Home | About | Journals | Submit | Contact Us | Français |

**|**HHS Author Manuscripts**|**PMC3008667

Formats

Article sections

Authors

Related links

Am J Physiol. Author manuscript; available in PMC 2010 December 22.

Published in final edited form as:

PMCID: PMC3008667

NIHMSID: NIHMS204002

Center for Bioengineering, University of Washington, Seattle, Washington 98195; Department of Medicine, University of Medicine and Dentistry of New Jersey-New Jersey Medical School, Newark, New Jersey 07103; Institute of Medical Physiology, University of Copenhagen, The Panum Institute, DK-2200 Copenhagen N, Denmark; Montreal General Hospital, Montreal, Quebec H3G 1A4, Canada; Bispebjerg Hospital, 2400 Copenhagen NV, Denmark; Department of Physiology, University of Limburg, 6200 MD Maastricht, the Netherlands; and Department of Physiology, The Johns Hopkins University School of Medicine, Baltimore, Maryland 21205

The publisher's final edited version of this article is available at Am J Physiol

See other articles in PMC that cite the published article.

Virtually all fields of physiological research now encompass various aspects of solute transport by convection, diffusion, and permeation across membranes. Accordingly, this set of terms, symbols, definitions, and units is proposed as a means of clear communication among workers in the physiological, engineering, and physical sciences. The goal is to provide a setting for quantitative descriptions of physiological transport phenomena.

THE SET OF SYMBOLS is an extension of those proposed by Wood (11), Gonzalez-Fernandez(3), Zierler (12), Kedem and Katchalsky (5), and Bassingthwaighte et al. (1). The extensions provide a set of symbols common to studies of transcapillary and cellular exchange and indicator-dilution studies. The rationale is to provide a self-consistent set of symbols covering broad aspects of circulatory flows, hydrodynamics, transcapillary and membrane transport. As the various previously rather separate aspects of these fields become intermeshed, the size of the required sets of symbols has enlarged to a point where the “standard” symbol for one group of users has a quite different “natural” meaning to another. This problem has necessitated some arbitrariness, but we have attempted to subscribe to the dominant usage so as to minimize changes in habits.

Care has been taken to provide each term with *1*) a name, *2*) a definition in words (and sometimes equations), *3*) a unique symbol whenever possible, and *4*) units mainly in centimeter-gram-second system but with some translation to approved International System of units (SI). Physical constants are listed separately.

An important feature of this list is the provision of operational terminology for the general description of the behavior of linear stationary systems. The use of the time-domain impulse response or transport function, *h*(*t*), etc., follows from the work of Stephenson (10), Meier and Zierler (6), and Zierler (12) and is reviewed by Bassingthwaighte and Goresky (2).

A system is diagramed in Figure 1. Most analysis is based on two fundamental assumptions, that the system is both linear and stationary. When both hold, superposition is applicable. In general, we also consider the system to be mass conservative; that is, indicator and solvent are neither formed nor consumed.

Block diagram of a linear stationary system. Response to ideal impulse input δ(*t*) at the entrance is *h*(*t*), the transport function. When input is of another form, C_{in}(*t*), then outflow response C_{out}(*t*) is the convolution of C_{in}(*t*) and *h*(*t*).

A linear system is one in which inputs and outputs are additive. Defining C_{in}(*t*), as concentration-time curve at the input to a segment of the circulation and C_{out}(*t*) as the concentration-time curve occurring in response to it at the outlet, the relationship is denoted by

$${\mathrm{C}}_{\text{in}}\left(t\right)\to {\mathrm{C}}_{\text{out}}\left(t\right)$$

Given a second pair with the same relationship ${\mathrm{C}}_{\text{in}}^{\prime}\left(t\right)\to {\mathrm{C}}_{\text{out}}^{\prime}\left(t\right)$, then in a linear system, these can be summed or multiplied by a scalar

$$\begin{array}{cc}\hfill {\mathrm{C}}_{\text{in}}\left(t\right)+{\mathrm{C}}_{\text{in}}^{\prime}\left(t\right)\to {\mathrm{C}}_{\text{out}}\left(t\right)& +{\mathrm{C}}_{\text{out}}^{\prime}\left(t\right)\phantom{\rule{1em}{0ex}}\text{or}\hfill \\ \hfill & k{\mathrm{C}}_{\text{in}}\left(t\right)\to k{\mathrm{C}}_{\text{out}}\left(t\right)\phantom{\rule{1em}{0ex}}\text{linearity}\hfill \end{array}$$

A stationary system is one in which the distribution of transit times through the system is constant from moment to moment; that is, flows and volumes are constant everywhere in the system. Stationarity implies that the response to a given input is independent of a shift in the timing of the input by an arbitrary time, *t*_{o},

$$\begin{array}{ccc}\text{If}\hfill & \hfill {\mathrm{C}}_{\text{in}}\left(t\right)\to {\mathrm{C}}_{\text{out}}\left(t\right)\hfill & \hfill \hfill \\ \text{then}\hfill & \hfill {\mathrm{C}}_{\text{in}}({t}_{\mathrm{o}}+t)\to {\mathrm{C}}_{\text{out}}({t}_{\mathrm{o}}+t)\hfill & \hfill \text{stationarity}\hfill \end{array}$$

When the input system is an ideal unit impulse, the Dirac delta function, *δ*(*t*), then the output is the transport function, *h*(*t*). When the input is of general form, C_{in}(*t*), and *h*(*t*) is known, then the form of the output, C_{out}(*t*), can be calculated using the convolution integral given in Fig. 1.

A probability density function *h*(*x*) or *w*(*x*) is a weighting function or a frequency function that gives the probability of occurrence of an observation or measure as a linear function of the quantitative measure, *x*. The sum of probabilities of all the observations is unity; therefore the units of the density function are fraction per unit of the measure [e.g., the transport function *h*(*t*)]. A typical form of *h*(*t*) for transport through an organ is given in Fig. 2, accompanied by closely related general functions.

A | Arterial |

B | Blood |

C or cap | Capillary, or the region of blood-tissue ex- change |

cell | Cell |

D | Diffusive, or indicating a permeant tracer |

ECF | Extracellular fluid |

F | Flow or filtration |

i,j
| Indices in series or summations or elements of arrays |

in or i | Into or inside or inflow |

ISF or I | Interstitial fluid space, the extravascular extracellular fluid |

m | Membrane |

out or o | Out of or outside or outflow |

P | Plasma |

RBC | Red blood cell |

R | Reference, nonpermeant tracer |

S | Solute |

T | Total |

v | Venous |

W | Water |

a | Activity, molar; a = C, an activity coefficienttimes a concentration |

A | Area of indicator concentration-time curve excluding recirculation $\text{A}={\int}_{0}^{\infty}\mathrm{C}\left(t\right)\mathrm{d}t,\text{mol}\xb7s\xb7{l}^{\mathrm{-1}}$ |

C | Concentration, mol/l; C_{c}(x, t) concentrationin the capillary plasma at position x at timet (mol·1^{−1}). Also [Na^{+}] = sodium concentra-tion. The relationship between an outflow concentration-time curve C _{out}(t) and theinflow curve C _{in}(t) in a stationary system isgiven by the convolution integral: ${\mathrm{C}}_{\text{out}}\left(t\right)={\int}_{0}^{t}h(t-\tau ){\mathrm{C}}_{\text{in}}\left(\tau \right)d\tau ={\mathrm{C}}_{\text{in}}\left(t\right)\ast h\left(t\right)$ where τ is a variable used in the integration. The asterisk denotes convolution |

C̄_{s} | Concentration of solute, the average of the concentrations on the two sides of a mem- brane, molal, used in irreversible thermo- dynamic equations. Note that this average does not represent the mean concentration within the membrane when both convection and diffusion occur through a channel of finite length |

CV | Coefficient of variation, dimensionless. See also RD; both are the standard deviation divided by the mean of a density function |

D | Diffusion coefficient, cm^{2}·s^{−1}; D_{o}, in free(aqueous) solution; D_{b} for observed bulkdiffusion coefficient through tissue; D_{cell} forintracellular; D_{I} for interstitial |

E | Electrical potential, volts; E_{m}, membrane po-tential; E_{N}, “Nernst” potential, occurringwith a difference in concentration of an ion on the two sides of a membrane, E_{N} =( RT/zF)log_{e}(C_{in},/C_{out}) |

E(t) | Extraction, dimensionless, is the fraction of a specific substance removed during transit through an organ. The calculation may be made relative to a reference substance that remains in the blood or relative to the inflow concentration. E( t) = [h_{R}(t) – h(_{o}t)]/h_{R}(t)and is the instantaneous apparent frac- tional extraction of a permeating species, subscripted D, relative to a nonpermeatingreference substance, subscripted R, at each time t, calculated from paired outflow dilu-tion curves. This differs from a steady-state extraction, E, calculated from the arterio- venous difference, E = (C _{A} – C_{v})/C_{A}, for asubstance that is consumed during transor- gan passage. E( t_{peak}) is the value of E(t)obtained at the time of the peak of the curve for the nonpermeating reference tracer, h_{R}(t). E_{max} is the maximum value of theinstantaneous extraction, E( t). E_{net}(t) is anintegral extraction, ${\int}_{0}^{t}({h}_{\mathrm{R}}-{h}_{\mathrm{D}})\mathrm{d}\tau \u2215{\int}_{0}^{t}{h}_{\mathrm{R}}\mathrm{d}\tau =({\mathrm{R}}_{D}-{\mathrm{R}}_{\mathrm{R}})\u2215(1-{\mathrm{R}}_{\mathrm{R}});$ when the reference tracer has all emerged, then E _{net}(t) = R_{D}(t), the retained fraction of apermeant solute |

ECF | Extracellular fluid, interstitial fluid + plasma |

f | Frictional coefficient, g·cm equals (g·cm^{2} s^{−1})/(cm·s ^{−1}), following Spiegler (9) |

f_{excl} | Excluded volume fraction, the fraction of sol- vent in a defined space that is not available to a particular solute, dimensionless |

f _{i} | Relative regional flow in the j^{th} region of anorgan divided by the mean flow for the organ per gram of tissue, dimensionless |

F | Flow, cm^{3}·s^{−1} or cm^{3}·min^{−1} |

F_{B} | Blood flow to an organ, cm^{3}·g^{−1}·min^{−1}(= F/W, where W = organ weight) |

F_{s}, F_{p} | Flow of solute-containing mother fluid, cm^{3}·g ^{−1}·min^{−1}. When solute is excluded fromred blood cells, F _{s} = F_{B}(1 – Hct) = F_{p}, theplasma flow. (In modeling analysis, this is the flow of fluid containing solute available for exchange.) |

FER(t) | Fractional escape rate at time t for indicatorcontained in a system regardless of time of entry, s ^{−1}. With an impulse input, δ(t), thenFER( t) = η(t), the emergence function. Ingeneral, FER = (dq/d t)/q = d log_{e}q/dt,where q is the system’s content of a sub- stance and dq/d t = F[C_{in}(t) – C_{out}(t)] |

h(t) | Transport function, fraction/unit time (s^{−1}),is the fraction of indicator injected at the inflow at t = 0, arriving at the outflow attime t. It is the unit impulse response, thefrequency function of transit times, or the probability density function of transit times. The transport function, h(t), has theshape of the concentration-time curve that would be obtained by flow-proportional sampling at the output if indicator were injected in ideal fashion into the inflow, i.e., across a cross section with indicator amount at each point in proportion to local flow, as defined by Gonzalez-Fernandez (3), and re- circulation absent. Under such conditions h(t) = F·C(t)/q_{o}, where q_{o} is the massinjected at t = 0. Subscripting denotes re-gion (e.g., A, V, or cap) or solute character- istic (R for intravascular or D for permeant) |

H(t) | Cumulative residence time distribution func- tion (dimensionless) of a system; it repre- sents the fraction of an ideally injected tracer that has exited from the system since t = 0. It is also the response to a step input.Formally, $H\left(t\right)={\int}_{0}^{t}h\left(\tau \right)\mathrm{d}\tau =1-R\left(t\right)$, where R(t) is the residue function |

Hct | Hematocrit, the fraction of the blood volume that is erythrocytes, dimensionless |

ISF | Interstitial fluid, the extravascular extracel- lular fluid |

J | Flux, usually moles per unit surface area of membrane per second, mol·s ^{−1}·cm^{−2}.J _{net 1→2} is net flux from side 1 to side 2. Inthe notation of irreversible thermody- namics the equations of Kedem and Katch- alsky (5) and Katchalsky and Curran (4) for water and solute transport across an ideal membrane composed of infinitely thin impermeant material pierced by aqueous channels (the K and K membrane) are $$\begin{array}{cc}\hfill & {J}_{\mathrm{V}}={L}_{\mathrm{p}}\Delta \mathrm{p}+{L}_{\mathrm{p}D}\Delta \pi \hfill \\ \hfill & {J}_{\mathrm{D}}={L}_{\mathrm{D}\mathrm{p}}\Delta \mathrm{p}+{L}_{D}\Delta \pi \hfill \end{array}$$ where J_{D} is a solute velocity relative to thesolvent velocity, J_{v}, which is in turn relativeto the membrane. [Although these expres- sions are incomplete in that the forces on the membrane, in effect a second solute, should also be considered (8), they provide an elementary conceptual approach to an idealized system.] J_{v} and J_{D} may beproperly regarded as flows rather than mass fluxes |

J
_{V}
| Solvent velocity or volume flux per unit mem- brane surface area relative to a membrane, cm·s ^{−1} or cm^{3} · s^{−1} per cm^{2} area$${J}_{\mathrm{V}}={J}_{\mathrm{W}}{\stackrel{~}{\mathrm{V}}}_{\mathrm{W}}+{J}_{\mathrm{S}}{\stackrel{~}{\mathrm{V}}}_{\mathrm{S}}\simeq {J}_{\mathrm{W}}{\stackrel{~}{\mathrm{V}}}_{\mathrm{W}}$$ |

J
_{D}
| Solute movement relative to solvent, cm^{3} ·s^{−1}per cm ^{2} surface area or cm·s^{−1}. For theKedem-Katchalsky (K-K) ideal membrane ${J}_{\mathrm{D}}={J}_{\mathrm{S}}\u2215{\stackrel{\u2012}{\mathrm{C}}}_{\mathrm{S}}-{\stackrel{~}{\mathrm{V}}}_{\mathrm{W}}{J}_{\mathrm{W}}$ See J_{S} |

J
_{W}
| Water flux across a membrane, mol·s^{−1}·cm^{−2}.For the K-K membrane ${J}_{\mathrm{W}}=-{\stackrel{~}{\mathrm{V}}}_{\mathrm{S}}{J}_{\mathrm{S}}\u2215{\stackrel{~}{\mathrm{V}}}_{\mathrm{W}}$ |

J
_{S}
| Solute flux across a membrane, mol·s^{−1}·cm^{−2}.For the K-K membrane ${J}_{\mathrm{S}}={\stackrel{\u2012}{\mathrm{C}}}_{\mathrm{S}}(1-\sigma ){\mathrm{J}}_{\mathrm{V}}$. Also $$\begin{array}{cc}\hfill {J}_{\mathrm{S}}\u2215{\stackrel{\u2012}{\mathrm{C}}}_{\mathrm{s}}& =({L}_{\mathrm{p}}+{L}_{{D}_{\mathrm{p}}})\Delta \mathrm{p}+({L}_{\mathrm{p}D}+{L}_{D})\Delta \pi ,\phantom{\rule{1em}{0ex}}\text{or}\hfill \\ \hfill {J}_{\mathrm{S}}& ={\stackrel{\u2012}{\mathrm{C}}}_{\mathrm{S}}(1-\sigma ){J}_{\mathrm{V}}+\omega \Delta \pi \hfill \end{array}$$ |

k
| Rate constant for an exchange process, usu- ally s ^{−1}; k(C) is concentration-dependentrate |

k
_{F}
| Filtration coefficient, cm^{3}·s^{−1}·cmm^{−2} (mmHg)^{−1};k_{F} = L_{p}. See L_{p} and also P_{F} |

K
_{m}
| Michaelis constant, molar. For a reaction $$\mathrm{E}+\mathrm{S}\underset{{k}_{-1}}{\overset{{k}_{1}}{\leftrightharpoons}}\mathrm{ES}\stackrel{{k}_{2}}{\to}\mathrm{E}+\mathrm{P}$$ then K_{m} = (k_{−1} + k_{2})/k_{1}, which in the limitwhere k_{2} << k^{−1} becomes the original appar-ent dissociation constant, k_{−1}/k_{1}, which atequilibrium = [E]·[S]/[ES]. (E, enzyme; S, substrate; P, product) |

l, L | Length, cm |

L
| Conductance (general) per unit area as in J =LX; flux = conductance times driving force |

L
_{p}
| Pressure filtration coefficient or hydraulic conductance; the flow of pure solvent across a membrane per unit area per unit pressure difference, e.g., cm·s ^{−1}(mmHg)^{−1}; also, ${L}_{\mathrm{p}}=\stackrel{~}{\mathrm{V}}{P}_{\mathrm{F}}\u2215RT={k}_{\mathrm{F}}$ |

L
_{pD}
| Osmotic coefficient; the flow of solution across a membrane per unit area per unit osmotic pressure difference. Same units as L_{p}; also,L_{pD} = σL_{p} |

L
_{Dp}
| Ultrafiltration coefficient; the conductance for the hydrostatically driven flow of solute relative to that of solvent, per unit area per unit hydrostatic pressure difference. Same units as L_{p}. By Onsager reciprocity, L_{Dp} =L_{pD}. (For an ideal semipermeable mem-brane, σ = 1, ω = 0, and − L_{pD} = L_{p} = L =_{D}− L_{Dp}) |

L
_{D} | Coefficient for diffusional mobility per unit osmotic pressure. Same units as L_{P}. See ωand P |

M | Molarity, moles of solute per liter of solution. Also mM, 10 ^{−3} M and μM, 10^{−6} M. (Molalityis moles of solute per kilogram of solvent. The use of molal units gives consistency in transient states; for example, the molal con- centration of solute 1 is not changed by the removal of solute 2, but the molar concen- tration may be raised or lowered) |

Mean | $\stackrel{\u2012}{X}$, the mean of a density function, ω(x), iscalculated by $$\begin{array}{c}\hfill \stackrel{\u2012}{x}={\int}_{0}^{\infty}x\cdot w\left(x\right)\mathrm{d}x/{\int}_{0}^{\infty}w\left(x\right)\mathrm{d}x\phantom{\rule{1em}{0ex}}\text{or}\hfill \\ \hfill =\sum _{i}{x}_{i}\cdot w\left({x}_{i}\right)\Delta {x}_{i}\u2215\sum _{i}w\left({x}_{i}\right)\Delta {x}_{i}\hfill \end{array}$$ Same as α _{1} |

n
_{i} | Same as α_{1} Moles of substance i in a solution. See molefraction x_{i} |

N
| Number of observations or number of ele- ments in a series, i = 1 to N |

p | pressure, mmHg or Pa (1 Torr = 1 mmHg). See osmotic pressure, π |

P
| Permeability coefficient for a solute traversing a membrane, cm·s ^{−1}; equivalent to a diffu-sion coefficient for a solute in a membrane divided by the thickness. P = ωRT. P_{0}, P_{L},permeabilities at the arterial and venous end of a capillary of length L, respectively.P(x) for 0 < x < L for permeability atposition x. (Usually observed as a product,PS, with the membrane surface area, S) |

Pe | Peclet number, ratio of a convective to a dif- fusive velocity, dimensionless |

P
_{F}
| Filtration permeability, ${L}_{\mathrm{p}}RT\u2215{\stackrel{~}{\mathrm{V}}}_{\mathrm{w}}$, cm·s^{−1}.[The conversion factor $RT\u2215{\stackrel{~}{\mathrm{V}}}_{\mathrm{w}}$ at 20°C, from the experimental units for L_{p} or k_{F}, is (18.36mmHg·cm ^{3}·mol^{−1})/( 18 cm^{3}·mol^{−1}) equals1.02 mmHg] |

PS
| Permeability-surface area product of a barrier, cm ^{3}·g^{−1}·s^{−1} or cm^{3}·g^{−1}·min^{−1}. PS_{cap} forcapillary (the same as capillary diffusion capacity), PS_{cell} for parenchymal cell |

q | Mass, g or mol. q(t) is mass (or content oftracer) in region or organ (at time t). q_{0},mass of indicator injected at t = 0 |

r, R
| Radius or radial distance, cm. R_{C}, capillaryradius |

RD | $\text{Relative dispersion}\phantom{\rule{thinmathspace}{0ex}}\left(\text{dimensionless}\right)=\mathrm{SD}\u2215\text{mean}=\sqrt{{\mu}_{2}\u2215{\alpha}_{1}}$. Same as coefficient of var- iation |

R(t) | Residue function (dimensionless) is the com- plement of H(t), i.e., R(t) = 1 – H(t). Itrepresents the fraction of injectate in the system at time t after an impulse input attime zero, i.e., the probability of a tracerresiding in the system for time t or greater |

S
| Surface area. S_{C} and S_{cell} for capillary and cellsurface areas, cm ^{2}·g tissue^{−1} |

SD | Standard deviation = square root of the vari- ance of a density function, ${\mu}_{2}^{1\u22152}$. Also $\mathrm{SD}=\sqrt{{\alpha}_{2}-{\alpha}_{1}^{2}}$ (units are those of the independent variable) |

SEM | Standard error of the $\mathrm{SD}=\sqrt{N}$, whereN = number of observations |

t, Δt | Time, s; Δt is a finite time interval |

$\stackrel{\u2012}{t}$ | Mean transit time, s. $\stackrel{\u2012}{t}={\int}_{0}^{\infty}t\cdot h\left(d\right)\mathrm{d}t={\int}_{0}^{\infty}R\left(t\right)\mathrm{d}t$ |

t
_{a}
| Appearance (a) time; the time at which the first detectable indicator (or a concentra- tion of, for example, 1% of the peak) passed through the system |

t
_{0}
| Zero time; midpoint of pulse injection for in- dicator-dilution studies or beginning of con- stant-rate injection |

t
_{peak}
| Time from injection to peak of indicator-di- lution curve (modal time) |

V | Volume, cm^{3} or ml; in a solution, $\mathrm{V}=\Sigma {n}_{i}{\stackrel{~}{\mathrm{V}}}_{i}$,the sum of the products of the mole fraction times the partial molar volume for each contained species |

V_{region} | Anatomic volumes within regions of an organ, i.e., V _{C}, capillary; V_{I}, interstitial fluid; V_{cell},parenchymal cells, cm ^{3}·(g tissue of theorgan) ^{−1} |

v_{region} | Fractional regional volumes of distribution available to a particular solute, i.e., v _{C},within the capillary; v _{I}, interstitial fluidspace; and v _{cell}, parenchymal cells. At equi-librium, for a substance passively exchang- ing between plasma and ISF, v _{I} is the ratioof the concentration in V _{I} to that in theplasma and is equal to the partition co- efficient λ = C _{I}/C_{p}. For steady-state proc-esses producing transmembrane fluxes, the effective volume of distribution is not the same as the equilibrium ratio, i.e., v _{I} ≠ λ |

ν
_{F}
| Velocity of fluid flow, cm·s^{−1} |

V’ | Volumes of distribution, cm^{3}·g^{−1}. ${\mathrm{V}}_{\mathrm{C}}^{\prime}$, in cap-illary; ${\mathrm{V}}_{\mathrm{I}}^{\prime}$, in ISF; and ${\mathrm{V}}_{\text{cell}}^{\prime}$, in parenchymal cell. These are the anatomic volumes times the fractional volume of distribution, e.g., ${\mathrm{V}}_{\mathrm{I}}^{\prime}={\mathrm{v}}_{\mathrm{I}}{\mathrm{V}}_{\mathrm{I}}$. Commonly used ratios are $\gamma ={\mathrm{V}}_{\mathrm{I}}^{\prime}\u2215{\mathrm{V}}_{\mathrm{c}}^{\prime}$ and $\u03f4={\mathrm{V}}_{\text{cell}}^{\prime}\u2215{\mathrm{V}}_{\mathrm{C}}^{\prime}$ |

${\stackrel{~}{\mathrm{V}}}_{i}$ | Partial molar volume of solute i, cm^{3}/mol; theincrement in the volume of a solution per mole of added solute, e.g., ${\stackrel{~}{\mathrm{V}}}_{\mathrm{w}}\simeq 18{\mathrm{cm}}^{3}\cdot {\mathrm{mol}}^{-1}$ |

W | Mass, g (“weight,” mass times gravitational acceleration) |

ω(x) | Weighting function or probability density function of variable x |

ω or_{i}ω(_{i}f) | Weighting or fraction of total in the i^{th} group.Units are fraction per unit of f. Given adensity function of regional flows, ω(f), inits finite histogram representation ωΔ_{i}f, is_{i}the fraction of the mass of the organ having a flow f, the average of the flows grouped_{i}as the i^{th} class. The fraction of the totalflow going to the regions falling into the i^{th}class is ωΔ_{i}f_{i}f_{i} |

x
| Distance, cm; e.g., distance along the capillary from inflow, x = 0, to outflow, x = L |

$\stackrel{\u2012}{x}$ | Mean of a density function, ω(x); see meanand moments, α |

X
| Generalized driving force |

x
_{i} | Mole fraction of component i; i.e., moles ofthe i^{th} component divided by the total molesin the system, = n, where _{i}/nn is the total |

z
| Valence of an ionic solute, number of unpaired electrons (or missing electrons) per molecule |

α_{0}, α_{1}, α_{n} | Moments about zero for a probability density function. (Units are t when ^{n}t is the inde-pendent variable.) [ α_{0} = area; α_{1} = mean;for the density function h(t), ${\alpha}_{n}={\int}_{-0}^{+\infty}{t}^{n}h\left(t\right)\mathrm{d}t$. See central moments, μ |

β
_{n–2}
| Dimensionless parameters of shape of density function calculated from the central mo- ments, $={\mu}_{n}\u2215{\mathrm{SD}}^{n}={\mu}_{n}\u2215{\mu}_{2}^{n\u22152}$. “ β_{1}” is skewness(or asymmetry); β_{1} is zero for all symmet-rical functions, positive for right skewness. “ β_{2}” is kurtosis (or flatness). β_{2} = 3.0 fornormal density function; β_{2} > 3 for lepto-kurtosis (highpeakedness), and <3 for platykurtosis |

γ
| Ratio of interstitial volume of distribution to intracapillary volume of distribution, ${\mathrm{V}}_{\mathrm{I}}^{\prime}\u2215{\mathrm{V}}_{\text{cap}}^{\prime}$ |

Δ | Difference |

δ(t) | Unit impulse func- tion, or Dirac delta function, has unity area, an infinite amplitude at t = 0, and is zero at all other times. It isthe limit of any symmetrical unimodal den- sity function of unity area as its width ap- proaches zero. For delta function occurring at a nonzero time t_{0}, it is written δ(t – t_{0}) |

Epsilon, vanishingly small difference | |

ζ | Tortuosity of diffusion pathway. ζ is ratio of apparent path length to measured length of diffusion pathway, dimensionless; thus the effective diffusion coefficient, D = D_{0}/ζ^{2}where D is the free aqueous diffusion coef-ficient |

η | Viscosity, poise (P) = dyn/cm^{2} = g·s^{−1}·cm^{−1}.Water viscosity = 0.01002 P at 20°C. Plasma viscosity ≈ 0.011 |

η(t) | Equals h(t)/R(t) (fraction/s); the emergencefunction, the specific fractional escape rate following an impulse input. Of the particles residing in the system for t seconds afterentering, η(t) is the fraction that will departor escape in the t^{th} second. In chemicalengineering it is known as the intensity function (7), and in population statistics as the risk function, the death rate of those living at age t. Also, η(t) = (dR/dt)/R(t) =−d log _{e}R(t)/dt. See FER(t) |

Θ | Ratio of intracellular volume of distribution to intracapillary volume of distribution, ${\mathrm{V}}_{\text{cell}}^{\prime}\u2215{\mathrm{V}}_{\text{cap}}^{\prime}$, dimensionless |

λ, λ
_{ij} | Partition coefficient, a dimensionless ratio of Bunsen solubility coefficients in two phases. λ is the ratio of solubility in region_{ij}or solvent i to the solubility in region j. Thereference region j is usually the plasma. Atequilibrium, λ is the ratio of concentra-_{ij}tions |

μ
| Chemical potential for a solute in a solution, N·m ^{−2}; μ = μ^{0} + RT In a, where the activitya is a concentration times an activity coef- ficient and μ^{0} is the potential at a referencestate of temperature and pressure |

μ
_{n} | n^{th} central moment of a density function,h(t), a moment around the mean, $\stackrel{\u2012}{t}$. ${\mu}_{n}={\int}_{-\infty}^{\infty}{(t-\stackrel{\u2012}{t})}^{n}h\left(t\right)\mathrm{d}t$. Units are those of t tothe n^{th} power |

μ_{2}, μ_{3}, μ_{4} | μ_{2} is variance, the second moment of a densityfunction around the mean, $={\alpha}_{2}-{\alpha}_{1}^{2}$. Also ${\mu}_{3}={\alpha}_{3}-3{\alpha}_{1}{\alpha}_{2}+2{\alpha}_{1}^{3}$ and ${\mu}_{4}={\alpha}_{4}-4{\alpha}_{1}{\alpha}_{3}+6{\alpha}_{1}^{2}{\alpha}_{2}$. See also β_{n} |

π | Osmotic pressure, Pa or N·m^{−2} or mmHg, isthe pressure that would have to be exerted on a solution to prevent pure water from entering it from across an ideal semiperme- able membrane, i.e., a membrane permeable to solvent only. π = CRT is Van’t Hoff’slaw for ideal dilute solutions, and across a membrane impermeable to solute. π = CRT is preferred to account for activitycoefficients less than unity. When the sol- ute can permeate the membrane, the effec- tive π = σ CRT. Osmotic pressure, a colli-gative property of solutions, is related to actual pressure rn the same fashion as a freezing point is to actual temperature. On- cotic pressure is a term, now obsolete al- though historically useful, for the osmotic pressure associated with the presence of large, relatively impermeant molecules such as plasma proteins. It should now be re- placed by more exact terms, e.g., across some specific membrane the effective Δπ equals $RT{\phi}_{i=1}^{i=N}{\sigma}_{i}{\varphi}_{i}\Delta {\mathrm{C}}_{i}$, where the effects of concentration differences for a set of Nsolutes are summed. |

ρ
| Density, g·cmm^{−3}. (Specific gravity is densityrelative to density of water) |

σ | Reflection coefficient, in notation of irrevers- ible thermodynamics, dimensionless; σ = –L _{pD}/L_{p} or, experimentally, σ = –JD/Jv forΔC _{8} = 0. The effective osmotic pressureacross a membrane is σΔπ, mmHg; i.e., σ = (observed osmotic pressure)/CRT |

τC | Capillary mean transit time, _{c}, used in Kroghcylinder capillary-tissue models with plug flow velocity profiles |

Activity coefficient, the ratio of apparent chemically effective concentration to the actual concentration in a solution, in the absence of chemical binding, dimensionless. The osmotic activity coefficient = π/CRT | |

ψ | Electrical potential, V |

ω | Solute permeability coefficient, ω = P/RT, mol·cm ^{−2}. s^{−1} ·(mmHg)^{−1}. In the notation ofirreversible thermodynamics ω = (L _{D} – RT/Fσ ^{2}L_{p})C̄_{s}, where C̄_{s}, is the average solute con-centration across the membrane |

A | Ampere, unit of electrical current, coulomb per second (C·s ^{−1}) |

Å | Ångstrom, 10^{−10} m or 0.l nm |

C | Charge, coulomb, ampere· second (A. s) |

°K | Degrees of temperature, Kelvin (absolute); °C for degrees Celsius = 273.15 + °K |

dyn | Dyne, force, g·cm·s^{−2} = 10^{−5} N (newton) |

eq | Equivalent weight = molecular weight/va- lence. One equivalent carries 9.65 × 10 ^{4} Cof charge |

e | Elementary charge, 1.6021892 × 10^{−19} C |

erg | Energy, dyn·cm = g·cm^{2}·s^{−2} = 10^{−7} J |

F | Faraday constant, 9.648456 × 10^{4} elementarycharge. eq ^{−1} = 96,484.6 C· mol^{−1} = N_{A}e |

g
| Acceleration due to gravity = 980.665 cm·s^{−2} |

h | Planck’s constant (energy quantum) = 6.626176 × 10 ^{−27} erg·s = 6.626 × 10^{−34} J·s |

η | Viscosity; 1 poise (P) = 1 cm^{−1}· g·s^{−1} = 0.1Pascal·second (Pa·s) |

I | Current, amperes |

J | Joule = Watt·second (W ·s) = ampere·volt· second (A·V·s) = 10 ^{7} erg = 10^{7} cm^{2}g·s^{−2} |

k | Boltzmann constant, 1.380662 × 10^{−23} J. °K^{−1}= R/N^{A}, the gas constant over Avogadro’snumber = 1.37900 × 10 ^{−16} cm^{2}·g·s^{−2}·°K^{−1} |

1, liter | Liter = 1 dm^{3} = 1,000 cm^{3}. Also milliliter (ml)and microliter ( μl) |

M | Mol/l (molarity) |

mol/kg | Mol solute/kg solvent (molality) |

N | Newton = 10^{5} dyn = 10^{5} cm·g·s^{−2} |

N
_{A}
| Avogadro’s number, 6.022045 × 10^{23} mol^{−1},the number of molecules contained in 1 mol |

n_{s}, n_{w} | Number of moles of solute and water |

P | Pressure (= force per unit area), N·m^{−2} or Pa(Pascal). (1 Pa 1 N·m ^{−2} 10 g·cm^{−1}·s^{−2}10 ^{−2} mbar 0.10197 mmH_{2}0 7.5 ×10 ^{−3} mmHg 9.869 × 10^{−6} atm; or 1 atm =101325 Pa = 760 Torr; 1 cmH _{2}0 (at density1 g·cm ^{−3}) = 98.0665 Pa = 981 g·cm^{−1}·s^{−2};1 mmHg = 1.00000014 Torr = 133.322 Pa = 1,333 g·cm ^{−1} · s^{−2} |

ρ
| Density, g·cm^{−3}. Water (3.98°C, 1 atm) =0.999972 g·cm ^{−3}. Mercury (0°C, 1 atm) =13.59508 g·c ^{−3} |

R
| Resistance, electrical (Ω), or electrophysiolog- ical (Ω/cm ^{2}) or vascular (a pressure dividedby a flow) |

R
| Universal gas constant = 8.31441 J.mol^{−1}°K ^{−1} = 8.3144 × 10^{7} cm^{2} ·g·s^{−2} ·mol^{−1} °K^{−1}= 0.082 1· atm· mol ^{−1} °K^{−1} = 0.0623 mmHg·mmol ^{−1}· °K^{−1} = 8.31441 × 10^{−7} erg·mol^{−1}·°K ^{−1} |

RT
| Energy/mol, gas constant × absolute temper- ature; e.g., at 37°C or 310.16°K, RT = 19.34× 10 ^{6} mmHg· cm^{3}· mol^{−1} |

RT/F
| 24.84 mV at 15°C.26.62 mV at 37°C. Values of log _{e},10 RT/F at 15, 20, 25, 30, and 37°Care 57.2, 58.2, 59.2, 60.2, and 61.3 mV |

STP | Standard temperature and pressure (ice point of water, 0°C = 273.16°K; 760 mmHg = 1 atm = 1.01325 × 10 ^{6} dyn·cm^{−2} = 1.013 ×10 ^{5} N· m^{−2}) |

T
| Temperature, absolute, in degrees Kelvin (°K); 0°C = 273.16 °K |

V | Volts; millivolt, mV; microvolt, μV |

_{i}
| Partial molar volume, ml/m01 = (V/n _{i})_{T,p,njj≠i} = change of volume of totalsystem per mole additional solute i, at T, p,and constant presence of other components j, and at the particular concentration n/V._{i}( _{W} is the partial molar volume of water;close to 18 ml/mol for physiological solu- tions) |

Watt | Unit of power, joules per second, J·s^{−1} |

Work | Work is energy × time or force × distance × time, erg·s or J· s or cm ^{2}g·s^{−1} |

Ω | Ohm, unit of electrical resistance; V/I |

The authors greatly appreciate the efforts of Geraldine Crooker in the preparation of this terminology.

1. Bassingthwaighte JB, Chinard FP, Crone C, Lassen NA, Perl W. Definitions and terminology for indicator dilution methods. In: Crone C, Lassen NA, editors. Capillary Permeability. Munksgaard; Copenhagen: 1970. pp. 665–669.

2. Bassingthwaighte JB, Goresky CA. Handbook of Physiology. Sect. 2, The Cardiovascular System. Vol. IV, The Microcirculation. Bethesda, MD: Modeling in the analysis of solute and water exchange in the microvasculature.Am. Physiol. Sot. 1984:549–626. chapt. 13.

3. Gonzalez-Fernandez JM. Theory of the measurement of the dispersion of an indicator in indicator-dilution studies. Circ. Res. 1962;10:409–428. [PubMed]

4. Katchalsky A, Curran PF. Nonequilibrium Thermody-namics in Biophysics. Harvard Univ. Press; Cambridge, MA: 1965.

5. Kedem O, Katchalsky A. Thermodynamic analysis of the permeability of biological membranes to non-electrolytes. Biochim. Biophys. Acta. 1958;27:229–246. [PubMed]

6. Meier P, Zierler KL. On the theory of the indicator-dilution method for measurement of blood flow and volume. J. Appl. Physiol. 1954;6:731–744. [PubMed]

7. Shinnar R, Naor P. Residence time distributions in systems with internal reflux. Chem. Eng. Sci. 1967;22:1369–1381.

8. Silberberg A. The mechanics and thermodynamics of separation flow through porous, molecularly disperse, solid media—the Poiseuille Lecture 1981. Biorheology. 1982;19:111–127. [PubMed]

9. Spiegler KS. Transport process in ionic membranes. Trans. Faraday Sot. 1958;54:1408–1428.

10. Stephenson JL. Theory of the measurement of blood flow by the dilution of an indicator. Bull. Math. Biophys. 1948;10:117–121. [PubMed]

11. WOOD EH. Definitions and symbols for terms commonly used in relation to indicator-dilution curves. Circ. Res. 1962;10:379–380.

12. Zierler KL. Equations for measuring blood flow by external monitoring of radioisotopes. Circ. Res. 1965;16:309–321. [PubMed]

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |