Fitting growth models in biogrowth

Fitting primary growth models

The most simple approach implemented in fit_growth() is fitting primary growth models to experimental data that includes the variation of a population during an experiment. This behaviour is defined when environment="constant". In this case, fit_growth() requires the definition of (at least) 4 arguments:

• fit_data to pass the experimental data
• model_keys defining the model equation
• start setting the initial values for the model parameters
• known defining the fixed model parameters

The experimental data must be defined as a tibble (or data frame). It must include two columns: one defining the elapsed time (by default, named time) and a second one defining the logarithm of the population size (by default, named logN). Note that the default names can be changed using the formula argument. In this example, we will define a tibble by hand.

my_data <- data.frame(time = c(0, 10, 15, 20, 25, 30, 35, 40, 45, 50), 
                      logN = c(1.2, 1, 1.7, 1.9, 2.3, 3.1, 3.3, 3.8, 4.3, 4.1)
                      )

ggplot(my_data) + geom_point(aes(x = time, y = logN))

Note that, by default, the population size is interpreted in log10 scale. The base of the logarithm can be modified using the logbase_logN argument. Please check the vignette dedicated to the topic for further details.

The next step in model fitting is the definition of the growth model to fit to the data using the model_keys argument. This argument is a named list assigning model equations to the different models. In the case when environment="constant", the functions only fits a primary model. Therefore, model_keys must have a unique entry named “primary” with the model key of a primary model. A list of valid keys can be retrieved calling primary_model_data() without any argument.

primary_model_data()
#>  [1] "modGompertz"          "Baranyi"              "Baranyi_noLag"       
#>  [4] "Baranyi_noStationary" "Trilinear"            "Logistic"            
#>  [7] "Richards"             "Loglinear"            "Bilinear_lag"        
#> [10] "Bilinear_stationary"

In this example, we will use the Baranyi growth model.

models <- list(primary = "Baranyi")

In the case of primary model fitting, fit_growth() uses non-linear regression through modFit(). This algorithm requires the definition of initial guesses for every model parameter. This is done in fit_growth() through the argument start, which is a named numeric vector (or list). The names of the vector must be parameter keys defined within biogrowth. These can be retrieved (as well as additional model meta-data) by passing a model key to primary_model_data().

primary_model_data("Baranyi")$pars
#> [1] "logN0"   "logNmax" "mu"      "lambda"

Therefore, the Baranyi model requires the definition of four model parameters. For a description of the model equations and the interpretation of the model parameters, please check the specific package vignette.

The biogrowth package includes several functions to aid in the definition of initial guesses for the model parameters. The function check_growth_guess() takes three arguments:

• fit_data defining the experimental data as in fit_growth()
• model_name, a character vector of length one defining the primary model (as per primary_model_data())
• guess, a numeric vector defining the initial guess as in fit_growth()
• environment, describing the type of environmental conditions ("constant" (default) or "dynamic")
• env_conditions defining the variation of the environmental conditions (only if environment="dynamic")

Then, the function creates a plot comparing the experimental data against the initial guess.

start <- c(logNmax = 6, lambda = 25, logN0 = 2, mu = .2)  # Initial guess

check_growth_guess(
  my_data,
  models,
  start
)

In this case, the plot shows that the initial guess is moderately far from the experimental data. For instance, the values of logN0 and logNmax are too high. These could potentially cause some convergence issues. Note that this function can use different log-scales using logbase_mu or logbase_logN.

In order to facilitate the definition of initial guesses, biogrowth includes make_guess_primary(). This function applies some heuristic rules to define guesses for the initial model parameters. It takes two arguments:

• fit_data that passes the experimental data as described above
• primary_model that defines the growth model

Furthermore, the function includes a logbase_mu argument to define the logbase of parameter \(\mu\).

Calling this function with our data and the Baranyi model returns a named vector with an initial guess of the model parameters.

auto_guess <- make_guess_primary(my_data, "Baranyi")
print(auto_guess)
#>       logN0          mu      lambda     logNmax 
#>  1.20000000  0.08857143 10.00000000  4.30000000

We can now use check_growth_guess() to assess the quality of the initial guess

check_growth_guess(
  my_data,
  models,
  auto_guess
)

From the plot, we can see that the initial guess is already quite close to the experimental data.

The last argument to be defined is known, which allows fixing any model parameter to a given value. This argument is also a named list with the same conventions as start. In a first example, we will define it as an empty vector, indicating that no model parameter is fixed (i.e. every one is estimated from the data).

known <- c()

Then, we can call fit_growth() with these arguments.

primary_fit <- fit_growth(my_data, 
                          models, 
                          auto_guess, 
                          known,
                          environment = "constant",
                          )

The function returns an instance of GrowthFit. It includes several S3 methods to facilitate the interpretation of the results of the model fit. For instance, the print() method provides a quick view of the model fitted:

print(primary_fit)
#> Primary growth model fitted to data
#> 
#> Growth model: Baranyi 
#> 
#> Estimated parameters:
#>      logN0         mu     lambda    logNmax 
#>  1.0987559  0.1104077 13.2555219  4.2695511 
#> 
#> Fixed parameters:
#> NULL
#> 
#> Parameter mu defined in log-10 scale
#> Population size defined in log-10 scale

The statistical summary of the fit can be retrieved using the summary() method.

summary(primary_fit)
#> 
#> Parameters:
#>         Estimate Std. Error t value Pr(>|t|)    
#> logN0    1.09876    0.16566   6.633 0.000566 ***
#> mu       0.11041    0.01511   7.306 0.000336 ***
#> lambda  13.25552    3.09831   4.278 0.005215 ** 
#> logNmax  4.26955    0.20220  21.116 7.35e-07 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.1925 on 6 degrees of freedom
#> 
#> Parameter correlation:
#>           logN0      mu  lambda logNmax
#> logN0    1.0000  0.3849  0.7906 -0.1485
#> mu       0.3849  1.0000  0.8020 -0.5394
#> lambda   0.7906  0.8020  1.0000 -0.3253
#> logNmax -0.1485 -0.5394 -0.3253  1.0000

And the plot() method compares the model fitted against the experimental data. This function takes additional arguments that enable editing several aesthetics of the plot. For a complete list, please check the help page of GrowthFit. Note that this method returns an instance of ggplot, so it can be edited using additional layers.

plot(primary_fit, line_col = "red", point_shape = 1)

The instance of GrowthFit includes additional methods to inspect and use the fitted model (predict, vcov, AIC…). For a complete list, please check its help page. Moreover, this instance is a subclass of list, making it easy to access several aspects of the data. Namely, it has the following entries:

• data with the experimental data
• start initial values of the model parameters
• known fixed model parameters
• best_prediction an instance of GrowthPrediction with the fitted curve
• logbase_mu the logbase for \(\mu\)
• logbase_logN the logbase for the population size
• environment the type of environment ("constant" for this type of fit)
• algorithm the fitting algorithm ("regression" for this type of fit)
• primary_model the primary model fitted
• fit_results an instance of modFit with the fitted model

As was mentioned above, the known argument allows fixing any model parameter to a given value. It must be a named vector with the same conventions as for the initial values. For instance, we can fix the maximum population size to 4.

known <- c(logNmax = 4)

If this argument was passed to fit_growth() together with auto_guess, the function would return a warning (as long as check=TRUE). The reason for this is that logNmax is defined both as an initial guess (i.e. a parameter to fit) and a fixed parameter. Therefore, the initial guess must be modified, including only the parameters to fit (i.e. logN0, mu and lambda)

new_guess <- auto_guess[c("logN0", "mu", "lambda")]

new_fit <-  fit_growth(my_data, 
                       models, 
                       new_guess, 
                       known,
                       environment = "constant",
                       )

The print method of the new_fit reflects the fact that logNmax was now fixed.

new_fit
#> Primary growth model fitted to data
#> 
#> Growth model: Baranyi 
#> 
#> Estimated parameters:
#>      logN0         mu     lambda 
#>  1.1178327  0.1194789 14.0961385 
#> 
#> Fixed parameters:
#> logNmax 
#>       4 
#> 
#> Parameter mu defined in log-10 scale
#> Population size defined in log-10 scale

And the summary method only includes the statistical information for the fitted parameters.

summary(new_fit)
#> 
#> Parameters:
#>        Estimate Std. Error t value Pr(>|t|)    
#> logN0   1.11783    0.17671   6.326 0.000394 ***
#> mu      0.11948    0.01813   6.590 0.000307 ***
#> lambda 14.09614    3.14958   4.476 0.002882 ** 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.2129 on 7 degrees of freedom
#> 
#> Parameter correlation:
#>         logN0     mu lambda
#> logN0  1.0000 0.3828 0.7810
#> mu     0.3828 1.0000 0.8007
#> lambda 0.7810 0.8007 1.0000

Another relevant function included in biogrowth is time_to_size. This function takes two arguments:

• model an instance of GrowthFit (or other object returned by fit_growth())
• size a target population size (in log units)

Then, it calculates the elapsed time required to reach the given size

time_to_size(primary_fit, 2)
#> [1] 20.93122

If the target population size is not reached within the fitted curve, the function returns NA.

time_to_size(primary_fit, 8)
#> [1] NA

This function can take two additional arguments. The argument logbase_logN allows defining size in different log-bases. By default, this function assumes the same base as the one used in the instance of GrowthFit. The second argument is type. By default, type="discrete", indicating that the function calculates a single value of the elapsed time required to reach the target population size. Setting type = "distribution" accounts for parameter uncertainty and calculates a distribution for the elapsed time. This is described in detail in the vignette about including uncertainty in growth predictions.

Fitting both primary and secondar models from a single experiment under dynamic environnental conditions

The fit_growth() function can also be used to fit a growth model based on both primary and secondary models to a set of data gathered under dynamic experimental conditions. This behavior is triggered by setting environment="dynamic". Then, fit_growth() supports two fitting approaches. The first one assumes that every data point was gathered (at different time points of) one or more growth experiments under a single, dynamic environmental profile. This method is set up setting approach="single". The second one, triggered when approach="global" considers that different experiments could have different (dynamic) environmental conditions. This section describes the former approach, whereas the latter will be described in the following section.

In this case, when environment="dynamic" and approach="single", the function requires the definition of the following arguments:

• fit_data to pass the experimental data
• model_keys defining the model equation
• start setting the initial values for the model parameters
• known defining the fixed model parameters
• algorithm selecting either regression or an Adaptive Monte Carlo algorithm
• env_conditions describing the variation of the environmental conditions during the experiment
• niter: number of iterations of the MC algorithm (only if algorithm="MCMC")

The arguments env_conditions and fit_data are tibbles (or data frames) that describe, respectively, the experimental design and the observations. The biogrowth package includes two example datasets to illustrate the input requirements for this function.

data("example_dynamic_growth")
data("example_env_conditions")

The tibble passed to the argument env_conditions must have a column defining the elapsed time (time by default) and as many additional columns as environmental factors. The fit_growth() function is totally flexible with respect to the number of factors or the way they are named. The only requirement is that the definition of every argument is consistent. In the case of example_env_conditions, this dataset considers two factors: temperature and water activity (aw).

head(example_env_conditions)
#> # A tibble: 3 × 3
#>    time temperature    aw
#>   <dbl>       <dbl> <dbl>
#> 1     0          20  0.99
#> 2     5          30  0.95
#> 3    15          35  0.9

Note that for the calculations this function joins the data points by linear interpolation, as shown in this plot:

ggplot(example_env_conditions, aes(x = time, y = temperature)) + 
  geom_line() +
  geom_point()

The tibble passed to the argument fit_data must have one column defining the elapsed time (time by default) and one defining the logarithm of the population size (logN by default). Different column names can be defined using the formula argument. For instance, formula=log_size ~ Time would mean that the elapsed time is called “Time” and the logarithm of the population size is called “log_size”. Note that the name of the column describing the elapsed time in fit_data must be identical to the one in env_conditions.

head(example_dynamic_growth)
#> # A tibble: 6 × 2
#>    time     logN
#>   <dbl>    <dbl>
#> 1 0      0.0670 
#> 2 0.517 -0.00463
#> 3 1.03  -0.0980 
#> 4 1.55  -0.0986 
#> 5 2.07   0.111  
#> 6 2.59  -0.0465

By default, the calculations are done considering that logN is defined in log10 scale. Nonetheless, this can be modified with the argument logbase_logN. Please check the vignette on the topic for additional details.

The type of secondary model for each environmental factor is defined by the argument model_names. This argument is a named vector where the names refer to the environmental factor and the value to the type of model. Supported models can be retrieved using secondary_model_data().

secondary_model_data()
#> [1] "CPM"           "Zwietering"    "fullRatkowsky" "Aryani"       
#> [5] "Rosso_aw"      "Inhibitory"

In this example we will use cardinal models for water activity and a Zwietering-type model for temperature. Note that the names of this vector must be identical to the columns in env_conditions.

sec_model_names <- list(temperature = "Zwietering", aw = "CPM")

This modelling approach implements two algorithms for model fitting: non-linear regression (FME::modFit()) and an adaptive Monte Carlo algorithm (FME:modMCMC()). This can be selected through the algorithm argument. If algorithm="regression" (default), the fitting is done with FME:modFit(). If algorithm="MCMC", the fitting is done with FME::modMCMC(). In this vignette, we focus on the former. For a description on MCMC fitting, please check the vignette on modelling growth including uncertainty.

Regardless of the fitting algorithm, fit_growth() enables to fit or fix any model parameter. This distinction is made using the arguments known (fixed parameters) and start (fitted parameters). Note that every parameter of the primary and secondary model must be in either of these arguments without duplication.

This function uses the Baranyi primary model. It has two variables that need initial values (N0 and Q0) and one primary model parameter (Nmax). The specific growth rate is described using the gamma concept. This requires the definition of its value under optimal conditions (mu_opt) as well as the cardinal parameters for each environmental factor. They must be defined as factor+_+parameter name. For instance, the minimum water activity for growth must be written as aw_xmin.

In this example we will consider the model parameters of the primary model as known. For the secondary model for water activity, we will only consider the optimum value for growth as unknown. Finally, for the effect of temperature, we will consider the order and xmax are known:

known_pars <- c(Nmax = 1e4,  # Primary model
                   N0 = 1e0, Q0 = 1e-3,  # Initial values of the primary model
                   mu_opt = 4, # mu_opt of the gamma model
                   temperature_n = 1,  # Secondary model for temperature
                   aw_xmax = 1, aw_xmin = .9, aw_n = 1  # Secondary model for water activity
                   )

Then, the remaining model parameters must be defined in start. Due to the use of non-linear regression for model fitting, it is required to define initial values for these parameters. In order to ease the definition of initial guesses for these parameters, biogrowth includes the function check_growth_guess(). As already mentioned, it takes five arguments:

• fit_data defining the experimental data as in fit_growth()
• model_name, a character vector of length one defining the primary model (as per primary_model_data())
• guess, a numeric vector defining the initial guess as in fit_growth()
• environment, describing the type of environmental conditions ("constant" (default) or "dynamic")
• env_conditions defining the variation of the environmental conditions (only if environment="dynamic")

All these arguments follow the same conventions as in fit_growth(). Furthermore, the function includes additional arguments to define the log-base of \(\mu\) (logbase_mu) and to define the column names for the population size and the elapsed time (formula).

Then, we can define an initial guess of the model parameters

my_start <-  c(temperature_xmin = 25, temperature_xopt = 35, aw_xopt = .95)

And pass it to check_growth_guess(), together with the values of the fixed parameters

check_growth_guess(
  example_dynamic_growth,
  sec_model_names,
  c(my_start, known_pars),
  environment = "dynamic",
  env_conditions = example_env_conditions
)

This plot shows that the initial guess of the model parameters is reasonably close to the experimental data. Then, once every argument has been defined, we can call fit_growth()

my_dyna_fit <- fit_growth(example_dynamic_growth, 
                          sec_model_names, 
                          my_start, known_pars,
                          environment = "dynamic",
                          env_conditions = example_env_conditions
                          ) 

Again, fit_growth() returns an instance of GrowthFit with the same S3 methods as for models fitted under constant environmental conditions.

print(my_dyna_fit)
#> Growth model fitted to data gathered under dynamic environmental conditions using regression
#> 
#> Environmental factors included: temperature, aw 
#> 
#> Secondary model for temperature: Zwietering
#> Secondary model for aw: CPM
#> 
#> Parameter estimates:
#> temperature_xmin temperature_xopt          aw_xopt 
#>       25.7926504       32.4099087        0.9886491 
#> 
#> Fixed parameters:
#>          Nmax            N0            Q0        mu_opt temperature_n 
#>         1e+04         1e+00         1e-03         4e+00         1e+00 
#>       aw_xmax       aw_xmin          aw_n 
#>         1e+00         9e-01         1e+00 
#> Parameter mu defined in log-10 scale
#> Population size defined in log-10 scale

summary(my_dyna_fit)
#> 
#> Parameters:
#>                  Estimate Std. Error t value Pr(>|t|)    
#> temperature_xmin 25.79265    0.62454   41.30  < 2e-16 ***
#> temperature_xopt 32.40991    5.46540    5.93 2.54e-06 ***
#> aw_xopt           0.98865    0.03988   24.79  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.1478 on 27 degrees of freedom
#> 
#> Parameter correlation:
#>                  temperature_xmin temperature_xopt aw_xopt
#> temperature_xmin           1.0000          -0.5433  0.4490
#> temperature_xopt          -0.5433           1.0000 -0.9939
#> aw_xopt                    0.4490          -0.9939  1.0000

Nonetheless, the plot() method can include the variation of one environmental factor alongside the growth curve. This is done by passing the name of an environmental factor to the add_factor argument. Note that this name must be identical to the ones used for model definition.

plot(my_dyna_fit, add_factor = "temperature")

Again, the time_to_size() function can be used to calculate the time required to reach a given population size (in log units):

time_to_size(my_dyna_fit, 3)
#> [1] 7.629713

Fitting both primary and secondary models from several experiments (global fitting)

As mentioned above, fit_growth() can also fit a unique growth model to a set of experimental data including the results of growth experiments under different (dynamic) environmental conditions. This is triggered passing approach="global". In this case, the function takes the same arguments as for approach="single".

• fit_data to pass the experimental data
• model_keys defining the model equation
• start setting the initial values for the model parameters
• known defining the fixed model parameters
• algorithm selecting either regression or an Adaptive Monte Carlo algorithm
• env_conditions describing the variation of the environmental conditions during the experiment
• niter: number of iterations of the MC algorithm (only if algorithm="MCMC")

However, the conventions for the definition of fit_data and env_conditions change when approach="global". Instead of being defined as a tibble (or data.frame), this information must be defined as a (named) list with as many elements as experiments. Then, each entry describes the particular information for each experiment. The biogrowth package includes the multiple_counts dataset as an example of this.

It is a list of two elements named exp1 and exp2. Then, each one of them is a tibble (it could also be a data.frame) describing the experimental observations for each experiment following the same conventions as when approach="single"

data("multiple_counts")
names(multiple_counts)
#> [1] "exp1" "exp2"
head(multiple_counts[[1]])
#>       time        logN
#> 1 0.000000 -0.20801574
#> 2 1.666667 -0.03630986
#> 3 3.333333 -0.29846914
#> 4 5.000000  0.35029686
#> 5 6.666667  0.14326140
#> 6 8.333333 -0.40357904

In a similar way, the variation of the experimental conditions during the experiment are also described as a (named) list. This list should have the same length (and names) as the experimental data, so each growth experiment is mapped to an environmental profile. The conditions during each experiment are described following the same conventions as when algorithm="single". That is, using a tibble (or data.frame) with one column describing the elapsed time and as many additional columns as environmental factors considered in the model. Note that, although the function admits any number of environmental conditions, these must be presnet in each experiment.

The multiple_conditions dataset included in the package is a convenient example of this format. It includes the results of two experiments, paired with multiple_counts, that consider two environmental factors: temperature and pH.

data("multiple_conditions")
names(multiple_conditions)
#> [1] "exp1" "exp2"
head(multiple_conditions[[1]])
#>   time temperature  pH
#> 1    0          20 6.5
#> 2   15          30 7.0
#> 3   40          40 6.5

The model equations, the initial guesses of the model parameters and the fixed parameters are defined in exactly the same way as when approach="single"

sec_models <- list(temperature = "CPM", pH = "CPM")

## Any model parameter (of the primary or secondary models) can be fixed

known_pars <- list(Nmax = 1e8, N0 = 1e0, Q0 = 1e-3,
                   temperature_n = 2, temperature_xmin = 20, 
                   temperature_xmax = 35,
                   pH_n = 2, pH_xmin = 5.5, pH_xmax = 7.5, pH_xopt = 6.5,
                   temperature_xopt = 30)
                   
## The rest, need initial guesses

my_start <- list(mu_opt = .8)

Again, check_growth_guess() can be used to assess the quality of the initial guess of the model parameters.

check_growth_guess(
  multiple_counts,
  sec_models,
  c(my_start, known_pars),
  environment = "dynamic",
  env_conditions = multiple_conditions,
  approach = "global"
)

Considering that the initial guess is relatively close to the observations, once all the arguments have been defined, we can call fit_growth()

global_fit <- fit_growth(multiple_counts, 
                         sec_models, 
                         my_start, 
                         known_pars,
                         environment = "dynamic",
                         algorithm = "regression",
                         approach = "global",
                         env_conditions = multiple_conditions
                         ) 

In this case, the function returns an instance of GlobalGrowthFit. The print method provides a summary of the model fitted.

print(global_fit)
#> Growth model fitted to data following a global approach conditions using regression
#> 
#> Number of experiments: 2
#> 
#> Environmental factors included: temperature, pH 
#> 
#> Secondary model for temperature: CPM
#> Secondary model for pH: CPM
#> 
#> Parameter estimates:
#>    mu_opt 
#> 0.5097542 
#> 
#> Fixed parameters:
#>             Nmax               N0               Q0    temperature_n 
#>          1.0e+08          1.0e+00          1.0e-03          2.0e+00 
#> temperature_xmin temperature_xmax             pH_n          pH_xmin 
#>          2.0e+01          3.5e+01          2.0e+00          5.5e+00 
#>          pH_xmax          pH_xopt temperature_xopt 
#>          7.5e+00          6.5e+00          3.0e+01 
#> Parameter mu defined in log-10 scale
#> Population size defined in log-10 scale

A more detailed output of the fitted model can be retrieved by summary()

summary(global_fit)
#> 
#> Parameters:
#>        Estimate Std. Error t value Pr(>|t|)    
#> mu_opt 0.509754   0.005812    87.7   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.463 on 49 degrees of freedom
#> 
#> Parameter correlation:
#>        mu_opt
#> mu_opt      1

In this case, plot() shows several subplots, one for each experiment

plot(global_fit)

As well as before, the plot can include the variation of any environmental factor alongside the growth curve, by passing the name of an environmental factor to add_factor. This function includes additional arguments for editing other aesthetics of the plot. For a complete list of options, please check the help page of GlobalGrowthFit.

plot(global_fit, add_factor = "temperature",
     label_y2 = "Temperature (ºC)",
     line_col2 = "green",
     line_type2 = "dotted")

Again, the time_to_size() function can be used to estimate the elapsed time required to reach a given population size (in log units). In this case, the function returns a named vector, with the elapsed time required to reach the target population size for each experiment

time_to_size(global_fit, 3)
#>     exp1     exp2 
#> 22.16405 15.76971

If the target population size was not reached for any of the experiments, the function returns NA

time_to_size(global_fit, 5)
#>     exp1     exp2 
#>       NA 23.44123