Fick's law of diffusion in tissue

Concept

Fick's law of diffusion in tissue

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Statement

In a tissue treated as a homogeneous medium at rest, the net molar flux of a dilute, non-reacting solute is proportional to minus its own concentration gradient, \(\mathbf{J}=-D\,\nabla C\), with \(D\) the diffusion coefficient of that solute in that medium at that temperature. Combined with conservation of matter this gives \(\partial C/\partial t = D\nabla^{2}C+q\); and for a barrier of area \(A\) and thickness \(L\) carrying no sources or sinks, at steady state, the transfer rate is \(\dot n = DA\,\Delta C/L\), or in the partial-pressure form used for respiratory gases \(\dot n = K A\,\Delta P/L\) with Krogh's diffusion constant \(K=D\alpha\), where \(\alpha\) is the solubility of the gas in the medium.

Why it matters

Diffusion is the only transport mechanism that costs a cell nothing and the only one that operates everywhere, all the time, without machinery. Every molecular process this unit describes is scheduled by it: a transcription factor finds its operator by three-dimensional diffusive search, a second messenger crosses a cell in milliseconds because the cell is small, and a morphogen sets up a spatial pattern precisely because diffusion spreads it while degradation consumes it. Diffusion and osmosis establishes the qualitative statistical picture; this page supplies the quantitative law, its derivation from that same statistical picture, and — crucially — the boundaries beyond which it stops describing anything real.

The law also fixes an architectural constraint that runs through all of physiology. Because the time to diffuse a distance \(L\) scales as \(L^{2}\), diffusion is superb over micrometres and hopeless over millimetres. That single scaling is why gas exchange surfaces are thin and enormous, why capillaries are never more than a few tens of micrometres apart, why anything larger than a flatworm needs a circulation (mass transport), and why the core of a solid tumour more than about \(200\ \mu\text{m}\) from a vessel is hypoxic. Clinically, the same equation is the definition of the lung's diffusing capacity, and the reason a thickened alveolar membrane impairs oxygen uptake.

Hypotheses

The solute is dilute and non-interacting, so that \(D\) does not depend on \(C\).Fick's first law is the leading term of a constitutive expansion about equilibrium. In concentrated solution the activity coefficient varies with concentration and the true driving force is the gradient of chemical potential, not of concentration; \(D\) then becomes a function of \(C\) and the flux is no longer linear in \(\nabla C\). Haemoglobin inside an erythrocyte, at roughly \(5\ \text{mM}\) and 33 g per 100 mL, is emphatically not in this regime.

There is no bulk flow of the medium relative to the frame in which \(\mathbf{J}\) is measured.Fick's law describes flux relative to the solvent. Where the solvent itself moves, total transport is \(\mathbf{J}=-D\nabla C + \mathbf{v}C\), and the convective term dominates once the Péclet number \(\mathrm{Pe}=vL/D\) exceeds unity. In a capillary, with \(v\approx 0.5\ \text{mm s}^{-1}\) and \(L\approx 1\ \text{mm}\), \(\mathrm{Pe}\approx 250\) for oxygen: along the vessel, transport is convection, and only across its wall is it diffusion.

The barrier itself neither produces nor consumes the solute.The linear concentration profile of the slab result follows from \(d^{2}C/dx^{2}=0\), which requires \(q=0\) inside the barrier. Living tissue consumes oxygen, so the profile through respiring tissue is parabolic, not linear, and the flux is not constant with depth — this is exactly the Krogh problem solved in Worked example 2.

The system is at steady state, and the medium is homogeneous and isotropic.Only then is a single \(L\) and a single \(D\) meaningful. After a step change in \(\Delta C\) the flux approaches its steady value over a time of order \(L^{2}/D\); and in structured tissue the diffusive path is longer than the straight-line distance, so the measured effective coefficient is \(D_{\text{eff}}=D/\lambda^{2}\) with tortuosity \(\lambda\) (about \(1.6\) in brain extracellular space, so \(D_{\text{eff}}\approx 0.4\,D\)).

Local equilibrium holds at every interface, and there is no cross-coupling between fluxes.What is continuous across a gas–liquid or water–lipid boundary is chemical potential, not concentration: concentration jumps by the partition or solubility coefficient, which is why respiratory gases are handled in partial pressures. Linear irreversible thermodynamics also permits cross terms (a temperature gradient driving mass flux, or solvent drag on solute); Fick's law is the statement that these are negligible.

Proof

The law is derived twice: once from the random walk, which shows where \(D\) comes from and what it means microscopically, and once from the chemical potential, which shows exactly which hypothesis makes the flux linear. Conservation of matter then converts the local law into the two forms actually used at the bench.

\[ J(x)=\frac{1}{2}\,\frac{\delta}{\tau}\,C\!\left(x-\tfrac{\delta}{2}\right)-\frac{1}{2}\,\frac{\delta}{\tau}\,C\!\left(x+\tfrac{\delta}{2}\right) \]

Idealise the thermal motion of Hypothesis 1 as a one-dimensional random walk: every \(\tau\) seconds each molecule steps a distance \(\delta\) left or right with probability \(\tfrac12\) each, independently of every other molecule and of its own history. In one interval \(\tau\), exactly half the molecules in the slab of thickness \(\delta\) immediately to the left of the plane at \(x\) cross it rightwards, i.e. \(\tfrac12 C(x-\delta/2)\,\delta\) molecules per unit area; the other term counts the leftward crossings. No molecule is doing anything directional; the asymmetry is entirely in how many are available on each side. A

\[ C\!\left(x\pm\tfrac{\delta}{2}\right)=C(x)\pm\frac{\delta}{2}\frac{\partial C}{\partial x}+O(\delta^{2}) \;\Longrightarrow\; J=-\frac{\delta^{2}}{2\tau}\frac{\partial C}{\partial x}\equiv -D\frac{\partial C}{\partial x},\qquad D=\frac{\delta^{2}}{2\tau} \]

Taylor-expand both terms about \(x\) and subtract; the zeroth-order terms cancel exactly, which is the formal content of “no net flux without a gradient”. Truncating after the first derivative is legitimate only in the continuum limit, where \(C\) changes negligibly over one step length \(\delta\) — the hidden smallness parameter of the whole theory. The step length and step time are not separately observable; only the combination \(\delta^{2}/2\tau\) survives, and that combination is \(D\), with units \(\text{m}^{2}\,\text{s}^{-1}\). B

\[ J=-uC\frac{\partial\mu}{\partial x},\qquad \mu=\mu^{0}+RT\ln\!\left(\frac{C}{C^{0}}\right) \;\Longrightarrow\; J=-uRT\frac{\partial C}{\partial x},\qquad D=uRT \]

The independent derivation. In linear irreversible thermodynamics a species drifts at velocity \(u\times(\text{force})\) under the thermodynamic force \(-\partial\mu/\partial x\) per mole, where \(u\) is the mobility; the flux is that velocity times the concentration. Substituting the ideal-dilute chemical potential and differentiating the logarithm gives \(C\,\partial\mu/\partial x=RT\,\partial C/\partial x\), and the concentration cancels. This is the Einstein relation \(D=uRT\), and it exposes Hypothesis 1 precisely: it is the ideality of \(\mu\), not anything about the random walk, that makes the flux strictly proportional to \(\nabla C\) with a constant coefficient. C

\[ \frac{\partial}{\partial t}\int_{V} C\,dV=-\oint_{\partial V}\mathbf{J}\cdot d\mathbf{A}+\int_{V} q\,dV \;\Longrightarrow\; \frac{\partial C}{\partial t}=-\nabla\cdot\mathbf{J}+q \]

Matter is conserved: the amount in any fixed control volume changes only by what crosses its surface plus what is made or destroyed inside, at local rate \(q\) (negative for consumption). The divergence theorem converts the surface integral to a volume integral, and since the control volume was arbitrary the integrands must agree pointwise. Nothing about diffusion has been used yet — this step is bookkeeping, and it is what makes the result independent of the microscopic model. A

\[ \frac{\partial C}{\partial t}=D\nabla^{2}C+q \qquad\text{(Fick's second law, with sources)} \]

Substitute the constitutive law of Step 2 into the conservation law of Step 4 and take \(D\) outside the divergence, which is legal exactly when the medium is homogeneous and \(D\) is independent of \(C\) (Hypotheses 1 and 4). The result is the diffusion equation; every transient statement made later — penetration depths, response times, morphogen gradients — is a solution of it under different boundary conditions. B

\[ \frac{\partial C}{\partial t}=0,\ q=0 \;\Longrightarrow\; \frac{d^{2}C}{dx^{2}}=0 \;\Longrightarrow\; C(x)=C_{1}+\left(C_{2}-C_{1}\right)\frac{x}{L} \]

Specialise to a plane barrier of thickness \(L\) with faces held at \(C_{1}\) and \(C_{2}\) (Hypotheses 3 and 4). A function whose second derivative vanishes on an interval is affine, and the two boundary values fix both constants: the steady profile through an inert slab is a straight line. Note what this forbids — a curved profile at steady state is a direct measurement of a source or sink inside the barrier. A

\[ J=-D\frac{dC}{dx}=\frac{D\,(C_{1}-C_{2})}{L}\ \ \text{(independent of }x\text{)},\qquad \dot n = JA=\frac{DA\,\Delta C}{L}=P A\,\Delta C,\qquad P\equiv\frac{D}{L} \]

Differentiating the affine profile gives a gradient that is the same everywhere in the slab, so the flux is uniform — as it must be, since any variation would accumulate matter somewhere inside and contradict steady state. Multiplying the per-unit-area flux by the area gives the transfer rate, and grouping \(D/L\) defines the permeability \(P\) in \(\text{m s}^{-1}\), the quantity actually measured for a membrane whose thickness is unknown. A

\[ C=\alpha P_{\text{gas}} \;\Longrightarrow\; \dot n=\frac{D\alpha A\,\Delta P}{L}=\frac{K A\,\Delta P}{L},\qquad K\equiv D\alpha \]

For a gas, local equilibrium at the interface (Hypothesis 5) means the dissolved concentration is Henry's-law-proportional to the partial pressure in the adjacent phase, with solubility \(\alpha\). Rewriting the driving force in partial pressures absorbs \(\alpha\) into Krogh's diffusion constant \(K=D\alpha\) and makes the two sides of a gas–liquid interface directly comparable, which raw concentrations are not. This is the form used throughout respiratory physiology; the lumped conductance \(D_{L}\equiv KA/L\) is the diffusing capacity, so that \(\dot n=D_{L}\Delta P\). B

\[ \langle x^{2}\rangle = n\delta^{2}=\frac{t}{\tau}\,\delta^{2}=2Dt \qquad\Longrightarrow\qquad t_{\text{diff}}\sim\frac{L^{2}}{2D} \]

Back to Step 1: after \(n=t/\tau\) independent \(\pm\delta\) steps the displacement is a sum of independent zero-mean terms, so variances add and \(\langle x^{2}\rangle=n\delta^{2}\); substituting \(D=\delta^{2}/2\tau\) gives the Einstein–Smoluchowski result. The same scaling falls out of Step 5 by dimensional analysis, since \(D\) has units of length squared per time and no other timescale exists. Distance therefore costs time quadratically, and that single fact, not the value of \(D\), is what forces the architecture of large organisms. C

Result

\[ \mathbf{J}=-D\,\nabla C,\qquad \frac{\partial C}{\partial t}=D\nabla^{2}C+q,\qquad \dot n=\frac{DA\,\Delta C}{L}=\frac{KA\,\Delta P}{L},\qquad K=D\alpha \]

Reading. The local law says flux is proportional to steepness of gradient and runs downhill; the minus sign carries no physics beyond that direction. The slab law says a barrier's transfer rate rises with area and with driving difference and falls with thickness, so the three design variables of any exchange surface are exactly \(A\), \(L\) and \(\Delta C\) (or \(\Delta P\)). The diffusion equation says the characteristic distance reached in time \(t\) is of order \(\sqrt{2Dt}\), equivalently that the time to cross \(L\) is of order \(L^{2}/2D\).

Units. \(J\) in \(\text{mol m}^{-2}\text{s}^{-1}\); \(D\) in \(\text{m}^{2}\text{s}^{-1}\); \(P=D/L\) in \(\text{m s}^{-1}\); \(\alpha\) in \(\text{mol m}^{-3}\text{Pa}^{-1}\); \(K=D\alpha\) in \(\text{mol m}^{-1}\text{s}^{-1}\text{Pa}^{-1}\); \(D_{L}=KA/L\) in \(\text{mol s}^{-1}\text{Pa}^{-1}\), usually quoted clinically as \(\text{mL min}^{-1}\text{mmHg}^{-1}\).

Scope. Dilute non-reacting solute, no bulk flow, inert homogeneous barrier, steady state (Hypotheses). Representative values at \(37\ ^\circ\text{C}\): \(D\approx 2\times10^{-9}\ \text{m}^{2}\text{s}^{-1}\) for oxygen in tissue water, \(\alpha_{\mathrm{O_2}}\approx 1.0\times10^{-5}\ \text{mol m}^{-3}\text{Pa}^{-1}\) (equivalently \(1.3\ \mu\text{mol L}^{-1}\text{mmHg}^{-1}\), the standard \(0.003\ \text{mL dL}^{-1}\text{mmHg}^{-1}\) of dissolved oxygen in plasma).

Corollaries & converses

Resistances in series add. For \(m\) stacked layers each at steady state the same \(J\) passes through all of them, so \(\Delta C_{\text{total}}=\sum_i J L_i/D_i\) and \(1/P_{\text{total}}=\sum_i L_i/D_i\). The slowest single layer dominates, which is why an unstirred water layer can outweigh the membrane it sits on.
Diffusing capacity. Writing \(D_{L}=KA/L\) turns the slab law into \(\dot n=D_{L}\Delta P\), a single measurable conductance. Its inverse partitions the same way: \(1/D_{L}=1/D_{M}+1/(\theta V_{c})\), separating the membrane component from the blood component (the erythrocyte reaction rate \(\theta\) times capillary blood volume \(V_{c}\)) — the Roughton–Forster partition.
Quadratic distance penalty. With \(D=2\times10^{-9}\ \text{m}^{2}\text{s}^{-1}\), \(t\sim L^{2}/2D\) gives \(25\ \text{ms}\) across a \(10\ \mu\text{m}\) cell, \(4\ \text{minutes}\) across \(1\ \text{mm}\), and about \(8\ \text{years}\) across \(1\ \text{m}\). Diffusion is not slow; distance is expensive.
Critical thickness with consumption. A slab supplied on one face and consuming at constant rate \(M\) is fully oxygenated only if \(L\le\sqrt{2DC_{0}/M}\) (Worked example 2). This is Krogh's tissue-cylinder criterion and the quantitative origin of intercapillary spacing and of tumour hypoxia.
Transient penetration. Into a semi-infinite medium held at \(C_{0}\) from time zero, \(C(x,t)=C_{0}\,\operatorname{erfc}\!\left(x/2\sqrt{Dt}\right)\); the front advances as \(\sqrt{Dt}\), never linearly in \(t\). For a slab, the classical time lag before steady flux is established is \(t_{\text{lag}}=L^{2}/6D\).
Converse. If measured transfer is not proportional to \(\Delta C\) — if it saturates, shows a \(K_{m}\), is stereoselective, or runs uphill — then it is not simple diffusion, and a carrier or pump is present (membrane transport). Linearity in \(\Delta C\) is the operational signature of the law.

Fails without

Sinks inside the barrier (consuming tissue): with \(q=-M\) constant, Step 6 becomes \(D\,d^{2}C/dx^{2}=M\) and the profile is parabolic, \(C(x)=C_{0}-(M/2D)\,x(2L-x)\). Flux now falls with depth instead of being uniform (\(J=M(L-x)\), vanishing at the sealed face), and this profile can be sustained only while \(L\le\sqrt{2DC_{0}/M}\); for a thicker slab oxygen penetrates just that far and everything beyond is anoxic (Problem 5). Applying the inert-slab formula to respiring tissue overestimates delivery to the far side by an unbounded factor.
Bulk flow (\(\mathrm{Pe}\gg1\)): in a capillary at \(v\approx0.5\ \text{mm s}^{-1}\) over a \(1\ \text{mm}\) path, \(\mathrm{Pe}=vL/D\approx250\) for oxygen, so axial transport is convective and the diffusive term is a \(0.4\%\) correction. Fick's law still governs the radial step across the wall, but using it for the axial step underestimates transport by more than two orders of magnitude. This is the entire reason circulatory systems exist.
Charge (ions across a membrane potential): for an ion the driving force includes the electric field, and the flux law becomes Nernst–Planck, \(J=-D\left(\partial C/\partial x+(zF/RT)\,C\,\partial\phi/\partial x\right)\), whose integrated form across a membrane is the Goldman–Hodgkin–Katz equation. Chloride can sit at a concentration difference of tenfold and carry zero net flux, because the two terms cancel — a flat-out contradiction of the uncharged law.
Concentrated or reacting solute: oxygen entering an erythrocyte binds haemoglobin, so the free concentration is buffered and the effective transport is facilitated: total oxygen flux exceeds \(-D_{\mathrm{O_2}}\,dC_{\mathrm{O_2}}/dx\) because oxymyoglobin and oxyhaemoglobin diffuse too (oxygen dissociation curve). Muscle myoglobin can add tens of percent to intracellular oxygen transport, which the bare law cannot produce.
Structured, anisotropic medium: in white matter or in packed extracellular space the straight-line distance is not the path length; substituting the geometric \(L\) with the free-water \(D\) overestimates flux by \(\lambda^{2}\approx2.6\) in brain extracellular space, and along versus across a fibre tract the effective coefficients differ by a factor of several.

Common errors

“Fick's law gives cardiac output.” That is the Fick principle, \(\dot Q=\dot V_{\mathrm{O_2}}/(C_{a}-C_{v})\), a mass-balance statement about a whole organ with no diffusion coefficient in it. The two results share only their author.
“Use the concentration difference across the alveolar membrane.” Concentration is discontinuous at a gas–liquid or water–lipid interface, by the factor \(\alpha\) or by the partition coefficient. Partial pressure (strictly, chemical potential) is what is continuous, which is why the \(K=D\alpha\) form exists.
“Doubling the thickness doubles the equilibration time.” It halves the steady-state flux (which goes as \(1/L\)) but quadruples the equilibration time (which goes as \(L^{2}\)). Two different exponents, routinely conflated.
“\(D\) is a property of the molecule.” It is a property of the molecule in a given medium at a given temperature. By Stokes–Einstein \(D=k_{B}T/6\pi\eta r\), so \(D\) falls with viscosity and rises with temperature both explicitly and through \(\eta\); oxygen's coefficient in tissue is roughly half its value in free water, and a large protein in crowded cytoplasm is slowed several-fold more again.
“Flux is the amount transferred per second.” \(J\) is per unit area. The transfer rate is \(JA\); forgetting \(A\) is the commonest unit error in this topic.
“Oxygen uptake in the healthy lung is diffusion-limited.” At rest it is perfusion-limited: equilibration is complete in roughly a third of the capillary transit time, and the barrier holds a large reserve (Worked example 1). Diffusion limitation appears with exercise, at altitude, with a thickened barrier — and always for carbon monoxide, which is why \(D_{L}\) is measured with CO.

Discussion

Adolf Fick published the law in 1855, and was explicit that he was transplanting the mathematics Fourier had built for heat conduction in 1822 into the problem of dissolved matter. The move was justified by analogy and by his own diffusion-cell experiments rather than by any microscopic argument, and it stayed that way for half a century: the law was a good empirical constitutive relation with an unexplained coefficient. Einstein's 1905 paper on Brownian motion supplied what was missing, deriving \(\langle x^{2}\rangle=2Dt\) and \(D=uRT\) (Steps 3 and 9) from the molecular hypothesis, and thereby turning a measurement of \(D\) into a measurement of Avogadro's number. The order matters pedagogically: the law is prior to and independent of its microscopic justification, which is why it survives unchanged in media where the random walk picture is crude.

The biology enters through the numbers, not the mathematics. With \(D\approx 2\times10^{-9}\ \text{m}^{2}\text{s}^{-1}\), the crossover at which diffusion stops being adequate falls at a few hundred micrometres for a respiring tissue, and essentially every large-scale feature of animal design sits at that boundary: alveolar walls a few tenths of a micrometre thick, capillaries spaced tens of micrometres apart, gut and gill epithelia folded to enormous area, and a pump to do the millimetre-and-above transport convectively. August Krogh's 1919 analysis of the oxygen supply of muscle was the first to run this argument quantitatively, and the tissue cylinder he introduced — a capillary supplying a coaxial sleeve of consuming tissue — is still the standard first model of oxygen delivery.

Two refinements matter at full rigour. First, in a mixture the flux of one species is defined only relative to a reference frame; Fick's law as written holds in the frame of zero volume flux, and in concentrated systems the distinction between the mutual diffusion coefficient, the self-diffusion coefficient measured by tracer or NMR, and the intrinsic coefficients of the Maxwell–Stefan formulation becomes material. Second, the linear law is the first term of a gradient expansion: it presumes the mean free path is small compared with the scale over which \(C\) varies (Step 2). In a synaptic cleft roughly \(20\ \text{nm}\) wide, or inside a narrow channel pore, that separation of scales is marginal, and continuum diffusion becomes an approximation to a stochastic first-passage problem rather than a description of it. Transmitter release into a cleft is better analysed as a small number of molecules executing individual walks with a capture probability than as a concentration field.

Common misconceptions. That diffusion is slow: it is extraordinarily fast at cellular scale — a small metabolite crosses a bacterium in under a millisecond — and only the quadratic scaling makes it useless at organism scale. That the minus sign expresses some attraction toward low concentration: it expresses only that the flux is opposite to the gradient vector, and the underlying molecular motion is unbiased (Step 1). That equilibrium means motion has stopped: at equilibrium the unidirectional fluxes in Step 1 are both still large and merely equal, which is precisely why isotope tracers reveal exchange across membranes that show no net transport. And that facilitated diffusion violates the law: it does not, because a carrier changes the pathway and hence the effective \(D\) and its saturability, not the thermodynamic direction — carrier-mediated transport still runs downhill in chemical potential.

Worked examples

Example 1. Estimate the maximum diffusive oxygen transfer of the human blood–gas barrier treated as a single inert slab, and compare with the measured diffusing capacity and with resting oxygen consumption. Use alveolar surface area \(A=70\ \text{m}^{2}\), barrier thickness \(L=0.6\ \mu\text{m}\), \(D=2.0\times10^{-9}\ \text{m}^{2}\text{s}^{-1}\), \(\alpha=1.0\times10^{-5}\ \text{mol m}^{-3}\text{Pa}^{-1}\), and the initial gradient \(\Delta P=100-40=60\ \text{mmHg}\).

\[ K=D\alpha=(2.0\times10^{-9})(1.0\times10^{-5})=2.0\times10^{-14}\ \text{mol m}^{-1}\text{s}^{-1}\text{Pa}^{-1} \]

Krogh's constant for oxygen in tissue water, from Step 8. Forming \(K\) first keeps the units transparent and lets the driving force stay in pressure. A

\[ D_{L}=\frac{KA}{L}=\frac{(2.0\times10^{-14})(70)}{6.0\times10^{-7}}=2.33\times10^{-6}\ \text{mol s}^{-1}\text{Pa}^{-1} \]

The lumped conductance of the barrier, symbols rearranged before any pressure is inserted. A

\[ \Delta P=60\ \text{mmHg}\times133.3\ \text{Pa mmHg}^{-1}=8.00\times10^{3}\ \text{Pa} \]

Convert the driving force to SI before multiplying; \(1\ \text{mmHg}=133.3\ \text{Pa}\) exactly enough for three figures. A

\[ \dot n=D_{L}\,\Delta P=(2.33\times10^{-6})(8.00\times10^{3})=1.87\times10^{-2}\ \text{mol s}^{-1} \]

The slab law of Step 8 evaluated at the initial gradient — the largest flux the barrier can carry, before capillary blood begins to equilibrate and \(\Delta P\) collapses. A

\[ \dot n=1.87\times10^{-2}\times60=1.12\ \text{mol min}^{-1}\times22.4\ \text{L mol}^{-1}=25\ \text{L min}^{-1}\ \text{(STPD)} \]

Converted to the volume units physiology quotes, using the molar volume \(22.4\ \text{L mol}^{-1}\) at standard temperature and pressure, dry. Resting oxygen consumption is about \(0.25\ \text{L min}^{-1}\), so the idealised barrier carries roughly a hundredfold reserve. B

\[ D_{L}=(2.33\times10^{-6})\times(133.3)\times(60)\times(22.4\times10^{3})\approx 4.2\times10^{2}\ \text{mL min}^{-1}\text{mmHg}^{-1} \]

The same conductance in clinical units, against a measured \(D_{L,\mathrm{O_2}}\) of order \(20\ \text{mL min}^{-1}\text{mmHg}^{-1}\) at rest: the geometric slab overestimates by about twentyfold, so the membrane itself contributes only a few percent of the real resistance. The remainder is plasma, the erythrocyte membrane and interior, and the finite rate of the haemoglobin reaction — exactly the blood component \(1/(\theta V_{c})\) of the Roughton–Forster partition (Corollaries). C

\[ \dot n_{\max}\approx1.9\times10^{-2}\ \text{mol s}^{-1}\approx25\ \text{L min}^{-1},\qquad D_{L}^{\text{slab}}\approx4\times10^{2}\ \text{mL min}^{-1}\text{mmHg}^{-1} \]

Reading. The geometry of the lung is not the limiting step in oxygen uptake at rest. The idealised slab carries a hundredfold margin over the resting demand of \(0.25\ \text{L min}^{-1}\); even the measured conductance, twentyfold smaller, would pass about \(1\ \text{L min}^{-1}\) at this gradient, so a several-fold reserve survives the correction. That is why healthy uptake is perfusion-limited and why substantial disease can be present before resting arterial oxygenation falls.

Scope. The estimate is deliberately naive — a single inert slab (Hypothesis 3) with the full alveolar area available and the initial rather than mean gradient. Its value is the comparison it forces: measured \(D_{L}\) is twentyfold smaller, which localises the true resistance to the blood phase rather than to the tissue barrier.

Example 2. A slab of tissue is supplied with oxygen from one face at \(P=40\ \text{mmHg}\) and is sealed at depth \(L\). It consumes oxygen at a constant rate \(M=1.0\ \text{mL O}_2\) per \(100\ \text{mL}\) tissue per minute (between resting skeletal muscle, of order \(0.1\)–\(0.5\), and brain, about \(3.5\)). With \(D=2.0\times10^{-9}\ \text{m}^{2}\text{s}^{-1}\) and \(\alpha=1.0\times10^{-5}\ \text{mol m}^{-3}\text{Pa}^{-1}\), how thick may the slab be before its far face goes anoxic?

\[ D\frac{d^{2}C}{dx^{2}}=M,\qquad C(0)=C_{0},\qquad \left.\frac{dC}{dx}\right|_{x=L}=0 \]

Steady state in Step 5 with a constant sink \(q=-M\) (zero-order kinetics is appropriate because mitochondrial respiration is saturated well above about \(1\ \text{mmHg}\)). The sealed face carries no flux, which is the boundary condition; by symmetry it also represents the midplane between two identical supply faces. B

\[ \frac{dC}{dx}=\frac{M}{D}(x-L)\;\Longrightarrow\; C(x)=C_{0}-\frac{M}{2D}\,x\,(2L-x) \]

Integrate twice, fixing the first constant with the no-flux condition at \(x=L\) and the second with \(C(0)=C_{0}\). The profile is parabolic, not linear — the signature of a sink inside the medium (Fails without). A

\[ C(L)=C_{0}-\frac{M L^{2}}{2D}\ \ge 0 \;\Longleftrightarrow\; L\le L_{\text{crit}}=\sqrt{\frac{2DC_{0}}{M}} \]

The minimum concentration is at the sealed face. Requiring it to remain non-negative and solving for \(L\) gives Krogh's critical thickness — all symbols, no numbers yet. B

\[ C_{0}=\alpha P=(1.0\times10^{-5})(40\times133.3)=5.33\times10^{-2}\ \text{mol m}^{-3} \]

Henry's law (Step 8) converts the supplied partial pressure into the dissolved concentration that actually appears in the diffusion equation: \(53\ \mu\text{mol L}^{-1}\), a strikingly small store. A

\[ M=\frac{1.0\ \text{mL}}{100\ \text{mL}\cdot\text{min}}=\frac{10\ \text{L m}^{-3}}{\text{min}}\times\frac{1}{22.4\ \text{L mol}^{-1}}\times\frac{1}{60\ \text{s min}^{-1}}=7.44\times10^{-3}\ \text{mol m}^{-3}\text{s}^{-1} \]

Convert the volumetric consumption rate to molar units: \(1\ \text{mL}\) per \(100\ \text{mL}\) is \(10\ \text{mL}\) per litre, i.e. \(10\ \text{L}\) of gas per cubic metre of tissue per minute, then divide by the molar volume and by \(60\). A

\[ L_{\text{crit}}=\sqrt{\frac{2(2.0\times10^{-9})(5.33\times10^{-2})}{7.44\times10^{-3}}}=\sqrt{2.87\times10^{-8}}=1.69\times10^{-4}\ \text{m} \]

Substituting the three numbers into Step 3. The square root is what matters: to supply tissue working ten times harder, the spacing must fall only by \(\sqrt{10}\approx3.2\). A

\[ L_{\text{crit}}=\sqrt{\frac{2DC_{0}}{M}}\approx1.7\times10^{-4}\ \text{m}=170\ \mu\text{m} \]

Reading. A capillary at \(40\ \text{mmHg}\) can oxygenate at most about \(170\ \mu\text{m}\) of tissue at this metabolic rate, so capillaries must be spaced no more than roughly \(340\ \mu\text{m}\) apart — and far closer in hard-working muscle or brain. The same number, arrived at from the same equation, is why solid tumours become hypoxic and necrotic beyond about \(150\)–\(200\ \mu\text{m}\) from the nearest vessel, and why hypoxia drives angiogenesis.

Scope. Planar geometry with zero-order consumption and a single supply face. The cylindrical Krogh version changes the numerical factor, not the \(\sqrt{DC_{0}/M}\) scaling; near the anoxic boundary the zero-order assumption also fails, because respiration becomes oxygen-limited below about \(1\ \text{mmHg}\).

Problems

Glucose diffuses across a \(50\ \mu\text{m}\) unstirred layer of water covering an epithelium, with \(D=6.7\times10^{-10}\ \text{m}^{2}\text{s}^{-1}\) and a concentration difference of \(5\ \text{mM}\) across the layer. Find the flux, and the transfer rate across \(1\ \text{cm}^{2}\).
Solution

Convert first: \(5\ \text{mM}=5\ \text{mol m}^{-3}\), \(L=5.0\times10^{-5}\ \text{m}\), \(A=1\ \text{cm}^{2}=1.0\times10^{-4}\ \text{m}^{2}\). The layer is inert and at steady state, so Step 7 applies with \(J=D\,\Delta C/L\).

\[ J=\frac{(6.7\times10^{-10})(5)}{5.0\times10^{-5}}=6.7\times10^{-5}\ \text{mol m}^{-2}\text{s}^{-1} \]

\[ \dot n=JA=(6.7\times10^{-5})(1.0\times10^{-4})=6.7\times10^{-9}\ \text{mol s}^{-1}=6.7\ \text{nmol s}^{-1} \]

Equivalently the layer's permeability is \(P=D/L=1.34\times10^{-5}\ \text{m s}^{-1}\). Note that \(A\) enters only in the second line: the commonest error here is to report \(J\) itself as the transfer rate.
Using \(D=2.0\times10^{-9}\ \text{m}^{2}\text{s}^{-1}\) and \(t\sim L^{2}/2D\), find the diffusion time over \(10\ \mu\text{m}\), \(100\ \mu\text{m}\), \(1\ \text{mm}\) and \(1\ \text{m}\). What does the pattern imply about the maximum size of an organism without a circulation?
Solution

\[ t=\frac{L^{2}}{2D}=\frac{L^{2}}{4.0\times10^{-9}} \]

\(L=10\ \mu\text{m}=1\times10^{-5}\ \text{m}\): \(t=1\times10^{-10}/4.0\times10^{-9}=2.5\times10^{-2}\ \text{s}=25\ \text{ms}\).

\(L=100\ \mu\text{m}\): \(t=1\times10^{-8}/4.0\times10^{-9}=2.5\ \text{s}\).

\(L=1\ \text{mm}=1\times10^{-3}\ \text{m}\): \(t=1\times10^{-6}/4.0\times10^{-9}=250\ \text{s}\approx4\ \text{min}\).

\(L=1\ \text{m}\): \(t=1/4.0\times10^{-9}=2.5\times10^{8}\ \text{s}\approx8\ \text{years}\).

Each tenfold increase in distance costs a hundredfold in time. Metabolic demand, by contrast, grows with volume, so beyond roughly a millimetre of unstirred tissue — the thickness of a flatworm, which is exactly why it has no circulation — diffusion cannot resupply fast enough and bulk convective transport becomes obligatory.
Compare the diffusive conductance of the alveolar barrier for carbon dioxide and oxygen, given \(D_{\mathrm{CO_2}}=1.6\times10^{-9}\), \(D_{\mathrm{O_2}}=2.0\times10^{-9}\ \text{m}^{2}\text{s}^{-1}\), \(\alpha_{\mathrm{CO_2}}=2.3\times10^{-4}\), \(\alpha_{\mathrm{O_2}}=1.0\times10^{-5}\ \text{mol m}^{-3}\text{Pa}^{-1}\). Physiological gradients are \(\Delta P_{\mathrm{O_2}}=60\ \text{mmHg}\) and \(\Delta P_{\mathrm{CO_2}}=6\ \text{mmHg}\). Why is the ratio of actual transfers not what your answer suggests?
Solution

Conductance per unit \(\Delta P\) is set by \(K=D\alpha\) (Step 8), since \(A\) and \(L\) are shared:

\[ \frac{K_{\mathrm{CO_2}}}{K_{\mathrm{O_2}}}=\frac{(1.6\times10^{-9})(2.3\times10^{-4})}{(2.0\times10^{-9})(1.0\times10^{-5})}=\frac{3.68\times10^{-13}}{2.0\times10^{-14}}=18.4 \]

Carbon dioxide is slightly the larger molecule and so diffuses \(20\%\) more slowly, but it is about \(23\) times more soluble; solubility wins, and the barrier is roughly twenty times more conductive for \(\mathrm{CO_2}\).

At the stated gradients the predicted flux ratio is \(18.4\times(6/60)=1.8\). The actual ratio of \(\mathrm{CO_2}\) output to \(\mathrm{O_2}\) uptake is the respiratory quotient, about \(0.8\). There is no contradiction: as Worked example 1 showed, both gases have diffusive conductance far in excess of demand, so neither flux is set by the barrier. Metabolism sets the fluxes; the barrier merely sets the gradients needed to carry them — which is exactly why \(\mathrm{CO_2}\) needs a tenfold smaller gradient than \(\mathrm{O_2}\).
A solute crosses an unstirred water layer \(100\ \mu\text{m}\) thick (\(D=2.0\times10^{-9}\ \text{m}^{2}\text{s}^{-1}\)) and then a membrane of permeability \(P_{m}=1.0\times10^{-5}\ \text{m s}^{-1}\). Find the overall permeability and the share of the total resistance held by each layer. What happens if the preparation is stirred so the layer falls to \(10\ \mu\text{m}\)?
Solution

The layer's permeability is \(P_{u}=D/L=(2.0\times10^{-9})/(1.0\times10^{-4})=2.0\times10^{-5}\ \text{m s}^{-1}\). Resistances in series add (Corollaries):

\[ \frac{1}{P}=\frac{1}{P_{u}}+\frac{1}{P_{m}}=\frac{1}{2.0\times10^{-5}}+\frac{1}{1.0\times10^{-5}}=5.0\times10^{4}+1.0\times10^{5}=1.5\times10^{5} \]

\[ P=6.7\times10^{-6}\ \text{m s}^{-1} \]

The unstirred layer holds \(5.0\times10^{4}/1.5\times10^{5}=33\%\) of the resistance and the membrane \(67\%\). Stirring to \(10\ \mu\text{m}\) gives \(P_{u}=2.0\times10^{-4}\ \text{m s}^{-1}\), so \(1/P=5.0\times10^{3}+1.0\times10^{5}=1.05\times10^{5}\) and \(P=9.5\times10^{-6}\ \text{m s}^{-1}\), now within \(5\%\) of the membrane's own value. The measured permeability rose by \(43\%\) although nothing about the membrane changed — which is why unstirred-layer artefacts are the standard trap in permeability measurement, and why a highly permeable membrane is the hardest case to measure.
Take the tissue of Worked example 2 (\(C_{0}=5.33\times10^{-2}\ \text{mol m}^{-3}\), \(M=7.44\times10^{-3}\ \text{mol m}^{-3}\text{s}^{-1}\), \(D=2.0\times10^{-9}\ \text{m}^{2}\text{s}^{-1}\)), but suppose the tissue extends \(400\ \mu\text{m}\) from the supply face — as in a tumour cord outgrowing its vessels. Locate the anoxic boundary, give the anoxic fraction, and find the oxygen partial pressure required to abolish it.
Solution

Because \(L=400\ \mu\text{m}\) exceeds \(L_{\text{crit}}=170\ \mu\text{m}\), the no-flux condition no longer sits at \(x=L\). Instead oxygen penetrates to some depth \(x_{p}\lt L\) at which both \(C\) and \(dC/dx\) vanish (no oxygen arrives beyond it, so none can flow past it). Re-solving Step 2 of Worked example 2 with the no-flux point at \(x_{p}\):

\[ C(x)=C_{0}-\frac{M}{2D}\,x\,(2x_{p}-x),\qquad C(x_{p})=C_{0}-\frac{Mx_{p}^{2}}{2D}=0 \]

\[ x_{p}=\sqrt{\frac{2DC_{0}}{M}}=1.69\times10^{-4}\ \text{m}=169\ \mu\text{m} \]

The penetration depth is numerically the critical thickness — the supply cannot reach further whatever lies beyond. The anoxic zone runs from \(169\ \mu\text{m}\) to \(400\ \mu\text{m}\), a fraction \((400-169)/400=58\%\) of the cord.

To oxygenate the full \(400\ \mu\text{m}\) we need \(L_{\text{crit}}=400\ \mu\text{m}\), so \(C_{0}'=ML^{2}/2D=(7.44\times10^{-3})(4.0\times10^{-4})^{2}/(2\times2.0\times10^{-9})=0.298\ \text{mol m}^{-3}\), and

\[ P=\frac{C_{0}'}{\alpha}=\frac{0.298}{1.0\times10^{-5}}=2.98\times10^{4}\ \text{Pa}=223\ \text{mmHg} \]

Since \(C_{0}\propto P\) and \(L_{\text{crit}}\propto\sqrt{C_{0}}\), covering \(2.4\) times the distance requires \(5.6\) times the partial pressure — \(223\ \text{mmHg}\) at the supply face, against the \(40\ \text{mmHg}\) a capillary normally offers and the \(159\ \text{mmHg}\) of oxygen in dry air at sea level. Air cannot deliver it. Pure or hyperbaric oxygen can push arterial \(P_{\mathrm{O_2}}\) well past \(223\ \text{mmHg}\), and that is precisely the rationale for hyperbaric oxygen and for oxygen as a radiosensitiser of hypoxic tumours; but oxygen toxicity caps both the pressure and the duration, so the gain is temporary. The quadratic penalty is why the durable biological answer is to shorten the distance — new vessels — rather than to raise the driving force.