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Model Equations Reference

Source: vignettes/model-equations.Rmd
model-equations.Rmd

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) V(t)

represents cumulative gas production at time:

t t

Single-Pool Models

Brody

Equation

V(t)=A(1−be−kt) 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(1−e−kt) 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)=Vf(1−e−kt) 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)=Vf(1−e−k(t−λ)) 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)=Aexp⁡[−exp(μeA(λ−t)+1)] 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)=A1+exp⁡[2+4k(λ−t)] 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[1−exp(−k(t−λ)−d(t+0.001−λ+0.001))] 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)=A(1−e−kt)1+exp⁡[ln(1d)−kt] 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)=A(1−e−k(t−λ))1+exp⁡[ln(1d)−k(t−λ)] 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)=Atctc+Kc 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)=VF1+(bt)k 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)=V1F1+exp⁡[2−4k1(t−λ)]+V2F1+exp⁡[2−4k2(t−λ)] 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 VF = A

b=K b = K

k=c 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:

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.