Equation Standards

Where appropriate, for consistency of presentation and to facilitate the understanding of the scientific principles detailed in this guide, the symbols/letters used to represent variables in many equations in the GastroPlus® Scientific Principles Guide have been changed from those referenced in the literature to the symbols /letters that adhere to Simulations Plus’s internal documentation standards for equations in GastroPlus®. This appendix details these internal documentation standards. See:

• Internal Documentation Standards for GastroPlus® Equations

Internal Documentation Standards for GastroPlus® Equations

The standards that are detailed here are ordered alphabetically by title/definition.

Compartments/Enzyme and Transporters/Sub-Layers

• "j" for compartment as a subscript.

• "i" for enzyme/transporter as a superscript.

• "k" for layer, sub-layer, sub-compartment as a subscript when used in conjunction with “j”.

• For Michelis-Menten, standardize to the following:

and

Clearance rate

See Rate/Rate constant.

Concentration subscripts

• “t” for total when used in conjunction with “C” for concentration and “V” for volume.

• “u” for unbound.

Concentration

“C” coupled with “t” or “u” has the following formats: C_n,u, C_n,t, or C^x_n,u, or C^x_n,t where:

• “n” is one of the following:

  • An abbreviation such as sal = saliva, depot = formulation depot, vasc = the plasma compartment in the tissue, etc.

  • Set to an absolute value of 1 for the first layer or sub-layer.

  • Remains “n” or “k” to represent 2 or greater for subsequent layers or sub-layers, respectively.

• “x” = “j” for the j^th compartment, or one of the following:

  • epi = epithelial cell layer

  • STR = stratum corneum

  • SBM = sebum

  • HL = hair lipid

  • HC = hair core

  • VE = viable epidermis

  • DE = dermis

• Information is read from right to left in a bottom to top order. For example, the concentration of unbound drug in the top-most skin compartment sub-layer is shown as .

Coefficients (Other than Diffusion Coefficient)

“K” represents a coefficient with the following caveats:

• K_x represents the standard format for a coefficient, where “x” is an abbreviation for the type of coefficient, such as K_mt for the mass transfer coefficient.

• Kp_x/x is the standard for the majority of partition coefficients, where:

  • “x” represents the two mediums involved in the coefficient such as tissue/plasma.

  • The mediums are either spelled in full, or a standard abbreviation is used, such as ext/water for the extracellular fluid to plasma partition coefficient.

• The octanol/water partition coefficient is to remain as logP_oct/w.

• The vegetable/octanol partition coefficient is to remain logP_vo/w.

D/Dose

• “d” for dissolved only.

• Spell out “Dose.” Do not use “d.”

• Other “d” variables to be defined as appropriate: degrad for degradation, death for cell death, drug for drug, and so on.

Diffusion coefficient/Diffusivity

“D” represents a diffusion coefficient or diffusivity. The format is D_x with the following caveats:

• "x" indicates the medium through which transport is taking place such as w = water, sol = solvent, air = air, and so on.

• The octanol/water distribution coefficient is to remain as logD_oct/w.

• The vegetable oil/water distribution coefficient is to remain as logD_vo/w.

Equation Direction

• If an equation represents a process that goes in one direction, use a single-headed arrow ---->.

• If an equation represents a process that goes in both directions, such as systemic absorption in the TCAT module, then use a double-headed arrow. < --- >.

f/fetal/fraction/free

• With one exception, use a lowercase “f” to represent a fraction. Do not use an uppercase “F.”

• The exception to using an uppercase “F” for fraction: Use “F” to represent the fraction of drug with specific charges defined for the Lukacova combined equation for Kp calculation.

• Do not use a lowercase “f” to represent “free,” and do not use “free” spelled out. Always use “u” for unbound instead.

• In the PEAR section of the PBPK chapter: “f” represents “fetal” with the pregnancy PEAR physiology for efficiency and clarity of reading.

Flow rate

“Q” = flow rate, with the following subscripts:

• hep for hepatic.

• b for local tissue blood flow.

• lymph for lymph flow.

• urine for urine flow.

Fluids/Spaces

• (int) = the intracellular space of a compartment such as the intracellular space of the tissue compartment for a permeability-limited (tissue(int)).

• (ext) = the extracellular space of a compartment such as the extracellular space of the tissue compartment for a permeability-limited (tissue(ext)).

Permeability-limited tissue is simply denoted as tissue without an int or ext clarifier, for example, C_tissue,u.

• extw = extracellular water

• intw = intracellular water

f_up/f_ut/f values

• f_up is the Simulations Plus standard for fraction of unbound drug in plasma.

• f_ut is the Simulations Plus standard for fraction of unbound drug in perfusion-limited tissue.

• f_u(int) is the Simulations Plus standard for fraction of unbound drug in the intracellular space of permeability-limited tissue.

• f_u(ext) is the Simulations Plus standard for fraction of unbound drug in the extracellular space of permeability-limited tissue.

h/Thickness

h = The thickness of a layer, where:

• A single entity is a subscript. For example, h_DE is the thickness of the dermis.

• A specific area of an entity is a subscript and the entity is a superscript. Information is read from right to left in a bottom to top order. For example,  is the thickness of the first sub-layer of the dermis.

Ions

Always lowercase, where:

• n = neutral

• a = anionic

• c = cationic

Lipids/Phospholipids

• nl = neutral lipids.

• apl = acidic phospholipids

• npl = neutral phospholipids

M/m

• Use only an upper case “M” for mass and do not spell out the word “mass.”

• Spell out “muscle” for muscle compartment. Do not use “m” or any abbreviation such as musc, mus, and so on.

• “m” for maternal with the pregnancy PEAR physiology for efficiency and clarity of reading.

Molecular weight

MW = drug molecular rate.

R/r and Ratio

• “r” = radius.

• “R” = Gas constant

• “R_x” represents a ratio, where “x” is a standard abbreviation such as R_b:p the blood/plasma drug concentration ratio.

Do not use any version of r/R for rate. For rate standards, see Rate/Rate constant.

Rate/Rate constant

Use “k” for rate/rate constant with different standard subscripts to define the rates, such as k_diss for dissolution rate, k_degrad for degradation rate, k_evap for evaporation rate, k_metab for metabolic rate, and so on.

• Systemic absorption rate is always k_sys,j.

• The one exception to this use of k for rate: The fixed (linear) intrinsic clearance rate in L/hr is CL_metab and CL(variable) is used as the standard for all variations of clearance rates, for example, CL_filt for the kidney filtration clearance rate.

• If a rate variable and a dissociation rate constant variable are used in the same equation, then use an uppercase “K” for the rate and the lowercase “k” for the rate constant.

Surface Area

Surface Area = SA_x, where::

• “x” is the surface area of the indicated entity, and: “x” = one of the following:

  • epi = epithelial cell layer

  • STR = stratum corneum

  • SBM = sebum

  • HL = hair lipid

  • HC = hair core

  • VE = viable epidermis

  • DE = dermis

• If the surface area is a radial or axial surface area, then use rad for radial and ax for axial, which is superscripted, and “x” is as before.

Time/tissue/total

• Unless noted otherwise, “t” is used for time, and it is neither superscripted nor subscripted. Instead, it is used either in the denominator of a derivative (dM/dt), or inline with a variable such as C(t) for concentration at time “t”, or V(t) for volume at time “t.”

• Do not confuse “t” for time with “t” for transit or use for tissue.

  • “t” for transit rate is to be used as explicitly defined in the variable table that accompanies an equation such as k_t,lymph for lymphatic transit rate.

  • With the exception of f_ut, do NOT use “t” for tissue. Spell out tissue. See fup/fut/f values.

• The one exception to using “t” for total is with the pregnancy PEAR physiology. For clarity and delineation in equation variables, where a “total” designation was used more than one time for a variable, “t” is used for total and can be either a subscript or a superscript and is used solely with either a maternal or fetal variable. For example, for the total mass of drug in the maternal placenta: ,

• “total” is used to represent across both fetal and maternal tissues with the pregnancy PEAR physiology. For example, the total concentration of drug in the fetal and maternal tissues is C_{tissue(total),t}.

V/Volume

• V = volume, where:

  • A single entity is a subscript. For example, VE_DE is the volume of the dermis.

  • A specific area of an entity is a subscript and the entity is a superscript. Information is read from right to left in a bottom to top order. For example, is the volume of the first sub-layer of the dermis.