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Abstract

The functioning of cerebral cortex depends on adequate tissue oxygenation. MRI-based techniques allow estimation of blood oxygen levels, tissue perfusion, and oxygen consumption rate (CMRO₂), but do not directly measure partial pressure of oxygen (PO₂) in tissue. To address the estimation of tissue PO₂, the oxygen mass transfer coefficient (KTO₂) is here defined as the CMRO₂ divided by the difference in spatially averaged PO₂ between blood and tissue, and is estimated by analyzing Krogh-cylinder type models. Resistance to radial diffusion of oxygen from microvessels to tissue is distributed within vessels and in the extravascular tissue. The value of KTO₂ is shown to depend strongly on vascular length density and also on microvessel tube hematocrits and diameters, but to be insensitive to blood flow rate and to transient changes in flow or oxygen consumption. Estimated values of KTO₂ are higher than implied by previous studies, implying smaller declines in PO₂ from blood to tissue. Average tissue PO₂ can be estimated from MRI-based measurements as average blood PO₂ minus the product of KTO₂ and CMRO₂. For oxygen consumption rates and vascular densities typical of mouse cortex, the predicted difference between average blood and tissue PO₂ is about 10 mmHg.

Keywords

Capillary density hematocrit magnetic resonance imaging oxygen transport theoretical models

Introduction

Maintenance of adequate tissue oxygenation is essential for normal function of the cerebral cortex, and inadequate oxygen supply can cause irreversible damage. Methods to measure or estimate oxygen levels are needed in order to evaluate the metabolic status of cortical tissue under various conditions, particularly when oxygen delivery is impaired. Under experimental conditions, direct measurements of tissue oxygen levels can be made^1,2 using microelectrodes^3–9 or phosphorescence lifetime imaging.^10–13 These techniques have the advantage of yielding measurements of local PO₂ (partial pressure of oxygen) on microscopic scales, but are invasive and technically demanding. Non-invasive techniques based on magnetic resonance imaging (MRI) provide data on blood oxygen levels¹⁴ and on tissue perfusion¹⁵ at millimeter scales.¹⁶ On the assumption that blood oxygen levels measured using BOLD (blood oxygen level dependent) MRI mainly reflect the oxygen content of venous blood, the rate of tissue oxygen consumption (CMRO₂) can be deduced as the product of the perfusion and the decrease in blood oxygen content relative to arterial blood.¹⁷ However, tissue oxygen levels are not directly measured by these techniques.

Theoretical analyses of oxygen transport to tissue can be used to deduce tissue oxygen levels from data on blood oxygen levels and oxygen consumption rate in tissue. Such an approach was introduced by Krogh¹⁸ a century ago. The Krogh cylinder model can be used to predict the variation of PO₂ with distance from a single capillary with known PO₂, assuming a fixed rate of oxygen consumption in the tissue. Numerous subsequent studies have extended this approach to take into account factors that were neglected in the original model, and to consider more complex microvascular geometries. This topic has been the subject of several reviews.^19–21

With regard to oxygen delivery in the cerebral cortex, a variety of theoretical approaches have been used. Some investigators have used Krogh-cylinder-type models,^22,23 while others have used models based on three-dimensional capillary network structures.^24–27 Such models can be used to predict spatial distributions of PO₂ in tissue on the scale of tens of µm. However, these approaches have not been widely adopted by investigators using MRI to study cerebral blood flow and metabolism, possibly because spatially resolved approaches are relatively complex to implement and yield predicted variations in PO₂ on length scales that do not match the resolution of MRI-based observations.

In several models for oxygen transport to cortex, the tissue domain is treated as a single well-mixed compartment with a uniform PO₂. In some of these analyses, the tissue PO₂ is assumed to be nearly zero, and oxygen efflux from the capillary is assumed to be proportional to the PO₂ of blood in the capillary.^28–31 However, the assumption of tissue PO₂ near zero is inconsistent with existing theoretical and experimental information. In other model studies,^17,32 a uniform non-zero tissue PO₂ is assumed. In these models, the oxygen content of the capillary is considered to be a function of position and time, and the rate of oxygen transport from blood to tissue is assumed to be proportional to the difference between capillary and tissue PO₂ levels.

In the models just described, a mass transfer coefficient is introduced to describe the proportionality between the oxygen efflux from blood and the vessel-to-tissue PO₂ difference. This coefficient has been defined in several studies^17,31,32 as the product (PS) of the permeability (P) and surface (S) area of the vessel wall. While the use of a PS product is appropriate for solutes for which the vascular endothelium is the main barrier to transport, this is not the case for oxygen, which passes readily through cell membranes, being a non-polar, lipid-soluble molecule. In reality, radial gradients of PO₂ are present within the microvessels and the surrounding tissue, and are not concentrated at the vessel wall.

The goal of the present study is to define an oxygen mass transfer coefficient (KTO₂) relating the rate of oxygen efflux from blood to the difference between average microvessel PO₂ and average tissue PO₂, to estimate its magnitude in the cerebral cortex based on the physics of oxygen diffusion in blood and in tissue, to determine its dependence on other physical, geometrical and mass transport parameters, and to show how it can be used to estimate average tissue PO₂ from average blood PO₂ and CMRO₂.

Methods

Definition of KTO₂

For the purposes of this analysis, the oxygen mass transfer coefficient is defined as

KT O_{2} = \frac{{JO}_{2}}{{\bar{P}}_{cap} - {\bar{P}}_{tiss}}

(1)

where JO₂ is the rate of oxygen efflux from vessels per volume of tissue,

{\bar{P}}_{cap}

is the average microvessel PO₂ and

{\bar{P}}_{tiss}

is the average tissue PO₂. More precisely,

{\bar{P}}_{cap}

is defined as the PO₂ corresponding to the oxyhemoglobin saturation averaged over the volume of red blood cells within the microvessels, according to the dissociation curve (see below). This definition facilitates calculations based on mass conservation for oxygen. Small arterioles, with diameters up to about 40 µm, have been shown to participate significantly in oxygen delivery in the brain.³³ Therefore, the terms

{JO}_{2}

and

{\bar{P}}_{cap}

in equation (1) are assumed to include small arterioles in addition to capillaries. The average

{\bar{P}}_{tiss}

is taken over the entire extravascular tissue volume.

Under steady-state conditions, the rate of oxygen efflux from blood vessels equals the rate of oxygen consumption in the tissue, and in that case

{KTO}_{2} = \frac{{CMRO}_{2}}{{\bar{P}}_{cap} - {\bar{P}}_{tiss}}

(2)

The oxygen consumption rate CMRO₂ is often expressed in units of cm³O₂ · (100 cm³ · min)⁻¹ or cm³O₂ · (100 g · min)⁻¹. For consistency, the same units are here used for JO₂, and KTO₂ values are given in units of cm³O₂ · (100 cm³ · min · mmHg)⁻¹.

Other studies^17,32 have used a definition of the mass transport coefficient based on the difference in oxygen concentrations between plasma and tissue. However, diffusive oxygen transport is driven by gradients in partial pressure, which are not necessarily equivalent to gradients in concentration (i.e. the product of oxygen solubility and partial pressure), because the solubility differs between plasma and tissue. A definition based on concentration difference therefore has limited applicability.

Blood and tissue mass transfer coefficients

In analyzing the determinants of KTO₂, the effects of the intravascular and extravascular domains can be quantified by defining ${\bar{P}}_{wall}$ as the mean PO₂ on the interface between the plasma and the vessel wall. Then it follows from equation (1) that

\frac{1}{{KTO}_{2}} = \frac{1}{{KTO}_{2}^{tiss}} + \frac{1}{{KTO}_{2}^{cap}}

(3)

where

{KTO}_{2}^{tiss} = \frac{{JO}_{2}}{{\bar{P}}_{wall} - {\bar{P}}_{tiss}} and {KTO}_{2}^{cap} = \frac{{JO}_{2}}{{\bar{P}}_{cap} - {\bar{P}}_{wall}}

(4)

are the mass transfer coefficients for oxygen transport in the tissue and in the microvessels. The inverses of these quantities represent the contributions of the intravascular and extravascular domains to the overall diffusive resistance to radial oxygen transport.

Estimation of tissue mass transfer coefficient using Krogh cylinder model

The Krogh cylinder model has been used in a number of analyses of oxygen transport in the microcirculation.^18,19,34 In this model, a single capillary with radius r_c is assumed to supply oxygen to a concentric cylindrical region of surrounding tissue with radius r_t (Figure 1(a) and Table 1). It is assumed that the tissue cylinder is surrounded by a parallel array of equivalent units, and so the diffusive flux of oxygen across the outer surface of the tissue cylinder is zero. For the present analysis, steady-state conditions are assumed and a fixed rate M₀ of oxygen consumption is assumed in the tissue region, implying that CMRO₂ = M₀(1 – r_c²/r_t²) where CMRO₂ is the average oxygen consumption rate expressed per unit volume including the intravascular volume. Diffusion in the axial direction is neglected, on the basis that the radius of the cylinder is much less than its length, and axial gradients and fluxes are therefore relatively small. Fick’s second law of diffusion implies that

\frac{D_{tiss} α_{tiss}}{r} \frac{d}{dr} (r \frac{dP}{dr}) = M_{0}

(5)

where P(r) is tissue PO₂, r is the distance from the cylinder axis, and D_tiss and α_tiss are the diffusivity and solubility of oxygen in tissue. The diffusivity, solubility and oxygen consumption rate are assumed to be uniform and represent spatial averages over the various cell types and extracellular tissue components. The boundary conditions are

P = {\bar{P}}_{wall}

when r = r_c and

dP / dt = 0

when r = r_t. The solution¹⁸ is well-known

P (r) = {\bar{P}}_{wall} - \frac{M_{0}}{4 D_{tiss} α_{tiss}} (2 r_{t}^{2} \ln \frac{r}{r_{c}} - r^{2} + r_{c}^{2})

(6)

Figure 1.

Schematic representations of vascular configurations analyzed. (a) Krogh-type model. The central cylinder represents the capillary, with radius r_c, length L, and blood flow rate Q_cap, surrounded by the tissue cylinder with radius r_t (light gray). Vertical arrows indicate the diffusive efflux JO₂ of oxygen from the capillary into the tissue. The flux across the outer tissue boundary is assumed to be zero. (b) Vascular network derived from observations of rat cortex.^25,35 Tissue domain is 150 × 160 × 140 µm. (c) Vascular network derived from observations of mouse cortex.³⁶ Overall dimensions of tissue domain are 609 × 609 × 662 µm; some avascular parts of the domain are excluded from the analysis.

Table 1.

Reference parameter values.

		Value	Reference
Krogh cylinder geometric parameters
r_c	Vessel radius	2 µm	²³
r_t	Tissue radius	25 µm	See text
L	Vessel length	500 µm	See text
N_cap	Capillary length density	509.3/mm²	Calculated
Microvascular network geometric parameters – rat
	Total vessel length	1.840 mm	²⁵
	Tissue volume	3.36 × 10⁻⁶ cm³	²⁵
N_cap	Vessel length density	547.7/mm²	²⁵
	Mean vessel radius	2.76 µm	²⁵
Microvascular network geometric parameters – mouse
	Total vessel length [excluding venules]	118.3 mm [108.6 mm]	³⁶
	Tissue volume	2.34 × 10⁻⁴ cm³	³⁶
N_cap	Vessel length density [excluding venules]	505.1/mm² [463.7/mm²]	³⁶
	Mean vessel radius [excluding venules]	2.88 µm [2.53 µm]	³⁶
Tissue oxygen transport and consumption
D	Diffusivity of oxygen	2410 µm²/s	²³
α	Solubility of oxygen	3.89 × 10⁻⁵ cm³O₂ ċ (cm³ ċ mmHg)⁻¹	²³
P₀	Michaelis constant for oxygen consumption	10.5 mmHg	⁴⁴
M₀	Baseline oxygen demand	7.5 cm³O₂ ċ (100 cm³ ċ min)⁻¹	See text
Blood flow and oxygen content
P₅₀	PO₂ at half maximal hemoglobin saturation	40.2 mmHg	⁶⁰
N	Hill equation exponent	2.59	⁶⁰
H_T	Tube hematocrit (µ_LD = 0.5)	0.281	²³
H_D	Discharge hematocrit	0.4
C₀	Oxygen binding capacity of red blood cells	0.516 cm³O₂/cm³	²³
K_0.5	Reference resistance parameter	2.52 mmHg ċ µm ċ s/µm³O₂	See text
P_in	PO₂ of inflowing blood	90 mmHg	²³
S_in	Saturation of inflowing blood	0.890	Calculated
v_RBC	Red blood cell velocity	1.46 mm/s	See text
CBF	Perfusion	78.76 cm³ ċ (100 cm³ ċ min)⁻¹	See text

The PO₂ averaged over the tissue region ( $r_{c} \leq r \leq r_{t}$ ) can be evaluated as

{\bar{P}}_{tiss} = {\bar{P}}_{wall} - \frac{M_{0} r_{t}^{2}}{8 D_{tiss} α_{tiss}} \frac{- 4 \ln δ - 3 + 4 δ^{2} - δ^{4}}{1 - δ^{2}}

(7)

where

δ = r_{c} / r_{t}

. Then equation (4) implies that

{KTO}_{2}^{tiss} = \frac{8 D_{tiss} α_{tiss}}{r_{t}^{2}} \frac{(1 - δ^{2}) 2}{- 4 \ln δ - 3 + 4 δ^{2} - δ^{4}}

(8)

which can also be written

\begin{matrix} {KTO}_{2}^{tiss} = 8 π D_{tiss} α_{tiss} N_{cap} f (δ) where \\ f (δ) = \frac{(1 - δ^{2}) 2}{- 4 \ln δ - 3 + 4 δ^{2} - δ^{4}} \end{matrix}

(9)

and where

N_{cap} = 1 / (π r_{t}^{2})

is the capillary density (capillary length per unit tissue volume).

Estimation of tissue mass transfer coefficient for realistic network structures

The Krogh cylinder model represents an idealized configuration of a capillary and surrounding tissue, and does not take into account the irregular geometry of the microvasculature and the contribution of arterioles to oxygen delivery. Therefore, additional simulations were performed with a small network derived²⁵ from corrosion casts of rat cortex³⁵ and an extensive network derived from two-photon microscopy of mouse cortex³⁶ (Figure 1(b) and (c) and Table 1). Within the tissue domain, the PO₂ satisfies

D_{tiss} α_{tiss} \nabla^{2} P = M_{0}

(10)

assuming steady-state conditions. This equation was solved numerically using the Green’s function method³⁷ in three-dimensional domains containing the microvessel networks, with a prescribed PO₂,

P = {\bar{P}}_{wall}

, at the vessel walls. To impose this boundary condition, the PO₂ of inflowing vessels was set to

{\bar{P}}_{wall}

, and very high flow rates were imposed in all vessels such that intravascular PO₂ was effectively constant, and intravascular diffusion resistance was set to zero. These artificial conditions were imposed for these calculations only, to allow estimation of the effect of vascular geometry on the tissue mass transfer coefficient

{KTO}_{2}^{tiss}

, without confounding effects of variations in intravascular PO₂. In all other calculations, physiologically realistic flow rates were used.

Estimation of capillary mass transfer coefficient

Hellums et al.^38,39 analyzed radial oxygen gradients within capillaries and showed the importance of intravascular resistance to oxygen diffusion. More recently, Lucker et al.²³ examined the effect of hematocrit and red blood cell velocity on tissue oxygen levels in a tapered tissue region with dimensions derived from the mouse cerebral cortex, with a central capillary of internal radius 2 µm. The results of detailed numerical simulations were closely approximated by an analytical model in which the intravascular mass transfer coefficient satisfies

{KTO}_{2}^{cap} = \frac{μ_{LD}}{0.5} \frac{N_{cap}}{K_{0.5}}

(11)

where µ_LD is the length density of red blood cells (total length of cells divided by capillary length) and K_0.5 is the coefficient of intravascular diffusion resistance when µ_LD = 0.5. In these simulations, red blood cells were represented as cylinders with radius 1.5 µm, and µ_LD = 16H_T/9 where H_T is the tube hematocrit. Assumed oxygen transport parameters are shown in Table 1. The value of K_0.5 given by Lucker et al.,²³ 4.98 mmHg µm s/µm³O₂, includes the contribution of vessel wall diffusion resistance, based on a wall thickness of 0.6 µm. For consistency with the definition in equation (4), the contribution of the wall to diffusion resistance must be subtracted, giving K_0.5 = 2.52 mmHg µm s/µm³O₂. Similar values were obtained in earlier analyses.^40,41

Simulations of nonlinear and time-dependent effects

The preceding analyses are for steady-state conditions, with uniform fixed oxygen consumption rates. Further numerical simulations were performed using a Krogh-cylinder geometry to examine other factors influencing KTO_2. A single equation was used to describe mass transport in the tissue and capillary domains. In cylindrical polar coordinates (r, θ, z), where z is the distance along the vessel in the flow direction

\begin{matrix} \frac{\partial C (P)}{\partial t} + v_{RBC} \frac{\partial C (P)}{\partial z} - D α [\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial P}{\partial r}) + \frac{\partial 2 P}{\partial z^{2}}] \\ = - M (P) \end{matrix}

(12)

where C is the concentration of oxygen, v_RBC is the red blood cell velocity, v_RBC = 0 in the tissue, D and α are the diffusivity and solubility of oxygen in blood (D_cap and α_cap) or tissue (D_tiss and α_tiss), M is the oxygen consumption rate, and M = 0 in the capillary. In blood, C is the sum of hemoglobin-bound and dissolved oxygen, i.e.

C (P) = H_{T} C_{0} S + α_{cap} P

where H_T is the tube hematocrit and C₀ is the oxygen binding capacity of red blood cells. The oxyhemoglobin saturation of the blood is described by the Hill equation:

S = P^{N} / (P^{N} + P_{50}^{N})

where P₅₀ is the PO₂ at 50% saturation and N is the Hill exponent. In the tissue,

C (P) = α_{tiss} P

. Oxygen consumption is described using Michaelis–Menten kinetics

M (P) = M_{0} P / (P + P_{0})

(13)

where M₀ is the oxygen demand and P₀ is the Michaelis constant.

Time-dependent effects might influence KTO₂ values. For simulations of oxygen transport during transient changes in blood flow rate and oxygen consumption, CBF and CMRO₂ are assumed to follow the typical time courses shown in Figure 2(c) of Buxton.⁴²

Figure 2.

Predicted tissue and capillary mass transfer coefficients. (a) Tissue mass transfer coefficient derived from Krogh cylinder model, as a function of capillary density, for three values of capillary radius as indicated. Inset shows function f(δ) defined in equation (9), where δ = r_c/r_t. Points a, b and c give numerical results for realistic microvascular network structures shown in Figure 1(b) and (c). For the larger network, b indicates the result including all vessels and c indicates the result when only arterioles and capillaries (diameter < 8 µm) are included. (b) Capillary mass transfer coefficient for capillary radius 2 µm, as a function of capillary density, for four values of tube hematocrit as indicated.

Intravascular oxygen transport processes are not represented in detail in these simulations. Instead, the intravascular diffusion coefficient D_cap is set to an artificially high value to ensure uniform PO₂ within each vessel cross-section, and the intravascular mass transport coefficient defined in equation (11) is applied in the form of a step change of PO₂ at r = r_c, proportional to the diffusive flux on that boundary. The assumed uniform velocity v_RBC within the capillary in equation (12) does not accurately represent plasma velocity, but resulting errors are negligible because oxygen dissolved in plasma is a very small fraction of the total. Equation (12) was solved using FlexPDE (Version 6.50, PDE Solutions Inc., Spokane WA), a finite-element-method solver.

Parameter values

Table 1 lists assumed parameter values, chosen to represent typical conditions in the normal mouse cerebral cortex. The tissue cylinder radius r_t = 25 µm corresponds to a capillary density N_cap = 509/mm², close to the value in the observed mouse network (Table 1). The assumed capillary radius of 2 µm is in the observed range and matches that used in the simulations of Lucker et al.²³ An overall capillary length of 500 µm is assumed; as discussed below, changing this value has little effect on the results if the red blood cell velocity is changed in proportion.

Geometrical parameters for the two observed network structures were computed from network data files. The assumed oxygen diffusivity and solubility in tissue are based on measurements made at body temperature.⁴³ In previous work,²⁵ the Michaelis constant for oxygen uptake was assumed to be 1 mmHg. In the present study, a much higher value, 10.5 mmHg, is assumed, based on more recent in vivo studies.⁴⁴ As a consequence, the oxygen consumption rate M according to Michaelis–Menten kinetics is significantly less than the oxygen demand M₀, even at typical levels of tissue PO₂. For example, at a tissue PO₂ of 40 mmHg, M/M₀ = 0.80 according to equation (13). The oxygen demand M₀ is chosen to yield oxygen consumption rates CMRO₂ of about 6 cm³O₂ · (100 cm³ · min)⁻¹, similar to the observed values in non-anesthetized mice.^45,46

As a consequence of the Fåhraeus effect, the tube hematocrit H_T is lower than the discharge hematocrit H_D. These quantities are related to the mean flow velocity $\bar{v}$ and the red blood cell velocity by⁴⁷ $H_{T} / H_{D} = \bar{v} / v_{RBC}$ . The flow rate in the capillary is given by $Q_{cap} = π r_{c}^{2} \bar{v} = (H_{T} / H_{D}) π r_{c}^{2} v_{RBC}$ . In 4 −µm diameter glass tubes, $H_{T} / H_{D} \approx 0.8$ , but this ratio is generally lower in vivo because the endothelial surface layer impedes plasma flow near the vessel wall.⁴⁷ The H_T assumed here (0.281) reflects these effects, relative to a typical H_D = 0.4.

To establish the red blood cell velocity, we assume that the oxygen extraction fraction (OEF) is approximately 0.4, based on observed values^46,48 ranging from 0.35 to 0.45. The relationship OEF = CMRO₂/(CBF · S_in · H_D · C₀) implies, with the above parameter values, that the tissue perfusion is CBF = 81.66 cm³ · (100 cm³ · min)⁻¹. When adjusted for the contribution of dissolved oxygen, the result is CBF = 78.76 cm³ · (100 cm³ · min)⁻¹. The perfusion satisfies $CBF = \bar{v} r_{c}^{2} / (r_{t}^{2} L)$ , and therefore $\bar{v}$ = 1.025 mm/s and v_RBC = 1.46 mm/s, similar to observed values.⁴⁸ The choice of a different capillary length L would imply a proportional change in $\bar{v}$ and v_RBC but would not otherwise change the results significantly. In the above analysis, parameter values are given to three or four significant figures to maintain internal consistency in the calculations. However, this does not imply that the parameters are known to corresponding accuracy.

Results

Equations (9) and (11) indicate that the length density of capillaries in tissue (N_cap) is a key determinant of the tissue and capillary mass transfer coefficients ${KTO}_{2}^{tiss}$ and ${KTO}_{2}^{cap}$ . Their dependence on N_cap is shown in Figure 2. The dependence of ${KTO}_{2}^{tiss}$ on N_cap is slightly nonlinear for a fixed capillary radius (Figure 2(a)). With increasing N_cap, the ratio $δ = r_{c} / r_{t}$ increases, resulting in an increase in the factor f(δ) appearing in equation (9) (see inset to Figure 2(a)). The coefficient ${KTO}_{2}^{tiss}$ shows a weak dependence on capillary radius in the range from 2 to 3 µm. Figure 2(b) shows predictions of ${KTO}_{2}^{cap}$ for a fixed capillary radius of 2 µm. This coefficient is linearly dependent on both the capillary density and the tube hematocrit H_T, which determines the length density of red blood cells within the capillary. It is noteworthy that the two coefficients ${KTO}_{2}^{tiss}$ and ${KTO}_{2}^{cap}$ are comparable in magnitude. According to equation (3), both intravascular and extravascular oxygen diffusion processes therefore significantly affect the overall mass transfer coefficient ${KTO}_{2}$ . For the reference conditions indicated in Table 1, the predicted values are ${KTO}_{2}^{tiss}$ = 0.997, ${KTO}_{2}^{cap}$ = 1.213 and ${KTO}_{2}$ = 0.547, all in units of cm³O₂ · (100 cm³ · min · mmHg)⁻¹.

Figure 2(a) also includes values of ${KTO}_{2}^{tiss}$ obtained from numerical simulations of oxygen diffusion in three-dimensional network structures derived from observations of rodent cortex microcirculation (see Figure 1 and Table 1). The point labeled a represents the network derived from rat cortex^25,35 for which the predicted ${KTO}_{2}^{tiss}$ from equation (9) based on the mean vessel radius and the vascular density is 1.317 (all in units of cm³O₂ · (100 cm³ · min · mmHg)⁻¹) and the computed ${KTO}_{2}^{tiss}$ for the network is 0.9766, i.e. 26% lower. The point labeled b represents the network derived from mouse cortex³⁶ for which the predicted ${KTO}_{2}^{tiss}$ from equation (9) is 1.216 and the computed ${KTO}_{2}^{tiss}$ for the network is 0.8934, i.e. 27% lower. The point labeled c is for the same network excluding venules (defined as vessels with diameters greater than 8 µm, fed by a converging bifurcation). The computed ${KTO}_{2}^{tiss}$ in this case is 0.7071, 30% below the value 1.008 from equation (9). Relative to the Krogh cylinder model, realistic network structures have more heterogeneous distributions of distances from tissue points to the nearest vessel, with some tissue regions more distant from a vessel than the corresponding Krogh cylinder radius based on the vascular density. Therefore, the mass transfer coefficient is lower for realistic networks, an effect that is quantified by the results presented here. The resulting overall ${KTO}_{2}$ values are close (within 10%) to those predicted by the Krogh model, because the larger (> 2 µm) average vessel radii in the networks largely compensate for the effects of heterogeneous network structure.

Results of steady-state simulations using the Krogh cylinder model with Michaelis–Menten oxygen uptake kinetics are shown in Figure 3. Results were obtained for a range of CBF values using the finite-element method. At relatively high CBF values, the CMRO₂ is insensitive to CBF. When CBF drops below a critical level (here about 25 cm³ (100 cm³ · min)⁻¹), convective oxygen delivery is no longer sufficient to meet demand and CMRO₂ declines. Oxygen extraction, defined as E = 1 – S_out/S_in where S_in and S_out are the inflowing and outflowing hemoglobin saturation values, declines as expected with increasing perfusion. The mass transfer coefficient is virtually constant, KTO₂ ≈ 0.54 cm³O₂ · (100 cm³ · min · mmHg)⁻¹, over this range of CBF values.

Figure 3.

Effect of cerebral blood flow (CBF) on oxygen transport parameters, based on Krogh cylinder model and Michaelis–Menten oxygen uptake kinetics in tissue, for parameters shown in Table 1. (a) Average tissue oxygen consumption rate. (b) Oxygen extraction fraction. Dashed curve shows prediction of Buxton and Frank formula.²⁸ (c) Oxygen mass transfer coefficient.

In several previous analyses of oxygen transport to cortical tissue, the effect of CBF (f) on extraction (E) has been modeled by the formula of Buxton and Frank^28,49,50

E = 1 - (1 - E_{0}) f_{0} / f

(14)

where E₀ and f₀ are the reference values. Figure 3(b) shows the behavior of this function for E₀ = 0.4 and f₀ = 78.77 cm³ (100 cm³ · min)⁻¹. This result assumes²⁸ that the rate of loss of oxygen from a capillary is proportional to the plasma oxygen concentration, as would be the case if oxygen uptake by tissue followed first-order kinetics. Uptake with Michaelis–Menten kinetics results in less reduction in consumption at low oxygen levels, compared to first-order kinetics. At low flow rates, such that low oxygen levels are present, this results in higher extraction with the present model than predicted by equation (14).

Figure 4 shows results of simulations using the Krogh cylinder model assuming time-dependent blood flow and oxygen demand.⁴² Two different scenarios are considered for the variation in oxygen demand, with same flow variation in both cases. In the first scenario (Figure 4(b)), the oxygen demand is assumed to vary in parallel with the flow, with the fractional increase in demand being 50% of the fractional flow increase. In the second scenario (Figure 4(c)), the oxygen demand is assumed to vary as shown by Buxton.⁴² With this assumption, the increase and decrease in oxygen demand precede the associated changes in blood flow rate. These scenarios give variations in flow and oxygen consumption that are similar in amplitude and time-scale to experimentally observed behaviors.⁵¹ The precise form of these variations is not important for the goal of assessing their effects on the mass transfer coefficient.

Figure 4.

Variation of oxygen levels during simulated neural stimulation, using Krogh cylinder model, assuming variations in blood flow rate and oxygen consumption.⁴² Solid bars represent the period of stimulation. (a) Variation of relative flow rate and oxygen demand. Oxygen demand curve (b): demand varies in parallel with flow, increase in demand is 50% of increase in flow. Oxygen demand curve (c): from Buxton.⁴² (b, c). Predicted variations in ${\bar{P}}_{cap}$ , ${\bar{P}}_{wall}$ and ${\bar{P}}_{tiss}$ , assuming oxygen demand curves (b) and (c) respectively. Also included is the average tissue PO₂, $P_{tiss}^{*}$ , derived from equation (2).

Before the onset of stimulation, the oxygen levels are at steady state, with ${\bar{P}}_{cap}$ = 58.75 mmHg, ${\bar{P}}_{wall}$ = 53.72 mmHg and ${\bar{P}}_{tiss}$ = 47.65 mmHg. The decline in PO₂ from blood to vessel wall is about 5 mmHg, and the decline from the wall to the average tissue level is about 6 mmHg. The mass transfer coefficients are ${KTO}_{2}^{tiss}$ = 0.990, ${KTO}_{2}^{cap}$ = 1.192 and KTO₂ = 0.541 cm³O₂ · (100 cm³ · min · mmHg)⁻¹, consistent with equations (2) and (4) with JO₂ = 6.01 cm³O₂ · (100 cm³ · min)⁻¹.

During the period of stimulation, the values of the mass transfer coefficients do not vary appreciably from the above values (∼0.1% variations, data not shown). The increase in CMRO₂ during stimulation results in proportional increases in ${\bar{P}}_{cap} - {\bar{P}}_{wall}$ and ${\bar{P}}_{wall} - {\bar{P}}_{tiss}$ . In the scenario with oxygen demand varying in parallel with blood flow (Figure 4(b)), the net effect is an almost constant ${\bar{P}}_{tiss}$ . In the scenario with oxygen demand varying with total neural excitation (Figure 4(c)), ${\bar{P}}_{tiss}$ shows significant fluctuations,⁴² including an “initial dip”⁵² during the initial increase in oxygen demand.

During transient changes in oxygen levels, the oxygen flux JO₂ through the vessel wall equals the sum of the rate of oxygen consumption in the tissue and the rate of change of the amount of oxygen dissolved in the tissue. However, the second contribution is relatively small for the rates of change of PO₂ assumed here. If a rate of change of 1 mmHg/s is assumed, then the ratio of second contribution to the first is approximately $(α / M) (dP / dt) \approx 0.04$ . Therefore, a good approximation $P_{tiss}^{*}$ to ${\bar{P}}_{tiss}$ can be obtained from the steady-state equation (2)

P_{tiss}^{*} = {\bar{P}}_{cap} - \frac{{CMRO}_{2}}{{KTO}_{2}}

(15)

The difference between

P_{tiss}^{*}

and

{\bar{P}}_{tiss}

is less than 1 mmHg (Figure 4).

Discussion

Under normal conditions, the oxygen supply available to intact brain cortex exceeds requirements²² and functional activity is essentially independent of tissue PO₂. But if blood perfusion and/or oxygenation are reduced, whether as a result of systemic hypoxia or local pathology, then the impact on cerebral function depends on the change in tissue PO₂. The overall goal of this work is to provide a basis for estimating tissue PO₂ from oxygen transport parameters that can be measured non-invasively, typically by MRI-based methods. The work focuses on the relationship between the average tissue PO₂ ( ${\bar{P}}_{tiss}$ ), the average blood PO₂ ( ${\bar{P}}_{cap}$ ) and the tissue oxygen consumption rate CMRO₂. Under steady-state conditions, this relationship can be expressed as

{\bar{P}}_{tiss} = {\bar{P}}_{cap} - \frac{{CMRO}_{2}}{{KTO}_{2}}

(16)

where KTO₂ is the oxygen mass transfer coefficient. As mentioned above, estimates of CMRO₂ and

{\bar{P}}_{cap}

can be obtained using MRI-based techniques. Equation (16) then provides a method to estimate

{\bar{P}}_{tiss}

A number of previous analyses have introduced analogous parameters to describe oxygen transport in the cerebral cortex. As shown in Table 2, the assumed values of these parameters show remarkably wide variations when expressed in consistent units, and all previous estimates are smaller than those obtained here. Many previous studies have assumed relatively low fixed levels of tissue PO₂, and even zero PO₂ in several cases. This is probably the main reason for the discrepant KTO₂ values obtained. Another possible reason for the discrepancy is the use of the formula²⁹

{\bar{P}}_{cap} = P_{50} (\frac{2}{E} - 1)^{1 / N}

(17)

to predict mean capillary PO₂. For the reference parameters assumed here, this yields

{\bar{P}}_{cap}

= 68.66 mmHg, whereas the computed value is 58.75 mmHg. Equation (17) gives an overestimate because it assumes 100% oxygen saturation of inflowing blood. This overestimate in turn implies an underestimate of KTO₂ for a given CMRO₂, according to equation (2).

Table 2.

Estimates of oxygen mass transfer coefficients.

Symbol	Published value	Equivalent KTO₂ value, cm³O₂ · (100 cm³ · min · mmHg)⁻¹	Reference
L	4.09 µmol/(100 g min mmHg)	0.1041	²⁹
EOD	0.0048 mL/mmHg/g/min	0.48	³¹
PS	3000 mL·min⁻¹(100 g)⁻¹	0.1167	¹⁷
PS	7020 mL·min⁻¹(100 g)⁻¹	0.2723	¹⁷
L	3.2 µmol/(100 g min mmHg)	0.0814	⁶¹
L	4.63 µmol/(100 g min mmHg)	0.1178	⁵⁷
L	5.45 µmol/(100 g min mmHg)	0.1387	⁶²
k	118 s⁻¹	0.4407^a	⁶³
KTO₂, present study, Krogh cylinder		0.54

Note: Unit conversions based on tissue solubility α_t = 3.89 × 10⁻⁵ cm³O₂ · (cm³ · mmHg)⁻¹ and molar volume = 2.545 × 10⁵ cm³ at 37℃. ^aBased on KTO₂ = 6000 · α_t · k · CBV′ where CBV′ = 0.016 is blood volume fraction

According to the present results, KTO₂ values are highly dependent on the vascular density in tissue, expressed as vessel length per unit volume, with a less sensitive dependence on vessel diameter (Figure 2). This implies that KTO₂ is not, as has sometimes been assumed, proportional to vessel surface area per tissue volume. Such an assumption would be justified if the vessel wall was the main barrier for oxygen transport, which is not the case; significant radial gradients in PO₂ are present both within capillaries and in extravascular tissue.

A further finding is that KTO₂ has a significant dependence on capillary tube hematocrit H_T (Figure 2(b)). With decreasing H_T, the length density of red blood cells within the capillary decreases. This reduction in the amount of capillary length available for oxygen delivery from red blood cells causes a corresponding reduction in the capillary mass transport coefficient ${KTO}_{2}^{cap}$ . The tube hematocrit is generally lower than systemic hematocrit as a consequence of the Fåhraeus effect.⁴⁷ This effect is accentuated by the presence of an endothelial surface layer of macromolecules that restricts the space available for red blood cells within capillaries.⁴⁷ For the reference H_T of 0.281, the estimates of ${KTO}_{2}^{cap}$ and ${KTO}_{2}^{tiss}$ are similar in magnitude, implying that the intravascular and extravascular domains contribute almost equally to the overall resistance to oxygen transport. As a consequence of unequal partition of hematocrit in diverging bifurcations,⁴⁷ some capillaries may have much lower H_T, in which case the intra-capillary domain contributes most of the resistance to oxygen diffusion.

The Krogh cylinder model is an idealized representation of microvascular geometry. In realistic microvascular networks, the distribution of distances from points in the tissue to the nearest vessel is broader than given by the Krogh model, and this would be expected to cause a decrease in ${KTO}_{2}^{tiss}$ relative to the Krogh result. Numerical simulations using a Green’s function method indicated that this reduction is 25 to 30% for the networks considered. These simulations include the contribution of arterioles to diffusive oxygen delivery. On the other hand, the vascular density assumed here (509/mm²) is at the low end of the observed range in murine cortex.⁵³ An increase in assumed vascular density would result in an increased estimate of ${KTO}_{2}^{tiss}$ .

Further numerical simulations using a finite-element method were performed to explore the effects of CBF, Michaelis–Menten oxygen uptake kinetics, and time-dependent CBF and CMRO₂ on KTO₂. Values of KTO₂ were found to be essentially unaffected by variations in CBF (Figure 3(c)). During transient variations in vessel PO₂ or oxygen demand, the radial profile of PO₂ differs from the steady-state profile, equation (6), and so KTO₂ values might be expected to fluctuate. However, for variations of CBF and CMRO₂ occurring over a period of seconds, KTO₂ values computed from equation (1) were virtually constant. Moreover, the difference between JO₂ and CMRO₂ was small, such that equation (16) provides a good approximation to ${\bar{P}}_{tiss}$ even during fluctuating conditions. This conclusion would not necessarily apply for larger or more rapid fluctuations, however.

The present study predicts higher values of KTO₂ than stated previously. Given that the decline in PO₂ from blood to tissue is inversely proportional to KTO₂, this finding has the important implication that tissue PO₂ levels do not differ as much from blood PO₂ as previous models have suggested. For example, for a typical cortical CMRO₂ value of 6 cm³O₂ · (100 cm³ · min)⁻¹, the present theory predicts a blood-to-tissue difference of 11 mmHg for reference parameter values.

Although the results presented here are based on parameter values for mice and rats, the approach is applicable to other species, including humans. Vascular length density in human cortex is about 30% lower than in mice,⁵⁴ implying a corresponding reduction in KTO₂ (Figure 2). However, CMRO₂ values are also lower in human than in mouse cortex by a similar factor,⁵⁵ and according to equation (1) the blood-to-tissue difference in PO₂ may therefore be similar in humans to that estimated above for mice. Pronounced variations in vascular density are observed through the successive layers of primate cortex, which are correlated with variations in metabolic demand⁵⁶ and imply corresponding variations in KTO₂.

Experimental measurements of tissue PO₂ in cerebral cortex vary considerably. Erecinska and Silver¹ cite values in the range 20 to 40 mmHg in various species. Other studies have shown baseline values of 18 mmHg,⁵⁷ 25 mmHg⁶ and 33 mmHg⁵¹ in rodent cortex. The present model predicts a higher value, ${\bar{P}}_{tiss}$ = 48 mmHg, for reference parameter values. One possible explanation for this difference is that the inflow PO₂ of 90 mmHg assumed here may be higher than values in other experimental systems. In rat cortex, Vovenko³ observed values of 81 mmHg at first-order branch arterioles and 58 mmHg at the arterial end of capillaries. Also, measurements of PO₂ using an oxygen microelectrode consume oxygen and inevitably cause some reduction of PO₂ in the neighborhood of the microelectrode, so measured values may be underestimates.

An important limitation of the present work is that it considers only the average value and not the distribution of tissue PO₂. The existence of radial gradients in the tissue implies non-uniformity in tissue PO₂. Moreover, the distributions of PO₂ levels in realistic network geometries are much broader than predicted from Krogh-cylinder type models.²⁵ Although hypoxia is not expected in healthy brain under normoxic conditions, deficits in oxygen supply or distribution may lead to regions of hypoxia in cortex even when the average tissue PO₂ is well above the hypoxic range. Heterogeneity in network structure and in blood flow distribution within microvascular networks tends to broaden the distribution of tissue PO₂ and increase the likelihood of tissue hypoxia. Although local acute regulation of blood flow is believed to be largely independent of tissue oxygen levels,^22,58 mechanisms of structural adaptation of vessel diameters may be oxygen dependent and may play a critical role in maintaining uniform oxygenation of the healthy cortex and avoiding maldistribution of flow that could lead to local tissue hypoxia even with adequate overall perfusion.⁵⁹

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Institutes of Health grants U01 HL133362 and R01 MH111359.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Authors’ contributions

TS conceived and carried out the study and wrote the text. SS and DB provided the vascular geometry of mouse cortex and conceptual guidance. KB developed the finite-element model. All authors edited the text.

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The mass transfer coefficient for oxygen transport from blood to tissue in cerebral cortex

Abstract

Keywords

Introduction

Methods

Definition of KTO2

Blood and tissue mass transfer coefficients

Estimation of tissue mass transfer coefficient using Krogh cylinder model

Estimation of tissue mass transfer coefficient for realistic network structures

Estimation of capillary mass transfer coefficient

Simulations of nonlinear and time-dependent effects

Parameter values

Results

Discussion

Funding

Declaration of conflicting interests

Authors’ contributions

References

Definition of KTO₂