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Publication date: 06/21/2023

Statistical Details for Fit Curve Models

The formulas for the models that are available in the Fit Curve red triangle menu are summarized in Table 14.1.

Table 14.1 Fit Curve Model Formulas

Model

Formula

Polynomials

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Equation shown here

where k is the order of the polynomial. These models can also be fit using the Fit Model and Fit Y by X platforms.

Power Model

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Equation shown here

a = Intercept

b = Slope

c = Power

Available only when both the response values and the x values are nonnegative.

Logistic 2P

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Equation shown here

a = Growth Rate

b = Inflection Point

Available only when all response values are between zero and one.

Logistic 3P

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Equation shown here

a = Growth Rate

b = Inflection Point

c = Asymptote

Logistic 4P

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Equation shown here

a = Growth Rate

b = Inflection Point

c = Lower Asymptote

d = Upper Asymptote

Logistic 4P Rodbard

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Equation shown here

a = Growth Rate

b = Inflection Point

c = Lower Asymptote

d = Upper Asymptote

Available only when the x values are positive.

Logistic 4P Hill

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Equation shown here

a = Growth Rate

b = Inflection Point

c = Lower Asymptote

d = Upper Asymptote

Logistic 5P

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Equation shown here

a = Growth Rate

b = Inflection Point

c = Asymptote 1

d = Asymptote 2

f = Power

Probit 2P

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Equation shown here

a = Growth Rate

b = Inflection Point

Φ = Normal Distribution CDF

Available only when all response values are between zero and one.

Probit 3P

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Equation shown here

a = Growth Rate

b = Inflection Point

c = Asymptote

Φ = Normal Distribution CDF

There is also an asymptote at zero.

Probit 4P

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Equation shown here

a = Growth Rate

b = Inflection Point

c = Lower Asymptote

d = Upper Asymptote

Φ = Normal Distribution CDF

Gompertz 3P

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Equation shown here

a = Asymptote

b = Growth Rate

c = Inflection Point

Gompertz 4P

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Equation shown here

a = Lower Asymptote

b = Upper Asymptote

c = Growth Rate

d = Inflection Point

Weibull Growth

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Equation shown here

a = Upper Asymptote

b = Growth Rate

c = Inflection Point

Available only when both the response values and the x values are nonnegative.

Exponential 2P

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Equation shown here

a = Scale

b = Growth Rate

Exponential 3P

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Equation shown here

a = Asymptote

b = Scale

c = Growth Rate

Biexponential 4P

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Equation shown here

a = Scale 1

b = Decay Rate 1

c = Scale 2

d = Decay Rate 2

Available only when the response values are positive.

Biexponential 5P

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Equation shown here

a = Asymptote

b = Scale 1

c = Decay Rate 1

d = Scale 2

f = Decay Rate 2

Mechanistic Growth

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Equation shown here

a = Asymptote

b = Scale

c = Growth Rate

Hybrid Exponential

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Equation shown here

a = Maximum

b = Scale

c = Rate

d = Power

Available only when both the response values and the x values are nonnegative.

Cell Growth 4P

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Equation shown here

a = Peak value if mortality rate, d, is zero

b = Response at time zero

c = Cell Division Rate

d = Cell Mortality Rate

Available only when the response values are positive and the x values are nonnegative.

Gaussian Peak

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Equation shown here

a = Peak Value

b = Critical Point

c = Growth Rate

ExGaussian Peak (Exponentially Modified Gaussian Peak)

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Equation shown here

a = AUC (area under the curve)

b = Location

c = Scale

d = Lambda

φ = the standard normal pdf

Lorentzian Peak

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Equation shown here

a = Peak Value

b = Growth Rate

c = Critical Point

One Compartment Oral Dose

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Image shown here

Equation shown here

a = Area Under Curve

b = Elimination Rate

c = Absorption Rate

Available only when the response values and the x values are all positive.

Two Compartment IV Bolus Dose

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Image shown here

Equation shown here

α =Equation shown here

β =Equation shown here

a = Initial Concentration

b = Transfer Rate In

c = Transfer Rate Out

d = Elimination Rate

Available only when the response values and the x values are all positive.

Michaelis-Menten

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Equation shown here

a = Maximum Velocity

b = Michaelis Constant

Available only when the response values and the x values are all positive.

Inverse Michaelis-Menten

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Equation shown here

a = Maximum Velocity

b = Michaelis Constant

Available only when the response values and the x values are all positive.

First Order Rate

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Equation shown here

a = Initial Value

b = Rate Constant

Available only if the x values are nonnegative.

First Order with Limits

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Equation shown here

a = Initial Value

b = Limiting Value

c = Rate Constant

Available only if the x values are nonnegative.

First Order with Equilibrium

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Equation shown here

a = Initial Value

b = Forward Rate

c = Backward Rate

Available only if the x values are nonnegative.

Second Order

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Equation shown here

a = Initial Value

b = Rate Constant

Available only if the x values are nonnegative.

Second Order with Two Components

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Equation shown here

a = Initial value

b = Offset

c = Rate Constant

Available only if all x values are nonnegative.

Antoine Equation

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Equation shown here

a = Component Specific Constant

b = Component Specific Constant

c = Component Specific Constant

Available only when the response values are greater than 0.

The vertical asymptote is at 10a.

The horizontal asymptote is at c.

Higuchi

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Equation shown here

a = Release Constant

Available only if all x and y values are nonnegative.

Higuchi with Lag

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Equation shown here

a = Release Constant

b = Time Lag

Available only if all x and y values are nonnegative.

Higuchi with Burst

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Equation shown here

a = Release Constant

b = Burst

Available only if all x and y values are nonnegative.

Hixson-Crowell

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Equation shown here

a = Release Constant

Available only if all x and y values are nonnegative.

Hixson-Crowell with Lag

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Equation shown here

a = Release Constant

Available only if all x and y values are nonnegative.

Korsmeyer-Peppas

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Equation shown here

a = Release Constant

b = Diffusional Exponent

Available only if all x and y values are nonnegative.

Korsmeyer-Peppas with Lag

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Equation shown here

a = Release Constant

b = Diffusional Exponent

c = Time Lag

Available only if all x and y values are nonnegative.

Korsmeyer-Peppas with Burst

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Equation shown here

a = Release Constant

b = Diffusional Exponent

c = Burst

Available only if all x and y values are nonnegative.

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