# Determination of Krogh Coefficient for Oxygen Consumption Measurement from Thin Slices of Rodent Cortical Tissue Using a Fick’s Law Model of Diffusion

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background: Model Equations and Solutions

#### 2.1. Diffusion Equations

**Figure 2.**Two models for oxygen consumption rate in tissue as a function of oxygen tension. Ivanova and Simeonov’s piecewise-linear model of Equation (6) (blue curve) approximates the first-order Michaelis–Menton form $Q\left(P\right)={Q}_{0}\phantom{\rule{0.166667em}{0ex}}P/({K}_{M}+P)$ with Michaelis constant ${K}_{M}={P}^{\ast}/2$ (red curve).

- (a)
- simple parabolic if minimum tension ${P}_{\mathrm{min}}$ exceeds a critical tension value ${P}^{\ast}$;
- (b)
- mixed parabolic/hyperbolic-cosine (cosh) if the $P\left(x\right)$ profile crosses the ${P}^{\ast}$ boundary;
- (c)
- hyperbolic-cosine if the tension at the $x=L$ tissue surface, ${P}_{\mathrm{s}}$, is less than ${P}^{\ast}$.

#### 2.2. Model for Oxygen Consumption vs. Oxygen Partial Pressure

#### 2.3. Solution of Ivanova & Simeonov Diffusion Equations

- (a)
- if the surface pressure ${P}_{\mathrm{s}}$ is sufficiently high such that tissue pressure everywhere exceeds the critical value, i.e., $P\left(x\right)>{P}^{\ast}$, then oxygen consumption rate is constant throughout the tissue: $Q\left(x\right)={Q}_{0}$;
- (b)
- if the surface oxygen tension is insufficiently high, then the declining internal tissue pressure may reach the critical value at some depth $x=\pm \delta $, i.e., $P\left(\right|\delta \left|\right)={P}^{\ast}$, in which case we separate the tissue into ‘outer’ layers $(\delta <x\le L)$ and $(-L\le x<-\delta )$ which are well supplied with oxygen so that $Q\left(x\right)={Q}_{0}$, and an ‘inner’ layer $(-\delta \le x\le \delta )$ with linearly restricted oxygen supply: $Q\left(P\right)=(P/{P}^{\ast}){Q}_{0}$;
- (c)
- if the surface pressure lies below the critical value, ${P}_{\mathrm{s}}<{P}^{\ast}$, then $P\left(x\right)<{P}^{\ast}$ everywhere in the slice, and consequently consumption rate scales linearly with pressure: $Q\left(P\right)=(P/{P}^{\ast}){Q}_{0}$.

**Case (a):**$P\left(x\right)>{P}^{\ast}\Rightarrow Q\left(x\right)={Q}_{0}=\mathrm{const}$$$\begin{array}{cc}\hfill P\left(x\right)\phantom{\rule{0.222222em}{0ex}}& =\phantom{\rule{0.222222em}{0ex}}{P}_{\mathrm{s}}-\frac{{Q}_{0}}{2K}\left({L}^{2}-{x}^{2}\right),\phantom{\rule{2.em}{0ex}}-L\le x\le L\hfill \end{array}$$$$\begin{array}{cc}\hfill q\left(x\right)\phantom{\rule{0.222222em}{0ex}}& =\phantom{\rule{0.222222em}{0ex}}{Q}_{0}x\hfill \end{array}$$This is the simplest case. The tension profile is parabolic throughout, and the flux profile is linear with zero flux at the slice centre.

**Case (b):**Profile crosses critical value at depth $x=\pm \delta $: i.e., $P\left(\right|\delta \left|\right)={P}^{\ast}$For inner layer $(-\delta \le x\le \delta )$:$$\begin{array}{cc}\hfill P\left(x\right)\phantom{\rule{0.222222em}{0ex}}& =\phantom{\rule{0.222222em}{0ex}}{P}^{\ast}\phantom{\rule{0.166667em}{0ex}}\frac{cosh\left(\alpha x\right)}{cosh\left(\alpha \phantom{\rule{0.166667em}{0ex}}\delta \right)},\phantom{\rule{2.em}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}\alpha \equiv \sqrt{\frac{{Q}_{0}}{{P}^{\ast}K}}\hfill \end{array}$$$$\begin{array}{cc}\hfill q\left(x\right)\phantom{\rule{0.222222em}{0ex}}& =\phantom{\rule{0.222222em}{0ex}}K\xb7\left(\alpha {P}^{\ast}\phantom{\rule{0.166667em}{0ex}}\frac{sinh\left(\alpha x\right)}{cosh\left(\alpha \phantom{\rule{0.166667em}{0ex}}\delta \right)}\right)\hfill \end{array}$$For outer layer $(\delta <x\le L)$:$$\begin{array}{cc}\hfill P\left(x\right)\phantom{\rule{0.222222em}{0ex}}& =\phantom{\rule{0.222222em}{0ex}}{P}^{\ast}\phantom{\rule{0.222222em}{0ex}}+\phantom{\rule{0.222222em}{0ex}}\left[\alpha {P}^{\ast}tanh\left(\alpha \phantom{\rule{0.166667em}{0ex}}\delta \right)\right](x-\delta )\phantom{\rule{0.222222em}{0ex}}+\phantom{\rule{0.222222em}{0ex}}\frac{{Q}_{0}}{2K}{(x-\delta )}^{2}\hfill \end{array}$$$$\begin{array}{cc}\hfill q\left(x\right)\phantom{\rule{0.222222em}{0ex}}& =\phantom{\rule{0.222222em}{0ex}}K\xb7\left(\phantom{\rule{-0.166667em}{0ex}}\alpha {P}^{\ast}tanh\left(\alpha \phantom{\rule{0.166667em}{0ex}}\delta \right)\phantom{\rule{0.222222em}{0ex}}+\phantom{\rule{0.222222em}{0ex}}\frac{{Q}_{0}}{K}(x-\delta )\phantom{\rule{-0.166667em}{0ex}}\right)\hfill \end{array}$$These equations for pressure and flux also apply to the mirror-image outer layer $(-L\le x<-\delta )$ after making the substitution $(\delta \to -\delta )$ in Equation (10). Note that the value of $\delta $ in Equations (9) and (10) is unknown in advance, and must be determined numerically by solving the nonlinear equation,$$\begin{array}{c}\hfill tanh\left(\alpha \phantom{\rule{0.166667em}{0ex}}\delta \right)\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}\frac{1-{P}_{\mathrm{s}}/{P}^{\ast}}{\alpha (\delta -L)}\phantom{\rule{0.222222em}{0ex}}+\phantom{\rule{0.222222em}{0ex}}{\textstyle \frac{1}{2}}\alpha (\delta -L)\end{array}$$

**Case (c):**${P}_{\mathrm{s}}<{P}^{\ast}\Rightarrow $ entire profile lies below critical oxygen tension$$\begin{array}{cc}\hfill P\left(x\right)\phantom{\rule{0.222222em}{0ex}}& =\phantom{\rule{0.222222em}{0ex}}{P}_{\mathrm{s}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{cosh\left(\alpha x\right)}{cosh\left(\alpha L\right)},\phantom{\rule{2.em}{0ex}}-L\le x\le L\hfill \end{array}$$$$\begin{array}{cc}\hfill q\left(x\right)\phantom{\rule{0.222222em}{0ex}}& =\phantom{\rule{0.222222em}{0ex}}K\xb7\left(\alpha {P}_{\mathrm{s}}\phantom{\rule{0.166667em}{0ex}}\frac{sinh\left(\alpha x\right)}{cosh\left(\alpha L\right)}\right)\hfill \end{array}$$

#### 2.4. Representative Oxygen Tension Profiles

#### 2.5. Choice of Units: SI vs. ‘Biological’

#### 2.6. Krogh Coefficient for Oxygen in Water

**Table 3.**Chemical composition of Normal and no-Magnesium (no-Mg) artificial cerebrospinal fluids (aCSF) used in mouse cortical thin-slice experiments. Last column lists chloride concentrations for the no-Mg fluid. Total chloride concentration $\mathsf{\Sigma}=4.928$ g/L corresponds to chlorinity $\left[{m}_{\mathrm{Cl}}\right]$ = 4.904 g/kg.

Species | Molar Mass | Normal aCSF | no-Magnesium aCSF | Chloride Content | |
---|---|---|---|---|---|

(g/mol) | (mM) | (mM) | (g/L) | (g/L) | |

NaCl | 58.44 | 130 | 130 | 7.597 | 4.609 |

KCl | 74.55 | 2.5 | 5 | 0.373 | 0.177 |

MgCl${}_{2}$ | 95.21 | 1 | — | — | — |

CaCl${}_{2}$ | 110.98 | 2 | 2 | 0.222 | 0.142 |

NaHCO${}_{3}$ | 84.01 | 2.5 | 2.5 | 0.210 | |

NaOH | 40.00 | 3.5 | 3.5 | 0.140 | |

HEPES | 238.30 | 10 | 10 | 2.383 | |

D-glucose | 180.16 | 20 | 20 | 3.603 | |

$\mathsf{\Sigma}$ = 4.928 |

#### 2.6.1. Diffusion Coefficient for Oxygen in Water

#### 2.6.2. Solubility of Oxygen in Water

#### 2.6.3. Glucose Effects on Oxygen Diffusion and Solubility

- O${}_{2}$ diffusion coefficient: 1% depression
- O${}_{2}$ solubility: 0.5% depression

#### 2.7. Krogh Coefficient for Tissue via Flux Conservation

**(a)****Parabolic profile**

**(b)****Parabolic–cosh mixed profile**

**(c)****Cosh profile**

## 3. Materials and Methods

#### 3.1. Tissue Preparation

#### 3.2. Data Recording

**Figure 5.**Perfusion bath and experimental setup for measuring oxygen-pressure profiles within a thin slice of mouse cortical tissue. (

**a**) Exploded CAD model showing the separate components of the dual-compartment slice perfusion apparatus. (

**b**) Assembled perfusion bath; arrows indicate direction of solution flow. (

**c**) Enlarged view of the brain slice (grey ‘pancake’ shape) and its support structures; the slice sits on a nylon net (shown as a black-shaded horizontal surface). (

**d**) Overview of experimental setup showing left and right pairs of micromanipulators, local-field potential (LFP) electrodes, oxygen probes. (

**e**) Zoomed view of in situ cortical slice sitting on nylon net, and probed by LFP (Ag/AgCl wires) and oxygen (glass) electrodes. (

**f**) Distribution of recording locations (black dots) within the slice cerebral cortex. For illustrative purposes, locations are shown on a single slice (but slices anterior to the one shown were used in some cases). Repeat profiles captured from some locations are not differentiated. (

**g**) Representative arrangement of LFP electrode and oxygen probe for a single recording location, expanded from the shadowed region in (

**f**).

#### 3.3. Experimental Protocol

#### 3.4. Numerical Methods and Curve Fitting

**Figure 6.**Representative ${\mathrm{O}}_{2}$ pressure profiles drawn from datasets v4 (first two columns) and v5 (third column). Profiles (

**a**,

**b**,

**d**,

**f**) show a gradient discontinuity at the lower boundary; profiles (

**c**,

**e**) do not. Negative/positive locations correspond to points lying above/below the $x=0$ line of symmetry. Key: × = sample values; ⊗ = samples selected for 9-point in-tissue curve-fit (parabola or paracosh); dashed-green = extrapolation of parabola/paracosh curve into fluid layers above (left) and below (right) the tissue–fluid interface; dashed-red = tangent to curve at boundary with slope ${(\partial P/\partial x)|}_{x=L}$ [mmHg/$\mathsf{\mu}$m]; solid-black linear segments identify linear pressure trends in proximal fluid layer; ‘Curv’ = ${10}^{3}$ × curvature = ${10}^{3}$ × $\left(2a\right)$ [mmHg/$\mathsf{\mu}$m${}^{2}$]. Gradient ratios ‘slopeL/tang’, ‘slopeR/tang’ give estimates for above-slice, below-slice $({K}_{\mathrm{t}}/{K}_{\mathrm{f}})$ Krogh ratios; ratios exceeding cutoff value 0.725 are rejected.

**Case (a):**If the profile shape is accurately modelled as parabolic, then Equation (20) applies, and the oxygen pressure gradient in tissue is given by

**Case (b):**If the pressure curve is ‘flat-bottomed’, then Equations (9) and (10) apply, indicating that the profile is a ‘paracosh’ mixed case with a hyperbolic-cosine (cosh) core for ($-\delta \le x\le \delta $), and parabolic wings for ($\left|x\right|>\delta $). Paracosh fitting proceeds by iterating curvature $(Q/{K}_{\mathrm{t}})=2a$ and critical pressure ${P}^{\ast}$ to maximise agreement between the paracosh curve and the subset of $({x}_{i},{P}_{i})$ data points lying within the tissue boundaries. Paracosh optimisation makes use of Matlab function fminsearchbnd [D’Errico (2022), www.mathworks.com/matlabcentral/fileexchange/8277-fminsearchbnd-fminsearchcon, (accessed 29 June 2022)] which adds bounded constraints to the standard fminsearch optimiser; in our case, we require that both $2a$ and ${P}^{\ast}$ be non-negative. The critical depth $\delta =\delta ({P}_{\mathrm{s}},{P}^{\ast},2a)$ is obtained via a separate iteration on Equation (11) with surface pressure ${P}_{\mathrm{s}}$ fixed by linear interpolation of the $({x}_{i},{P}_{i})$ data pairs bracketing the $x=\pm L$ boundaries. Once the paracosh curve parameters $(2a,{P}^{\ast},\delta )$ have been established, the pressure gradient at the $x=L$ boundary is given by Equation (25).

**Case (c):**The cosh-only profile (${P}_{\mathrm{s}}<{P}^{\ast}$) was never encountered in any of our oxygen tension soundings; nevertheless, the case-(b) curve-fitting algorithm should work equally well here.

## 4. Results

**Dataset v3**(10 mL/min): Recorded from 19 cortical locations from 4 slices (1 animal); $n=19$ profiles

**Dataset v4**(1 and 2 mL/min): Recorded from 8 locations from 4 slices (1 animal), repeated at 1 and 2 mL/min for each location; $n=16$ profiles

**Dataset v5**(0.5 mL/min): Recorded from 6 cortical locations from 2 slices (1 animal), each profile repeated once, giving 6 profile pairs; $n=12$ profiles

#### 4.1. Oxygen Tension Profiles in Fluid and Tissue

- candidate ratios larger than unity are biologically disallowed (oxygen permeability in tissue cannot be greater than permeability in fluid);
- candidate ratios larger than 0.73 appear to form part of the ${K}_{\mathrm{t}}/{K}_{\mathrm{f}}>1.0$ ‘disallowed’ cluster (e.g., see panel (a) of Figure 7);
- the aggregated histogram ratios of Figure 9 suggest a clear break between ‘allowed’ and ‘disallowed’ clusters if we set the cutoff at ${K}_{\mathrm{t}}/{K}_{\mathrm{f}}=0.725$.

#### 4.2. Scatter Plots of Krogh Ratio vs. Curvature

- Krogh ratio requires separate tissue and fluid curve fits, so variance in the ratio will be the sum of the individual curve-fitting variances for tissue and fluid gradients;
- the flux conservation argument used to derive Equation (24) is invalid if the proximal fluid layer is not stationary (hence the need to impose a Krogh ratio cutoff);
- formation of a local stagnant layer is not guaranteed, even within the closely woven structure of the nylon net that supports the slice.

**Table 4.**Sensitivity of profile curvature and Krogh ratio to variations in aCSF flow rate. Column headings identify the datasets used for computing statistics; the v4-dataset is partitioned into its high- and low-flow subsets. Perfusion rates decrease from left to right across the columns. Krogh statistics summarise the $x=L$ (lower interface) retrievals, but note that candidate Krogh ratios that exceed the ${({K}_{\mathrm{f}}/{K}_{\mathrm{f}})}^{\mathrm{max}}=0.725$ cutoff have been excluded (see Table A1).

v3 | v4 (hi) | v4 (lo) | v5 | All | |
---|---|---|---|---|---|

Flow (mL/min) | 10.0 | 2.0 | 1.0 | 0.5 | |

Curvature, $\left(2a\right)$ | |||||

${10}^{3}$$\times \phantom{\rule{3.33333pt}{0ex}}\mathrm{mean}$ | 6.55 | 5.02 | 4.23 | 3.14 | |

${10}^{3}$$\times \phantom{\rule{3.33333pt}{0ex}}\mathrm{stdev}$ | 1.38 | 0.85 | 0.80 | 0.90 | |

N | 19 | 8 | 8 | 12 | |

Krogh ratio, $({K}_{\mathrm{t}}/{K}_{\mathrm{f}})$ | |||||

mean | 0.553 | 0.595 | 0.538 | 0.568 | 0.562 |

stdev | 0.083 | 0.140 | 0.082 | 0.072 | 0.088 |

N | 11 | 5 | 5 | 9 | 30 |

**Figure 7.**Distribution of Krogh ratios as a function of curvature of fitted parabolic/paracosh function. Results are clustered by dataset (three columns: v3/v4/v5) and interface (two rows: upper/lower). Dashed-red horizontal marks the selected cut-off between accepted (below red line) and rejected (above line) Krogh ratios. Scanning from left to right, lower aCSF flow rates are generally associated with reduced curvature values, implying increasingly constrained ${\mathrm{O}}_{2}$ consumption. Linked pairs show repeated sampling at the same location. For v4, flow rate was set at 1 (open circles) or 2 mL/min (filled circles); for v5, flow rate was fixed at 0.5 mL/min. Outlier pairs 414/416 (v4) and 507/508 (v5) have very discrepant Krogh ratio estimates at the lower interface, possibly due to mechanical disturbance of the slice during withdrawal of ${\mathrm{O}}_{2}$ probe prior to repeat sounding. See Figure 6 and Table A1.

**Figure 8.**Flow rate clustering and curvature dependence aggregated across (lower interface) of Figure 7d–f. (

**a**) Aggregated scatterplot of candidate Krogh ratio $({K}_{\mathrm{t}}/{K}_{\mathrm{f}})$ vs. curvature of fitted parabolic/paracosh function, clustered by flow rate [10, 2, 1, or 0.5] mL/min, as indicated by shaded convex-hull polygons (computed via Matlab function convhull). Qualitatively, the polygon centroids move to the left as flow rate decreases, implying that curvature decreases (profiles become flatter) as perfusion flow rate is reduced. This trend is made quantitative in (

**b**) with a sigmoid fit to the Table 4 curvature means (magenta asterisks) at each flow rate. The fitted curve is $y={y}^{\mathrm{max}}\xb7{x}^{n}/({K}^{n}+{x}^{n})$ with [${y}^{\mathrm{max}}=7.5\times {10}^{-3}$ mmHg/$\mathsf{\mu}$m${}^{2}$; $K=0.75$ mL/min; $n=0.75$].

#### 4.3. Histograms for Krogh Ratio and Profile Curvature

**Figure 9.**Histograms for Krogh ratios aggregated over [v3, v4, v5] datasets illustrated in Figure 7, but restricted to domain $({K}_{\mathrm{t}}/{K}_{\mathrm{f}})\le 1.0$. Red-dashed line marks the accept/reject boundary set at 0.725: only Krogh ratios below cutoff are associated with a well-defined stationary layer. Comparing panels (

**a**,

**b**), the lower tissue–fluid interface is more likely to form a nonflowing boundary layer. In panel (

**c**), for each profile, the smaller of the [upper interface, lower interface] Krogh ratio is selected.

#### 4.4. Possible Linkage between SLE Activity and Formation of Stationary Boundary Layer

**Figure 10.**Curvature histograms for each of the [v3, v4, v5] datasets illustrated in Figure 7. As flow rate decreases from (

**a**) 10 → (

**b**) [1 or 2] → (

**c**) 0.5 mL/min, average curvature decreases, meaning that the parabolic/paracosh curves become ‘flatter’ with shallower wings. For fixed ${K}_{\mathrm{t}}$, a flatter curvature implies reduced metabolism.

#### 4.5. Estimation of Krogh Coefficient for Cortical Tissue at Room Temperature

## 5. Discussion

(mL O_{2})/(cm·min·atm): | Ganfield et al. [8] | |

(mmol O_{2})/(cm·min·mmHg): | Ivanova & Simeonov [10] |

- 0.59 (kidney), 0.65 (liver), 1.35 (brain), 1.44 (heart),

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

1D | one-dimensional |

aCSF | artificial cerebral spinal fluid |

Ag/AgCl | silver/silver chloride (electrode) |

I & S | Ivanova and Simeonov (2012) |

IUPAC | International Union of Pure and Applied Chemistry |

LFP | local-field potential |

NIST | National Institute of Standards and Technology |

NMDA | N-methyl-D-aspartate |

no-Mg | no magnesium |

PDE | partial differential equation |

pO${}_{2}$ | partial pressure of oxygen |

rmse | root-mean-square error |

SD | standard deviation |

SLE | seizure-like event |

## Appendix A. Curve-Fitting Summary

**Table A1.**Summary table for 47 pO${}_{2}$ pressure profiles. ‘Ref’ = vnn where v = [3, 4, 5] identifies dataset, nn = profile index; ‘Flow’ = aCSF flow rate [mL/min]; ‘Flat’ = flag indicating paracosh (1) or parabolic (0) fit; ‘rmse’ = rms error for curve fit [mmHg]; ‘Curvature’ = ${10}^{3}$ × $\left(2a\right)$ [mmHg/$\mathsf{\mu}$m${}^{2}$]; ${P}_{\mathrm{s}}$ = fitted surface pressure at $x=L$ boundary [mmHg]; ${P}_{\mathrm{min}}$ = fitted pressure at $x=0$ [mmHg]; ‘Tangent’ = ${(\partial P/\partial x)|}_{x=L}$ [mmHg/$\mathsf{\mu}$m]; ‘Ratio (top)’, ‘Ratio (bot)’ = Krogh ratio (${K}_{\mathrm{t}}/{K}_{\mathrm{f}}$) at upper, lower interface. NB: Values in [brackets] are to be disregarded (existence of stationary layer is implausible).

Ref | Flow | Flat? | rmse | Curvature | ${\mathit{P}}_{\mathbf{s}}$ | ${\mathit{P}}_{\mathbf{min}}$ | Tangent | Ratio (top) | Ratio (bot) |
---|---|---|---|---|---|---|---|---|---|

301 | 10.0 | 0 | 4.62 | 5.97 | 120.2 | 0.8 | 1.194 | [0.999] | [1.018] |

302 | 10.0 | 1 | 1.15 | 6.92 | 124.2 | 0.0 | 1.311 | [0.835] | 0.505 |

303 | 10.0 | 1 | 3.20 | 9.34 | 83.3 | 0.0 | 1.247 | [0.993] | [0.981] |

304 | 10.0 | 0 | 1.20 | 5.04 | 131.7 | 31.0 | 1.007 | [0.789] | [1.032] |

305 | 10.0 | 0 | 3.40 | 6.04 | 165.8 | 45.0 | 1.208 | [1.006] | [0.930] |

306 | 10.0 | 0 | 3.26 | 5.17 | 170.5 | 67.0 | 1.035 | [0.763] | 0.640 |

307 | 10.0 | 0 | 5.86 | 8.97 | 205.6 | 26.2 | 1.795 | [0.806] | 0.608 |

308 | 10.0 | 0 | 3.65 | 6.05 | 154.1 | 33.1 | 1.210 | [0.892] | 0.649 |

309 | 10.0 | 0 | 3.18 | 5.96 | 134.4 | 15.2 | 1.193 | [0.923] | 0.538 |

310 | 10.0 | 1 | 2.38 | 9.13 | 134.9 | 0.0 | 1.570 | [0.839] | 0.462 |

311 | 10.0 | 1 | 4.17 | 6.65 | 121.4 | 0.0 | 1.270 | [1.037] | 0.563 |

312 | 10.0 | 0 | 3.31 | 6.69 | 191.1 | 57.3 | 1.338 | [1.379] | [0.908] |

313 | 10.0 | 1 | 1.64 | 5.05 | 93.1 | 1.3 | 0.962 | [1.001] | [1.049] |

314 | 10.0 | 0 | 5.13 | 7.87 | 187.2 | 29.8 | 1.575 | [0.951] | [0.764] |

315 | 10.0 | 1 | 2.46 | 5.13 | 47.5 | 0.0 | 0.698 | [1.132] | [1.314] |

316 | 10.0 | 0 | 2.13 | 6.22 | 133.4 | 8.9 | 1.245 | [1.154] | 0.675 |

317 | 10.0 | 1 | 2.31 | 7.03 | 95.0 | 0.0 | 1.155 | [1.235] | 0.479 |

318 | 10.0 | 0 | 2.68 | 6.12 | 135.1 | 12.6 | 1.225 | [0.884] | 0.551 |

319 | 10.0 | 1 | 1.63 | 5.12 | 49.2 | 0.0 | 0.710 | [1.640] | 0.417 |

401 | 1.0 | 1 | 1.51 | 4.50 | 60.6 | 0.0 | 0.739 | [0.947] | 0.461 |

402 | 1.0 | 1 | 2.15 | 3.90 | 73.6 | 0.0 | 0.758 | [0.740] | 0.565 |

403 | 2.0 | 0 | 2.48 | 5.09 | 119.8 | 18.0 | 1.018 | [1.022] | 0.428 |

404 | 2.0 | 0 | 1.47 | 4.56 | 115.0 | 23.8 | 0.912 | [1.132] | 0.704 |

405 | 1.0 | 0 | 1.36 | 4.49 | 93.6 | 3.8 | 0.897 | [0.845] | 0.582 |

406 | 1.0 | 0 | 5.27 | 5.49 | 128.3 | 18.5 | 1.098 | [0.894] | [0.766] |

407 | 2.0 | 0 | 4.18 | 5.51 | 145.0 | 34.8 | 1.102 | [1.013] | 0.458 |

408 | 2.0 | 0 | 3.78 | 5.62 | 171.3 | 59.0 | 1.123 | [0.856] | [0.998] |

409 | 1.0 | 0 | 1.43 | 2.81 | 58.3 | 2.1 | 0.562 | [1.378] | [1.151] |

410 | 1.0 | 1 | 0.90 | 3.69 | 40.0 | 0.0 | 0.543 | [1.250] | [1.185] |

411 | 2.0 | 0 | 2.77 | 3.72 | 89.6 | 15.1 | 0.745 | [1.365] | [1.355] |

412 | 2.0 | 0 | 2.00 | 3.98 | 89.3 | 9.7 | 0.796 | [1.226] | 0.717 |

413 | 1.0 | 0 | 2.85 | 4.82 | 99.1 | 2.7 | 0.965 | 0.570 | 0.637 |

414 | 1.0 | 1 | 1.39 | 4.16 | 68.2 | 0.0 | 0.753 | [0.949] | 0.447 |

415 | 2.0 | 0 | 3.83 | 5.61 | 140.7 | 28.6 | 1.121 | 0.656 | 0.668 |

416 | 2.0 | 1 | 3.74 | 6.09 | 78.7 | 0.0 | 0.979 | [1.102] | [1.229] |

501 | 0.5 | 1 | 0.58 | 4.57 | 49.9 | 0.0 | 0.675 | [0.949] | 0.541 |

502 | 0.5 | 1 | 1.33 | 3.55 | 50.0 | 0.0 | 0.596 | [1.070] | 0.553 |

503 | 0.5 | 0 | 1.64 | 3.05 | 92.3 | 31.3 | 0.610 | [0.752] | [1.150] |

504 | 0.5 | 0 | 1.11 | 3.16 | 90.8 | 27.7 | 0.631 | 0.614 | [1.103] |

505 | 0.5 | 1 | 2.82 | 3.95 | 71.8 | 3.7 | 0.730 | [0.958] | 0.675 |

506 | 0.5 | 0 | 4.69 | 3.72 | 79.4 | 4.9 | 0.745 | [0.830] | 0.629 |

507 | 0.5 | 0 | 0.61 | 2.31 | 50.7 | 4.5 | 0.462 | [0.849] | [1.093] |

508 | 0.5 | 0 | 1.04 | 2.52 | 58.6 | 8.2 | 0.504 | 0.685 | 0.607 |

509 | 0.5 | 0 | 3.77 | 3.94 | 99.7 | 21.0 | 0.788 | 0.488 | 0.430 |

510 | 0.5 | 0 | 2.78 | 3.41 | 110.2 | 41.9 | 0.683 | 0.560 | 0.512 |

511 | 0.5 | 0 | 1.47 | 1.86 | 185.9 | 148.7 | 0.372 | [0.835] | 0.608 |

512 | 0.5 | 0 | 1.56 | 1.66 | 162.8 | 129.6 | 0.332 | [0.985] | 0.557 |

## References

- Yamamoto, C.; McIlwain, H. Electrical activities in thin sections from the mammalian brain maintained in chemically-defined media in vitro. J. Neurochem.
**1966**, 13, 1333–1343. [Google Scholar] [CrossRef] [PubMed] - Fountain, S.B.; Hennes, S.K.; Teyler, T.J. Aspartame exposure and in vitro hippocampal slice excitability and plasticity. Fundam. Appl. Toxicol.
**1988**, 11, 221–228. [Google Scholar] [CrossRef] [PubMed] - Lipton, P.; Aitken, P.G.; Dudek, F.E.; Eskessen, K.; Espanol, M.T.; Ferchmin, P.A.; Kelly, J.B.; Kreisman, N.R.; Landfield, P.W.; Larkman, P.M. Making the best of brain slices: Comparing preparative methods. J. Neurosci. Methods
**1995**, 59, 151–156. [Google Scholar] [CrossRef] - Voss, L.J.; van Kan, C.; Envall, G.; Lamber, O. Impact of variation in tissue preparation methodology on the functional outcome of neocortical mouse brain slices. Brain Res.
**2020**, 1747, 147043. [Google Scholar] [CrossRef] - Ramirez, O.T.; Mutharasan, R. Cell cycle- and growth phase-dependent variations in size distribution, antibody productivity, and oxygen demand in hybridoma cultures. Biotechnol. Bioeng.
**1990**, 36, 839–848. [Google Scholar] [CrossRef] [PubMed] - Deshpande, R.R.; Wittmann, C.; Heinzle, E. Microplates with integrated oxygen sensing for medium optimization in animal cell culture. Cytotechnology
**2004**, 46, 1–8. [Google Scholar] [CrossRef][Green Version] - Eyer, K.; Oeggerli, A.; Heinzle, E. On-line gas analysis in animal cell cultivation: II. Methods for oxygen uptake rate estimation and its application to controlled feeding of glutamine. Biotechnol. Bioeng.
**1995**, 45, 54–62. [Google Scholar] [CrossRef] - Ganfield, R.A.; Nair, P.; Whalen, W.J. Mass transfer, storage, and utilization of O
_{2}in cat cerebral cortex. Am. J. Physiol.**1970**, 219, 814–821. [Google Scholar] [CrossRef][Green Version] - Fujii, T.; Buerk, D.G.; Whalen, W.J. Activation energy in the mammalian brain slice as determined by oxygen micro-electrode measurements. Jpn. J. Physiol.
**1981**, 31, 279–283. [Google Scholar] [CrossRef] - Ivanova, R.; Simeonov, G. A formula for the oxygen uptake of thin tissue slice in terms of its surface oxygen tension. Comput. Math. Appl.
**2012**, 64, 322–336. [Google Scholar] [CrossRef][Green Version] - Chen, P.Y.; Van Liew, H.D. Krogh constants for diffusion of nitrogen and carbon monoxide in bladder tissue. Respir. Physiol.
**1975**, 24, 43–49. [Google Scholar] [CrossRef] - Kawashiro, T.; Scheid, P. Measurement of Krogh’s diffusion constant of CO
_{2}in respiring muscle at various CO_{2}levels: Evidence for facilitated diffusion. Pflüg. Arch.**1976**, 362, 127–133. [Google Scholar] [CrossRef] [PubMed] - van der Laarse, W.J.; des Tombe, A.L.; van Beek-Harmsen, B.J.; Lee-de Groot, M.B.E.; Jaspers, R.T. Krogh’s diffusion coefficient for oxygen in isolated Xenopus skeletal muscle fibers and rat myocardial trabeculae at maximum rates of oxygen consumption. J. Appl. Physiol.
**2005**, 99, 2173–2180. [Google Scholar] [CrossRef][Green Version] - Sasaki, N.; Horinouchi, H.; Ushiyama, A.; Minamitani, H. A new method for measuring the oxygen diffusion constant and oxygen consumption rate of arteriolar walls. Keio J. Med.
**2012**, 61, 57–65. [Google Scholar] [CrossRef] [PubMed][Green Version] - Poole, D.C.; Pittman, R.N.; Musch, T.I.; Østergaard, L. August Krogh’s theory of muscle microvascular control and oxygen delivery: A paradigm shift based on new data. J. Physiol.
**2020**, 598, 4473–4507. [Google Scholar] [CrossRef] [PubMed] - Krogh, A. The rate of diffusion of gases through animal tissues, with some remarks on the coefficient of invasion. J. Physiol.
**1919**, 52, 391–408. [Google Scholar] [CrossRef] [PubMed] - Forstner, H.; Gnaiger, E. Calculation of Equilibrium Oxygen Concentration. In Polarographic Oxygen Sensors; Gnaiger, E., Forstner, H., Eds.; Springer: Berlin/Heidelberg, Germany, 1983; pp. 321–333. [Google Scholar] [CrossRef]
- Han, P.; Bartels, D.M. Temperature dependence of oxygen diffusion in H
_{2}O and D_{2}O. J. Phys. Chem.**1996**, 100, 5597–5602. [Google Scholar] [CrossRef] - Green, E.J.; Carritt, D.E. New tables for oxygen saturation of seawater. J. Mar. Res.
**1967**, 25, 140–147. [Google Scholar] - van Stroe, A.; Janssen, L.J.J. Determination of the diffusion coefficient of oxygen in sodium chloride solutions with a transient pulse technique. Anal. Chim. Acta
**1993**, 279, 213–219. [Google Scholar] [CrossRef][Green Version] - Burkholder, J.B.; Sander, S.P.; Abbatt, J.; Barker, J.R.; Cappa, C.; Crounse, J.D.; Dibble, T.S.; Huie, R.E.; Kolb, C.E.; Kurylo, M.J.; et al. Chemical Kinetics and Photochemical Data for Use in Atmospheric Studies; Evaluation No. 19; JPL Publication 19-5; Jet Propulsion Laboratory: Pasadena, CA, USA, 2019. [Google Scholar]
- Sander, R. Compilation of Henry’s law constants (version 4.0) for water as solvent. Atmos. Chem. Phys.
**2015**, 15, 4399–4981. [Google Scholar] [CrossRef][Green Version] - van Stroe-Biezen, S.A.M.; Janssen, A.A.; Janssen, L.J.J. Solubility of oxygen in glucose solutions. Anal. Chim. Acta
**1993**, 280, 217–222. [Google Scholar] [CrossRef][Green Version] - Thomas, M.G.; Covington, J.A.; Wall, M.J. A chamber for the perfusion of in vitro tissue with multiple solutions. J. Neurophysiol.
**2013**, 110, 269–277. [Google Scholar] [CrossRef] [PubMed][Green Version] - Aram, J.A.; Lodge, D. Validation of a neocortical slice preparation for the study of epileptiform activity. J. Neurosci. Methods
**1988**, 23, 211–224. [Google Scholar] [CrossRef] - Anderson, W.W.; Lewis, D.V.; Swartzwelder, H.S.; Wilson, W.A. Magnesium-free medium activates seizure-like events in the rat hippocampal slice. Brain Res.
**1986**, 398, 215–219. [Google Scholar] [CrossRef] [PubMed] - Buerk, D.G.; Saidel, G.M. Local kinetics of oxygen metabolism in brain liver tissues. Microvasc. Res.
**1978**, 16, 391–405. [Google Scholar] [CrossRef]

**Figure 1.**Representative $P\left(x\right)$ solutions of Ivanova & Simeonov diffusion–consumption Equations (4) and (5), assuming a piecewise linear model (6) for consumption rate (see blue curve in Figure 2). Shading represents thin slab of tissue which extends from $-L$ to L. Here, ${P}_{\mathrm{s}}=$ surface pressure; ${P}^{\ast}=$ critical pressure below which oxygen consumption is restricted; $\delta =$ depth at which pressure reaches critical value. (

**a**) If minimum tension exceeds ${P}^{\ast}$, pressure profile is a simple parabolic function of depth. (

**b**) If the pressure profile crosses the ${P}^{\ast}$ boundary, the central portion forms a flattened hyperbolic-cosine (cosh) ‘basin’ that smoothly merges with parabolic ‘wings’ for $P>{P}^{\ast}$. (

**c**) If the surface tension falls below the critical value, the cosh basin extends to the tissue boundaries.

**Figure 3.**Parabolic and para–cosh curve fits for four representative tension vs. depth profiles measured from healthy slices of mouse brain tissue at room temperature (∼20 ${}^{\xb0}$C). The grey shading indicates the 400-$\mathsf{\mu}$m extent of the slice for $-200\le x/\mathsf{\mu}\mathrm{m}\le 200$, with $x=0$ $\mathsf{\mu}$m marking the central plane of symmetry of the slice. (

**a**) Parabolic profiles showing no restriction of metabolic rate. (

**b**) Insufficient oxygen tension leads to flattened profiles with a cosh-modulated central basin joined to parabolic wings to the left and right.

**Figure 4.**Temperature dependence of oxygen diffusion, solubility, and permeability in pure water, and in no-Mg artificial cerebral spinal fluid (aCSF). (

**a**) Oxygen diffusion in aCSF (dashed-red trace) is depressed by ∼1% relative to pure water (solid blue trace) by small viscosity increase caused by presence of 0.02 mol/L glucose. (

**b**) Oxygen solubility in aCSF is reduced by ∼5% by ‘salting out’ effect of chloride salts (dashed-blue trace) with chlorinity $\left[{m}_{\mathrm{Cl}}\right]=4.904$ g/kg; and by a further ∼0.5% by glucose effect. (

**c**) Oxygen Krogh coefficient in fluid is given by the product of diffusion and solubility, $K=DS$. Glucose effect is assumed to be cumulative, resulting in a ∼1.5% depression below the saline curve. Asterisk (*) marks the value (${K}^{\ast}=2.5\times {10}^{-14}$ mol/(m·s·Pa)) used by Ganfield et. al. (1970) [8] for Krogh coefficient for fluid at 20 ${}^{\xb0}\mathrm{C}$.

**Table 1.**Selection of non-SI units for oxygen Krogh coefficient appearing in the physiology literature. The numerator represents quantity of gas, e.g., (mmol O${}_{2}$), (mL O${}_{2}$), (cm${}^{3}$ O${}_{2}$), etc. Bracketed rows of the table indicate equivalent units.

Unit | Reference | ||
---|---|---|---|

1 | $\frac{\mathrm{mol}}{\mathrm{m}\xb7\mathrm{s}\xb7\mathrm{Pa}}$ | SI | |

2 | $\frac{\mathrm{mmol}}{\mathrm{cm}\xb7\mathrm{min}\xb7\mathrm{mmHg}}$ | Ivanova & Simeonov (2012) [10] | $\left.\begin{array}{c}\\ \end{array}\right\}$ |

3 | $\frac{\mathrm{mmol}}{\mathrm{cm}\xb7\mathrm{min}\xb7\mathrm{torr}}$ | Kawashiro & Scheid (1976) [12] | |

4 | $\frac{\mathrm{nM}\xb7{\mathrm{mm}}^{3}}{\mathrm{mm}\xb7\mathrm{s}\xb7\mathrm{mmHg}}$ | van der Laarse et al. (2005) [13] | |

5 | $\frac{\mathrm{mL}}{\mathrm{cm}\xb7\mathrm{min}\xb7\mathrm{atm}}$ | Ganfield et al. (1970) [8] | $\left.\begin{array}{c}\\ \end{array}\right\}$ |

6 | $\frac{\mathrm{cm}{}^{3}}{\mathrm{cm}\xb7\mathrm{min}\xb7\mathrm{atm}}$ | Chen & Liew (1975) [11]; Poole et al. (2020) [15] | |

7 | $\frac{\mathrm{mL}}{\mathrm{cm}\xb7\mathrm{s}\xb7\mathrm{mmHg}}$ | Sasaki et al. (2012) [14] |

**Table 2.**Ideal-gas molar volumes (L/mol) for STP and NTP temperature and pressure definitions. Bracketed value in first row is the actual (non-ideal) molar volume for O${}_{2}$ at STP (pre-1982).

Standard | Temperature, T | Pressure, p | Molar Volume, ${\mathit{V}}_{\mathbf{m}}$ |
---|---|---|---|

STP (IUPAC to 1982) | 273.15 K (0 ${}^{\xb0}\mathrm{C}$) | 1 atm = 101.325 kPa | 22.414 (22.392) |

STP (IUPAC after 1982) | 273.15 K (0 ${}^{\xb0}\mathrm{C}$) | 100 kPa | 22.711 |

NTP (NIST) | 293.15 K (20 ${}^{\xb0}\mathrm{C}$) | 1 atm | 24.055 |

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**MDPI and ACS Style**

Steyn-Ross, D.A.; Steyn-Ross, M.L.; Sleigh, J.W.; Voss, L.J. Determination of Krogh Coefficient for Oxygen Consumption Measurement from Thin Slices of Rodent Cortical Tissue Using a Fick’s Law Model of Diffusion. *Int. J. Mol. Sci.* **2023**, *24*, 6450.
https://doi.org/10.3390/ijms24076450

**AMA Style**

Steyn-Ross DA, Steyn-Ross ML, Sleigh JW, Voss LJ. Determination of Krogh Coefficient for Oxygen Consumption Measurement from Thin Slices of Rodent Cortical Tissue Using a Fick’s Law Model of Diffusion. *International Journal of Molecular Sciences*. 2023; 24(7):6450.
https://doi.org/10.3390/ijms24076450

**Chicago/Turabian Style**

Steyn-Ross, D. Alistair, Moira L. Steyn-Ross, Jamie W. Sleigh, and Logan J. Voss. 2023. "Determination of Krogh Coefficient for Oxygen Consumption Measurement from Thin Slices of Rodent Cortical Tissue Using a Fick’s Law Model of Diffusion" *International Journal of Molecular Sciences* 24, no. 7: 6450.
https://doi.org/10.3390/ijms24076450