Starling equation#The equation

The Starling equation as applied to a blood vessel wall reads as

:$\backslash \; J\_v\; =\; L\_\backslash mathrm\{p\}\; S\; (\; [P\_\backslash mathrm\{c\}\; -\; P\_\backslash mathrm\{i\}]\; -\; \backslash sigma[\backslash pi\_\backslash mathrm\{p\}\; -\; \backslash pi\_\backslash mathrm\{g\}]\; )$

where:{{anchor|variables}}

• $J\_v$ is the trans endothelial solvent filtration volume per second.
• $[P\_\backslash mathrm\{c\}\; -\; P\_\backslash mathrm\{i\}]\; -\; \backslash sigma[\backslash pi\_\backslash mathrm\{p\}\; -\; \backslash pi\_\backslash mathrm\{g\}]$ is the net driving force,
• $P\_c$ is the capillary hydrostatic pressure
• $P\_i$ is the interstitial hydrostatic pressure
• $\backslash pi\_p$ is the plasma protein oncotic pressure
• $\backslash pi\_g$ is the subglycocalyx oncotic pressure, which varies inversely with $J\_v$ and so stabilises $J\_v$.
• $L\_p$ is the hydraulic conductivity of the membrane
• $S$ is the surface area for filtration, determined by gaps in the "tight junction" glue that binds endothelial cells at their edges.
• $\backslash sigma$ is Staverman's reflection coefficient, determined by the condition of the endothelial glycocalyx over the junction gaps.

Pressures are customarily measured in millimetres of mercury (mmHg), and the filtration coefficient in millilitres per minute per millimetre of mercury (ml·min⁻¹·mmHg⁻¹).

The rate at which fluid is filtered across vascular endothelium (transendothelial filtration) is determined by the sum of two outward forces, capillary pressure ($P\_c$) and colloid osmotic pressure beneath the endothelial glycocalyx ($\backslash pi\_g$), and two absorptive forces, plasma protein osmotic pressure ($\backslash pi\_p$) and interstitial pressure ($P\_i$). The Starling equation is the first of two Kedem–Katchalski equations which bring nonsteady state thermodynamics to the theory of osmotic pressure across membranes that are at least partly permeable to the solute responsible for the osmotic pressure difference.{{Cite journal |last=Staverman |first=A. J. |date=1951 |title=The theory of measurement of osmotic pressure |url=https://onlinelibrary.wiley.com/doi/abs/10.1002/recl.19510700409 |journal=Recueil des Travaux Chimiques des Pays-Bas |language=en |volume=70 |issue=4 |pages=344–352 |doi=10.1002/recl.19510700409 |issn=0165-0513}}{{Cite journal |last1=Kedem |first1=O. |last2=Katchalsky |first2=A. |date=February 1958 |title=Thermodynamic analysis of the permeability of biological membranes to non-electrolytes |url=https://dx.doi.org/10.1016%2F0006-3002%2858%2990330-5 |journal=Biochimica et Biophysica Acta |language=en |volume=27 |issue=2 |pages=229–246 |doi=10.1016/0006-3002(58)90330-5 |issn=0006-3002 |pmid=13522722}} The second Kedem–Katchalsky equation explains the trans endothelial transport of solutes, $J\_s$.

It is now known that the average colloid osmotic pressure of the interstitial fluid has no effect on $J\_v$. The colloid osmotic pressure difference that opposes filtration is now known to be π'_p minus the subglycocalyx $\backslash pi\_g$.The subglycocalyx space is a very small but vitally important micro domain of the total interstitial fluid space. The concentration of soluble proteins in that microdomain, which determines $\backslash pi\_g$, is close to zero while there is adequate filtration to flush them out of the interendothelial clefts. For this reason $J\_v$ is much less than previously calculated and is tightly regulated . Any transient rise in plasma colloid osmotic pressure or fall in capillary hydrostatic pressure sufficient to allow reverse (negative) $J\_v$ causes unopposed diffusion of interstitial proteins to the subglycocalyx space, reducing the colloid osmotic pressure difference that was driving absorption of fluid to the capillary. The dependence of $\backslash pi\_g$ upon the local $J\_v$ has been called The Glycocalyx Model or the Michel-Weinbaum model, in honour of two scientists who, independently, described the filtration function of the glycocalyx. The Michel-Weinbaum Model explains how most continuous capillaries are in a steady state of filtration along their entire length most of the time. Transient disturbances of the Starling forces return rapidly to steady state filtration.

{{Main article|Filtration coefficient}}

In some texts the product of hydraulic conductivity and surface area is called the filtration co-efficient K_fc.{{cn|date=March 2021}}

Staverman's reflection coefficient, σ, is a unitless constant that is specific to the permeability of a membrane to a given solute.{{Cite journal|date=1972-04-01|title=Membrane Permeability: Generalization of the Reflection Coefficient Method of Describing Volume and Solute Flows|url= |journal=Biophysical Journal|language=en|volume=12|issue=4|pages=414–419|doi=10.1016/S0006-3495(72)86093-4|issn=0006-3495|pmc=1484119|last1=Zelman|first1=A.|pmid=5019478|bibcode=1972BpJ....12..414Z}}

The Starling equation, written without σ, describes the flow of a solvent across a membrane that is impermeable to the solutes contained within the solution.{{Cite journal|last1=Michel|first1=C. Charles|last2=Woodcock|first2=Thomas E.|last3=Curry|first3=Fitz-Roy E.|date=2020|title=Understanding and extending the Starling principle|journal=Acta Anaesthesiologica Scandinavica|language=en|volume=64|issue=8|pages=1032–1037|doi=10.1111/aas.13603|pmid=32270491|issn=1399-6576|doi-access=free}}

σ_n corrects for the partial permeability of a semipermeable membrane to a solute n.

Where σ is close to 1, the plasma membrane is less permeable to the denotated species (for example, larger molecules such as albumin and other plasma proteins), which may flow across the endothelial lining, from higher to lower concentrations, more slowly, while allowing water and smaller solutes through the glycocalyx filter to the extravascular space.

• Glomerular capillaries have a reflection coefficient close to 1 as normally no protein crosses into the glomerular filtrate.
• In contrast, hepatic sinusoids have no reflection coefficient as they are fully permeable to protein. Hepatic interstitial fluid within the Space of Diss has the same colloid osmotic pressure as plasma and so hepatocyte synthesis of albumin can be regulated.

Following are typical values for the variables in the Starling equation which regulate net $J\_v$ to about 0.1ml per second, 5-6 ml per minute or about 8 litres per day.

Glomerular capillaries have a continuous glycocalyx layer in health and the total transendothelial filtration rate of solvent ($J\_v$) to the renal tubules is normally around 125 ml/ min (about 180 litres/ day). Glomerular capillary $J\_v$ is more familiarly known as the glomerular filtration rate (GFR).

The Starling equation can describe the movement of fluid from pulmonary capillaries to the alveolar air space.{{Citation |last1=Pal |first1=Pramod K. |title=Chapter 1 - Breathing and the Nervous System |date=2014-01-01 |url=http://www.sciencedirect.com/science/article/pii/B9780124077102000011 |work=Aminoff's Neurology and General Medicine (Fifth Edition) |pages=3–23 |editor-last=Aminoff |editor-first=Michael J. |place=Boston |publisher=Academic Press |language=en |doi=10.1016/b978-0-12-407710-2.00001-1 |isbn=978-0-12-407710-2 |access-date=2020-11-28 |last2=Chen |first2=Robert |editor2-last=Josephson |editor2-first=S. Andrew |s2cid=56748572}}{{Citation |last1=Nadon |first1=A. S. |title=Pathobiology of the Acute Respiratory Distress Syndrome |date=2014-01-01 |url=http://www.sciencedirect.com/science/article/pii/B9780123864567053090 |work=Pathobiology of Human Disease |pages=2665–2676 |editor-last=McManus |editor-first=Linda M. |place=San Diego |publisher=Academic Press |language=en |doi=10.1016/b978-0-12-386456-7.05309-0 |isbn=978-0-12-386457-4 |access-date=2020-11-28 |last2=Schmidt |first2=E. P. |editor2-last=Mitchell |editor2-first=Richard N.}}

Woodcock and Woodcock showed in 2012 that the revised Starling equation (steady-state Starling principle) provides scientific explanations for clinical observations concerning intravenous fluid therapy.{{cite journal|last=Woodcock|first=T. E.|author2=Woodcock, T. M. |title=Revised Starling equation and the glycocalyx model of transvascular fluid exchange: an improved paradigm for prescribing intravenous fluid therapy|journal=British Journal of Anaesthesia|date=29 January 2012|volume=108|issue=3|pages=384–394|doi=10.1093/bja/aer515|pmid=22290457|doi-access=free}} Traditional teaching of both filtration and absorption of fluid occurring in a single capillary has been superseded by the concept of a vital circulation of extracellular interstitial fluid running parallel to the circulation of blood. Infusing intravenous fluids that raise plasma colloid osmotic pressure (colloid therapy) has much less effect on plasma volume than originally expected, in part because the initially reduced filtration rate allows the concentration of proteins in the subglycocalx spaces to rise, returning the colloid osmotic pressure difference and trans endothelial solvent filtration rate to their steady state levels within an hour. Prevention and treatment of oedema (excess interstitial fluid) depends on normalisation of $P\_c$ and optimisation of the flow rate of lymph.

The Starling equation is named for the British physiologist Ernest Starling, who is also recognised for the Frank–Starling law of the heart.{{Cite journal |last=Starling |first=Ernest H. |date=1896-05-05 |title=On the Absorption of Fluids from the Connective Tissue Spaces |url= |journal=The Journal of Physiology |language=en |volume=19 |issue=4 |pages=312–326 |doi=10.1113/jphysiol.1896.sp000596 |pmc=1512609 |pmid=16992325}} Starling can be credited with identifying that the "absorption of isotonic salt solutions (from the extravascular space) by the blood vessels is determined by this osmotic pressure of the serum proteins" in 1896.

• Renal function

• Derangedphysiology.com: Starling's Principle of Transvascular Fluid Dynamics [https://derangedphysiology.com/main/cicm-primary-exam/required-reading/cardiovascular-system/Chapter%20471/starling-forces-and-fluid-exchange-microcirculation Starling's principle of transvascular fluid dynamics | Deranged Physiology]

Category:Eponymous equations of physics

Category:Equations of fluid dynamics

Category:Cardiovascular physiology

Category:Mathematics in medicine