| Title: | Nonlinear Regression in Predictive Microbiology |
|---|---|
| Description: | Data sets and nonlinear regression models dedicated to predictive microbiology. |
| Authors: | Florent Baty [aut], Marie-Laure Delignette-Muller [aut], Aurelie Siberchicot [aut, cre] |
| Maintainer: | Aurelie Siberchicot <[email protected]> |
| License: | GPL-3 |
| Version: | 1.0-0 |
| Built: | 2024-09-17 05:12:22 UTC |
| Source: | https://github.com/lbbe-software/nlsmicrobio |
Kinetics of simultaneous growth of Escherichia coli O157:H7 and ground beef background microflora in enrichment broth
data(competition1) data(competition2)data(competition1) data(competition2)
Data frames with 3 columns (t: time, flora : 1 for the first flora and 2 for the second one, LOG10N: decimal logarithm of bacterial density)
Two of the kinetics used in Vimont et al. (2006)
Vimont A, Vernozy-Rozand C, Montet MP, Lazizzera C, Bavai C and Delignette-Muller ML (2006) Modeling and predicting the simultaneous growth of Escherichia coli O157:H7 and ground beef background microflora in various enrichment protocols. Applied and Environmental Microbiology 72, 261-268.
data(competition1) data(competition2) def.par <- par(no.readonly = TRUE) par(mfrow = c(1,2)) twocolors <- c("red","blue") plot(competition1$t,competition1$LOG10N,col=twocolors[competition1$flora]) plot(competition2$t,competition2$LOG10N,col=twocolors[competition2$flora]) par(def.par)data(competition1) data(competition2) def.par <- par(no.readonly = TRUE) par(mfrow = c(1,2)) twocolors <- c("red","blue") plot(competition1$t,competition1$LOG10N,col=twocolors[competition1$flora]) plot(competition2$t,competition2$LOG10N,col=twocolors[competition2$flora]) par(def.par)
Formulas of primary growth models used in predictive microbiology to model the simultaneous growth of two competitive bacterial flora assuming a Jameson effect
jameson_buchanan jameson_baranyi jameson_without_lagjameson_buchanan jameson_baranyi jameson_without_lag
These models describe the simultaneous evolution of the decimal logarithm of
the microbial counts of two flora (LOG10N) as a function of the time (t) and of the flora (flora)
coded as 1 for counts of flora 1 and 2 for counts of flora 2. These three
models assume independent lag and growth parameters for flora 1 and 2, except for
the saturation which is supposed to be governed by the Jameson effect and modelled
by a common parameter (tmax) which represents the time at which both flora stop to
multiply. Modelling the simultaneous saturation by this way enables the model
to be fitted by nls, as an analytical form of the model is available.
jameson_buchanan is based on the model of Buchanan et al. (1997) for lag phase modelling
and is characterized by seven parameters
(LOG10N0_1, mumax_1, lag_1, LOG10N0_2, mumax_2, lag_2 and the common saturation
time tmax). This model was described and used in Vimont et al. (2006).jameson_baranyi is based on the model of Baranyi and Roberts (1994) for lag phase modelling
and is characterized by seven parameters
(LOG10N0_1, mumax_1, lag_1, LOG10N0_2, mumax_2, lag_2 and the common saturation
time tmax)jameson_without_lag is based on the exponential model without lag phase
and is thus characterized by five parameters
(LOG10N0_1, mumax_1, LOG10N0_2, mumax_2 and the common saturation time tmax)
A formula
Florent Baty, Marie-Laure Delignette-Muller
Baranyi J and Roberts, TA (1994) A dynamic approach to predicting bacterial growth in food, International Journal of Food Microbiology, 23, 277-294.
Buchanan RL, Whiting RC, Damert WC (1997) When is simple good enough: a comparison of the Gompertz, Baranyi, and three-phase linear models for fitting bacterial growth curves. Food Microbiology, 14, 313-326.
Vimont A, Vernozy-Rozand C, Montet MP, Lazizzera C, Bavai C and Delignette-Muller ML (2006) Modeling and predicting the simultaneous growth of Escherichia coli O157:H7 and ground beef background microflora in various enrichment protocols.
Applied and Environmental Microbiology 72, 261-268.
options(digits = 3) ### Example 1: fit of model jameson_buchanan data(competition1) nls1 <- nls(jameson_buchanan, competition1, list(lag_1 = 2, mumax_1 = 1, LOG10N0_1 = 1, tmax = 12, lag_2 = 2, mumax_2 = 1, LOG10N0_2 = 4)) overview(nls1) # Plot of theoretical curves with data twocolors <- c("red","blue") npoints <- 100 seq.t <- seq(0,max(competition1$t),length.out=npoints) prednls1.1 <- predict(nls1,data.frame(t=seq.t,flora=rep(1,npoints))) prednls1.2 <- predict(nls1,data.frame(t=seq.t,flora=rep(2,npoints))) plot(competition1$t,competition1$LOG10N,col=twocolors[competition1$flora],xlab="t",ylab="LOG10N") lines(seq.t,prednls1.1,col=twocolors[1]) lines(seq.t,prednls1.2,col=twocolors[2]) ### Example 2 : fit of model jameson_baranyi data(competition1) nls2 <- nls(jameson_baranyi, competition1, list(lag_1 = 2, mumax_1 = 1, LOG10N0_1 = 1, tmax = 12, lag_2 = 2, mumax_2 = 1, LOG10N0_2 = 4)) overview(nls2) plotfit(nls2) # Plot of theoretical curves with data twocolors <- c("red","blue") npoints <- 100 seq.t <- seq(0,max(competition1$t),length.out=npoints) prednls2.1 <- predict(nls2,data.frame(t=seq.t,flora=rep(1,npoints))) prednls2.2 <- predict(nls2,data.frame(t=seq.t,flora=rep(2,npoints))) plot(competition1$t,competition1$LOG10N,col=twocolors[competition1$flora],xlab="t",ylab="LOG10N") lines(seq.t,prednls2.1,col=twocolors[1]) lines(seq.t,prednls2.2,col=twocolors[2]) ### Example 3: fit of model jameson_without_lag data(competition2) nls3 <- nls(jameson_without_lag, competition2, list(mumax_1 = 1, LOG10N0_1 = 1, tmax = 12, mumax_2 = 1, LOG10N0_2 = 4)) overview(nls3) plotfit(nls3) # Plot of theoretical curves with data twocolors <- c("red","blue") npoints <- 100 seq.t <- seq(0,max(competition2$t),length.out=npoints) prednls3.1 <- predict(nls3,data.frame(t=seq.t,flora=rep(1,npoints))) prednls3.2 <- predict(nls3,data.frame(t=seq.t,flora=rep(2,npoints))) plot(competition2$t,competition2$LOG10N,col=twocolors[competition2$flora],xlab="t",ylab="LOG10N") lines(seq.t,prednls3.1,col=twocolors[1]) lines(seq.t,prednls3.2,col=twocolors[2])options(digits = 3) ### Example 1: fit of model jameson_buchanan data(competition1) nls1 <- nls(jameson_buchanan, competition1, list(lag_1 = 2, mumax_1 = 1, LOG10N0_1 = 1, tmax = 12, lag_2 = 2, mumax_2 = 1, LOG10N0_2 = 4)) overview(nls1) # Plot of theoretical curves with data twocolors <- c("red","blue") npoints <- 100 seq.t <- seq(0,max(competition1$t),length.out=npoints) prednls1.1 <- predict(nls1,data.frame(t=seq.t,flora=rep(1,npoints))) prednls1.2 <- predict(nls1,data.frame(t=seq.t,flora=rep(2,npoints))) plot(competition1$t,competition1$LOG10N,col=twocolors[competition1$flora],xlab="t",ylab="LOG10N") lines(seq.t,prednls1.1,col=twocolors[1]) lines(seq.t,prednls1.2,col=twocolors[2]) ### Example 2 : fit of model jameson_baranyi data(competition1) nls2 <- nls(jameson_baranyi, competition1, list(lag_1 = 2, mumax_1 = 1, LOG10N0_1 = 1, tmax = 12, lag_2 = 2, mumax_2 = 1, LOG10N0_2 = 4)) overview(nls2) plotfit(nls2) # Plot of theoretical curves with data twocolors <- c("red","blue") npoints <- 100 seq.t <- seq(0,max(competition1$t),length.out=npoints) prednls2.1 <- predict(nls2,data.frame(t=seq.t,flora=rep(1,npoints))) prednls2.2 <- predict(nls2,data.frame(t=seq.t,flora=rep(2,npoints))) plot(competition1$t,competition1$LOG10N,col=twocolors[competition1$flora],xlab="t",ylab="LOG10N") lines(seq.t,prednls2.1,col=twocolors[1]) lines(seq.t,prednls2.2,col=twocolors[2]) ### Example 3: fit of model jameson_without_lag data(competition2) nls3 <- nls(jameson_without_lag, competition2, list(mumax_1 = 1, LOG10N0_1 = 1, tmax = 12, mumax_2 = 1, LOG10N0_2 = 4)) overview(nls3) plotfit(nls3) # Plot of theoretical curves with data twocolors <- c("red","blue") npoints <- 100 seq.t <- seq(0,max(competition2$t),length.out=npoints) prednls3.1 <- predict(nls3,data.frame(t=seq.t,flora=rep(1,npoints))) prednls3.2 <- predict(nls3,data.frame(t=seq.t,flora=rep(2,npoints))) plot(competition2$t,competition2$LOG10N,col=twocolors[competition2$flora],xlab="t",ylab="LOG10N") lines(seq.t,prednls3.1,col=twocolors[1]) lines(seq.t,prednls3.2,col=twocolors[2])
Bacterial kinetics data sets
data(growthcurve1) data(growthcurve2) data(growthcurve3) data(growthcurve4)data(growthcurve1) data(growthcurve2) data(growthcurve3) data(growthcurve4)
Data frames with 2 columns (t: time, LOG10N: decimal logarithm of bacterial density)
Data obtained by Florent Baty and Marie-Laure Delignette-Muller
data(growthcurve1) data(growthcurve2) data(growthcurve3) data(growthcurve4) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,2)) plot(growthcurve1) plot(growthcurve2) plot(growthcurve3) plot(growthcurve4) par(def.par)data(growthcurve1) data(growthcurve2) data(growthcurve3) data(growthcurve4) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,2)) plot(growthcurve1) plot(growthcurve2) plot(growthcurve3) plot(growthcurve4) par(def.par)
Formulas of primary growth models commonly used in predictive microbiology
baranyi baranyi_without_Nmax baranyi_without_lag buchanan buchanan_without_Nmax buchanan_without_lag gompertzmbaranyi baranyi_without_Nmax baranyi_without_lag buchanan buchanan_without_Nmax buchanan_without_lag gompertzm
These models describe the evolution of the decimal logarithm of the microbial count (LOG10N) as a function of the time (t).
baranyi is the model of Baranyi and Roberts (1994) with four parameters (LOG10N0, mumax, lag, LOG10Nmax)baranyi_without_Nmax is the model of Baranyi and Roberts (1994) with three parameters (LOG10N0, mumax, lag), without brakingbaranyi_without_lag is the model of Baranyi and Roberts (1994) with three parameters (LOG10N0, mumax, LOG10Nmax), without lagbuchanan is the three-phase linear model proposed by Buchanan et al. (1997)buchanan_without_Nmax is the two-phase linear model with three parameters (LOG10N0, mumax, lag), without brakingbuchanan_without_lag is the two-phase linear model with three parameters (LOG10N0, mumax, LOG10Nmax), without laggompertzm is the modified Gompertz model introduced by Gibson et al. (1988) and reparameterized by Zwietering et al. (1990)
A formula
Florent Baty, Marie-Laure Delignette-Muller
Baranyi J and Roberts, TA (1994) A dynamic approach to predicting bacterial growth in food, International Journal of Food Microbiology, 23, 277-294.
Buchanan RL, Whiting RC, Damert WC (1997) When is simple good enough: a comparison of the Gompertz, Baranyi, and three-phase linear models for fitting bacterial growth curves. Food Microbiology, 14, 313-326.
Gibson AM, Bratchell N, Roberts TA (1988) Predicting microbial growth: growth responses of salmonellae in a laboratory medium as affected by pH, sodium chloride and storage temperature. International Journal of Food Microbiology, 6, 155 -178.
Zwietering MH, Jongenburger I, Rombouts FM, Van't Riet K (1990) Modeling of the bacterial growth curve. Applied and Environmental Microbiology, 56, 1875-1881.
# Example 1 data(growthcurve1) nls1 <- nls(baranyi, growthcurve1, list(lag=4, mumax=1, LOG10N0 = 4, LOG10Nmax = 9)) nls2 <- nls(gompertzm,growthcurve1, list(lag = 4, mumax = 1, LOG10N0 = 4, LOG10Nmax = 9)) nls3 <- nls(buchanan, growthcurve1, list(lag = 4, mumax = 1, LOG10N0 = 4, LOG10Nmax = 9)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,2)) plotfit(nls1, smooth = TRUE) plotfit(nls2, smooth = TRUE) plotfit(nls3, smooth = TRUE) par(def.par) # Example 2 data(growthcurve2) nls4 <- nls(baranyi_without_Nmax, growthcurve2, list(lag = 2, mumax = 0.4, LOG10N0 = 7.4)) nls5 <- nls(buchanan_without_Nmax,growthcurve2, list(lag = 2, mumax = 0.4, LOG10N0 = 7.4)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,1)) plotfit(nls4, smooth = TRUE) plotfit(nls5, smooth = TRUE) par(def.par) # Example 3 data(growthcurve3) nls6 <- nls(baranyi_without_lag, growthcurve3, list(mumax = 1, LOG10N0 = 0, LOG10Nmax = 5)) nls7 <- nls(buchanan_without_lag, growthcurve3, list(mumax = 1, LOG10N0 = 0, LOG10Nmax = 5)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,1)) plotfit(nls6, smooth = TRUE) plotfit(nls7, smooth = TRUE) par(def.par)# Example 1 data(growthcurve1) nls1 <- nls(baranyi, growthcurve1, list(lag=4, mumax=1, LOG10N0 = 4, LOG10Nmax = 9)) nls2 <- nls(gompertzm,growthcurve1, list(lag = 4, mumax = 1, LOG10N0 = 4, LOG10Nmax = 9)) nls3 <- nls(buchanan, growthcurve1, list(lag = 4, mumax = 1, LOG10N0 = 4, LOG10Nmax = 9)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,2)) plotfit(nls1, smooth = TRUE) plotfit(nls2, smooth = TRUE) plotfit(nls3, smooth = TRUE) par(def.par) # Example 2 data(growthcurve2) nls4 <- nls(baranyi_without_Nmax, growthcurve2, list(lag = 2, mumax = 0.4, LOG10N0 = 7.4)) nls5 <- nls(buchanan_without_Nmax,growthcurve2, list(lag = 2, mumax = 0.4, LOG10N0 = 7.4)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,1)) plotfit(nls4, smooth = TRUE) plotfit(nls5, smooth = TRUE) par(def.par) # Example 3 data(growthcurve3) nls6 <- nls(baranyi_without_lag, growthcurve3, list(mumax = 1, LOG10N0 = 0, LOG10Nmax = 5)) nls7 <- nls(buchanan_without_lag, growthcurve3, list(mumax = 1, LOG10N0 = 0, LOG10Nmax = 5)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,1)) plotfit(nls6, smooth = TRUE) plotfit(nls7, smooth = TRUE) par(def.par)
A data frames describing the specific growth rate of Escherichia coli as a function of various environmental factors
data(ross)data(ross)
A data frame with five columns (author: the author of the paper from which the data was extracted, T: the temperature in Celsius, aw: the water activity, pH: the pH value, sqrtmumax: the square root of the maximum specific growth rate)
Ross T, Ratkowsky DA, Mellefont LA, McMeekin TA (2003) Modelling the effects of temperature, water activity, pH and lactic acid concentration on the growth rate of Escherichia coli. International of Food Microbiology, 82, 33-43.
data(ross) def.par <- par(no.readonly = TRUE) par(mfrow = c(2, 2)) plot(ross[c("T", "sqrtmumax")]) plot(ross[c("pH", "sqrtmumax")]) plot(ross[c("aw", "sqrtmumax")]) par(def.par)data(ross) def.par <- par(no.readonly = TRUE) par(mfrow = c(2, 2)) plot(ross[c("T", "sqrtmumax")]) plot(ross[c("pH", "sqrtmumax")]) plot(ross[c("aw", "sqrtmumax")]) par(def.par)
Formulas of secondary growth models commonly used in predictive microbiology
cpm_T cpm_pH_4p cpm_pH_3p cpm_aw_3p cpm_aw_2p cpm_T_pH_awcpm_T cpm_pH_4p cpm_pH_3p cpm_aw_3p cpm_aw_2p cpm_T_pH_aw
All the models describe the evolution of the square root of the maximum specific growth rate (sqrtmumax) as a function of one or more environmental factors among temperature (T), pH (pH) and water activity (aw). Each model must be fitted to a data frame including at least two columns, the last one named "sqrtmumax" and the first ones named "T", "pH" or "aw" according to the model.
cpm_T is the cardinal temperature model with inflection (Rosso et al., 1993) with four parameters (Tmin, Topt, Tmax,muopt)cpm_pH_4p is the cardinal pH model (Rosso et al., 1995) with four parameters (pHmin, pHopt, pHmax, muopt)cpm_pH_3p is a symetric cardinal pH model with three parameters (pHmin, pHopt, muopt), obtained by fixing pHmax to 2pHopt-pHmin in the cpm_pH_4p modelcpm_aw_3p is the cardinal aw model (Rosso and Robinson, 2001) with three parameters (awmin, awopt, muopt)cpm_aw_2p is a simplified cardinal aw model (Rosso and Robinson, 2001) with two parameters (awmin, muopt) obtained by fixing awopt to 1 in the cpm_aw_3p modelcpm_T_pH_aw is the cardinal model based on the gamma concept (Pinon et al., 2004) with 9 parameters (Tmin, Topt, Tmax, pHmin, pHopt, pHmax, awmin, awopt, muopt)
A formula
Florent Baty, Marie-Laure Delignette-Muller
Pinon A, Zwietering M, Perrier L, Membr\'e J, Leporq B, Mettler E, Thuault D, Coroller L, Stahl V, Vialette M (2004) Development and validation of experimental protocols for use of cardinal models for prediction of microorganism growth in food products. Applied Environmental Microbiology, 70, 1081-1087.
Rosso L, Robinson TP (2001) A cardinal model to describe the effect of water activity on the growth of moulds. International Journal of Food Microbiology, 63, 265-273.
Rosso L, Lobry JR, Bajard S, Flandrois JP (1995) Convenient model to describe the combined effects of temperature and pH on microbial growth. Applied Environmental Microbiology, 61, 610-616.
Rosso L, Lobry JR, Flandrois JP (1993) An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model. Journal of Theoretical Biology 162, 447-463.
data(ross) # Example for the cpm_T model d1 <- subset(ross, author == "salter" & aw == 0.997, select = c(T, sqrtmumax)) nls1 <- nls(cpm_T, d1, list(muopt = 1.7, Tmin = 4, Topt = 40, Tmax = 47)) plotfit(nls1, smooth = TRUE) overview(nls1) # Example for the cpm_pH_4p model d2 <- subset(ross, author == "presser" & aw > 0.99, select = c(pH, sqrtmumax)) nls2 <- nls(cpm_pH_4p, d2, list(muopt = 0.5, pHmin = 4, pHopt = 6.5, pHmax = 9)) plotfit(nls2, smooth = TRUE) overview(nls2) # Example for the cpm_pH_3p model d3 <- subset(ross, author == "presser" & aw == 0.997, select = c(pH, sqrtmumax)) nls3 <- nls(cpm_pH_3p, d3, list(muopt = 0.5, pHmin = 4, pHopt = 6.5)) plotfit(nls3, smooth = TRUE) overview(nls3) # Example for the cpm_aw_3p model d4<-subset(ross, author == "mellefont", select = c(aw, sqrtmumax)) nls4 <- nls(cpm_aw_3p, d4, list(muopt = 0.6, awmin = 0.95, awopt = 0.995)) plotfit(nls4, smooth = TRUE) overview(nls4) # Example for the cpm_aw_2p model d5 <- subset(ross, author == "mellefont" & aw < 0.99, select = c(aw, sqrtmumax)) nls5 <- nls(cpm_aw_2p, d5, list(muopt = 0.6, awmin = 0.95)) plotfit(nls5, smooth = TRUE) overview(nls5) # Examples for the cpm_T_pH_aw model d6 <- subset(ross, select = c(T, pH, aw, sqrtmumax)) nls6 <- nls(cpm_T_pH_aw, d6, list(muopt = 2, Tmin = 4, Topt = 40, Tmax = 49, pHmin = 4, pHopt = 6.5, pHmax = 9, awmin = 0.95, awopt = 0.995)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2, 2)) plotfit(nls6, variable = 1) plotfit(nls6, variable = 2) plotfit(nls6, variable = 3) overview(nls6) par(def.par)data(ross) # Example for the cpm_T model d1 <- subset(ross, author == "salter" & aw == 0.997, select = c(T, sqrtmumax)) nls1 <- nls(cpm_T, d1, list(muopt = 1.7, Tmin = 4, Topt = 40, Tmax = 47)) plotfit(nls1, smooth = TRUE) overview(nls1) # Example for the cpm_pH_4p model d2 <- subset(ross, author == "presser" & aw > 0.99, select = c(pH, sqrtmumax)) nls2 <- nls(cpm_pH_4p, d2, list(muopt = 0.5, pHmin = 4, pHopt = 6.5, pHmax = 9)) plotfit(nls2, smooth = TRUE) overview(nls2) # Example for the cpm_pH_3p model d3 <- subset(ross, author == "presser" & aw == 0.997, select = c(pH, sqrtmumax)) nls3 <- nls(cpm_pH_3p, d3, list(muopt = 0.5, pHmin = 4, pHopt = 6.5)) plotfit(nls3, smooth = TRUE) overview(nls3) # Example for the cpm_aw_3p model d4<-subset(ross, author == "mellefont", select = c(aw, sqrtmumax)) nls4 <- nls(cpm_aw_3p, d4, list(muopt = 0.6, awmin = 0.95, awopt = 0.995)) plotfit(nls4, smooth = TRUE) overview(nls4) # Example for the cpm_aw_2p model d5 <- subset(ross, author == "mellefont" & aw < 0.99, select = c(aw, sqrtmumax)) nls5 <- nls(cpm_aw_2p, d5, list(muopt = 0.6, awmin = 0.95)) plotfit(nls5, smooth = TRUE) overview(nls5) # Examples for the cpm_T_pH_aw model d6 <- subset(ross, select = c(T, pH, aw, sqrtmumax)) nls6 <- nls(cpm_T_pH_aw, d6, list(muopt = 2, Tmin = 4, Topt = 40, Tmax = 49, pHmin = 4, pHopt = 6.5, pHmax = 9, awmin = 0.95, awopt = 0.995)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2, 2)) plotfit(nls6, variable = 1) plotfit(nls6, variable = 2) plotfit(nls6, variable = 3) overview(nls6) par(def.par)
Bacterial kinetics data sets
data(survivalcurve1) data(survivalcurve2) data(survivalcurve3)data(survivalcurve1) data(survivalcurve2) data(survivalcurve3)
Data frames with 2 columns (t: time, LOG10N: decimal logarithm of bacterial density)
Data obtained by Florent Baty and Marie-Laure Delignette-Muller
data(survivalcurve1) data(survivalcurve2) data(survivalcurve3) def.par <- par(no.readonly = TRUE) par(mfrow=c(2,2)) plot(survivalcurve1, type="b") plot(survivalcurve2, type="b") plot(survivalcurve3, type="b") par(def.par)data(survivalcurve1) data(survivalcurve2) data(survivalcurve3) def.par <- par(no.readonly = TRUE) par(mfrow=c(2,2)) plot(survivalcurve1, type="b") plot(survivalcurve2, type="b") plot(survivalcurve3, type="b") par(def.par)
Formulas of primary survival models commonly used in predictive microbiology
geeraerd geeraerd_without_Nres geeraerd_without_Sl mafart albert trilinear bilinear_without_Nres bilinear_without_Slgeeraerd geeraerd_without_Nres geeraerd_without_Sl mafart albert trilinear bilinear_without_Nres bilinear_without_Sl
These models describe the evolution of the decimal logarithm of the microbial count (LOG10N) as a function of the time (t).
geeraerd is the model of Geeraerd et al. (2005) with four parameters (LOG10N0, kmax, Sl, LOG10Nres)geeraerd_without_Nres is the model of of Geeraerd et al. (2005) with three parameters (LOG10N0, kmax, Sl), without tailgeeraerd_without_Sl is the model of of Geeraerd et al. (2005) with three parameters (LOG10N0, kmax, Nres), without shouldermafart is the Weibull model as parameterized by Mafart et al. (2002) with three parameters (p, delta, LOG10N0)albert is the modified Weibull model proposed by Albert and Mafart (2005) with four parameters (p, delta, LOG10N0, LOG10Nres)trilinear is the three-phase linear model with four parameters (LOG10N0, kmax, Sl, LOG10Nres)bilinear_without_Nres is the two-phase linear model with three parameters (LOG10N0, kmax, Sl), without tailbilinear_without_Sl is the two-phase linear model with three parameters (LOG10N0, kmax, LOG10Nres), without shoulder
A formula
Florent Baty, Marie-Laure Delignette-Muller
Albert I, Mafart P (2005) A modified Weibull model for bacterial inactivation. International Journal of Food Microbiology, 100, 197-211.
Geeraerd AH, Valdramidis VP, Van Impe JF (2005) GInaFiT, a freeware tool to assess non-log-linear microbial survivor curves. International Journal of Food Microbiology, 102, 95-105.
Mafart P, Couvert O, Gaillard S, Leguerinel I (2002) On calculating sterility in thermal preservation methods : application of the Weibull frequency distribution model. International Journal of Food Microbiology, 72, 107-113.
# Example 1 data(survivalcurve1) nls1a <- nls(geeraerd, survivalcurve1, list(Sl = 5, kmax = 1.5, LOG10N0 = 7, LOG10Nres = 1)) nls1b <- nls(trilinear, survivalcurve1, list(Sl = 5, kmax = 1.5, LOG10N0 = 7, LOG10Nres = 1)) nls1c <- nls(albert,survivalcurve1, list(p = 1.2, delta = 4, LOG10N0 = 7, LOG10Nres = 1)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,2)) overview(nls1a) plotfit(nls1a, smooth = TRUE) overview(nls1b) plotfit(nls1b, smooth = TRUE) overview(nls1c) plotfit(nls1c, smooth = TRUE) par(def.par) # Example 2 data(survivalcurve2) nls2a <- nls(geeraerd_without_Nres, survivalcurve2, list(Sl = 10, kmax = 1.5, LOG10N0 = 7.5)) nls2b <- nls(bilinear_without_Nres, survivalcurve2, list(Sl = 10, kmax = 1.5, LOG10N0 = 7.5)) nls2c <- nls(mafart, survivalcurve2, list(p = 1.5, delta = 8, LOG10N0 = 7.5)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,2)) overview(nls2a) plotfit(nls2a, smooth = TRUE) overview(nls2b) plotfit(nls2b, smooth = TRUE) overview(nls2c) plotfit(nls2c, smooth = TRUE) par(def.par) # Example 3 data(survivalcurve3) nls3a <- nls(geeraerd_without_Sl, survivalcurve3, list(kmax = 4, LOG10N0 = 7.5, LOG10Nres = 1)) nls3b <- nls(bilinear_without_Sl, survivalcurve3, list(kmax = 4, LOG10N0 = 7.5, LOG10Nres = 1)) nls3c <- nls(mafart, survivalcurve3, list(p = 0.5, delta = 0.2, LOG10N0 = 7.5)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,2)) overview(nls3a) plotfit(nls3a, smooth = TRUE) overview(nls3b) plotfit(nls3b, smooth = TRUE) overview(nls3c) plotfit(nls3c, smooth = TRUE) par(def.par)# Example 1 data(survivalcurve1) nls1a <- nls(geeraerd, survivalcurve1, list(Sl = 5, kmax = 1.5, LOG10N0 = 7, LOG10Nres = 1)) nls1b <- nls(trilinear, survivalcurve1, list(Sl = 5, kmax = 1.5, LOG10N0 = 7, LOG10Nres = 1)) nls1c <- nls(albert,survivalcurve1, list(p = 1.2, delta = 4, LOG10N0 = 7, LOG10Nres = 1)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,2)) overview(nls1a) plotfit(nls1a, smooth = TRUE) overview(nls1b) plotfit(nls1b, smooth = TRUE) overview(nls1c) plotfit(nls1c, smooth = TRUE) par(def.par) # Example 2 data(survivalcurve2) nls2a <- nls(geeraerd_without_Nres, survivalcurve2, list(Sl = 10, kmax = 1.5, LOG10N0 = 7.5)) nls2b <- nls(bilinear_without_Nres, survivalcurve2, list(Sl = 10, kmax = 1.5, LOG10N0 = 7.5)) nls2c <- nls(mafart, survivalcurve2, list(p = 1.5, delta = 8, LOG10N0 = 7.5)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,2)) overview(nls2a) plotfit(nls2a, smooth = TRUE) overview(nls2b) plotfit(nls2b, smooth = TRUE) overview(nls2c) plotfit(nls2c, smooth = TRUE) par(def.par) # Example 3 data(survivalcurve3) nls3a <- nls(geeraerd_without_Sl, survivalcurve3, list(kmax = 4, LOG10N0 = 7.5, LOG10Nres = 1)) nls3b <- nls(bilinear_without_Sl, survivalcurve3, list(kmax = 4, LOG10N0 = 7.5, LOG10Nres = 1)) nls3c <- nls(mafart, survivalcurve3, list(p = 0.5, delta = 0.2, LOG10N0 = 7.5)) def.par <- par(no.readonly = TRUE) par(mfrow = c(2,2)) overview(nls3a) plotfit(nls3a, smooth = TRUE) overview(nls3b) plotfit(nls3b, smooth = TRUE) overview(nls3c) plotfit(nls3c, smooth = TRUE) par(def.par)