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 |
Several gas-production models commonly used in rumen fermentation research and nonlinear modeling are mathematically equivalent.
\[ V(t) = \frac{VF} { 1+\left(\frac{b}{t}\right)^k } \]
\[ V(t) = A \frac{t^{c}} { t^{c}+K^{c} } \]
\[ V(t) = VF \frac{(rt)^a} { 1+(rt)^a } \]
The Log-logistic formulation can be rewritten as:
\[ V(t) = VF \frac{t^a} { t^a+(1/r)^a } \]
which is identical to the generalized Michaelis-Menten formulation.
| Groot | Generalized Michaelis-Menten | Log-logistic |
|---|---|---|
| VF | A | VF |
| b | K | 1/r |
| k | c | a |
Therefore:
\[ \text{Groot} \equiv \text{Generalized Michaelis-Menten} \equiv \text{Log-logistic} \]
These formulations produce identical:
when equivalent parameter transformations are used.
Researchers may therefore choose the parameterization most commonly used within their field while obtaining identical fitted curves.
For this reason, rumenGP does not currently implement a separate Log-logistic fitting routine. The Log-logistic formulation is already represented through the existing Groot and generalized Michaelis-Menten parameterizations.
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: