rumenGP implements a collection of nonlinear models for describing cumulative gas production during in vitro rumen fermentation.
This vignette summarizes:
Throughout this vignette:
\[ V(t) \]
represents cumulative gas production at time:
\[ t \]
\[ V(t) = A \left( 1 - b e^{-kt} \right) \]
| Parameter | Description |
|---|---|
| A | Asymptotic gas production |
| b | Integration constant |
| k | Fractional rate constant |
\[ V(t) = VF + b \left( 1-e^{-kt} \right) \]
| Parameter | Description |
|---|---|
| VF | Initial gas volume (intercept) |
| b | Fermentable fraction |
| k | Fractional rate constant |
\[ V(t) = V_f \left( 1-e^{-kt} \right) \]
| Parameter | Description |
|---|---|
| Vf | Asymptotic gas production |
| k | Fractional rate constant |
\[ V(t) = V_f \left( 1-e^{-k(t-\lambda)} \right) \]
| Parameter | Description |
|---|---|
| Vf | Asymptotic gas production |
| k | Fractional rate constant |
| \(\lambda\) | Lag time |
\[ V(t) = A \exp \left[ - \exp \left( \frac{\mu e}{A} (\lambda-t) + 1 \right) \right] \]
| Parameter | Description |
|---|---|
| A | Asymptotic gas production |
| \(\mu\) | Maximum gas production rate |
| \(\lambda\) | Lag time |
\[ V(t) = \frac{A} { 1+\exp \left[ 2+ 4k(\lambda-t) \right] } \]
| Parameter | Description |
|---|---|
| A | Asymptotic gas production |
| k | Fractional rate constant |
| \(\lambda\) | Lag time |
\[ V(t) = A \left[ 1 - \exp \left( -k(t-\lambda) - d \left( \sqrt{t+0.001} - \sqrt{\lambda+0.001} \right) \right) \right] \]
| Parameter | Description |
|---|---|
| A | Asymptotic gas production |
| k | Fractional rate constant |
| d | Shape parameter |
| \(\lambda\) | Lag time |
\[ V(t) = \frac{ A \left( 1-e^{-kt} \right) } { 1+\exp \left[ \ln\left(\frac{1}{d}\right)-kt \right] } \]
| Parameter | Description |
|---|---|
| A | Asymptotic gas production |
| k | Fractional rate constant |
| d | Shape parameter |
\[ V(t) = \frac{ A \left( 1-e^{-k(t-\lambda)} \right) } { 1+\exp \left[ \ln\left(\frac{1}{d}\right) - k(t-\lambda) \right] } \]
| Parameter | Description |
|---|---|
| A | Asymptotic gas production |
| k | Fractional rate constant |
| d | Shape parameter |
| \(\lambda\) | Lag time |
\[ V(t) = A \frac{t^{c}} { t^{c}+K^{c} } \]
| Parameter | Description |
|---|---|
| A | Asymptotic gas production |
| K | Half-time parameter |
| c | Shape parameter |
\[ V(t) = \frac{VF} { 1+\left(\frac{b}{t}\right)^k } \]
| Parameter | Description |
|---|---|
| VF | Asymptotic gas production |
| b | Half-time parameter |
| k | Shape parameter |
\[ V(t) = VF \left[ 1 - \left( 1+(rt)^a \right)^{-p} \right] \]
| Parameter | Description |
|---|---|
| VF | Asymptotic gas production |
| r | Rate parameter |
| a | Shape parameter |
| p | Shape parameter |
\[ a > 1 \]
\[ p > 0 \]
The Burr XII model is a highly flexible sigmoidal model capable of describing a broad range of fermentation profiles.
The parameter \(r\) controls gas-production rate, while \(a\) and \(p\) jointly influence curve shape, asymmetry, and inflection behavior.
Recent comparative studies have identified Burr XII among the strongest-performing models across diverse feed categories when evaluated using information-criterion-based selection methods.
\[ V(t) = VF \left[ 1 + (rt)^{-a} \right]^{-a} \]
| Parameter | Description |
|---|---|
| VF | Asymptotic gas production |
| r | Rate parameter |
| a | Shape parameter |
\[ a > 1 \]
The Inverse Paralogistic model is a flexible sigmoidal model capable of describing a wide range of cumulative gas-production curves.
The parameter \(r\) controls production speed, while \(a\) controls curve shape and steepness.
Recent comparative studies have identified Inverse Paralogistic among the strongest-performing models across diverse feed categories.
\[ V(t) = \frac{V_{1F}} { 1+\exp \left[ 2-4k_1(t-\lambda) \right] } + \frac{V_{2F}} { 1+\exp \left[ 2-4k_2(t-\lambda) \right] } \]
| Parameter | Description |
|---|---|
| \(V_{1F}\) | Gas volume from rapidly fermentable fraction |
| \(V_{2F}\) | Gas volume from slowly fermentable fraction |
| \(k_1\) | Rate constant of rapid fraction |
| \(k_2\) | Rate constant of slow fraction |
| \(\lambda\) | Lag time |
The Groot and generalized Michaelis-Menten models are mathematically equivalent for positive values of time.
Parameter correspondence:
VF = Ab = Kk = cBoth formulations describe the same underlying sigmoidal gas production curve but differ in parameterization.
The Groot model contains the term
\[ \left(\frac{b}{t}\right)^k \]
and is therefore undefined at \(t =
0\). Consequently, observations collected at
Time_h = 0 are excluded when fitting the Groot model.
Researchers may choose either formulation depending on the terminology commonly used in their field.
The Log-logistic formulation
\[ V(t) = VF \frac{(rt)^a} { 1 + (rt)^a } \]
can be rewritten as
\[ V(t) = VF \frac{t^a} { t^a + (1/r)^a } \]
which is mathematically equivalent to both the Groot and generalized Michaelis-Menten models for positive values of time.
Parameter correspondence:
| Groot | Generalized Michaelis-Menten | Log-logistic |
|---|---|---|
| VF | A | VF |
| b | K | 1/r |
| k | c | a |
Therefore, the Groot, Generalized Michaelis-Menten, and Log-logistic models represent mathematically equivalent parameterizations of the same underlying sigmoidal gas production curve.
These formulations differ only in how their parameters are expressed.
Unlike the Groot model, the generalized Michaelis-Menten and Log-logistic formulations are mathematically defined at \(t = 0\) and therefore retain the initial observation during model fitting.
Consequently, parameter estimates and goodness-of-fit statistics may differ slightly among parameterizations when time-zero observations are present.
rumenGP implements all three parameterizations, allowing researchers to choose the formulation that is most familiar within their field while recognizing their underlying mathematical equivalence.
A practical progression is:
Use when:
Use when:
Use when:
Use when:
No single gas-production model should be considered universally superior.
Model performance depends on:
A practical workflow is: