Starling forces and capillary exchange

Concept

Starling forces and capillary exchange

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Statement

Let a microvessel wall separate flowing plasma from the interstitial fluid of a tissue, and let it behave as a passive, isothermal, partially selective membrane with hydraulic conductivity \(L_p\), exchange area \(S\) and Staverman reflection coefficient \(\sigma\) for the plasma proteins. If the wall performs no work on the fluid, if the small solutes cross it so freely that their reflection coefficients are effectively zero, if the protein solutions are dilute enough for their osmotic pressures to be treated as state functions of local concentration, and if the local driving forces are evaluated at the same point of the wall, then the volume flux of water and small solutes across it is linear in the four Starling pressures, \[ J_v \;=\; L_p S\big[(P_c-P_i)-\sigma(\pi_c-\pi_i)\big] \;=\; K_f\,\mathrm{NFP}, \] where \(P_c,P_i\) are the hydrostatic pressures in capillary and interstitium, \(\pi_c,\pi_i\) the corresponding colloid osmotic (oncotic) pressures, \(K_f=L_pS\) the filtration coefficient and \(\mathrm{NFP}\) the net filtration pressure; the hydrostatic difference enters undiscounted while the oncotic difference is discounted by \(\sigma\), and in the revised form the oncotic term is evaluated not against bulk interstitial fluid but against the small space immediately beneath the endothelial glycocalyx, \(\pi_i \to \pi_g\), which changes the prediction of sustained venular reabsorption into a prediction of near-zero steady flux.

Why it matters

Every gram of tissue is fed by diffusion, but the water it sits in is placed there by filtration, and the two obey different laws. Diffusion and osmosis tell you how a molecule crosses a short distance; Starling's principle tells you how much fluid the circulation deposits in the interstitium per minute, and therefore how much the lymphatics must carry back. It is the arithmetic behind oedema, the reason a burn or a septic patient loses litres of plasma into their tissues, the equation that sets glomerular filtration rate and so the entire scale of renal function — the input to everything the countercurrent multiplier then does with that filtrate — and the constraint that fixes how much crystalloid or colloid stays in the vascular compartment after an infusion. It is also where the bulk-flow logic of mass transport hands over to the local, gradient-driven exchange that keeps a tissue alive.

It is also the cleanest example in physiology of a linear phenomenological law derived from irreversible thermodynamics rather than from a mechanism. Nothing in the derivation says what the pathway through the wall looks like — interendothelial cleft, fenestra, or transcytotic vesicle — and that is a feature: the same four pressures govern a muscle capillary, a glomerulus and a lung, and the anatomy enters only through the two coefficients \(L_p\) and \(\sigma\). Measuring those two numbers in a tissue is then a complete description of its fluid exchange, which is exactly what the isogravimetric and osmotic-transient techniques were built to do. Finally, the page is a case study in how a correct equation can be applied with the wrong argument for a century: the classical “filter at the arteriolar end, reabsorb at the venular end” picture uses the right formula with the wrong value of \(\pi_i\), and the correction — the glycocalyx model — changes clinical practice in fluid resuscitation.

Hypotheses

The wall is passive: it does no work on the filtrate.Starling flux is downhill in the combined potential. Where an epithelium actively pumps solute and water follows — proximal tubule, gall bladder, choroid plexus — fluid moves against the net Starling pressure and the equation predicts the wrong direction. Capillary endothelium has no such pump, which is why the passive law works there and not across a transporting epithelium.

Exactly one class of solute is appreciably reflected: the plasma proteins.The full driving force is \(\sum_k \sigma_k \Delta\pi_k\). Across a continuous capillary the small solutes have \(\sigma \approx 0.02\)–\(0.1\) and, more decisively, diffusive permeabilities so high that no gradient of them survives: a \(10\,\mathrm{mM}\) sodium chloride step (\(20\,\mathrm{mosmol\,L^{-1}}\), worth \(387\,\mathrm{mmHg}\) if fully reflected) drives a large transient flux and then nothing, because \(\Delta\pi_k\) itself decays to zero within seconds. Only the proteins, which cannot equilibrate, hold a standing difference. Across the blood–brain barrier \(\sigma_{\mathrm{Na}}\) is close to \(1\) and the permeability is low, the term does not vanish, and osmotherapy works precisely because this hypothesis fails there.

The four pressures are evaluated locally, at the same point of the wall.\(P_c\) falls by tens of millimetres of mercury from the arteriolar to the venular end, so \(J_v\) is a local flux density, and any statement about a whole capillary requires integration along its length (Steps 9–10). Using the mid-capillary value as if it were the mean is exact only when the pressure profile is linear.

The oncotic pressure that opposes filtration is the one adjacent to the selective layer.The classical model inserts the bulk interstitial value \(\pi_i\). If the selective barrier is the glycocalyx and the fluid emerging from the junctional breaks beneath it is nearly protein-free, the relevant value is \(\pi_g \lt \pi_i\), the opposing force is larger, and steady venular reabsorption disappears. This is the single substitution that separates the classical from the revised principle.

The protein solutions are dilute enough to be treated by a state equation in concentration alone.Plasma is not ideal-dilute. At \(70\,\mathrm{g\,L^{-1}}\) total protein the van ’t Hoff value is roughly \(16\,\mathrm{mmHg}\) whereas the measured colloid osmotic pressure is \(25\)–\(28\,\mathrm{mmHg}\) (Step 8). The law survives because \(\pi\) is still a well-defined, monotone function of concentration — only its ideal form is wrong.

The coefficients \(L_p\), \(\sigma\) and the exchange area are constant over the range of forces applied.Linear irreversible thermodynamics guarantees linearity only near equilibrium. Raising venous pressure recruits previously unperfused capillaries and so raises \(S\); inflammation raises \(L_p\) and lowers \(\sigma\) within minutes; a large distending pressure stretches interendothelial gaps. All three make \(K_f\) a function of the very pressures it multiplies.

The interstitium is a well-mixed compartment with a defined pressure and protein concentration.It is a gel: glycosaminoglycan and collagen networks in which a large fraction of the water is bound, in which protein is excluded from part of the volume, and whose compliance is strongly non-linear — nearly rigid below atmospheric pressure and very compliant above it. That compliance curve, not the Starling equation, is what determines how much swelling a given filtration rate produces.

Proof

Steps 1–2 fix the two elementary driving forces. Steps 3–6 are the real content: linear irreversible thermodynamics forces the two into a single expression with exactly one extra coefficient, the reflection coefficient, and shows why the small solutes drop out of it. Steps 7–8 assemble the physiological form and correct the oncotic term for non-ideality. Steps 9–11 take the local flux to a whole vessel, impose steady state, and derive the glycocalyx correction.

\[ J_v \;=\; L_p S\,\Delta P, \qquad L_p \;\equiv\; \frac{n\pi r^{4}}{8\eta \ell\, S} \ \ \text{(cylindrical-pore illustration)} \]

With no solute gradient, flow through a porous wall is Darcy's law: volume flux proportional to hydrostatic pressure difference, with the constant of proportionality written as hydraulic conductivity \(L_p\) times area \(S\). The bracketed expression is only an illustration of what \(L_p\) contains — \(n\) pores of radius \(r\) and length \(\ell\) spread over the exchange area \(S\), viscosity \(\eta\), by Poiseuille — and none of the argument depends on the pathway being cylindrical pores. A

\[ \mu_w = \mu_w^{\circ} + \overline{V}_w P + RT\ln a_w, \qquad \text{equal } \mu_w \Longrightarrow \Delta P = -\frac{RT}{\overline{V}_w}\ln a_w \equiv \pi \;\simeq\; RT\!\sum_k c_k \]

Osmotic pressure is defined by the pressure that must be applied to a solution to raise the chemical potential of its water back to that of pure water at the same temperature. Expanding \(\ln a_w \approx -\overline{V}_w\sum_k c_k\) for a dilute solution gives van ’t Hoff's law, with \(R=8.314\,\mathrm{J\,K^{-1}mol^{-1}}\). Note that \(\pi\) is a property of the solution, not of any membrane. B

\[ \Phi = J_w\,\Delta\mu_w + J_s\,\Delta\mu_s \;=\; J_v\,\Delta P + J_D\,\Delta\pi, \qquad J_v = J_w\overline{V}_w + J_s\overline{V}_s,\quad J_D = \frac{J_s}{\bar c_s}-J_w\overline{V}_w \]

The rate of free-energy dissipation across the wall is the sum over species of flux times conjugate force. Substituting \(\Delta\mu_w=\overline{V}_w(\Delta P-\Delta\pi)\) and \(\Delta\mu_s=\overline{V}_s\Delta P+\Delta\pi/\bar c_s\) and collecting terms re-expresses the same dissipation in a new pair of conjugate variables: the total volume flux \(J_v\) driven by \(\Delta P\), and the exchange (relative velocity) flux \(J_D\) driven by \(\Delta\pi\). This is the Kedem–Katchalsky transformation; nothing has been assumed except that \(\Phi\) is a bilinear form. C

\[ \begin{aligned} J_v &= L_p\,\Delta P + L_{pD}\,\Delta\pi, \qquad J_D = L_{Dp}\,\Delta P + L_{D}\,\Delta\pi, \qquad L_{pD}=L_{Dp}\ \ \text{(Onsager)}\\[3pt] \sigma &\equiv -\frac{L_{pD}}{L_p} \qquad\Longrightarrow\qquad J_v = L_p\big(\Delta P - \sigma\,\Delta\pi\big) \end{aligned} \]

Close to equilibrium each flux is a linear function of both forces, and Onsager reciprocity makes the cross-coefficients equal. Defining their ratio as \(\sigma\) is a definition, not an assumption; the physical content is the reciprocity, which forces the same coefficient to appear in the solute equation of Step 5 and so makes \(\sigma\) measurable in two independent ways. Thermodynamic stability of the coefficient matrix bounds \(\sigma\) to \([0,1]\) for a membrane that does not solvent-drag against its own gradient. C

\[ J_s = \omega\,\Delta\pi_s + (1-\sigma)\,\bar c_s\,J_v \qquad\Longrightarrow\qquad \sigma = 1-\frac{J_s^{\,\text{convective}}}{\bar c_s J_v} \]

The same transformation applied to the solute flux gives diffusive permeation (permeability \(\omega\)) plus convective drag. The drag term carries the solute at a concentration \((1-\sigma)\bar c_s\) rather than \(\bar c_s\): a fraction \(\sigma\) of the solute is “reflected” by the wall and left behind. Hence \(\sigma=1\) means total exclusion (full osmotic effect), \(\sigma=0\) means the solute rides through with the water and exerts no sustained osmotic force at all. C

\[ J_v = L_pS\Big[\Delta P - \sum_k \sigma_k\,\Delta\pi_k\Big], \qquad \sigma_{\text{small}}\approx 0 \ \Longrightarrow\ \sum_k \sigma_k\Delta\pi_k \;\to\; \sigma\,(\pi_c-\pi_i) \]

With several solutes the osmotic terms add, each weighted by its own reflection coefficient. Across continuous capillary endothelium the interendothelial cleft is wide compared with a hydrated sodium or glucose molecule, so \(\sigma_k\approx 0\) for every crystalloid, and their high diffusive permeability drives \(\Delta\pi_k\) itself to zero in seconds; only the plasma proteins, with \(\sigma\approx 0.9\) and no route of escape, hold a standing difference. This is why the relevant quantity is the colloid osmotic (oncotic) pressure of about \(25\,\mathrm{mmHg}\) and not the total osmotic pressure of roughly \(5600\,\mathrm{mmHg}\) that plasma's \(290\,\mathrm{mosmol\,L^{-1}}\) would exert against a perfect membrane. B

\[ J_v = \underbrace{L_pS}_{K_f}\Big[\underbrace{(P_c-P_i)}_{\text{out}} - \underbrace{\sigma(\pi_c-\pi_i)}_{\text{in}}\Big], \qquad \mathrm{NFP} \equiv (P_c-P_i)-\sigma(\pi_c-\pi_i) \]

Write the pressures with the physiological sign convention: positive \(J_v\) is filtration out of the vessel. \(P_c\) pushes fluid out, \(P_i\) pushes it back in, \(\pi_c\) pulls it in, \(\pi_i\) pulls it out. Grouping \(L_pS\) into a single measurable filtration coefficient \(K_f\) is what makes the law usable, because \(L_p\) and \(S\) are almost never separable in an intact organ — a weight-gain experiment measures only their product. A

\[ \pi = RT\!\left(c + B_2c^{2} + B_3c^{3}\right) + \pi_{\text{Donnan}}, \qquad \pi_{\text{Donnan}} \approx RT\,\frac{(z_pc_p)^{2}}{4c_{\text{salt}}} \]

Two corrections lift the measured colloid osmotic pressure well above the ideal value. First, albumin at pH \(7.4\) carries a net charge near \(-17\), and electroneutrality retains an excess of small counter-ions on the plasma side; those retained ions are themselves osmotically active, and if they are reflected along with the protein they add the Gibbs–Donnan term. Second, the virial coefficients \(B_2,B_3\) are positive and not small at \(70\,\mathrm{g\,L^{-1}}\), making \(\pi\) rise supralinearly with concentration. Neither correction touches the structure of the law: \(\pi_c\) and \(\pi_i\) are simply measured, not computed from van ’t Hoff. C

\[ P_c(x) = P_a - (P_a-P_v)\,x, \quad x\in[0,1] \qquad\Longrightarrow\qquad \mathrm{NFP}(x)=0 \ \ \text{at}\ \ x^{*}=\frac{P_a-\big[P_i+\sigma(\pi_c-\pi_i)\big]}{P_a-P_v} \]

Along the vessel only \(P_c\) changes appreciably, and to first approximation it falls linearly with fractional distance \(x\). \(\mathrm{NFP}\) is therefore linear in \(x\) and crosses zero at most once, at the point where \(P_c\) equals the “isogravimetric” pressure \(P_i+\sigma(\pi_c-\pi_i)\). If \(x^{*}\gt 1\) the whole vessel filters; if \(0\lt x^{*}\lt 1\) the distal fraction \(1-x^{*}\) reabsorbs. This single expression contains the entire classical filtration–reabsorption picture. B

\[ \dot{Q}_{\text{lymph}} = \int_{0}^{1} J_v(x)\,\mathrm{d}x \;\ge\; 0 \qquad\text{in any steady state} \]

The interstitium neither accumulates nor loses volume indefinitely, so whatever is filtered net must leave by the only other route: the lymphatics. Because lymph flow cannot be negative, a steady state with net reabsorption over a whole tissue is impossible unless some other source supplies the fluid — which is exactly the situation in the renal peritubular bed and the absorbing intestinal mucosa, where tubular or luminal uptake supplies it. Everywhere else, the integral is a small positive number. B

\[ J_v = K_f\big[(P_c-P_i)-\sigma(\pi_c-\pi_g)\big], \qquad V\frac{\mathrm{d}\pi_g}{\mathrm{d}t} = \begin{cases} -J_v\,\pi_g, & J_v\gt 0 \quad \text{(protein-free filtrate sweeps in)}\\[2pt] -J_v\,\pi_i, & J_v\lt 0 \quad \text{(interstitial fluid drawn in)} \end{cases} \]

If the selective layer is the glycocalyx, the fluid the oncotic force actually sees is that of the narrow subglycocalyx space — volume \(V\), colloid osmotic pressure \(\pi_g\) — and a volume balance on that space fixes \(\pi_g\) instead of letting it be read off the bulk interstitium. The two cases are not symmetric, and the asymmetry is the whole argument. Absorption is self-extinguishing: inflow of interstitial fluid raises \(\pi_g\) toward \(\pi_i\), which shrinks the opposing oncotic force; if \(P_c-P_i\lt\sigma\pi_c\) the flux vanishes at the attainable value \(\pi_g^{*}=\pi_c-(P_c-P_i)/\sigma\), lying in \((0,\pi_i)\), and because \(\partial J_v/\partial\pi_g=+K_f\sigma\gt0\) that zero-flux state is a stable attractor (Problem 5 does the linearisation). Filtration is not: the sweep can drive \(\pi_g\) no lower than zero, so the flux settles at \(K_f\big[(P_c-P_i)-\sigma\pi_c\big]\), which is positive wherever \(P_c-P_i\gt\sigma\pi_c\) and makes the opposing oncotic force \(\sigma\pi_c\), its largest possible value. The revised principle therefore permits a sustained filtering steady state and forbids a sustained absorbing one — Step 10's global conclusion, recovered locally. C

Result

\[ J_v \;=\; \underbrace{L_pS}_{K_f}\Big[(P_c-P_i)-\sigma\big(\pi_c-\pi_i\big)\Big], \qquad 0\le\sigma\le 1 \]

Reading. Fluid leaves a microvessel at a rate proportional to the excess of the hydrostatic push over the oncotic pull, with the oncotic pull — and only the oncotic pull — discounted by the wall's reflection coefficient for protein. Everything anatomical about the vessel is compressed into the two numbers \(L_pS\) and \(\sigma\).

Units check. \(K_f\) in \(\mathrm{mL\,min^{-1}mmHg^{-1}}\) times a pressure in \(\mathrm{mmHg}\) gives \(\mathrm{mL\,min^{-1}}\); as a flux density, \(L_p\) in \(\mathrm{cm\,s^{-1}mmHg^{-1}}\) times \(\mathrm{mmHg}\) gives \(\mathrm{cm\,s^{-1}}\), a velocity across unit wall area. \(\sigma\) and the ratio inside any logarithm are dimensionless. Conversions: \(1\,\mathrm{mmHg}=133.3\,\mathrm{Pa}=1.36\,\mathrm{cmH_2O}\).

Scope. A passive, isothermal, linear wall with the crystalloids unreflected; local, not whole-vessel. In the revised form \(\pi_i\) is replaced by the subglycocalyx value \(\pi_g\), which is what forbids sustained reabsorption in most tissues. It gives a flux, never a volume: the swelling that results depends on interstitial compliance and lymphatic reserve.

Corollaries & converses

Isogravimetric capillary pressure. Setting \(J_v=0\) gives \(P_c^{*}=P_i+\sigma(\pi_c-\pi_i)\), the capillary pressure at which an isolated perfused organ neither gains nor loses weight. Measuring \(P_c^{*}\) while varying arterial and venous pressures independently — the Pappenheimer–Soto-Rivera method — is how capillary pressure was first determined in an intact tissue, and it is a direct converse of the law.
The filtration coefficient is the slope, not the intercept. \(\partial J_v/\partial P_c = K_f\) is independent of every oncotic term, so \(K_f\) can be measured from a step change in venous pressure without knowing any protein concentration. This is the basis of every capillary filtration coefficient measurement.
Reflection coefficient from an osmotic transient. Applying a step \(\Delta\pi_k\) of a test solute and reading the initial transient volume flux gives \(\sigma_k=-(\partial J_v/\partial\Delta\pi_k)/(\partial J_v/\partial\Delta P)\) — the ratio of osmotic to hydraulic effectiveness, the minus sign because an osmotic difference drives flux the opposite way to a hydrostatic one. Small solutes score near zero across muscle capillaries and near one across the blood–brain barrier.
Safety factors against oedema. Differentiating \(\mathrm{NFP}\) exhibits three of the negative feedbacks that resist a rise in \(P_c\): \(P_i\) rises as the interstitium fills, \(\pi_i\) falls as filtrate washes protein out, and lymph flow increases. Together they can absorb a rise in \(P_c\) of order \(15\)–\(20\,\mathrm{mmHg}\) before free interstitial fluid accumulates.
Lymph is the residue. By Step 10 the steady lymph flow from a tissue equals the net filtration, so lymph protein concentration reports \(\pi_i\) and lymph flow reports \(\int J_v\). Lymph that is nearly as protein-rich as plasma indicates \(\sigma\to 0\) — the signature of the hepatic sinusoid.
Genuine absorbers exist. The renal peritubular capillary has a low \(P_c\) (about \(13\,\mathrm{mmHg}\)) and a high \(\pi_c\) (about \(32\,\mathrm{mmHg}\), raised by glomerular filtration upstream), and is continuously supplied with fluid by tubular reabsorption. Step 10's ban on sustained absorption does not apply because the fluid source is not the interstitial store.
Crystalloid versus colloid. An infused crystalloid distributes across the wall in minutes because \(\sigma\approx0\), so it expands plasma volume only in proportion to the plasma fraction of extracellular fluid; a colloid with \(\sigma\approx 1\) raises \(\pi_c\) and is retained — but only while the endothelial barrier is intact, which is why the advantage collapses in sepsis.
Converse for \(\sigma=0\). If nothing is reflected, \(J_v=K_f(P_c-P_i)\): fluid exchange becomes purely hydrostatic and no protein infusion can retain water. Hepatic sinusoids approximate this, which is why portal hypertension produces ascites from a modest pressure rise.

Fails without

Intact reflection coefficient dropped (inflammation, sepsis, burns): cytokine-driven retraction of endothelial junctions and shedding of the glycocalyx raise \(L_p\) several-fold and drop \(\sigma\) from about \(0.9\) toward \(0.5\) or below. With \(P_c=17\,\mathrm{mmHg}\), \(P_i=-2\), \(\pi_c=25\), \(\pi_i=8\), the net filtration pressure rises from \((19)-0.9(17)=+3.7\,\mathrm{mmHg}\) to \((19)-0.5(17)=+10.5\,\mathrm{mmHg}\), and \(K_f\) has trebled at the same time: filtration rises roughly eightfold. Worse, protein now follows the water, \(\pi_i\) climbs, and the oncotic defence is lost altogether — which is why albumin infusion does not correct capillary leak oedema.
Bulk \(\pi_i\) used where \(\pi_g\) belongs (the classical venular-reabsorption error): with \(\sigma=0.9,\ \pi_c=25,\ \pi_i=8,\ P_i=-2\) the classical crossover sits at \(P_c^{*}=13.3\,\mathrm{mmHg}\), predicting reabsorption over any part of the capillary where pressure falls below that. Substituting the subglycocalyx value \(\pi_g\approx0\) moves the crossover to \(P_c^{*}=20.5\,\mathrm{mmHg}\), so that the same vessel is predicted to filter along its whole length and any absorption is transient. Direct measurements of steady-state flux in single perfused microvessels follow the second prediction, and the classical one overestimates whole-body filtration by roughly an order of magnitude against measured lymph flow.
Small-solute reflection no longer negligible (blood–brain barrier): tight junctions give \(\sigma\approx1\) for sodium and mannitol and their diffusive permeability is low, so Step 6's cancellation fails twice over. A mannitol bolus that raises plasma osmolality by \(20\,\mathrm{mosmol\,L^{-1}}\) is worth \(\Delta\pi = RT\Delta c \approx 2578\,\mathrm{J\,mol^{-1}}\times20\,\mathrm{mol\,m^{-3}} = 5.2\times10^{4}\,\mathrm{Pa}=387\,\mathrm{mmHg}\), which dwarfs every colloid term and is why osmotherapy dehydrates the brain. Applying the muscle-capillary form of the law here predicts, wrongly, that mannitol does nothing.
Linearity and constant coefficients dropped: raising venous pressure in a muscle bed recruits closed capillaries, so \(S\) — and hence \(K_f\) — is itself a function of pressure. The measured weight-gain-versus-pressure curve is then not a straight line, and a \(K_f\) read off its slope at one pressure does not predict flux at another. Similarly, at high filtration rates the concentration polarisation of protein against the wall makes the local \(\pi_c\) exceed the bulk plasma value, so the law becomes implicit in \(J_v\).
Passivity dropped (transporting epithelia): the proximal tubule reabsorbs about two thirds of \(180\,\mathrm{L}\) of filtrate a day across an epithelium whose net Starling pressure is small and, in places, adverse; the flux is set by active sodium transport with water following osmotically through aquaporins. Feeding tubular Starling pressures into this equation predicts neither the direction nor the magnitude.
Steady state dropped: after acute haemorrhage \(P_c\) falls abruptly while \(\pi_c\) is momentarily unchanged, so \(\mathrm{NFP}\) turns negative and interstitial fluid is genuinely absorbed into the circulation — the “autotransfusion” that can recover several hundred millilitres over the following hour. This is real and clinically important, and it is not a counterexample to Step 11 but an illustration of it: the absorption decays as \(\pi_g\) rises and as interstitial pressure falls, and it is a transient, not a new steady state.

Common errors

“The venular end reabsorbs most of what the arteriolar end filters.” In steady state that would require the interstitium to be an inexhaustible fluid source, and Step 10 forbids it. Most capillary beds filter along their entire length and return the surplus as lymph; the classical picture is an artefact of using the bulk interstitial oncotic pressure in place of the subglycocalyx value.
“Plasma osmotic pressure is thousands of millimetres of mercury, so it dominates everything.” That is the total osmotic pressure against a perfect membrane, about \(5600\,\mathrm{mmHg}\) for \(290\,\mathrm{mosmol\,L^{-1}}\). Against a capillary wall each solute is weighted by \(\sigma_k\), and the crystalloids, which supply over \(99\%\) of the osmoles, are weighted by nearly zero and cannot hold a gradient anyway. Only the milliosmole or so per litre contributed by the proteins and the counter-ions they retain counts, and it is worth about \(25\,\mathrm{mmHg}\).
“Discount both differences by \(\sigma\).” \(\sigma\) multiplies the oncotic term alone. It is the fraction of solute reflected, and hydrostatic pressure is transmitted whatever the wall reflects; writing \(\sigma[(P_c-P_i)-(\pi_c-\pi_i)]\) is dimensionally fine and physically wrong.
“\(P_i\) is zero or positive.” In loose subcutaneous tissue and in lung the measured interstitial hydrostatic pressure is subatmospheric, around \(-2\) to \(-8\,\mathrm{mmHg}\); subtracting a negative number adds to the outward force. In encapsulated organs — kidney, brain, skeletal muscle within tight fascia — it is positive. Sign errors here are worth \(10\,\mathrm{mmHg}\) or more.
“Compute \(\pi_c\) from van ’t Hoff.” Ideal theory gives about \(16\,\mathrm{mmHg}\) for normal plasma against a measured \(25\)–\(28\,\mathrm{mmHg}\). Colloid osmotic pressures are measured with an oncometer or read from an empirical concentration relation, never predicted from \(RTc\).
“Low albumin therefore oedema.” Falling \(\pi_c\) raises \(\mathrm{NFP}\), but filtration also washes interstitial protein out, so \(\pi_i\) falls and much of the change is cancelled; substantial oedema in nephrotic syndrome additionally requires renal sodium retention to raise \(P_c\). A single Starling term is rarely the whole clinical story.
“\(K_f\) and \(L_p\) are interchangeable.” \(K_f=L_pS\) has units of \(\mathrm{mL\,min^{-1}mmHg^{-1}}\) (an organ property, quoted per \(100\,\mathrm{g}\)), while \(L_p\) is \(\mathrm{cm\,s^{-1}mmHg^{-1}}\) (a wall property). Comparing a glomerulus with a muscle bed by \(K_f\) alone conflates a leakier wall with a larger surface.
“A negative \(\mathrm{NFP}\) means the tissue is dehydrating.” \(J_v\) is a flux across one wall; net tissue volume also depends on lymph drainage and on interstitial compliance, which is nearly flat at negative pressures and very steep once \(P_i\) passes zero. That non-linearity, not the Starling equation, is why oedema appears suddenly rather than gradually.

Discussion

Ernest Starling proposed the balance in 1896, on the basis of a strikingly simple experiment: he showed that serum injected into a limb's tissue spaces was reabsorbed into the blood while saline was not, and concluded that the plasma proteins exert an osmotic force across the capillary wall that opposes the filtering effect of blood pressure. Almost everything quantitative came later. Eugene Landis measured capillary pressure directly by micropuncture of single frog mesenteric vessels in the 1920s and confirmed that filtration and absorption reverse as that pressure crosses the oncotic value; Pappenheimer and Soto-Rivera developed the isogravimetric method in 1948, which turned the corollary above into the standard tool for measuring \(P_c\) and \(K_f\) in intact organs; Staverman introduced the reflection coefficient in 1951 for non-ideal membranes generally; and Kedem and Katchalsky in 1958 showed that Starling's law, the reflection coefficient and the solute-drag equation all follow from one application of linear irreversible thermodynamics, which is the derivation given above.

The century-long error was not in the equation but in one of its inputs. If the interstitial oncotic pressure is genuinely felt at the barrier, then a capillary whose pressure falls below the isogravimetric value must reabsorb, and the textbook diagram of filtration at one end and reabsorption at the other follows. Levick pointed out in 1991 that the classical parameters, applied honestly, predict a whole-body filtration rate far larger than the measured lymph flow can carry, and that in most tissues the venular pressure never falls low enough for reabsorption anyway. The resolution came from ultrastructure: the selective barrier is the endothelial glycocalyx, a fibre matrix a few hundred nanometres deep on the luminal face, and the ultrafiltrate that emerges beneath it enters a small protected space before it reaches the interstitium proper. Because filtration continuously sweeps that space clear of protein, the oncotic force opposing filtration is close to its maximum value \(\sigma\pi_c\), and any absorption is self-extinguishing. Michel and Weinbaum formalised this in the 1990s and Levick and Michel drew the physiological consequences together in 2010; the practical upshot — that infused colloid does not stay in the circulation as classical theory predicts, particularly when the glycocalyx is damaged — has changed how fluid resuscitation is taught.

Two subtleties reward attention at full rigour. The first is that the reflection coefficient is not a single number for a wall but one per solute, and for a heteroporous wall it is a weighted average over pathways: a wall with a small-pore population of \(\sigma=1\) and a large-pore population of \(\sigma=0\) carrying a fraction \(f\) of the hydraulic conductance has an effective \(\sigma = 1-f\), so the same measured \(\sigma=0.9\) may mean “uniformly slightly leaky” or “ten per cent of the water goes through holes that reflect nothing”. The two are distinguishable only by their solute permeabilities, which is what the large-pore/small-pore analysis of lymph protein data was built to do. The second is that the linear phenomenological equations are guaranteed only near equilibrium, and the capillary wall operates with driving forces of tens of millimetres of mercury; the empirical fact that \(J_v\) is linear in \(P_c\) over a wide range is a measured result, not a theorem, and it does fail once concentration polarisation builds a protein layer against the wall at high filtration rates.

Common misconceptions. That Starling forces determine tissue water content: they determine a flux, and content follows only after lymphatic capacity and interstitial compliance are accounted for. That oncotic pressure is a property of the membrane: \(\pi\) belongs to the solution, \(\sigma\) belongs to the wall, and only their product is a force. That the law is specific to capillaries: it is the general linear law for a passive selective barrier, and the same expression with different coefficients governs a dialysis filter, a plant cell wall and a reverse-osmosis membrane. And that the revised principle overturns Starling: it uses Starling's equation unchanged and corrects only the value substituted for one of its four pressures.

Worked examples

Example 1. A skeletal-muscle capillary has hydrostatic pressure \(35\,\mathrm{mmHg}\) at its arteriolar end falling linearly to \(15\,\mathrm{mmHg}\) at its venular end. Take the standard values \(P_i=-2\,\mathrm{mmHg}\), \(\pi_c=25\,\mathrm{mmHg}\), \(\pi_i=8\,\mathrm{mmHg}\) and \(\sigma=0.9\). Find the net filtration pressure at each end, locate the point at which flux reverses, and test how sensitive that point is to \(\sigma\) and \(\pi_i\).

\[ \mathrm{NFP}(x) = \big(P_c(x)-P_i\big)-\sigma(\pi_c-\pi_i), \qquad \sigma(\pi_c-\pi_i) = 0.9\,(25-8)\,\mathrm{mmHg} = 15.3\,\mathrm{mmHg} \]

Evaluate the oncotic term first: it is the same everywhere along the vessel, because neither \(\pi_c\) nor \(\pi_i\) changes appreciably over one capillary transit (only about \(0.5\%\) of the plasma water is filtered). All the variation lives in \(P_c\). A

\[ \mathrm{NFP}_{\text{art}} = \big(35-(-2)\big)-15.3 = 37-15.3 = +21.7\,\mathrm{mmHg} \]

The interstitial pressure is subatmospheric, so subtracting it adds \(2\,\mathrm{mmHg}\) to the outward force — a term students routinely drop or sign wrongly. The positive result means filtration. A

\[ \mathrm{NFP}_{\text{ven}} = \big(15-(-2)\big)-15.3 = 17-15.3 = +1.7\,\mathrm{mmHg} \]

Still positive: with these standard values the venular end of this capillary filters as well, weakly. The textbook picture of venular reabsorption has not appeared, and no approximation has been made to suppress it. B

\[ P_c^{*} = P_i + \sigma(\pi_c-\pi_i) = -2 + 15.3 = 13.3\,\mathrm{mmHg}, \qquad x^{*} = \frac{35-13.3}{35-15} = \frac{21.7}{20} = 1.09 \]

Solve \(\mathrm{NFP}=0\) for the capillary pressure, then convert to fractional distance using the linear profile of Step 9. Since \(x^{*}\gt1\) the crossover lies beyond the venular end: the vessel filters over its whole length, and would need to be \(9\%\) longer, or its venous pressure \(1.7\,\mathrm{mmHg}\) lower, before any reabsorption occurred. B

\[ \frac{\partial P_c^{*}}{\partial \sigma} = \pi_c-\pi_i = 17\,\mathrm{mmHg}, \qquad \frac{\partial P_c^{*}}{\partial \pi_i} = -\sigma = -0.9 \]

Differentiate the crossover expression before substituting numbers. A change of \(0.1\) in \(\sigma\) moves the crossover pressure by \(1.7\,\mathrm{mmHg}\), which on a \(20\,\mathrm{mmHg}\) axial gradient is \(8.5\%\) of the capillary length; a change of \(3\,\mathrm{mmHg}\) in \(\pi_i\) moves it by \(2.7\,\mathrm{mmHg}\), or \(14\%\) of the length. The classical picture is therefore not robust — it is a knife-edge conclusion drawn from parameters known to about that precision. B

\[ \sigma=1.0,\ \pi_i=5\,\mathrm{mmHg}: \quad P_c^{*} = -2+20 = 18\,\mathrm{mmHg}, \quad x^{*} = \frac{35-18}{20} = 0.85 \]

Take the other end of the plausible parameter range and the textbook result reappears: the last \(15\%\) of the vessel reabsorbs. So the classical diagram is not wrong arithmetic; it is a particular parameter choice. Deciding between them requires an independent measurement, which is what the lymph-flow comparison of Problem 2 and the glycocalyx argument of Problem 5 supply. C

\[ \mathrm{NFP}_{\text{art}} = +21.7\,\mathrm{mmHg}, \qquad \mathrm{NFP}_{\text{ven}} = +1.7\,\mathrm{mmHg}, \qquad P_c^{*}=13.3\,\mathrm{mmHg}\ (x^{*}=1.09) \]

Reading. With the standard parameter set the capillary filters along its whole length, most strongly at the arteriolar end. The zero-flux point sits just past the venular end, and it moves by \(8.5\%\) of the capillary length for every \(0.1\) change in \(\sigma\).

Scope. Textbook values for a resting muscle capillary; individual beds differ, and the linear pressure profile is an idealisation. The calculation gives local flux densities, not a filtration rate — that needs \(K_f\), which Problem 2 supplies.

Example 2. Compute the glomerular filtration rate from Starling forces, allowing for the fact that filtration itself concentrates the plasma protein and so raises \(\pi\) along the glomerular capillary. Take \(P_{GC}=60\,\mathrm{mmHg}\), Bowman's space pressure \(P_{BS}=18\,\mathrm{mmHg}\), afferent plasma oncotic pressure \(\pi_a=28\,\mathrm{mmHg}\), \(\sigma=1\) for albumin, filtration coefficient \(K_f=12.5\,\mathrm{mL\,min^{-1}mmHg^{-1}}\) for both kidneys and renal plasma flow \(\mathrm{RPF}=625\,\mathrm{mL\,min^{-1}}\). Then find what happens when ureteric obstruction raises \(P_{BS}\) to \(30\,\mathrm{mmHg}\).

\[ \mathrm{GFR} = K_f\big[(P_{GC}-P_{BS}) - \bar\pi\big], \qquad \pi_{BS}=0 \]

The filtrate is essentially protein-free, so the oncotic pressure on the Bowman's-space side is zero and only the plasma value appears. \(\sigma=1\) because the glomerular barrier reflects albumin almost completely — a normal filtrate carries under \(30\,\mathrm{mg\,L^{-1}}\) of protein against \(70\,\mathrm{g\,L^{-1}}\) in plasma. \(\bar\pi\) is the mean plasma oncotic pressure along the capillary, which is not \(\pi_a\). A

\[ c_e = \frac{c_a}{1-\mathrm{FF}}, \qquad \mathrm{FF} \equiv \frac{\mathrm{GFR}}{\mathrm{RPF}} \qquad\Longrightarrow\qquad \pi_e \simeq \frac{\pi_a}{1-\mathrm{FF}} \]

Protein is conserved while water is removed, so the efferent protein concentration exceeds the afferent by the reciprocal of the fraction of plasma not filtered. Treating \(\pi\) as proportional to \(c\) is the ideal (van ’t Hoff) approximation of Step 2; the true relation is supralinear (Step 8), so this underestimates \(\pi_e\) slightly. B

\[ \bar\pi \simeq \tfrac{1}{2}\left(\pi_a+\pi_e\right) = \frac{\pi_a}{2}\left(1+\frac{1}{u}\right), \qquad u \equiv 1-\mathrm{FF} \]

Approximate the axial mean by the arithmetic mean of the two ends, exact if \(\pi\) rises linearly along the capillary. Introducing \(u\) as the unfiltered fraction turns the problem into one unknown, because \(\mathrm{GFR}=\mathrm{RPF}\,(1-u)\) as well. The calculation is implicit: GFR sets \(\bar\pi\), which sets GFR. B

\[ \begin{aligned} \mathrm{RPF}\,(1-u) &= K_f\left[\Delta P - \frac{\pi_a}{2}\left(1+\frac{1}{u}\right)\right] \\[3pt] \mathrm{RPF}\,u(1-u) &= K_f\left[\left(\Delta P-\tfrac{\pi_a}{2}\right)u - \tfrac{\pi_a}{2}\right] \\[3pt] \mathrm{RPF}\,u^{2} - \left[\mathrm{RPF}-K_f\!\left(\Delta P-\tfrac{\pi_a}{2}\right)\right]u - \tfrac{K_f\pi_a}{2} &= 0 \end{aligned} \]

Substitute \(\mathrm{GFR}=\mathrm{RPF}(1-u)\), multiply through by \(u\) to clear the reciprocal, and collect. Doing this symbolically before any number is inserted shows the structure: the self-consistent state is the root of a quadratic in the unfiltered fraction, with \(\Delta P \equiv P_{GC}-P_{BS}\). Only the root in \((0,1)\) is physical. C

\[ 625\,u^{2}-\big[625-12.5(42-14)\big]u-\tfrac{12.5\times28}{2}=0 \;\Longrightarrow\; 25u^{2}-11u-7=0 \;\Longrightarrow\; u=\frac{11+\sqrt{121+700}}{50}=0.793 \]

With \(\Delta P = 60-18 = 42\,\mathrm{mmHg}\) and \(\pi_a/2 = 14\,\mathrm{mmHg}\): \(625-350=275\) and \(175\), and dividing through by \(25\) gives the tidy quadratic. \(\sqrt{821}=28.65\); the negative root is rejected as unphysical. A

\[ \mathrm{FF}=1-0.793=0.207, \qquad \mathrm{GFR}=625\times0.207 = 129\,\mathrm{mL\,min^{-1}}, \qquad \pi_e=\frac{28}{0.793}=35.3\,\mathrm{mmHg} \]

Check against the original equation: \(\bar\pi=(28+35.3)/2=31.6\,\mathrm{mmHg}\), \(\mathrm{NFP}=42-31.6=10.4\,\mathrm{mmHg}\), \(\mathrm{GFR}=12.5\times10.4=130\,\mathrm{mL\,min^{-1}}\) — consistent to rounding. The filtration fraction of \(0.21\) and the GFR of about \(130\,\mathrm{mL\,min^{-1}}\) are the measured human values, and the net filtration pressure of about \(10\,\mathrm{mmHg}\) is an output of the calculation, not an input. B

\[ P_{BS}=30\,\mathrm{mmHg}:\quad \Delta P = 30, \quad 625u^{2}-\big[625-12.5(30-14)\big]u-175=0 \;\Longrightarrow\; 25u^{2}-17u-7=0 \;\Longrightarrow\; u=\frac{17+\sqrt{989}}{50}=0.969 \]

Only \(\Delta P\) changes. \(625-200=425\), and dividing by \(25\) gives \(17\); \(\sqrt{989}=31.45\). The unfiltered fraction rises almost to one. A

\[ \mathrm{GFR}=625\times(1-0.969)=19.4\,\mathrm{mL\,min^{-1}}, \qquad \frac{129}{19.4}=6.7 \]

A \(12\,\mathrm{mmHg}\) rise in Bowman's-space pressure — a \(29\%\) fall in \(\Delta P\) — cuts GFR by a factor of \(6.7\). The amplification comes from the feedback in Step 3: as GFR falls, \(\mathrm{FF}\) falls, \(\pi_e\) falls from \(35.3\) to \(28/0.969=28.9\,\mathrm{mmHg}\), and \(\bar\pi\) drops to \(28.4\,\mathrm{mmHg}\) — which defends GFR. Without that defence, \(\mathrm{NFP}\) would have been \(30-31.6 = -1.6\,\mathrm{mmHg}\) and filtration would have stopped altogether. C

\[ \mathrm{GFR}=129\,\mathrm{mL\,min^{-1}},\quad \mathrm{FF}=0.21,\quad \pi_e=35\,\mathrm{mmHg}; \qquad P_{BS}\!:\,18\to30\,\mathrm{mmHg} \ \Longrightarrow\ \mathrm{GFR}=19\,\mathrm{mL\,min^{-1}} \]

Reading. The Starling equation plus conservation of protein reproduces the measured human GFR and filtration fraction from four pressures and two flows, with no fitted parameter. The rising oncotic pressure along the glomerular capillary is a negative feedback that both limits filtration in health and cushions it when the driving pressure falls.

Scope. The arithmetic mean for \(\bar\pi\) and the linear \(\pi(c)\) relation are both approximations that slightly overestimate GFR; the true relation is supralinear and in some species the capillary reaches filtration equilibrium (\(\mathrm{NFP}\to0\)) before its end, in which case GFR becomes flow-limited and \(K_f\) drops out of the answer entirely.

Problems

An intestinal mucosal capillary has \(P_c=30\,\mathrm{mmHg}\) at its arteriolar end and \(18\,\mathrm{mmHg}\) at its venular end, with \(P_i=0\,\mathrm{mmHg}\), \(\pi_c=25\,\mathrm{mmHg}\), \(\pi_i=6\,\mathrm{mmHg}\) and \(\sigma=0.9\). Find the net filtration pressure at both ends and the capillary pressure at which flux reverses. What would \(\pi_i\) have to become for the venular end to reabsorb?
Solution
The oncotic term is \(\sigma(\pi_c-\pi_i)=0.9\,(25-6)=0.9\times19=17.1\,\mathrm{mmHg}\), constant along the vessel.
Arteriolar end: \(\mathrm{NFP}=(30-0)-17.1=+12.9\,\mathrm{mmHg}\), filtration. Venular end: \(\mathrm{NFP}=(18-0)-17.1=+0.9\,\mathrm{mmHg}\), still filtration, but only just.
Reversal requires \(\mathrm{NFP}=0\), i.e. \(P_c^{*}=P_i+\sigma(\pi_c-\pi_i)=0+17.1=17.1\,\mathrm{mmHg}\), which lies \(0.9\,\mathrm{mmHg}\) below the venular pressure; the vessel filters over its whole length.
For the venular end to reabsorb we need \(P_c^{*}\gt18\), i.e. \(0.9(25-\pi_i)\gt18\), so \(25-\pi_i\gt20\) and \(\pi_i\lt5\,\mathrm{mmHg}\). A fall of just over one millimetre of mercury in interstitial oncotic pressure flips the sign of flux at the venular end — and that is precisely what the glycocalyx model says happens locally, since the fluid at the barrier is far more dilute than bulk interstitial fluid. Note also the physiological point: during active water absorption from the gut lumen, the mucosal interstitium is supplied with fluid from the lumen, so this bed is one of the few where sustained capillary absorption is genuinely available.
An isolated hindlimb preparation weighing \(250\,\mathrm{g}\) is perfused at constant arterial pressure. Raising venous pressure by \(20\,\mathrm{mmHg}\) raises capillary pressure by \(16\,\mathrm{mmHg}\) and produces a steady weight gain of \(0.35\,\mathrm{g\,min^{-1}}\). (a) Compute the capillary filtration coefficient in \(\mathrm{mL\,min^{-1}}\,(100\,\mathrm{g})^{-1}\mathrm{mmHg^{-1}}\). (b) Taking the capillary exchange area of skeletal muscle as \(70\,\mathrm{cm^{2}}\) per gram, estimate \(L_p\) in \(\mathrm{cm\,s^{-1}mmHg^{-1}}\). (c) If the whole-body muscle mass is \(30\,\mathrm{kg}\) and the resting mean net filtration pressure is \(1\,\mathrm{mmHg}\), estimate the daily filtration and compare it with a thoracic-duct lymph flow of a few litres per day.
Solution
(a) Weight gain is filtration, and \(1\,\mathrm{g}\) of filtrate is \(1\,\mathrm{mL}\) to better than a percent. The extra flux is \(\Delta J_v = 0.35\,\mathrm{mL\,min^{-1}}\) for \(\Delta P_c = 16\,\mathrm{mmHg}\), and \(K_f=\partial J_v/\partial P_c\) is the slope, independent of every oncotic term: \[ K_f = \frac{0.35}{16}=0.0219\,\mathrm{mL\,min^{-1}mmHg^{-1}}\ \text{for }250\,\mathrm{g}. \] Per \(100\,\mathrm{g}\): \(0.0219/2.5 = 8.8\times10^{-3}\,\mathrm{mL\,min^{-1}}(100\,\mathrm{g})^{-1}\mathrm{mmHg^{-1}}\), which sits in the usual reported range of about \(5\times10^{-3}\) to \(1\times10^{-2}\) for skeletal muscle.
(b) Area for \(100\,\mathrm{g}\): \(S = 70\,\mathrm{cm^{2}\,g^{-1}}\times100\,\mathrm{g}=7.0\times10^{3}\,\mathrm{cm^{2}}\). Then \[ L_p = \frac{K_f}{S} = \frac{8.8\times10^{-3}\,\mathrm{mL\,min^{-1}mmHg^{-1}}}{7.0\times10^{3}\,\mathrm{cm^{2}}} = 1.25\times10^{-6}\,\mathrm{cm\,min^{-1}mmHg^{-1}}, \] and dividing by \(60\) gives \(L_p = 2.1\times10^{-8}\,\mathrm{cm\,s^{-1}mmHg^{-1}}\) — an order-\(10^{-8}\) value, at the tight end of the range reported for continuous capillary endothelium, as expected for muscle rather than for the much leakier mesenteric or hepatic beds.
(c) \(30\,\mathrm{kg}\) is \(300\) units of \(100\,\mathrm{g}\), so the whole-muscle \(K_f\) is \(300\times8.8\times10^{-3}=2.6\,\mathrm{mL\,min^{-1}mmHg^{-1}}\). At \(\mathrm{NFP}=1\,\mathrm{mmHg}\), filtration is \(2.6\,\mathrm{mL\,min^{-1}} = 3.8\,\mathrm{L\,day^{-1}}\) — the right order of magnitude for whole-body lymph return, and reassuring.
Now repeat with the classical mean net filtration pressure of Example 1. Averaging its two ends gives \(\overline{\mathrm{NFP}}=(21.7+1.7)/2=11.7\,\mathrm{mmHg}\), whence \(2.6\times11.7=30\,\mathrm{mL\,min^{-1}}=44\,\mathrm{L\,day^{-1}}\) from muscle alone. That is an order of magnitude more than the lymphatics carry, and the tissue would swell visibly within an hour. Something in the classical parameter set must be wrong, and the candidate is the oncotic pressure actually felt at the barrier: raising the effective opposing term from \(\sigma(\pi_c-\pi_i)=15.3\) to \(\sigma\pi_c=22.5\,\mathrm{mmHg}\) reduces the mean net filtration pressure to \(25+2-22.5=+4.5\,\mathrm{mmHg}\) and, at lower resting capillary pressures, to about \(1\,\mathrm{mmHg}\). This is the quantitative argument that motivated the revised Starling principle.
Plasma contains albumin at \(42\,\mathrm{g\,L^{-1}}\) (molar mass \(66.5\,\mathrm{kg\,mol^{-1}}\)) and globulins at \(28\,\mathrm{g\,L^{-1}}\) (take \(150\,\mathrm{kg\,mol^{-1}}\)). (a) Compute the van ’t Hoff colloid osmotic pressure at \(37\,{}^{\circ}\mathrm{C}\) and compare it with the measured value of \(25\,\mathrm{mmHg}\). (b) In nephrotic syndrome albumin falls to \(20\,\mathrm{g\,L^{-1}}\) with globulins unchanged; estimate the new colloid osmotic pressure. (c) Using the muscle values \(P_c=17\,\mathrm{mmHg}\), \(P_i=-2\,\mathrm{mmHg}\), \(\pi_i=8\,\mathrm{mmHg}\), \(\sigma=0.9\), find the new net filtration pressure, and then find what \(\pi_i\) would have to fall to in order to restore the original value.
Solution
(a) Molar concentrations: \[ c_{\text{alb}}=\frac{42\,\mathrm{g\,L^{-1}}}{66\,500\,\mathrm{g\,mol^{-1}}}=6.32\times10^{-4}\,\mathrm{mol\,L^{-1}}=0.632\,\mathrm{mol\,m^{-3}}, \] \[ c_{\text{glob}}=\frac{28}{150\,000}=1.87\times10^{-4}\,\mathrm{mol\,L^{-1}}=0.187\,\mathrm{mol\,m^{-3}}. \] With \(RT=(8.314)(310.15)=2579\,\mathrm{J\,mol^{-1}}\): \(\pi_{\text{alb}}=2579\times0.632=1630\,\mathrm{Pa}\) and \(\pi_{\text{glob}}=2579\times0.187=482\,\mathrm{Pa}\), total \(2112\,\mathrm{Pa}\). Converting, \(2112/133.3=15.8\,\mathrm{mmHg}\).
The measured value is \(25\,\mathrm{mmHg}\), a factor of \(1.6\) larger. Two effects account for the gap (Step 8): the Gibbs–Donnan retention of small counter-ions by albumin's net charge of about \(-17\) at pH \(7.4\), and positive virial (excluded-volume) terms that make \(\pi\) rise faster than linearly at \(70\,\mathrm{g\,L^{-1}}\). Note also that albumin supplies \(1630/2112=77\%\) of the ideal colloid osmotic pressure despite being only \(60\%\) of the protein mass, because it is the smaller molecule and osmotic pressure counts particles.
(b) Scaling the measured value by the ideal composition, the albumin contribution falls by \(20/42=0.476\): \[ \pi_c' \approx 25\big[0.23+0.77(0.476)\big]=25(0.597)=14.9\,\mathrm{mmHg}. \] Because \(\pi(c)\) is supralinear, the true fall is somewhat greater; take \(\pi_c'\approx14\,\mathrm{mmHg}\).
(c) Before: \(\mathrm{NFP}=(17+2)-0.9(25-8)=19-15.3=+3.7\,\mathrm{mmHg}\). After: \(\mathrm{NFP}=19-0.9(14-8)=19-5.4=+13.6\,\mathrm{mmHg}\), a \(3.7\)-fold rise in filtration at unchanged \(K_f\).
To restore \(\mathrm{NFP}=3.7\,\mathrm{mmHg}\) we need \(0.9(14-\pi_i)=15.3\), i.e. \(14-\pi_i=17\), i.e. \(\pi_i=-3\,\mathrm{mmHg}\) — impossible, since an oncotic pressure cannot be negative. Even complete washout of interstitial protein (\(\pi_i\to0\)) leaves \(\mathrm{NFP}=19-12.6=+6.4\,\mathrm{mmHg}\), still \(1.7\) times the original. So the washout safety factor absorbs roughly three quarters of the insult but cannot cancel it, which is why hypoalbuminaemia predisposes to oedema without reliably producing it: the remaining excess is small enough that lymphatic reserve and a rise in \(P_i\) can often cope, and clinically evident nephrotic oedema usually needs renal sodium retention to raise \(P_c\) as well.
Pulmonary capillary values in a healthy adult are \(P_c=7\,\mathrm{mmHg}\), \(P_i=-8\,\mathrm{mmHg}\), \(\pi_c=28\,\mathrm{mmHg}\), \(\pi_i=14\,\mathrm{mmHg}\); take \(\sigma=1\). (a) Find the net filtration pressure and comment on the sign. (b) Acute left ventricular failure raises \(P_c\) to \(25\,\mathrm{mmHg}\); find the immediate net filtration pressure and the factor by which filtration rises. (c) Model the two extravascular safety factors by letting the rising filtration wash interstitial protein out, \(\pi_i\to6\,\mathrm{mmHg}\), and fill the interstitium, \(P_i\to+1\,\mathrm{mmHg}\); recompute. (d) With these safety factors in place, at what \(P_c\) does the net filtration pressure reach ten times its normal value?
Solution
(a) \(\mathrm{NFP}=(P_c-P_i)-(\pi_c-\pi_i)=(7-(-8))-(28-14)=15-14=+1\,\mathrm{mmHg}\). It is positive: even the healthy lung filters continuously, and the filtrate is carried away by a pulmonary lymph flow that keeps the alveolar interstitium and the alveoli themselves dry. The margin is only \(1\,\mathrm{mmHg}\), which is why the lung is the organ where Starling arithmetic becomes clinically urgent fastest.
(b) Immediately: \(\mathrm{NFP}=(25+8)-14=33-14=+19\,\mathrm{mmHg}\), nineteen times the normal value. At unchanged \(K_f\) the filtration rate rises nineteen-fold, far beyond any plausible lymphatic reserve.
(c) With washout and interstitial filling: \[ \mathrm{NFP}=(25-1)-(28-6)=24-22=+2\,\mathrm{mmHg}. \] The two safety factors have absorbed \(17\) of the \(18\,\mathrm{mmHg}\) insult, leaving only a doubling of filtration. Washout contributed \(8\,\mathrm{mmHg}\) and the rise in interstitial pressure \(9\,\mathrm{mmHg}\).
(d) With \(\pi_i=6\) and \(P_i=+1\) held, \(\mathrm{NFP}=(P_c-1)-22=P_c-23\). Setting this to \(10\,\mathrm{mmHg}\) gives \(P_c=33\,\mathrm{mmHg}\). So the compensated lung tolerates a capillary pressure of the mid-twenties with only a modest rise in filtration, and deteriorates steeply beyond about \(30\,\mathrm{mmHg}\) — which is why frank pulmonary oedema is associated with pulmonary capillary wedge pressures above roughly \(25\,\mathrm{mmHg}\) in an acute presentation, and why chronically elevated pressures are tolerated better, the lymphatics having had time to proliferate. Note that the safety factors are themselves exhaustible: \(\pi_i\) cannot fall below zero, and once \(P_i\) rises past the point where the interstitial gel is fully hydrated, compliance rises steeply and the same filtration produces far more accumulation.
Take a muscle capillary with \(P_i=-2\,\mathrm{mmHg}\), \(\pi_c=25\,\mathrm{mmHg}\), bulk \(\pi_i=8\,\mathrm{mmHg}\) and \(\sigma=0.9\). In the revised principle the oncotic force is evaluated against the subglycocalyx colloid osmotic pressure \(\pi_g\), which filtration sweeps toward \(0\) and absorption drags toward the bulk value \(\pi_i\). (a) Find the capillary pressure below which the classical model predicts absorption, and the corresponding value for the revised model with \(\pi_g=0\). (b) At \(P_c=15\,\mathrm{mmHg}\) the revised model with \(\pi_g=0\) predicts absorption; find the value of \(\pi_g\) at which flux ceases, and check it is attainable. (c) Show that this state is stable. (d) Name two tissues in which sustained absorption nevertheless occurs, and say why they escape the argument.
Solution
(a) Classical: \(P_c^{*}=P_i+\sigma(\pi_c-\pi_i)=-2+0.9(17)=-2+15.3=13.3\,\mathrm{mmHg}\). Revised with \(\pi_g=0\): \(P_c^{*}=P_i+\sigma\pi_c=-2+0.9(25)=-2+22.5=20.5\,\mathrm{mmHg}\). The threshold has moved up by \(\sigma\pi_i=7.2\,\mathrm{mmHg}\), so vessels with capillary pressures between \(13.3\) and \(20.5\,\mathrm{mmHg}\) — which is most of the venular microcirculation — are predicted to filter by the classical model and to absorb by the swept-clean revised model. Neither prediction survives, because \(\pi_g\) is not a constant.
(b) Solve \(\mathrm{NFP}=0\) for \(\pi_g\): \[ (P_c-P_i)=\sigma(\pi_c-\pi_g) \;\Longrightarrow\; \pi_g=\pi_c-\frac{P_c-P_i}{\sigma}=25-\frac{15+2}{0.9}=25-18.89=6.1\,\mathrm{mmHg}. \] This is attainable because \(0\lt6.1\lt\pi_i=8\): the subglycocalyx space needs to reach only about three quarters of the bulk interstitial protein concentration, and absorption of interstitial fluid is exactly what carries it there. Substituting back: \(\mathrm{NFP}=17-0.9(25-6.1)=17-17.0=0\).
(c) Write \(J_v=K_f[(P_c-P_i)-\sigma(\pi_c-\pi_g)]\), so \(\partial J_v/\partial\pi_g=+K_f\sigma\gt0\), and take the volume balance of Step 11. Below the steady state, \(\pi_g\lt\pi_g^{*}\) makes \(J_v\lt0\), the space fills with interstitial fluid, \(V\,\mathrm{d}\pi_g/\mathrm{d}t=-J_v\pi_i\gt0\), and \(\pi_g\) rises; above it, \(\pi_g\gt\pi_g^{*}\) makes \(J_v\gt0\), protein-free filtrate sweeps through, \(V\,\mathrm{d}\pi_g/\mathrm{d}t=-J_v\pi_g\lt0\), and \(\pi_g\) falls. The balance is not smooth at \(\pi_g^{*}\), so linearise each branch; on the absorbing side, with \(\pi_g=\pi_g^{*}+\delta\), \[ V\frac{\mathrm{d}\delta}{\mathrm{d}t}=-\pi_i\,\frac{\partial J_v}{\partial\pi_g}\,\delta=-\pi_i K_f\sigma\,\delta, \] a negative coefficient, so \(\delta\) decays exponentially with time constant \(V/(K_f\sigma\pi_i)\); the filtering side decays likewise with \(\pi_g^{*}\) in place of \(\pi_i\). Physically, too little protein beneath the glycocalyx means absorption, which imports protein; too much means filtration, which flushes it out. The zero-flux state is an attractor, and sustained absorption is not an available steady state — consistent with Step 10's requirement that lymph flow be non-negative.
(d) The renal peritubular capillaries (low \(P_c\approx13\,\mathrm{mmHg}\), high \(\pi_c\approx32\,\mathrm{mmHg}\) because \(20\%\) of the plasma water was removed upstream) and the intestinal mucosa during water absorption. Both escape the argument for the same reason: the interstitium is being continuously refilled from another source — tubular reabsorbate in one case, luminal absorbate in the other — so the fluid the capillary takes up is not drawn from a finite interstitial store, and Step 10's steady-state ban does not bite. Lymph nodes, whose sinuses receive afferent lymph continuously, are a third example. In every other tissue the observed steady state is small net filtration balanced by lymph flow, which is the revised principle's central claim.