--- 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 | | \(\lambda\) | 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 | | \(\mu\) | Maximum gas production rate | | \(\lambda\) | 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 | | \(\lambda\) | 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 | | \(\lambda\) | 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 | | \(\lambda\) | Lag time | ### Advantages - Flexible shape - Explicit lag phase ### Limitations - Additional complexity may affect convergence --- ## Generalized 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 sigmoidal behavior - Strong biological interpretation - Widely applicable across fermentation studies ### 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 --- ## Burr XII ### Equation \[ V(t) = VF \left[ 1 - \left( 1+(rt)^a \right)^{-p} \right] \] ### Parameters | Parameter | Description | |------------|------------| | VF | Asymptotic gas production | | r | Rate parameter | | a | Shape parameter | | p | Shape parameter | ### Parameter Constraints \[ a > 1 \] \[ p > 0 \] ### Interpretation 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. ### Advantages - Highly flexible curve shape - Accommodates diverse fermentation profiles - Often provides excellent goodness of fit - Useful for broad model comparison studies ### Limitations - More parameters than simpler models - Greater risk of overfitting - Parameter interpretation may be less intuitive ### Typical Applications - Diverse feed datasets - Comparative model-selection studies - Sigmoidal fermentation profiles ### Literature Context Recent comparative studies have identified Burr XII among the strongest-performing models across diverse feed categories when evaluated using information-criterion-based selection methods. --- ## Inverse Paralogistic ### Equation \[ V(t) = VF \left[ 1 + (rt)^{-a} \right]^{-a} \] ### Parameters | Parameter | Description | |------------|------------| | VF | Asymptotic gas production | | r | Rate parameter | | a | Shape parameter | ### Parameter Constraints \[ a > 1 \] ### Interpretation 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. ### Advantages - Flexible sigmoidal behavior - Relatively simple parameterization - Performs well across diverse gas-production datasets ### Limitations - Less common in rumen literature - Shape parameter can be difficult to interpret biologically - Requires strictly positive incubation times ### Typical Applications - Flexible nonlinear gas-production modeling - Comparative model-selection studies - Curves exhibiting sigmoidal behavior ### Literature Context Recent comparative studies have identified Inverse Paralogistic among the strongest-performing models across diverse feed categories. --- # 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 | |------------|------------| | \(V_{1F}\) | 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 | ### Advantages - Represents multiple fermentation pools - Biologically meaningful decomposition ### Limitations - More parameters - Greater convergence challenges --- ## Model Equivalence ### Groot and Generalized Michaelis-Menten The Groot and generalized Michaelis-Menten models are mathematically equivalent for positive values of time. Parameter correspondence: - `VF = A` - `b = K` - `k = c` Both 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. ### Groot, Generalized Michaelis-Menten, and Log-logistic 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. --- # 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 - Generalized Michaelis-Menten - Burr XII - Inverse Paralogistic - Log-logistic Use when: - Fermentation profiles display sigmoidal behavior - Greater flexibility is needed - Multiple biologically plausible models are being compared - Information-criterion-based selection (AIC or BIC) is desired --- ## 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. --- # Model Selection Guidance No single gas-production model should be considered universally superior. Model performance depends on: - Feed type - Experimental design - Data quality - Fermentation profile characteristics - Model-selection criteria A practical workflow is: 1. Fit several biologically plausible candidate models. 2. Verify convergence. 3. Examine fitted curves visually. 4. Examine residual patterns. 5. Compare RMSE. 6. Compare AIC and BIC. 7. Evaluate parameter plausibility. 8. Select the model most appropriate