Fitting complex non-linear function with nls() in R gives singular gradient error

Question

I am trying to fit oral pharmacokinetic function using R nls. The function can be seen here:

I tried the following:

dat <- data.frame(
        Time = c(10, 15, 20, 30, 40, 60, 90, 120, 180, 210, 240, 300, 360),
        C_PO = c(0,0.28,0.55,1.2,2,1.95,1.85,1.6,
                 0.86,0.78,0.6,0.21,0.18))

plot(dat$C_PO ~ dat$Time, data = dat, log = "y")

fit <- nls(C_PO~ka*100* (exp(-k*Time) - exp(-ka*Time))/(v*(ka- k)),
    data = dat,
    start = list(ka = 0.09, k = 0.01, v = 50))
#> Error in nls(C_PO ~ ka * 100 * (exp(-k * Time) - exp(-ka * Time))/(v * : singular gradient

The code is trying to fit a non-linear model according to the concentration values C_PO over a time period Time. The initial values were calculated using graph stripping methods. However, the error message reads "singular gradient." How can this be fixed?

Answer 1

You can use gslnls package for fitting non-linear model like

library(gslnls)

dat <- data.frame(
  Time = c(10, 15, 20, 30, 40, 60, 90, 120, 180, 210, 240, 300, 360),
  C_PO = c(0,0.28,0.55,1.2,2,1.95,1.85,1.6,
           0.86,0.78,0.6,0.21,0.18))

plot(dat$C_PO ~ dat$Time, data = dat, log = "y")

fit <- gsl_nls(C_PO ~ ka*100* (exp(-k*Time) - exp(-ka*Time))/(v*(ka- k)), 
                    data = dat, 
                    start = c(ka = 0.09, k = 0.01, v = 50), 
                    control = gsl_nls_control(maxiter = 500))

summary(fit)

#> Formula: C_PO ~ ka * 100 * (exp(-k * Time) - exp(-ka * Time))/(v * (ka - 
#>     k))
#> 
#> Parameters:
#>     Estimate Std. Error t value Pr(>|t|)
#> ka    0.0132     4.0027   0.003    0.997
#> k     0.0132     4.0025   0.003    0.997
#> v    21.0181  6372.7704   0.003    0.997
#> 
#> Residual standard error: 0.3363 on 10 degrees of freedom
#> 
#> Number of iterations to convergence: 18 
#> Achieved convergence tolerance: 0