In this example, we use a generic, one-compartment PK model from httk package (Pearce et al. 2017) to demonstrate how pksensi can be applied to pharmacokinetic studies.
The differential equations for the one-compartment PK model can be written as:
$$\frac{dA_{gutlumen}}{dt} = -k_{gutabs} \cdot A_{gutlumen} + g(t)$$ $$\frac{dC_{rest}}{dt} = \frac{k_{gutabs}}{V_{dist}}-k_{elim} \cdot C_{rest}$$
where Agutlumen is the state variable that describes the quantity of compound in the gut lumen (mg) and Arest is the quantity of compound in rest of body and blood (mg). The parameter kgutabs is the absorption rate constant that describes the chemical absorption from the gut lumen into gut tissue through first-order processes (/h) and kelim is the elimination rate constant (/h), which is equal to the total clearance divided by the volume of distribution. The time-dependent function g(t) is used to describe the oral dosing schedule.
The concentration of the chemical in the rest of body and blood (Crest, mg/L) can be calculated as
Crest = Arest/Vdist ⋅ BW
where Vdist is the volume of distribution (L/kg BW) and BW is the body weight (kg). The Crest can also be seen as the chemical concentration in plasma that can be further used to compare with observed results in a pharmacokinetic experiment. The bioavailability is assumed to be 100% in this model.
To start, we implemented the one-compartment PK model in R. The pksensi allows users select the preferred method to solve the PK model, either with the deSolve (Soetaert, Petzoldt, and Setzer 2010) package or with GNU MCSim (Bois and Maszle 1997) through the compile function. This section will show how to conduct global SA with pure R and GNU MCSim model code.
The one-compartment PK model can describe the quantity of compound in
the gut lumen (Agutlument) and the rest of body
(Acompartmant). The Ametabolized is the
quantity of compound transform and metabolize through hepatic clearance.
The model mainly includes two state variables that are the quantity of
compound in the gut lumen (Agutlument) and the rest of body
(Acompartmant). The Ametabolized is the
quantity of compound transform and metabolize through hepatic
clearance.
pbtk1cpt <- function(t, state, parameters) {
with(as.list(c(state, parameters)), {
dAgutlument = - kgutabs * Agutlument
dAcompartment = kgutabs * Agutlument - ke * Acompartment
dAmetabolized = ke * Acompartment
Ccompartment = Acompartment / vdist * BW;
list(c(dAgutlument, dAcompartment, dAmetabolized),
"Ccompartment" = Ccompartment)
})
}The parameter values and initial states need to be assigned to
specific values before simulation. Here, we use the corresponding
parameter value of acetaminophen in this example. These model parameters
are derived from the in-vivo or in-vitro experiment
results. The parameter value can be generated from
parameterize_1comp function in httk
package as:
parms <- c(vdist = pars1comp$Vdist,
ke = pars1comp$kelim,
kgutabs = pars1comp$kgutabs,
BW = pars1comp$BW)
initState <- c(Agutlument = 10, Acompartment = 0, Ametabolized = 0)
parmsThe given value of vdist, ke, and
kgutabs in httk are 1.1 (L/kg BW), 0.23
(/h), and 2.18 (/h), respectively. The body weight is assumed to be 70
(kg).
Here shows the given parameter value (parms) and initial
state condition (initState) that need to specify in model
solving. Both parms and initState are “numeric
variables” that contain the value of parameter and initial state
condition.
Using the ode() function in deSolve
package, we can visualize the PK profile according to the given
parameter baseline and the time points (t).
library(deSolve)
t <- seq(from = 0.01, to = 24.01, by = 1)
y <- ode(y = initState, times = t, func = pbtk1cpt, parms = parms)Setting up the parameter distributions. The distribution of parameter is taken to be uniform with bounds corresponding to 50% and 200% of the nominal value.
params <- c("vdist", "ke", "kgutabs", "BW")
q <- c("qunif", "qunif", "qunif", "qnorm")
q.arg <- list(list(min = pars1comp$Vdist / 2, max = pars1comp$Vdist * 2),
list(min = pars1comp$kelim / 2, max = pars1comp$kelim * 2),
list(min = pars1comp$kgutabs / 2, max = pars1comp$kgutabs * 2),
list(mean = pars1comp$BW, sd = 5))
q.argThe parameter ranges are assumed to be 0.55 and 2.2 L/kg BW for
vdist. The ke are ranged from 0.12 to 0.47 /h,
corresponding to half-times of 1.5 and 5.8 hr. The ka are
ranged 1.09 to 4.36 /h. The BW is assumed to a normal
distribution with mean = 70 kg and sd = 5 kg.
Here, we set a sample size of 200 with 10 replications. Through
rfast99() function, a S3 object with class
rfast99 will be created. The set.seed() can
use to reproduce the same parameter matrix in the random sampling. The
sample size determines the robustness of the result of SA. Higher number
of sample size lead to narrower confidence intervals for sensitivity
measurements across different replications. However, it will take a
longer time in computation.
The generated parameters are stored as a 3-D array under the named
a, with the dimension of sample size, the number of
replications, and the number of parameters, respectively.
The sample number is 200, with 4 model parameters, which generates 800 model evaluations. The replication is set to 10. Therefore, the total of 8,000 parameter sequence will be used to compute the corresponding outputs. Figure plotted the sampling process for each parameter from the first 3 replications. The search curves show the different intensity of sampling patterns in each segment.
par(mfrow=c(4,4),mar=c(0.8,0.8,0.8,0),oma=c(4,4,2,1), pch =".")
for (j in c("vdist", "ke", "kgutabs", "BW")) {
if ( j == "BW") {
plot(x$a[,1,j], ylab = "BW")
} else plot(x$a[,1,j], xaxt="n", ylab = "")
for (i in 2:3) {
if ( j == "BW") {
plot(x$a[,i,j], ylab = "", yaxt="n")
} else plot(x$a[,i,j], xaxt="n", yaxt="n", ylab = "")
}
hist <- hist(x$a[,,j], plot=FALSE,
breaks=seq(from=min(x$a[,,j]), to=max(x$a[,,j]), length.out=20))
barplot(hist$density, axes=FALSE, space=0, horiz = T, main = j)
}
mtext("Model evaluation", SOUTH<-1, line=2, outer=TRUE)Because the PK model is being used to describe a continuous process
for the chemical concentration over time, the sensitivity measurements,
therefore, have the time-dependent relationships for each model
parameter. Here we use the defined output time points (t)
to examine the change of the parameter sensitivity over time. To solve
the model through deSolve, we need to provide the
details of the argument, which include time (t), initial
conditions of state variable (initState), output variables
(outnames), and name of the model function
(func). To create the time-dependent sensitivity
measurement, we set the time duration from 0.01 to 24.01 hours with the
time segment of 1 hour as the above definition in ode()
function in this example. The initial time point should avoid 0 to
prevent computational error in misconduct. The outnames is
based on the arguments from the ode() function in
deSolve package.
outputs <- c("Ccompartment", "Ametabolized")
out <- solve_fun(x, time = t, func = pbtk1cpt, initState = initState, outnames = outputs)The output result out is an S3 object of
rfast99() as well, which can link with
print(), plot(), and check()
method to examine the sensitivity measurements. The print()
function gives the sensitivity and convergence indices for main,
interaction, and total order at each time point. In addition to print
out the result of SA, the more efficient way to distinguish the
influence of model parameter is to visualize these indices.
The SI has computed range from 0 (no impact) to 1 (high impact) and represents the contribution percentage of output variance under the given parameter distributions. The solid line represents the total (black) and first (red) order SI with 95% confidence interval (polygon). The dashed line is the cut-off with the default value of 0.05.
Here, we can see that vdist and ke dominate
the plasma concentration before and after 5-hr post chemical intake,
respectively. The parameter kgutabs only plays a crucial
role to determine the plasma concentration in the first hour. However,
the current result only based on the distribution of model parameters
for APAP. Given different input conditions (e.g., range of parameter
uncertainty, chemical-dependent parameter value) the result can of
course change (result not shown).
The default output in the plotting is setting at the first variable.
To exam the time-dependent SI of other variables, such as
Ametabolized in this case, we need to assign the variable
name vars = "Ametabolized" in plot()
function.
The amount of metabolized is also determined by parameter
ke. Same as Ccompartment, the
kgutabs contribute about 30 - 40% variation of model output
in the first hour. The BW is the least important parameter
in the current analysis, and therefore, can be fixed in the model
fitting to data and additional applications.
In addition to using the time-SI profile to investigate the parameter impact on model output, we can directly examine the relationship between parameters and model output graphically.
par(mfcol=c(4,4),mar=c(0.8,0.8,0,0),oma=c(4,4,2,1), pch = ".")
plot(x$a[,1,"vdist"], out$y[,1,"0.01",1], xaxt="n", main = "\nvdist")
plot(x$a[,1,"vdist"], out$y[,1,"2.01",1], xaxt="n")
plot(x$a[,1,"vdist"], out$y[,1,"6.01",1], xaxt="n")
plot(x$a[,1,"vdist"], out$y[,1,"24.01",1])
for (j in c("ke", "kgutabs", "BW")){
for (k in c("0.01", "2.01", "6.01", "24.01")){
if (k == "0.01") {
plot(x$a[,1,j], out$y[,1,k,1], yaxt = "n", xaxt="n", main = paste0("\n", j))
} else if (k == "24.01") {
plot(x$a[,1,j], out$y[,1,k,1], yaxt = "n")
} else plot(x$a[,1,j], out$y[,1,k,1], xaxt = "n", yaxt = "n")
}
}
mtext("Parameter", SOUTH<-1, line=2, outer=TRUE)
mtext("Ccompartment", WEST<-2, line=2, outer=TRUE)Here shows the relationship between the concentration of the rest of
the body (Ccompartment) and the model parameters at times
0.01, 2.01, 6.01, and 24.01 hr (top to bottom), respectively. We can
find that kgutabs and vdist have higher
correlation with Ccompartment in the beginning (t = 0.01 h)
of post-intake duration compared with other parameters, suggesting that
the parameters have high impact on the modeling result. The
ke shows a high correlation at the later time period (t =
24.01 h). The parameter BW does not show any obvious
relationship with Ccompartment.
The output variable out containing all the input
arguments detailed above and the calculated SI of first order
(mSI), interaction (iSI), and total order
(tSI). Convergence indices are also stored in the list
named mCI, iCI, and tCI. The
outputs are formatted as 4-D array in y with the dimension
name of model evaluation, number of replications, number of time points,
and number of output variables, respectively.
Some functions in pksensi provide efficient ways to
check the result from global SA. The check() can determine
which parameters have relatively lowered sensitivity measurement across
the given time points and model outputs, and therefore can be applied
parameter fixing in model calibration. The check() also
provides an argument to specify the target output or change the cut-off
value. The argument of cutoff set for example at 0.05, is
used to detect the relative non-influential parameters as default, in
this case representing a 5% change of the output is contributed from the
specific parameter variation.
Based on the sensitivity measurement of the total order, the result
shows that BW has a relative lower measurement of SI.
However, all parameters do not converge to the setting cut-off, which
means the larger sample size is required in further sensitivity testing.
Similar to the plot() function that can assign specific
output variable in the examination, the check() function
can also use the assignment (vars) to examine a given
output.
Running model under GNU MCSim native code can have a faster speed to obtain the model outputs. This subsection will show how to conduct global SA with deSolve package with GNU MCSim model code.
The code must first be compiled to run the model. After create the
model file, we can use compile_model() function to generate
the file that has dynamic-link library (DLL) or share object (SO)
extension and can be linked dynamically into an R session
(pbtk1cpt.dll on Windows or pbtk1cpt.so on
other systems) and R file (pbtk1cpt_inits.R) with default
input parameters and initial state settings with the definition of
application = "R".
The pbtk1cpt_inits.R file includes
initParms and initStates functions and
Outputs variable. The created function have default value
of model parameters and initial conditions that can further use to
customize in the simulation.
The parameter values and initial states can be customized to the
specific condition. It can also schedule for the given dosing scenario.
In the current setting, we assumed the initial condition of the intake
dose to be 1000 mg. We can use initParms and
initStates functions to customize the parameter values and
the initial state that will be used in the following modeling and SA.
These additional functions are generated when compiling the model file.
We used the the same parameter values in this section.
parms <- initParms()
parms["vdist"] <- pars1comp$Vdist
parms["ke"] <- pars1comp$kelim
parms["kgutabs"] <- pars1comp$kgutabs
parms["BW"] <- pars1comp$BW
initState <- initStates(parms=parms)
initState["Agutlument"] <- 10Here shows the given parameter value (parms), initial
state condition (initStat), and the output variable
(Outputs) that need to specify in model solving.
Using the ode() function in deSolve
package. Unlike above example, we have to assign additional arguments,
such as dllname, initfunc, nout,
and outnames.
t <- seq(from = 0.01, to = 24.01, by = 1)
y <- ode(y = initState, times = t, func = "derivs", parms = parms,
dllname = mName, initfunc = "initmod",
nout = 1, outnames = Outputs)Same as above, use the following code to define the parameter distributions and generate parameter matrix.
# Define parameter distribution
q <- c("qunif", "qunif", "qunif", "qnorm")
q.arg <- list(list(min = parms["vdist"] / 2, max = parms["vdist"] * 2),
list(min = parms["ke"] / 2, max = parms["ke"] * 2),
list(min = parms["kgutabs"] / 2, max = parms["kgutabs"] * 2),
list(mean = parms["BW"], sd = 5))
params <- c("vdist", "ke", "kgutabs", "BW")
# Generate parameter matrix
set.seed(1234)
x <- rfast99(params, n = 200, q = q, q.arg = q.arg, replicate = 10)Run simulation and check result.
outputs <- c("Ccompartment", "Ametabolized")
out <- solve_fun(x, time = t, initState = initState, outnames = outputs, dllname = mName)
check(out)The simulation time had huge improvement when using GNU MCSim model code.
In addition to use deSolve to solve differential
equations in PK model, the GNU MCSim can provide better
computational efficiency. To solve ODE through GNU
MCSim, we need to change the argument to
application = mcsim in compile_model()
function. The computing time of using solve_fun() function
in SA is estimated as,
Then, before we conduct the SA through GNU MCSim, The following code is used to compile the GNU MCSim model code to the executable program.
Similar to solve_fun() function that can define the
initial value of parameter and state variable through generated
functions, the solve_mcsim() also has a
condition argument that can be used to give the specific
input value such as exposure dose, fixed parameter value or initial
condition of state variable.
conditions <- c("Agutlument = 10")
system.time(out <- solve_mcsim(x, mName = mName, params = params,
vars = Outputs, time = t,
condition = conditions))After solving the equations under the same given condition, we can find that GNU MCSim provide faster speed in computing performance than using deSolve. In this case, we only focus on performing the global SA alone for generic PK model without additional comparison with experiment data. The PBPK example will display and reproduce our previous published result (Hsieh et al. 2018) with full global SA workflow.