Cerebral Microcirculation and Oxygen Tension in the Human Secondary Cortex

Abstract

The three-dimensional spatial arrangement of the cortical microcirculatory system is critical for understanding oxygen exchange between blood vessels and brain cells. A three-dimensional computer model of a 3 × 3 × 3 mm³ subsection of the human secondary cortex was constructed to quantify oxygen advection in the microcirculation, tissue oxygen perfusion, and consumption in the human cortex. This computer model accounts for all arterial, capillary and venous blood vessels of the cerebral microvascular bed as well as brain tissue occupying the extravascular space. Microvessels were assembled with optimization algorithms emulating angiogenic growth; a realistic capillary bed was built with space filling procedures. The extravascular tissue was modeled as a porous medium supplied with oxygen by advection–diffusion to match normal metabolic oxygen demand. The resulting synthetic computer generated network matches prior measured morphometrics and fractal patterns of the cortical microvasculature. This morphologically accurate, physiologically consistent, multi-scale computer network of the cerebral microcirculation predicts the oxygen exchange of cortical blood vessels with the surrounding gray matter. Oxygen tension subject to blood pressure and flow conditions were computed and validated for the blood as well as brain tissue. Oxygen gradients along arterioles, capillaries and veins agreed with in vivo trends observed recently in imaging studies within experimental tolerances and uncertainty.

Appendices

Appendix 1: Generation of Mesh-Like Capillary Bed

We invented a novel computational method to synthesize mesh-like capillary networks. A mesh-like capillary bed connected to the arterial and venous trees is essential for the anatomically consistent closure of hemodynamic simulations with the following key properties: (i) even random spacing of desired average segment length (ii) connection to arteriole and venule terminals (iii) adjustable degree of tortuosity, and (iv) smooth diameter transitions from the pre-capillaries to post-capillary venules. To the best of our knowledge, this algorithm for capillary synthesis has never been proposed in the literature.

The methodology encompasses a six step procedure illustrated Fig. 9 frames a–e. In the first step, the computational domain between the terminal nodes of the arterial (a₁, a₂) and venous (v₁) trees is evenly divided by Delaunay triangulation.8 The Delaunay mesh creates an even partition of the intravascular space. This partition is shown in Fig. 9a by triangles for clarity. The new Delaunay nodes are more densely spaced than the arterial and venous terminals (a₁, a₂, v₁). The number of Delaunay nodes controls the desired average capillary segment length in the emerging capillary bed. It is tempting to use the Delaunay edges directly as capillary segments. However, the branching factor of three-dimensional Delaunay meshes is not physiological, easily exceeding 50 edges per node in complex brain geometries, while real capillary meshes have only bifurcations (branching factor of two).

Figure 9

Stepwise construction of capillary beds. (a) Delaunay triangulation of the space between the terminal nodes of the arteriole tree (a₁, and a₂) and the venous tree (v₁). Mesh density (shown as triangles for clarity) controls the segment length of the desired capillary bed. (b) Construction of Voronoi tessellation, which is the dual of the Delaunay mesh. (c) Connection of arterial and venous terminal to the closest neighbor capillary node. (d) Bezier curve approximation of capillary segments to adjust desired degree of tortuosity. (e) Network flow simulation yields a pressure field to determine the mean capillary pressure, P. Segments within a desired pressure range, P ± 2σ _F, are assigned the smallest capillary diameter, D _min = 3 μm. (f) Diameters of capillary segments are according to an iterative averaging scheme, producing smooth transitions form the pre-capillary arterioles to the post–capillary venules. Steps E to F can be repeated to eliminate sharp diameter transitions at the inlets and outlets of the capillary bed

Therefore, we construct the dual mesh, know as a Voronoi tessellation,72 as depicted in Fig. 9b. The Voronoi cells have branching factors of two in two dimensions, and three in for three dimensions. To provide closure, each arterial terminal is connected to the closest capillary node. Similarly, each terminal node of the venous tree is linked to the Voronoi mesh without allowing a venous and an arterial terminal to connect to the same capillary node. In effect, the arterial and the venous trees are linked through the capillary bed creating a closed network structure without dangling segments. The crude Voronoi capillary mesh depicted in Fig. 9c requires three more refinement steps for better physiological match with real capillary beds.

In every capillary node with more than three segment connections, one superfluous segment can be deleted. This process we term pruning is required only three dimensional Voronoi meshes (not shown). We choose to eliminate the shortest edge, whose removal does not leave another node connected to the deleted edge dangling. We refrained from pruning Voronoi trifurcations in which no edge can be removed without causing dangling capillary nodes.

Several researchers have pointed out the significance of the tortuosity in hemodynamic simulations.18 Optionally, we can control tortuosity of capillary segments by Bezier curve interpolation, as depicted in Fig. 9D with detailed mathematical description given in Appendix 2.

The pruned, tortuous capillary bed still requires the reasonable diameters for each segment. A simple choice is constant diameters for all segments, but has the disadvantage of sharp transitions from the arteriole tree to the capillary bed, and from the capillary bed to the venous tree. We propose to enforce smooth diameter transitions with slight tapering coming from the arteriole side and slow diameter increases towards the venous side as follows: We perform blood flow simulations as in Eq. (2)–(3) through the network as a function of segment length and diameter. The mean pressure, P, is defined as the arithmetic mean of all capillary node pressures; σ _F is its standard deviation. All segments within the pressure range, P ± 2σ _F, were assigned the minimum diameter, D _min = 3 μm. Pre- and post capillary segments connecting to arteriole and venule terminals were assigned a diameter commensurate with their parent terminal. For all segments between the terminals and the minimum diameter group, we computed their diameters by iteratively executing the averaging scheme given in Eq. (8). This method for ensuring smooth diameter transitions is similar to physiological regularization techniques. Our level-set type approach gives rise to smooth diameter transitions, which monotonically decreases from inlet pre-capillary vessels to the smallest capillaries, then increase to the post-capillary veins.

$$ {\text{Capillary diameter smoothing}}\;d_{i} = \sum\limits_{j = 1}^{N} {\frac{{d_{i,j} }}{N}} $$

(8)

where d _i , _j is the diameter of segment j connected to segment i

If needed to further reduce sharp diameter transitions, the flow computations can be repeated with the adjusted diameters. Additional passes of diameter averaging can be performed until a sufficiently smooth diameter transition and spectrum is achieved.

Appendix 2: Cubic Bezier Curves

A cubic Bezier curve is a vector function in terms of the scalar parameter t as defined in Eq. (9). It is defined by two end points, $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{P}}_{0} }}$ and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{P}}_{1}}}$ and two control points, $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{C}}_{0}}}$ and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{C}}_{1}}}$ as shown in Panel a of Fig. 10. The endpoints delineate the curve; the control points dictate the shape of the curve.

Figure 10

Bezier Curve adjustment of capillary tortuosity. (a) Depiction of a Bezier curve between endpoints, $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{\text{P}}_{0} }} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{\text{P}}_{1} }}$ and control points, $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{\text{C}}_{0} }} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{\text{C}}_{1} }}$. (b) Automatic generation of control points based on segments connected to the current Bezier segment. (c) Straight line segments. (d) Tortuous capillary segment with Bezier curve parameter, α = 0.25

$$ \overrightarrow {B} \left( t \right) = (1 - t)^{3} \cdot\overrightarrow {{P_{0} }} + 3t\left( {1 - t} \right)^{2} \cdot\overrightarrow {{C_{0} }} + 3t^{2} \left( {1 - t} \right)\cdot\overrightarrow {{C_{1} }} + t^{3} \cdot\overrightarrow {{P_{1} }} t \in [0,1] $$

(9)

For each capillary segment between points $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{P}}_{0} }}$ and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{P}}_{1}}}$, a 3D Bezier curve with desired degree of tortuosity can be constructed. To ensure reasonable transitions between different segments, we derive the control points for a given segment as a function of its neighboring segments. The first step is to find the auxiliary point $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{A}}_{0}}}$, defined as the average of the set of points, $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{P}}_{0,1} }}$ connected to $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{P}}_{0} }}$ exclusive of $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{P}}_{1}}}$ as given in Eq. (10). Similarly, auxiliary point $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{A_{1}}}$, is found from the set of points, $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{P}}_{1,1} }}$ connected to $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{P}}_{1} }}.$ except for $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{P}}_{0} }}.$

$$ \begin{gathered} \overrightarrow {{A_{0} }} = \frac{{\mathop \sum \nolimits_{i = 1}^{N} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{P}_{0,i}}}{N} \quad \forall \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{P}_{0,i} \; {\text{connect to}}\; \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{P}_{0} , {\text{except for}}\; \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{P}_{1} \hfill \\ \overrightarrow {{A_{1} }} = \frac{{\mathop \sum \nolimits_{i = 1}^{N} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{P}_{1,i}}}{N} \quad \forall \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{P}_{1,i} \; {\text{connect to}}\; \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{P}_{1} , {\text{except}}\; {\text{for}}\; \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{P}_{0} \hfill \\ \end{gathered} $$

(10)

Finally, the desired control $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{C}}_{0}}}$ and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{{{C}}_{1}}}$ are calculated with Eq. (11), where the scalar α controls the degree of tortuosity: α = 0 produces a straight line, α = 1.0 gives in the most tortuous curve. We recommend a value of α = 0.25 for natural capillary mesh appearance. Sample meshes with and without Bezier vessels are depicted in panels C and D of Fig. 10. Note that arc lengths of Bezier curves given in Eq. (12) in are always greater than the straight line counterpart.

$$ \begin{gathered} \rightharpoonup {{C_{0} }} = \rightharpoonup {{P_{0} }} + \alpha \cdot\left( {\rightharpoonup {{P_{0} }} - \rightharpoonup {{A_{0} }} } \right) \quad \alpha \in [0,1] \hfill \\ \rightharpoonup {{C_{1} }} = \rightharpoonup {{P_{1} }} + \alpha \cdot\left( {\rightharpoonup {{P_{1} }} - \rightharpoonup {{A_{1} }} } \right) \quad \alpha \in [0,1] \hfill \\ \end{gathered} $$

(11)

$$ \begin{gathered} {\text{Analytically arc length}}: \hfill \\ \mathop \int \limits_{0}^{1} \left| {\dot{B}(t)} \right| dt = \mathop \int \limits_{0}^{1} \sqrt {\sum\nolimits_{i = x,y,z} {\left[ {\frac{{dB_{i} (t)}}{dt}} \right]^{2} } } dt \quad {\text{where}}\; B_{i} \left( t \right) {\text{ are the components of}}\;B\left( {\text{t}} \right) \hfill \\ {\text{Numerically}} \hfill \\ \mathop \int \limits_{0}^{1} \sqrt {\sum\nolimits_{i = x,y,z} {\left[ {\frac{{dB_{i} (t)}}{dt}} \right]^{2} } } dt \approx \mathop \sum \limits_{{t_{j} }}^{M} \left\{ \begin{gathered} \left. {\root{{}} \of {{\sum\nolimits_{i = x,y,z} {\left[ {\frac{{\Updelta B_{i} \left( t \right)}}{{\Updelta t_{j} }}} \right]^{2} } }} \Updelta t_{j} } \right\} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$

(12)

where $\frac{{\Updelta B_{i} (t)}}{{\Updelta t_{j} }}$ are the numerical derivatives of the components of B(t) along the arc t

Appendix 3: Krogh Cylinder: Discrete and Analytical Solution

At the request of one reviewer, we also included a comparison of our dual mesh technique with an analytical Krogh model. Even though the Krogh cylinder offers an analytical solution to the oxygen diffusion with 0th order reaction, it has several limitations including the lack of axial dispersion which makes it imperfect for numerical validation studies.36 Several extension of the Krogh model have been proposed by Secomb,25 but approaches like Green’s function are beyond the intended scope of numerical validation of our network.

We plotted the results for a cylindrical domain with a single blood vessel in its center as described in Krogh’s original work.37 The analytical radial concentration profiles, C(r,z) at different levels of the axial coordinate are given Eq. (13) and Eq. (14) as a function of the product of the inlet concentration, C ₀, and the bulk velocity, V, the radius of the vessel, r _C, the thickness of the wall t _m, the radius of the tissue cylinder, r _T, and the zero order rate of solute destruction in the tissue, R ₀, and the diffusion of the solute in tissue, D. The concentration of solute in the tissue at the wall boundary, $\overline{C}_{{r_{c + tm} }}$ equal to the flux of transport through the vessel governed by the mass transport coefficient, K ₀, as shown in Eq. (14). Finally, Eq. (14) includes the axial concentration profile inside the blood vessel, C(z), at the axial coordinate, z. The radial and axial solute gradients were computed for the analytic solution as shown in Fig. 11.

Figure 11

Krogh cylinder solution of tissue oxygen perfusion from a single vessel, comparison of analytical and discrete methods. (a) Analytical solution of a Krogh cylinder with capillary radius (r _C = 5 μm) and capillary length (L = 1000 μm) and surrounding tissue radius (r _T = 40 μm) and a constant injection of solute (C ₀ = 5 μM/mL). This radial distribution was determined at z = 570 μm. (b) Solution of the same system using the discrete approach described in this paper. (c) Discrete radial concentration of solute compared to realizations of the analytical solution for z = 569 mm and z = 571 mm (d) axial distribution of oxygen between the analytical and the discrete solution

$$ \begin{gathered} C\left( {z,r} \right) = \overline{C}_{{r_{\text{C}} + t_{\text{m}} }} + \frac{{R_{0} }}{{4D_{T} }}\left[ {r^{2} - \left( {r_{\text{c}} + t_{\text{m}} } \right)^{2} } \right] - \frac{{R_{0} r_{\text{T}}^{2} }}{{2D_{\text{T}} }}\ln \frac{r}{{r_{C} + t_{m} }} \hfill \\ {\text{with}}\quad \overline{C}_{{r_{\text{C}} + t_{\text{m}} }} = C(z) - \frac{{R_{0} }}{{2r_{\text{C}} K_{0} }}r_{*}^{2} \hfill \\ \end{gathered} $$

(13)

$$ C\left( z \right) = C_{0} - \frac{{R_{0} }}{{Vr_{c}^{2} }}r_{*}^{2}\;z,\quad r_{*}^{2} = \left[ {r_{\text{T}}^{2} - (r_{\text{c}} + t_{\text{m}} )^{2} } \right] $$

(14)

We compare oxygen extraction numerically computer with our model to the analytical Krogh model.15 The central blood vessel was discretized with cylindrical segments of Δz = 8 μm in length. A relatively coarse tetrahedral mesh to represent the extravascular space was constructed with the same dimensions, a mesh edge length of 34 μm, which is the same resolution as used the main paper. Flow, oxygen inlet concentration, mass transfer and 0th order oxygen consumption were implemented with the same parameters as in the Krogh model listed compactly in Table 4. Figure 11 summarize the results for the analytical and the numerical methods described in this work. Radial oxygen profiles of all tetrahedral cells falling with at an axial range of z = 570 ± 1 mm were plotted as dots in Fig. 11. For the analytical solution, two profiles were drawn for z = 569 mm and z = 571 mm, corresponding to the extreme axial positions of the tetrahedral elements in that zone. The analytical and our numerical results are in excellent agreement as expected.

Table 4 Parameters of distribution using the Krogh model

For completeness, we also checked total oxygen exchange and plotted the radial oxygen profiles. Although the total oxygen exchange matches exactly within numerical tolerances, the extraction profile along the axis cannot be identical as expected. This result is valid, as the Krogh model neglects axial dispersion, while our model does not. Therefore the slight mismatch is actually a confirmation of the quality of the proposed technique.

Additionally, a molar balance was constructed to determine the soundness of the discrete method and enforcement of mass conservation. The rate of 5 μmole/cm³ entering the cylinder at a velocity of 500 μm/s with a cross sectional area of 78.5 μm² gives an inlet flux of 19.63 × 10⁻⁸ μmole/s. The outlet concentration from cylinder was determined as 3.74 μmole/cm³ in the analytical, and 3.72 μmole/cm³ in the discrete model. These outlet concentrations give rise to an outlet flux of 14.69 × 10⁻⁸ μmole/s in the analytical solution and 14.61 × 10⁻⁸ μmole/s in the discrete model. The volume of the tissue cylinder (total tissue cylinder volume minus the vessel volume) is 4.93 × 10⁶ μm³ in the analytical domain and 4.98 × 10⁶ μm³ in the discrete domain, giving a volumetric 0th order reaction rate of 4.93 × 10⁻⁸ μmole/s and 4.98 × 10⁻⁸ μmole/s, respectively. This compares to the 0th destruction rate computed by the analytical and discrete methods with a reported numeric error <0.002. This error scales accordingly for the large microvessel subsection model presented in this paper.