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Model Equations Reference
In rumenGP: Rumen Gas Production Modeling, Comparison, and Visualization
title: "Model Equations Reference"
output: rmarkdown::html_vignette
vignette: >
%\VignetteIndexEntry{Model Equations Reference}
%\VignetteEngine{knitr::rmarkdown}
\usepackage[utf8]{inputenc}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
Introduction
rumenGP implements a collection of nonlinear models for
describing cumulative gas production during in vitro rumen
fermentation.
This vignette summarizes:
- Model equations
- Parameter definitions
- Biological interpretation
- Advantages
- Limitations
- Typical applications
Throughout this vignette:
[
V(t)
]
represents cumulative gas production at time:
[
t
]
Single-Pool Models
Brody
Equation
[
V(t)
=
A
\left(
1
-
b
e^{-kt}
\right)
]
Parameters
| Parameter | Description |
|------------|------------|
| A | Asymptotic gas production |
| b | Integration constant |
| k | Fractional rate constant |
Advantages
- Simple and robust
- Stable convergence
- Easy interpretation
Limitations
- No lag parameter
- Limited flexibility
Ørskov and McDonald
Equation
[
V(t)
=
VF
+
b
\left(
1-e^{-kt}
\right)
]
Parameters
| Parameter | Description |
|------------|------------|
| VF | Initial gas volume (intercept) |
| b | Fermentable fraction |
| k | Fractional rate constant |
Advantages
- Widely used in ruminant nutrition
- Simple biological interpretation
Limitations
- No explicit lag phase
EXP0
Equation
[
V(t)
=
V_f
\left(
1-e^{-kt}
\right)
]
Parameters
| Parameter | Description |
|------------|------------|
| Vf | Asymptotic gas production |
| k | Fractional rate constant |
Advantages
- Very simple
- Fast convergence
Limitations
- No lag phase
- Limited flexibility
EXPL
Equation
[
V(t)
=
V_f
\left(
1-e^{-k(t-\lambda)}
\right)
]
Parameters
| Parameter | Description |
|------------|------------|
| Vf | Asymptotic gas production |
| k | Fractional rate constant |
| λ | Lag time |
Advantages
- Explicit lag parameter
- Easy interpretation
Limitations
- Less flexible than sigmoidal models
Gompertz
Equation
[
V(t)
=
A
\exp
\left[
-
\exp
\left(
\frac{\mu e}{A}
(\lambda-t)
+
1
\right)
\right]
]
Parameters
| Parameter | Description |
|------------|------------|
| A | Asymptotic gas production |
| μ | Maximum gas production rate |
| λ | Lag time |
Advantages
- Explicit lag and growth-rate parameters
- Excellent flexibility
- Widely used in gas production studies
Limitations
- More complex than exponential models
Logistic
Equation
[
V(t)
=
\frac{A}
{
1+\exp
\left[
2+
4k(\lambda-t)
\right]
}
]
Parameters
| Parameter | Description |
|------------|------------|
| A | Asymptotic gas production |
| k | Fractional rate constant |
| λ | Lag time |
Advantages
- Sigmoidal behavior
- Stable convergence
Limitations
- Assumes symmetric sigmoid shape
Mitscherlich
Equation
[
V(t)
=
A
\left[
1
-
\exp
\left(
-k(t-\lambda)
-
d
\left(
\sqrt{t+0.001}
-
\sqrt{\lambda+0.001}
\right)
\right)
\right]
]
Parameters
| Parameter | Description |
|------------|------------|
| A | Asymptotic gas production |
| k | Fractional rate constant |
| d | Shape parameter |
| λ | Lag time |
Advantages
- Flexible curve shape
- Explicit lag phase
Limitations
- More parameters
- Increased parameter correlation
LE0 (Logistic-Exponential Without Lag)
Equation
[
V(t)
=
\frac{
A
\left(
1-e^{-kt}
\right)
}
{
1+\exp
\left[
\ln\left(\frac{1}{d}\right)-kt
\right]
}
]
Parameters
| Parameter | Description |
|------------|------------|
| A | Asymptotic gas production |
| k | Fractional rate constant |
| d | Shape parameter |
Advantages
- Flexible shape
- No lag parameter required
Limitations
- More complex than simple exponential models
LEL (Logistic-Exponential With Lag)
Equation
[
V(t)
=
\frac{
A
\left(
1-e^{-k(t-\lambda)}
\right)
}
{
1+\exp
\left[
\ln\left(\frac{1}{d}\right)
-
k(t-\lambda)
\right]
}
]
Parameters
| Parameter | Description |
|------------|------------|
| A | Asymptotic gas production |
| k | Fractional rate constant |
| d | Shape parameter |
| λ | Lag time |
Advantages
- Flexible shape
- Explicit lag phase
Limitations
- Additional complexity may affect convergence
Michaelis-Menten
Equation
[
V(t)
=
A
\frac{t^{c}}
{
t^{c}+K^{c}
}
]
Parameters
| Parameter | Description |
|------------|------------|
| A | Asymptotic gas production |
| K | Half-time parameter |
| c | Shape parameter |
Advantages
- Flexible
- Strong biological interpretation
Limitations
- Shape parameter may be difficult to interpret biologically
Groot
Equation
[
V(t)
=
\frac{VF}
{
1+\left(\frac{b}{t}\right)^k
}
]
Parameters
| Parameter | Description |
|------------|------------|
| VF | Asymptotic gas production |
| b | Half-time parameter |
| k | Shape parameter |
Advantages
- Excellent flexibility
- Widely used in rumen gas production studies
Limitations
- Requires positive incubation times
Multi-Pool Models
Dual Logistic
Equation
[
V(t)
=
\frac{V_{1F}}
{
1+\exp
\left[
2-4k_1(t-\lambda)
\right]
}
+
\frac{V_{2F}}
{
1+\exp
\left[
2-4k_2(t-\lambda)
\right]
}
]
Parameters
| Parameter | Description |
|------------|------------|
| V1F | Gas volume from rapidly fermentable fraction |
| V2F | Gas volume from slowly fermentable fraction |
| k1 | Rate constant of rapid fraction |
| k2 | Rate constant of slow fraction |
| λ | Lag time |
Advantages
- Represents multiple fermentation pools
- Biologically meaningful decomposition
Limitations
- More parameters
- Greater convergence challenges
Model Equivalence
Groot and Michaelis-Menten
The Groot and generalized Michaelis-Menten models are
mathematically equivalent.
Parameter correspondence:
[
VF = A
]
[
b = K
]
[
k = c
]
Both formulations produce identical fitted values and
model diagnostics when convergence is achieved.
Researchers may select either model according to the
terminology commonly used in their field.
Choosing a Model
A practical progression is:
Simple Models
- EXP0
- Brody
- Ørskov and McDonald
Use when:
- Data show monotonic behavior
- Lag is negligible
- Simplicity is preferred
Lag Models
- EXPL
- Logistic
- Gompertz
- Mitscherlich
Use when:
- A lag phase is biologically expected
- Initial microbial adaptation is important
Flexible Sigmoidal Models
- LE0
- LEL
- Groot
- Michaelis-Menten
Use when:
- Fermentation profiles display sigmoidal behavior
- Greater flexibility is needed
Multi-Pool Models
- Dual Logistic
Use when:
- Fast and slow fermenting fractions are expected
- Substrate heterogeneity is important
Custom Models
Researchers can also define their own equations using:
fit_custom()
See:
vignette("custom-models")
for additional details.
Summary
rumenGP provides a diverse collection of nonlinear
kinetic models ranging from simple exponential
equations to flexible multi-pool formulations.
Model choice should be guided by:
- Biological plausibility
- Goodness of fit
- Parameter interpretability
- Convergence stability
- Research objectives
Researchers are encouraged to compare multiple models
before selecting a final representation of fermentation
kinetics.
Try the rumenGP package in your browser
Any scripts or data that you put into this service are public.
rumenGP documentation built on Oct. 2, 2026, 5:09 p.m.
R Package Documentation
Browse R Packages
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title: "Model Equations Reference" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Model Equations Reference} %\VignetteEngine{knitr::rmarkdown} \usepackage[utf8]{inputenc}
knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
Introduction
rumenGP implements a collection of nonlinear models for describing cumulative gas production during in vitro rumen fermentation.
This vignette summarizes:
- Model equations
- Parameter definitions
- Biological interpretation
- Advantages
- Limitations
- Typical applications
Throughout this vignette:
[ V(t) ]
represents cumulative gas production at time:
[ t ]
Single-Pool Models
Brody
Equation
[ V(t) = A \left( 1 - b e^{-kt} \right) ]
Parameters
| Parameter | Description | |------------|------------| | A | Asymptotic gas production | | b | Integration constant | | k | Fractional rate constant |
Advantages
- Simple and robust
- Stable convergence
- Easy interpretation
Limitations
- No lag parameter
- Limited flexibility
Ørskov and McDonald
Equation
[ V(t) = VF + b \left( 1-e^{-kt} \right) ]
Parameters
| Parameter | Description | |------------|------------| | VF | Initial gas volume (intercept) | | b | Fermentable fraction | | k | Fractional rate constant |
Advantages
- Widely used in ruminant nutrition
- Simple biological interpretation
Limitations
- No explicit lag phase
EXP0
Equation
[ V(t) = V_f \left( 1-e^{-kt} \right) ]
Parameters
| Parameter | Description | |------------|------------| | Vf | Asymptotic gas production | | k | Fractional rate constant |
Advantages
- Very simple
- Fast convergence
Limitations
- No lag phase
- Limited flexibility
EXPL
Equation
[ V(t) = V_f \left( 1-e^{-k(t-\lambda)} \right) ]
Parameters
| Parameter | Description | |------------|------------| | Vf | Asymptotic gas production | | k | Fractional rate constant | | λ | Lag time |
Advantages
- Explicit lag parameter
- Easy interpretation
Limitations
- Less flexible than sigmoidal models
Gompertz
Equation
[ V(t) = A \exp \left[ - \exp \left( \frac{\mu e}{A} (\lambda-t) + 1 \right) \right] ]
Parameters
| Parameter | Description | |------------|------------| | A | Asymptotic gas production | | μ | Maximum gas production rate | | λ | Lag time |
Advantages
- Explicit lag and growth-rate parameters
- Excellent flexibility
- Widely used in gas production studies
Limitations
- More complex than exponential models
Logistic
Equation
[ V(t) = \frac{A} { 1+\exp \left[ 2+ 4k(\lambda-t) \right] } ]
Parameters
| Parameter | Description | |------------|------------| | A | Asymptotic gas production | | k | Fractional rate constant | | λ | Lag time |
Advantages
- Sigmoidal behavior
- Stable convergence
Limitations
- Assumes symmetric sigmoid shape
Mitscherlich
Equation
[ V(t) = A \left[ 1 - \exp \left( -k(t-\lambda) - d \left( \sqrt{t+0.001} - \sqrt{\lambda+0.001} \right) \right) \right] ]
Parameters
| Parameter | Description | |------------|------------| | A | Asymptotic gas production | | k | Fractional rate constant | | d | Shape parameter | | λ | Lag time |
Advantages
- Flexible curve shape
- Explicit lag phase
Limitations
- More parameters
- Increased parameter correlation
LE0 (Logistic-Exponential Without Lag)
Equation
[ V(t) = \frac{ A \left( 1-e^{-kt} \right) } { 1+\exp \left[ \ln\left(\frac{1}{d}\right)-kt \right] } ]
Parameters
| Parameter | Description | |------------|------------| | A | Asymptotic gas production | | k | Fractional rate constant | | d | Shape parameter |
Advantages
- Flexible shape
- No lag parameter required
Limitations
- More complex than simple exponential models
LEL (Logistic-Exponential With Lag)
Equation
[ V(t) = \frac{ A \left( 1-e^{-k(t-\lambda)} \right) } { 1+\exp \left[ \ln\left(\frac{1}{d}\right) - k(t-\lambda) \right] } ]
Parameters
| Parameter | Description | |------------|------------| | A | Asymptotic gas production | | k | Fractional rate constant | | d | Shape parameter | | λ | Lag time |
Advantages
- Flexible shape
- Explicit lag phase
Limitations
- Additional complexity may affect convergence
Michaelis-Menten
Equation
[ V(t) = A \frac{t^{c}} { t^{c}+K^{c} } ]
Parameters
| Parameter | Description | |------------|------------| | A | Asymptotic gas production | | K | Half-time parameter | | c | Shape parameter |
Advantages
- Flexible
- Strong biological interpretation
Limitations
- Shape parameter may be difficult to interpret biologically
Groot
Equation
[ V(t) = \frac{VF} { 1+\left(\frac{b}{t}\right)^k } ]
Parameters
| Parameter | Description | |------------|------------| | VF | Asymptotic gas production | | b | Half-time parameter | | k | Shape parameter |
Advantages
- Excellent flexibility
- Widely used in rumen gas production studies
Limitations
- Requires positive incubation times
Multi-Pool Models
Dual Logistic
Equation
[ V(t) = \frac{V_{1F}} { 1+\exp \left[ 2-4k_1(t-\lambda) \right] } + \frac{V_{2F}} { 1+\exp \left[ 2-4k_2(t-\lambda) \right] } ]
Parameters
| Parameter | Description | |------------|------------| | V1F | Gas volume from rapidly fermentable fraction | | V2F | Gas volume from slowly fermentable fraction | | k1 | Rate constant of rapid fraction | | k2 | Rate constant of slow fraction | | λ | Lag time |
Advantages
- Represents multiple fermentation pools
- Biologically meaningful decomposition
Limitations
- More parameters
- Greater convergence challenges
Model Equivalence
Groot and Michaelis-Menten
The Groot and generalized Michaelis-Menten models are mathematically equivalent.
Parameter correspondence:
[ VF = A ]
[ b = K ]
[ k = c ]
Both formulations produce identical fitted values and model diagnostics when convergence is achieved.
Researchers may select either model according to the terminology commonly used in their field.
Choosing a Model
A practical progression is:
Simple Models
- EXP0
- Brody
- Ørskov and McDonald
Use when:
- Data show monotonic behavior
- Lag is negligible
- Simplicity is preferred
Lag Models
- EXPL
- Logistic
- Gompertz
- Mitscherlich
Use when:
- A lag phase is biologically expected
- Initial microbial adaptation is important
Flexible Sigmoidal Models
- LE0
- LEL
- Groot
- Michaelis-Menten
Use when:
- Fermentation profiles display sigmoidal behavior
- Greater flexibility is needed
Multi-Pool Models
- Dual Logistic
Use when:
- Fast and slow fermenting fractions are expected
- Substrate heterogeneity is important
Custom Models
Researchers can also define their own equations using:
fit_custom()
See:
vignette("custom-models")
for additional details.
Summary
rumenGP provides a diverse collection of nonlinear kinetic models ranging from simple exponential equations to flexible multi-pool formulations.
Model choice should be guided by:
- Biological plausibility
- Goodness of fit
- Parameter interpretability
- Convergence stability
- Research objectives
Researchers are encouraged to compare multiple models before selecting a final representation of fermentation kinetics.
Try the rumenGP package in your browser
Any scripts or data that you put into this service are public.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
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