growth_sigmoid: Sigmoid growth function

In maomlab/BayesPharma: Tools for Bayesian Analysis of Non-Linear Pharmacology Models

Sigmoid growth function

Description

Functional form for the sigmoid growth model, related to the Richards growth model by setting nu = 1.

The parameterization follows (Zwietering, 1990) and grofit:

K      = **carrying capacity**, `K = response(time = Inf)`. The
         \pkg{grofit} package calls this parameter `A`. `K` has the same
         units as the `response`.
K0     = **initial population size** `K0 = response(time = 0)`. The
         \pkg{grofit} package assumes `K0=0`. `K0` has the same units as
         the `response`.
rate   = **maximum growth rate** `rate = max[d(response)/d(time)]`. The
         \pkg{grofit} package calls this `mu`. `rate` has the units of
         `response/time`
lambda = **duration of the lag-phase** the time point at which the
         tangent through the growth curve when it achieves the maximum
         growth rate crosses the initial population size `K0`. (see
         Figure 2 in (Kahm et al., 2010)).

See the vignettes(topic = "derive_growth_model", package = "BayesPharma") for more details.

Usage

growth_sigmoid(K, K0, rate, lambda, time)

Arguments

K	numeric, the carrying capacity
K0	numeric, initial population size
rate	numeric, maximum growth rate
lambda	numeric, duration of the lag-phase
time	numeric, time point at which to evaluate the response

Details

The Richards growth model is given by

response(time) = K0 + (K - K0)/(1 + nu * exp( 1 + nu + rate/(K - K0) * (1 + nu)^(1 + 1/nu) * (lambda - time))) ^ (1/nu)

Setting nu = 1, and simplifying gives

response(time) = K0 + (K - K0)/(1 + 1 * exp( 1 + 1 + rate/(K - K0) * (1 + 1)^(1 + 1/1) * (lambda - time))) ^ (1/1)

response(time) = K0 + (K - K0)/(1 + exp( 2 + rate/(K - K0) * (2)^(2) * (lambda - time))) ^ (1/1)

response(time) = K0 + (K - K0)/(1 + exp(2 + rate/(K - K0) * 4 * (lambda - time)))

response(time) = K0 + (K - K0)/(1 + exp(4 * rate/(K - K0) * (lambda - time) + 2))

The sigmoid growth curve is related to the sigmoid agoinst model by setting top = K, bottom = K0, hill = 4 * rate / (K - K0) / log10(e), AC50 = lambda + 2 * log10(e) / hill, and log_dose = time, where e is exp(1):

response(time) = bottom + (top - bottom)/(1 + exp(hill/log10(e) * (lambda - time) + 2))

response(time) = bottom + (top - bottom)/(1 + exp(hill/log10(e) * (AC50 - time)))

response(log_dose) = bottom + (top - bottom)/(1 + 10^((ac50 - log_dose) * hill))

Value

numeric, response given the time and parameters

References

Zwietering M. H., Jongenburger I., Rombouts F. M., van 't Riet K., (1990) Modeling of the Bacterial Growth Curve. Appl. Environ. Microbiol., 56(6), 1875-1881 https://doi.org/10.1128/aem.56.6.1875-1881.1990

Kahm, M., Hasenbrink, G., Lichtenberg-Fraté, H., Ludwig, J., & Kschischo, M. (2010). grofit: Fitting Biological Growth Curves with R. J. Stat. Softw., 33(7), 1–21. https://doi.org/10.18637/jss.v033.i07

Examples

## Not run: 
 # Generate Sigmoid growth curve
 data <- data.frame(
   time = seq(0, 2, length.out = 101)) |>
     dplyr::mutate(
       response = stats::rnorm(
         n = length(time),
         mean = BayesPharma::growth_sigmoid(
           K = 1,
           K0 = 0,
           rate = 2,
           lambda = 0.5,
           time = time),
       sd = .2))

## End(Not run)

maomlab/BayesPharma documentation built on Aug. 24, 2024, 8:45 a.m.