Starling equation

The classic Starling equation reads as follows:^[4]

${\displaystyle \ J_{v}=L_{\mathrm {p} }S([P_{\mathrm {c} }-P_{\mathrm {i} }]-\sigma [\pi _{\mathrm {p} }-\pi _{\mathrm {i} }])}$ 

where:

• ${\displaystyle J_{v}}$  is the trans endothelial solvent filtration volume per second (SI units of m³·s⁻¹).
• ${\displaystyle [P_{\mathrm {c} }-P_{\mathrm {i} }]-\sigma [\pi _{\mathrm {p} }-\pi _{\mathrm {i} }]}$  is the net driving force (SI units of Pa = kg·m⁻¹·s⁻², often expressed as mmHg),
  • ${\displaystyle P_{c}}$  is the capillary hydrostatic pressure
  • ${\displaystyle P_{i}}$  is the interstitial hydrostatic pressure
  • ${\displaystyle \pi _{p}}$  is the plasma protein oncotic pressure
  • ${\displaystyle \pi _{i}}$  is the interstitial oncotic pressure
  • ${\displaystyle L_{p}}$  is the hydraulic conductivity of the membrane (SI units of m²·s·kg⁻¹, equivalent to m·s⁻¹·mmHg⁻¹)
  • ${\displaystyle S}$  is the surface area for filtration (SI units of m²)
    • the product ${\displaystyle L_{p}}$ ·${\displaystyle S}$  is defined as the filtration coefficient (SI units of m⁴·s·kg⁻¹, or equivalently in m³·s⁻¹·mmHg⁻¹)
  • ${\displaystyle \sigma }$  is Staverman's reflection coefficient (adimensional)

By convention, outward force is defined as positive, and inward force is defined as negative. If J_v is positive, solvent is leaving the capillary (filtration). If negative, solvent is entering the capillary (absorption).

Applying the classic Starling equation, it had long been taught that continuous capillaries filter out fluid in their arteriolar section and reabsorb most of it in their venular section, as shown by the diagram.^[4]

However, empirical evidence shows that, in most tissues, the flux of the intraluminal fluid of capillaries is continuous and, primarily, effluent. Efflux occurs along the whole length of a capillary. Fluid filtered to the space outside a capillary is mostly returned to the circulation via lymph nodes and the thoracic duct.^[5]

A mechanism for this phenomenon is the Michel-Weinbaum model, in honour of two scientists who, independently, described the filtration function of the glycocalyx. Briefly, the colloid osmotic pressure π_i of the interstitial fluid has been found to have no effect on Jv and the colloid osmotic pressure difference that opposes filtration is now known to be π'_p minus the subglycocalyx π, which is close to zero while there is adequate filtration to flush interstitial proteins out of the interendothelial cleft. Consequently, Jv is much less than previously calculated, and the unopposed diffusion of interstitial proteins to the subglycocalyx space if and when filtration falls wipes out the colloid osmotic pressure difference necessary for reabsorption of fluid to the capillary.^[4]

The revised Starling equation is compatible with the steady-state Starling principle:

${\displaystyle \ J_{v}=L_{\mathrm {p} }S([P_{\mathrm {c} }-P_{\mathrm {i} }]-\sigma [\pi _{\mathrm {p} }-\pi _{\mathrm {g} }])}$ 

where:

• ${\displaystyle J_{v}}$  is the trans endothelial solvent filtration volume per second.
• ${\displaystyle [P_{\mathrm {c} }-P_{\mathrm {i} }]-\sigma [\pi _{\mathrm {p} }-\pi _{\mathrm {i} }]}$  is the net driving force,
  • ${\displaystyle P_{c}}$  is the capillary hydrostatic pressure
  • ${\displaystyle P_{i}}$  is the interstitial hydrostatic pressure
  • ${\displaystyle \pi _{p}}$  is the plasma protein oncotic pressure
  • ${\displaystyle \pi _{g}}$  is the subglycocalyx oncotic pressure
  • ${\displaystyle L_{p}}$  is the hydraulic conductivity of the membrane
  • ${\displaystyle S}$  is the surface area for filtration
  • ${\displaystyle \sigma }$  is Staverman's reflection coefficient

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

In some texts the product of hydraulic conductivity and surface area is called the filtration co-efficient K_fc.^{[citation needed]}

Staverman's reflection coefficient, σ, is a unitless constant that is specific to the permeability of a membrane to a given solute.^[6]

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.^[7]

σ_n corrects for the partial permeability of a semipermeable membrane to a solute n.^[7]

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.^[7]

• 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. Albumin and other proteins in the interstitial spaces return to the circulation via lymph.^[8]

Following are typically quoted values for the variables in the classic Starling equation:

It is reasoned that some albumin escapes from the capillaries and enters the interstitial fluid where it would produce a flow of water equivalent to that produced by a hydrostatic pressure of +3 mmHg. Thus, the difference in protein concentration would produce a flow of fluid into the vessel at the venous end equivalent to 28 − 3 = 25 mmHg of hydrostatic pressure. The total oncotic pressure present at the venous end could be considered as +25 mmHg.^{[citation needed]}

In the beginning (arteriolar end) of a capillary, there is a net driving force (${\displaystyle [P_{\mathrm {c} }-P_{\mathrm {i} }]-\sigma [\pi _{\mathrm {c} }-\pi _{\mathrm {i} }]}$ ) outwards from the capillary of +9 mmHg. In the end (venular end), on the other hand, there is a net driving force of −8 mmHg.^{[citation needed]}

Assuming that the net driving force declines linearly, then there is a mean net driving force outwards from the capillary as a whole, which also results in that more fluid exits a capillary than re-enters it. The lymphatic system drains this excess.^{[citation needed]}

J. Rodney Levick argues in his textbook that the interstitial force is often underestimated, and measurements used to populate the revised Starling equation show the absorbing forces to be consistently less than capillary or venular pressures.

Glomerular capillaries have a continuous glycocalyx layer in health and the total transendothelial filtration rate of solvent (${\displaystyle J_{v}}$ ) to the renal tubules is normally around 125 ml/ min (about 180 litres/ day). Glomerular capillary ${\displaystyle J_{v}}$  is more familiarly known as the glomerular filtration rate (GFR). In the rest of the body's capillaries, ${\displaystyle J_{v}}$  is typically 5 ml/ min (around 8 litres/ day), and the fluid is returned to the circulation via afferent and efferent lymphatics.^{[citation needed]}

The Starling equation can describe the movement of fluid from pulmonary capillaries to the alveolar air space.^[10][11]

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.^[12] 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 fluid running parallel to the circulation of blood. New approaches to the treatment of oedema (tissue swelling) are suggested.

The Starling equation is named for the British physiologist Ernest Starling, who is also recognised for the Frank–Starling law of the heart.^[13] 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.^[13]

• Renal function

• Derangedphysiology.com: Starling's Principle of Transvascular Fluid Dynamics Starling's principle of transvascular fluid dynamics | Deranged Physiology