate the depth-dependent dilation properties of pericytes, the total range was divided into 4 categories: 0–50, 50–150, 150–250 and 250–350 μm. Figures 12–15 in the appendix show the imaged capillary dilation for each vessel depth category. Figure 6 shows that there are slight differences in peak dilation at different imaging depths. It is worth noting that the vessel number and diameter differ for each depth category.
Figure 6. Model fit for depth-dependent capillary dilation where: (a) 0–50 μm, (b) 50–150 μm, (c) 150–250 μm and (d) 250–350 μm.
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Standard image High-resolution imageTable 1 Summarizes the depth-dependent vessel classification and table 2. Summarizes the respective imaging depth time constant parameters, the mean values, standard deviations and error bars. The linear regression, p-value and R-squared values of each time constant are also included.
Table 1. Imaging depth-dependent vessel classification.
| Imaging depth | 0–50 μm | 50–150 μm | 150–250 μm | 250–350 μm |
| Vessel | — | V4, V15, V22 | V11, V16, V23 | V1, V18 |
| Peak dilation (%) | — | 2.42 | 2.37 | 3.2 |
| Diameter (2–6 μm) | ||||
| Imaging depth | 0–50 μm | 50–150 μm | 150–250 μm | 250–350 μm |
| Vessel | V13 | V2, V7, V24 | — | — |
| Peak dilation (%) | 2.1 | 2.79 | — | — |
3.1.3. Statistical significance
A statistical analysis was performed to investigate the differences in response to Ca2+ stimuli, between large capillaries (6–10 μm diameter) and small capillaries (2–6 μm diameter) and imaging depths:
- Diameter significance A statistical analysis (two sample t-test) showed that there is a statistically significant difference p=0.034 between the peak dilation of the large and small capillary vessels.
- Imaging depth significance It was shown that the response of vessels also differs for various imaging depths. To compare four groups of imaging depths the ANOVA test was applied. A statistically significant difference was proven p < 0.001. A post-hoc analysis showed that all groups are statistically different from each other.
- Time constants variation
The t-test was performed in order to investigate the statistical significance of the difference of the time constant parameters in the small and large capillary vessels. The comparison of the time constants revealed no statistical significance (p = 0.53), yielded a positive statistical significance (p = 0.025), whereas resulted in no statistical significance (p = 0.35). This suggests that compared to large capillaries (6–10 μm diameter), the smaller capillaries (2–6 μm diameter) have an overall slower response time.
3.1.4. Calcium trace
An analogous analysis was performed for the Ca2+ stimuli peak detection and response time. Figure 7(a) shows the Ca2+ trace for the eight selected vessels and figure 7(b) illustrates the function fit for the Ca2+ data points.
Figure 7. (a) Averaged Ca2+trace; (b) fitted function to the Ca2+trace for the eight vessels selected to be included in the computational model.
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Standard image High-resolution image3.1.5. Capillary CMRO2, CBF and CBV
As capillaries increase in diameter, so does the blood flow and volume. Figure 8 illustrates how the 2.5% increase in vessel diameter corresponds to a 5.53% increase in CBF, a 3.25% increase in CBV and a 6.09% increase in CMRO2. The relationship between CBF and CBV was defined by Grubb as:
where α is Grubb's exponent whose original value is 0.38. Given that this value was derived from rhesus monkeys, a number of studies have attempted to find a better fit for relating the two parameters in humans. At present, there is still a lack of consensus and the proposed values range from 0.29 (Ito et al 2005) to 0.73 (Rostrup et al 2005). Here, the value used for α is 0.73, to best fit the relationship between CBV and CBF, where the increase in volume corresponds to about half the increase in flow.
Figure 8. As a response to neural activation, signaled by the increase in Ca2+ in the tissue surrounding the vessel, CBV increases by 3.25% (yellow); CMRO2 increases by 6.09% (blue); CBF increases by 5.53% (red).
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Standard image High-resolution imageTo investigate the CMRO2 change in the parenchyma around the capillary, resulting from the visual stimuli-induced increase in Ca2+ concentration, an average response was computed from the total of all stimuli across the 36 vessels. As outlined in section 2.1, this second data set was used to study changes in oxygen metabolism, as changes in CBF and changes in vessel diameter cannot be measured simultaneously.
3.2. Model simulations
The ability of the four feedback mechanisms to physiologically accurately replicate CBF regulation for variations of ABP, was evaluated under static and dynamic conditions. In the steady state, CBF was maintained constant over a wide range of ABP (40–150 mmHg) while in the dynamic state, a 10% increase in ABP resulted in a 20%–35% temporary increase in CBF, depending on the feedback mechanism that was used.
3.2.1. Steady state cerebral autoregulation (CA)
Solving the equations under steady state conditions, the change of CBF was observed over a wide range of ABP (40–150 mmHg). All four feedback mechanisms proved to be in good agreement with the literature (Ursino et al 1998) as CBF was maintained constant between 60 and 120 mmHg (see figure 9). The models showed however a clear difference in the range of the upper and lower autoregulation limit. The arterial compliance of the flow feedback model clearly saturated more promptly, resulting in a sudden drop in flow below 60 mmHg of ABP and a rapid increase above 120 mmHg of ABP. In comparison, the range of autoregulation of the oxygen and pericyte feedback model is larger, stretching from about 50 mmHg to 140 mmHg. Frequently cited data of cats and mice (Ursino et al 1998) were used to validate the steady state (F1 + F2 + F3) feedback model. It should be noted that the upper and lower autoregulation limits strongly depend on the parameters of the compliance equation. The passive presence of pericytes was also studied by imposing a maximum increase in capillary diameter (2.5%), derived from the fitted function in section 3.1. The steady state analysis showed no statistically significantly different behavior compared to the active based one.
Figure 9. (a) Flow feedback-based CA model; (b) oxygen, pericytes and neural feedback-based (F1 + F2 + F3) CA model fitted against experimental data reported by Ursino (1998).
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Standard image High-resolution image3.2.2. Dynamic CBF regulation
To further investigate the physiological accuracy of the proposed computational model, the time response was evaluated for each of the four feedback mechanisms. Indeed, the rapidity of the response can serve as an indicator for the overall ability of each regulatory variable (oxygen, pericytes and neurons) in regulating CBF when ABP changes. To investigate the time response to changes in oxygen partial tissue pressure, the governing equations of the model were solved under dynamic conditions and a 10% increase in ABP was imposed at time t = 0 s of the solver (see figure 10).
Figure 10. CBF variation upon a 10% increase in ABP for 4 scenarios: a flow feedback (F0), an oxygen-feedback (F1), an oxygen and pericytes feedback (F1 + F2), and an oxygen, pericytes and neural feedback (F1 + F2 + F3).
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Standard image High-resolution imageAt the time of the impulse, CBF increased by 23.79% in the flow-based model (F0), by 30.58% in the oxygen-based model (F1), by 34.33% in the model including pericytes (F1 + F2) and by 26.32% in the one including neural activation (F1 + F2 + F3). The time responses of CBF are 2.3 s for the flow based, 10.8 s for the oxygen based, 5.8 s for the oxygen and pericyte driven model and 4.1 s when neural activity is included as a further CBF regulator.
The dynamic response of the blood volume compartments was evaluated under the same conditions. The computational model has shown that the arterial volume VA rises by 11.98% at the moment of the impulse, while the venous volume VV rises by 2.76% before settling to baseline condition. The sum of both volume compartments yields the total CBV whose impulse response shows an increase of 14.88%. Finally, as a result of the 10% increase in ABP which resulted in a 26.32% increase in CBF, the model calculated a 9.10% increase in CMRO2 (see figure 11).
Figure 11. CBF, CMRO2 and CBV variation upon a 10% increase in ABP.
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Standard image High-resolution imageA new mathematical model of cerebral autoregulation was presented here. Four feedback mechanisms modeling approaches (F0, F1, F1 + F2 and F1 + F2 + F3) were compared for the modeling of CBF regulation. The first is a flow feedback-based model, the second a solely oxygen sensitivity signaling, the third is an oxygen and pericyte based feedback mechanism and the fourth included neural activity as a further CBF regulator. The peak dilation for capillaries with a large diameter (6–10 μm diameter) was found to be a 2.51% increase from the vessel's baseline value, the response time being about 1 s. In comparison, smaller capillaries (2–6 μm diameter) responded with a peak dilation of 2.34% and a time response of about 2 s. The comparison of the response of both vessel groups proved to be statistically significant. The differences in capillary dilation for vessels imaged at different depths was also studied and was found to be statistically significant. Indeed, capillary vessels found in deeper cortical layers expressed a longer response time and a greater dilation (see figure 6). This difference in dilation magnitude could be attributed to the fact the deeper cortical layers are more active, therefore capillaries might be required to dilate more in order to supply the tissue with a high nutrient demand. To validate this hypothesis, further analysis with a higher number of vessels should be conducted, in order to be able to surely conclude that larger capillaries found in the deeper layers of the cortex (150–350 μm depth) have a greater dilation potential than capillaries found in more superficial layers of the cortex (0–150 μm depth).
Under steady state conditions, all feedback scenarios proved to be in good agreement with experimental data (see figure 9). The differences in feedback mechanisms were highlighted by the dynamic analysis (see figure 10). It was found that metabolism alone (i.e. oxygen sensitivity) is insufficient for a rapid CBF regulation, and that including pericytes and neural activation improved agreement physiological reality. In a healthy brain, CBF is maintained by the autoregulatory mechanism of vasoconstriction and vasodilation when the cerebral perfusion pressure (CPP) is increased or decreased respectively. Hence, during autoregulation while CBF is maintained constant, the drop in CBV is compensated by the cerebrovascular reserve. When CPP surpasses the upper limit of autoregulation, CBF increases as a result. Alternatively, when CPP decreases below the lower limit of autoregulation, CBF decreases. Low CPP below the limit of autoregulation results in a decrease in CBF with an increase in the oxygen extraction fraction, which is necessary to maintain CMRO2.
CMRO2 and CBF changes were also quantified experimentally. As a result of neural activation, CMRO2 was found to increase by 6.09% from its baseline value within 2 s of the stimulation. CBF increased by 5.53% from baseline after about 3 s since the start of the stimulation (see figure 8).
Changes in oxygen content within the tissue have shown to be slow relative to the changes in blood flow. Within the macro-vasculature, this indicates a low sensitivity of SMC to changes in oxygen concentration of the surrounding tissue. Moreover, in a whole brain mathematical model, it was found that for each percentage increase in CBF, about a third of the increase in CMRO2 is shown (see figure 11). This result differs when considering an individual vessel of the microvasculature experimentally and possibly even the microvasculature network in its entirety. Here, the changes of oxygen metabolism occur at a much faster rate (2 s compared to the 10.8 s in a whole brain model) and the ratio of CBF to CMRO2 change is 1:1. This could be justified by the activity of pericyte cells, whose proximity to the neurons ensures a fast change in vessel diameter. Moreover, a faster response of microvasculature compared to large arteries and veins, supports the now widespread notion that the first stage of blood flow regulation in response to metabolic and myogenic stimuli occurs within the microvasculature (Blanco et al 2017, Sweeney et al 2018).
A limitation of the experimental validation of this study is the impossibility of measuring changes in CMRO2 in the tissue and changes in capillary diameter simultaneously. Indeed, the experimental methods available for the study of the effect of visual stimulation in mice, can only image either CMRO2/Hb or Ca2+/vasomotion. This results in two sets of data, obtained in different instances, to be compared with each other. Although the experimental procedure is the same in both scenarios, the inability of monitoring all the variables within the same measurement is to be noted.
Finding the right balance between physiological accuracy and computational feasibility is always a challenge. Hence, three computational limitations of the model need to be mentioned. Firstly, no oxygen diffusion between tissue compartments was considered. Each vessel supplies an individual and independent tissue volume. Including nutrient exchange between volume voxels would possibly decrease the response time further yet this is still to be explored and is significantly harder to implement and validate with adequate experimental data. Secondly, the number of pericytes considered for the model fit is very low and these are from animal models. While the pericyte to endothelial cell ratio (1:3) is the same for mice and humans (Hall et al 2014), the use of animal data in a human brain model is to be noted. Thirdly, there is no spatial information on the oxygen distribution within the microvasculature. Studies (Shaw 2019) have shown that there are regional differences in oxygen distribution within the tissue, suggesting that some parts of the capillary network might be more active than others depending on their location within the brain.
Currently existing mathematical models of cerebral blood flow autoregulation are flow feedback based. Despite the fact that such models provide results that can be validated with experimental data, the nature of the feedback is not physiologically accurate. In order to be clinically significant, new mathematical models need to be designed. Such models could indeed be of aid for patients under continuous monitoring, as they could provide clinicians with a clearer insight into which of the numerous parameters influencing CBF regulation need adjustment.
Numerous studies have suggested that pericytes may be an additional regulating factor of blood perfusion. They act on the capillary bed, constricting or dilating capillary vessels in response to chemical stimuli (e.g. Ca2+ concentration variations). Data on Ca2+ concentration-induced capillary diameter variation were collected and subsequently analyzed here. It was found that the average maximum dilation of capillaries is of 2.51% from their baseline diameter.
In the steady state analysis, all four feedback mechanisms were found to be maintaining CBF over a wide range of ABP. Their differences, however, were highlighted when the dynamic response of CBF to changes in ABP was analyzed. An increase of 10% of ABP was induced and the response of the vasculature was observed. It was found the response time of the flow feedback model is 2.3 s whilst that of the oxygen feedback based one is 10.8 s, that of the oxygen and pericytes based one is 5.8 s and that by including neural activity as an additional CBF regulator, the response time decreases to 4.1 s. These findings highlight the essential role of an active microvasculature for a rapid CBF regulation. In particular, this study indicates that pericytes do indeed play a significant role in CBF regulation, as oxygen sensitivity alone results in a response that is too slow to be physiologically accurate.
Figure 12. Capillary vessel imaged at 0–50 μ depth, where (a) shows the vessel at rest and (b) shows the vessel dilation. To facilitate visualization, the red line outlines the vessel wall at rest and the red triangles direct attention towards the areas of dilation.
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Standard image High-resolution imageFigure 13. Capillary vessel imaged at 50–150 μ depth, where (a) shows the vessel at rest and (b) shows the vessel dilation. To facilitate visualization, the red line outlines the vessel wall at rest and the red triangles direct attention towards the areas of dilation.
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Standard image High-resolution imageFigure 14. Capillary vessel imaged at 150–250 μ depth, where (a) shows the vessel at rest and (b) shows the vessel dilation. To facilitate visualization, the red line outlines the vessel wall at rest and the red triangles direct attention towards the areas of dilation.
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Standard image High-resolution imageFigure 15. Capillary vessel imaged at 250–350 μ depth, where (a) shows the vessel at rest and (b) shows the vessel dilation. To facilitate visualization, the red line outlines the vessel wall at rest and the red triangles direct attention towards the areas of dilation.
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