Fick’s principle

Fick’s principle states that tissue uptake j (mass/time) of substance equals the blood flow F (volume/time) multiplied by the difference between arterial (CA) and venous (CV) concentrations (both in mass/volume) of the substance in steady state:

The net extraction fraction, En, is the net fraction of the radioligand that is extracted into tissue during a single capillary pass:

While En represents the difference of inward and outward fluxes between blood and tissue, unidirectional extraction fraction, Eu, refers only to the flux from blood to tissue. After bolus infusion of radioligand, during the first pass of blood containing radioligand the concentration in tissue is zero, CT=0, and therefore also the flux from tissue to blood is effectively zero. In this situation, Eu=En, and is often referred to as the first-pass extraction fraction, with simple notation E:

and the flux can be written as:

In compartmental models, the differential equation for the concentration change over time in the first (possibly only) tissue compartment, C1, is:

During the first pass of the radioligand, with C1=0, the (net) flux into tissue is:

, where K1 represents the unidirectional transport rate of the radioligand from the blood to the tissue (delivery rate), with meaning

The first-pass extraction fraction E (Eu) can be derived from the Renkin-Crone capillary model, which assumes that capillary is a rigid cylindrical tube (Renkin, 1959; Crone, 1963). Based on this model, E depends on perfusion, and the product of capillary permeability P (cm/min) and and capillary surface area S (cm2/cm3), PS:

When PS is high compared to F, E approaches its maximal value 1, and K1 ≅ F, which is used in PET for measuring perfusion.

Based on Fick’s equation, highly diffusive (E ≅ 1) and inert indicators can be used to measure regional blood flow (Kety and Schmidt, 1945). Lassen and Munck (1955) introduced the usage of radioactive inert gases for cerebral perfusion measurement. In this setting, the radioactivity concentration in arterial and venous (internal jugular vein) blood were followed during 14 min of inhaling 85Kr-containing air. Effective biological half-life of 85Kr was ∼11 min, and concentration in arterial blood reached a steady level during the study, which could be used in calculation of the cerebral blood flow. When A-V difference in O2 is measured, too, cerebral oxygen consumption can be calculated (Lassen and Munck, 1955). The same method was applied to assess renal blood flow and oxygen consumption (Brun et al., 1955).

In gas washout method, bolus injection of radioactive 85Kr or 133Xe is administered via arterial catheter to the organ of interest. After intra-arterial injection, the gas diffuses rapidly into the perfusable tissue, and instantaneous equilibrium between tissue and blood is assumed. The washout curve is monitored by scintillation detector with collimator. The radioactive gas that leaves the tissue with venous blood is almost completely removed in the lungs, and any remaining gas is distributed into the whole body; thus the arterial gas concentration after bolus injection can be assumed to be zero:

The concentration of radioactive gas in the venous blood leaving the tissue is directly relational to the concentration in the tissue (CT):

, where p is the partition coefficient of the gas between tissue and blood. If blood flow is represented in blood volume per (time × tissue volume), f, after substitution of CV:

If tissue is perfused uniformly, the decreasing tissue concentration curve can be represented by a single exponential function,

, and in case of heterogeneously distributed tissue perfusion, with n parallel tissue compartments, with sum of exponentials:

The sum of exponential functions can be fitted to the washout curve, and the exponential coefficients provide information on the flow in the individual tissue compartments. Mean tissue blood flow has been commonly estimated by integration method:

Even intra-arterial bolus administration cannot be absolutely instantaneous, and defining the zero time will introduce errors in the washout method. Early studies were analyzed using peeling method on data plotted on semi-logarithmic paper, which introduced subjective bias. Compartmental blood flow rates (exponential constants) must differ at least by a factor of 3-4 to allow reliable separation. Additionally, quantification of perfusion with washout method requires that the p of the tissue is known.

Fick’s principle in PET

Fick’s principle is frequently used in validation of PET methods and to calculate additional physiological parameters which are not available from the PET study alone. Arteriovenous (A-V) concentration difference can be used to study the metabolism of a substance at organ level, when the system is in steady state: arterial and venous blood concentrations of the substance are constant, perfusion (blood flow) is constant, and metabolism rate of substance in the organ is constant.

PET can be used to measure perfusion (f) in units mL blood/min/mL tissue (for example using [15O]H2O), and substance uptake in units mmol/min/mL tissue (for example tissue glucose uptake using [18F]FDG, or oxygen consumption using [15O]O2). PET scanner provides the concentrations in units of radioactivity per tissue volume, where radioactivity is directly relational to the mass. Thus, when perfusion is measured with PET and arteriovenous difference in oxygen concentration is measured using traditional methods from blood samples, the metabolic rate of oxygen, (MRO2) can be calculated by multiplying arteriovenous difference in oxygen concentration by perfusion (Heinonen et al., 2011):

Alternatively, PET results can be used to calculate substrate net extraction (arteriovenous difference), for example glucose extraction in mmols per litre (or percentage) when both tissue perfusion and metabolic rate of glucose (regional glucose uptake, rGU) is measured using PET (Nuutila et al., 1996):

Blood versus plasma

It must be kept in mind that the Fick’s equation contains blood flow, and concentrations in arterial and venous blood.

However, substrate concentration is often measured in plasma, instead of blood, and concentrations in blood and plasma are usually not equal. Substrate concentration in blood (BL) is the fractional volume (hematocrit, HCT) weighted average of substrate concentrations in red blood cells (RBC) and plasma (PL):

If substrate does not pass the RBC membrane (CARBC=CVRBC=0), then

and Fick’s equation with plasma concentrations instead of blood concentration becomes:

Note that in the above equation f*(1-HCT) equals plasma flow.

If red blood cells contain substrate (CRBC>0), but substrate transfer from RBC to plasma (and vice versa) is slow compared to the time that it takes for the blood to flow from the arterial side to the venous sampling site, and to the transfer rate between plasma and tissue, then we can assume that CVRBC=CARBC. The terms including CRBC cancel out, and we get again the same equation (11).

For example, RBC-plasma transfer of neutral amino acids (Ellison & Pardridge, 1990) is relatively slow, and the red cells are not available for transport into/out of tissue.

If substrate transfer between RBC to plasma is (nearly) instantaneous, all substrate in the blood is available for the tissue. Then the use of plasma concentrations will not give correct results, but the concentrations in whole blood must be either measured, or calculated based on an empirical function for conversion of plasma concentrations to blood concentrations.

For example, in humans D-glucose has a RBC-plasma equilibration time of about 4 s, and thus blood flow and blood concentration must be used; glucose level is 15% higher in plasma than in blood under typical conditions of hematocrit, proteinemia, and erythrocyte volume (Ferrannini & DeFronzo, 2004). Using plasma concentrations and plasma flow with D-glucose would lead to 31% underestimation (Ferrannini & DeFronzo, 2004). However, there are marked species differences and even age-effects (Jacquez 1984).

Lactate equilibration between RBC and plasma is also fast, and ratio of lactate concentrations in RBC and plasma water is normally about 0.5 (which means that blood/plasma ratio is around 0.7), but the ratio is dependent on pH and thus may change for example during exercise (Goodwin et al., 2007; Wahl et al., 2010).

Small vessel hematocrit

Hematocrit is smaller in capillaries and smaller vessels in the tissue than in the large arteries and veins from where the blood is sampled, but this is irrelevant to application of Fick equation when both arterial and venous blood samples are taken from larger vessels, both having the same haematocrit.


The above mentioned methods are applicable only during systemic steady-state, but methods exist for nonsteady-state situations as well, see for example Manesso et al., 2011.

See also:


Ellison S, Pardridge WM. Red cell phenylalanine is not available for transport through the blood-brain barrier. Neurochem Res. 1990; 15(8): 769-772.

Ferrannini E. and DeFronzo RA. 2004. Insulin Actions In Vivo: Glucose Metabolism. International Textbook of Diabetes Mellitus. Wiley Online Library. doi: 10.1002/0470862092.d0305.

Goodwin ML, Harris JE, Hernández A, Gladded LB. Blood lactate measurements and analysis during exercise: a guide for clinicians. J Diabetes Sci Technol. 2007; 1(4): 558-569.

Heinonen I, Bengt S, Kemppainen J, Sipilä HT, Oikonen V, Nuutila P, Knuuti J, Kalliokoski K, Hellsten Y. Skeletal muscle blood flow and oxygen uptake at rest and during exercise in humans: a pet study with nitric oxide and cyclooxygenase inhibition. Am J Physiol Heart Circ Physiol. 2011; 300: H1510-H1517.

Hinderling PH. Red blood cells: a neglected compartment in pharmacokinetics and pharmacodynamics. Pharmacol Rev. 1997; 49(3): 279-295.

Jacquez JA. Red blood cell as glucose carrier: significance for placental and cerebral glucose transfer. Am J Physiol. 1984; 246: R289-R298.

Kety SS, Schmidt CF. The determination of cerebral blood flow in man by the use of nitrous oxide in low concentrations. Am J Physiol. 1945; 143: 53-66. doi: 10.1152/ajplegacy.1945.143.1.53.

Kety SS. The theory and applications of the exchange of inert gas at the lungs and tissues. Pharmacol Rev. 1951; 3(1): 1-41.

Lassen NA, Munck O. The cerebral blood flow in man determined by the use of radioactive krypton. Acta Physiol Scand. 1955; 33(1): 30-49. doi: 10.1111/j.1748-1716.1955.tb01191.x.

Lorthois S, Duru P, Billanou I, Quintard M, Celsis P. Kinetic modeling in the context of cerebral blood flow quantification by H215O positron emission tomography: The meaning of the permeability coefficient in Renkin-Crone’s model revisited at capillary scale. J Theor Biol. 2014; 353: 157-69.

Manesso E, Toffolo GM, Basu R, Rizza RA, Cobelli C. Modeling nonsteady-state metabolism from arteriovenous data. IEEE Trans Biomed Eng. 2011; 58(5): 1253-1259.

Morris ED, Endres CJ, Schmidt KC, Christian BT, Muzic RF Jr, Fisher RE (2004): Kinetic modeling in positron emission tomography. In: Emission Tomography: The Fundamentals of PET and SPECT. (Eds: Wermick MN, Aarsvold JN). Elsevier Inc., pp 499-540.

Nuutila P, Raitakari M, Laine H, Kirvelä O, Takala T, Utriainen T, Mäkimattila S, Pitkänen O-P, Ruotsalainen U, Iida H, Knuuti J, Yki-Järvinen H. Role of blood flow in regulating insulin-stimulated glucose uptake in humans. J Clin Invest. 1996; 97(7): 1741-1747.

Wahl P, Zinner C, Yue Z, Bloch W, Mester J. Warming-up affects performance and lactate distribution between plasma and red blood cells. J Sports Sci Med. 2010; 9: 499-507.

Fick principle in Wikipedia:

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Created at: 2011-05-11
Updated at: 2018-11-20
Written by: Vesa Oikonen