--- title: "Model Equations Reference" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Model Equations Reference} %\VignetteEngine{knitr::rmarkdown} \usepackage[utf8]{inputenc} --- ```{r, include = FALSE} 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: ```r fit_custom() ``` See: ```r 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.