The aim of this vignette is to
reproduce our previous published (Hsieh et al.
2018) result of global sensitivity analysis for acetaminophen
PBPK model through pksensi. The model codes are
included in this package and can be generated through
pbpk_apap_model(). We applied the global sensitivity
analysis workflow to the original published model with 21 model
parameters (Zurlinden and Reisfeld 2016).
The descriptions of each parameter and the sampling ranges are list in
Table 1.
Same as the example of one-compartment PK model. The model parameter and the corresponding sampling range should be defined to create the parameter matrix. Previously, the probability distributions of model parameters were set to either truncated normal or uniform distribution when the parameters have informative prior information or not. To rapidly reach the acceptance convergence, we apply uniform distribution for all testing parameters. The ranges of informative parameters are set to 1.96-times difference for single side under log-scaled (approximate 54.6 times difference between minimum and maximum in natural scaled). The nominal values of informative model parameters were defined as:
# Nominal value
Tg <- log(0.23)
Tp <- log(0.033)
CYP_Km <- log(130)
SULT_Km_apap <- log(300)
SULT_Ki <- log(526)
SULT_Km_paps <- log(0.5)
UGT_Km <- log(6.0e3)
UGT_Ki <- log(5.8e4)
UGT_Km_GA <-log(0.5)
Km_AG <- log(1.99e4)
Km_AS <- log(2.29e4)
rng <- 1.96 Generally, wide range of parameter value might cause the computing
error when solving the differential equation. One of the effective ways
to prevent this problem is to adjust the value of relative and absolute
error tolerance to control the error appearance by resetting these
parameters in a lower value. The generate_infile() and
solve_mcsim() provide the arguments of rtol
and atol that adjust the error tolerance to prevent the
unwanted error. However, the modification will decrease the computing
speed. Therefore, the alternative method to prevent this issue is to
detect the crucial parameter range that causes the problem. Also,
setting the maximum number of steps to higher value instead of using the
default value (500) in GNU MCSim can prevent this
problem (internally defined). The maximum number of step is set to 5000
in this case. Here we separate the global SA of APAP-PBPK model process
to several steps.
The model code needs to be prepared in the following global SA
workflow. After creating the pbpk_apap.model file in the
working directory, the next step is to generate the executable program
(mcsim.pbpk_apap) through compile_model()
function.
The 21 testing model parameters are defined in this part, including
parameter name, probability distribution, and distributed parameter
value. To prevent the computing error, the range of
SULT_VmaxC and UGT_VmaxC need to adjust from
U(0, 15) (Zurlinden and Reisfeld 2016) to U(0, 10) (Hsieh et al. 2018). The objects q
and dist are set to the type of distribution that will use
to generate the parameter matrix in GNU MCSim (for
uncertainty analysis) and R (for SA).
params <- c("lnTg", "lnTp", "lnCYP_Km","lnCYP_VmaxC",
"lnSULT_Km_apap","lnSULT_Ki","lnSULT_Km_paps","lnSULT_VmaxC",
"lnUGT_Km","lnUGT_Ki","lnUGT_Km_GA","lnUGT_VmaxC",
"lnKm_AG","lnVmax_AG","lnKm_AS","lnVmax_AS",
"lnkGA_syn","lnkPAPS_syn", "lnCLC_APAP","lnCLC_AG","lnCLC_AS")
dist <- rep("Uniform", 21)
q <- rep("qunif", 21)
q.arg <-list(list(Tg-rng, Tg+rng), list(Tp-rng, Tp+rng),
list(CYP_Km-rng, CYP_Km+rng), list(-2., 5.),
list(SULT_Km_apap-rng, SULT_Km_apap+rng),
list(SULT_Ki-rng, SULT_Ki+rng),
list(SULT_Km_paps-rng, SULT_Km_paps+rng),
list(0, 10), list(UGT_Km-rng, UGT_Km+rng),
list(UGT_Ki-rng, UGT_Ki+rng),
list(UGT_Km_GA-rng, UGT_Km_GA+rng),
list(0, 10), list(Km_AG-rng, Km_AG+rng),
list(7., 15), list(Km_AS-rng, Km_AS+rng),
list(7., 15), list(0., 13), list(0., 13),
list(-6., 1), list(-6., 1), list(-6., 1))To optimize the computing speed, this case only uses GNU
MCSim to estimate the concentration of APAP and its metabolites
glucuronide (APAP-G) and sulfate (APAP-S) in plasma. The setting oral
dose of APAP is 20 mg/kg in this example. Generally, the input dosing
method can be defined through the condition argument. Since
the unit of the given dose is mg/kg, the mgkg_flag is set
to 1. More definition of input schedule functions can be found in the
section of input functions in GNU MCSim User’s Manual
(https://www.gnu.org/software/mcsim/mcsim.html#Input-functions).
We apply uncertainty analysis through the solve_mcsim()
and visualize the result by pksim() function. Some example
data are included in the pksensi with experiment time
(h) and concentration (mg/L).
In the setting condition of simulation, The relative and absolute
error tolerance (rtol & atol) were set to
1e-7 and 1e-9, respectively, to prevent the computing error. The Monte
Carlo simulation is run for 1000 iteration as the assignment of
monte_carlo. The input file (‘sim.in’) and output file
(‘simmc.out’) will be generated under the standard ASCII format.
set.seed(1111)
out <- solve_mcsim(mName = mName, params = params, vars = vars,
monte_carlo = 1000, dist = dist, q.arg = q.arg,
time = times, condition = conditions,
rtol = 1e-7, atol = 1e-9)par(mfrow = c(1,3), mar = c(4,4,1,1))
pksim(out, xlab = "Time (h)", ylab = "Conc. (ug/L)", main = "APAP")
points(APAP$Time, log(APAP$APAP * 1000))
pksim(out, vars = "lnCPL_AG_mcgL", xlab = "Time (h)", main = "APAP-G",
ylab = " ", legend = FALSE)
points(APAP$Time, log(APAP$AG * 1000))
pksim(out, vars = "lnCPL_AS_mcgL", xlab = "Time (h)", main = "APAP-S",
ylab = " ", legend = FALSE)
points(APAP$Time, log(APAP$AS * 1000))Here shows the coverage checks of prior PBPK model predictions with calibrated APAP data. For parent compound, all data points are located in the simulated interval of 25-75%. Through this result, we can determine that the simulated outputs can accurately generate the same concentration profile as the in-vivo experiment under the setting of parameter ranges for APAP. The simulated result of metabolites APAP-G shows the different pharmacokinetic profile with experiment data. However, all data points are located in the simulated interval.
In global SA, we have to additionally generate the parameter matrix from the eFAST method. The current setting uses 512 sample size with 10 replication.
To conduct the global SA with GNU MCSim and
pksensi, the input file with given “setpoint” condition
should be generated before modeling. The file can create by
generate_infile function. The solve_mcsim can
also automatically create the input file and compute the output.
The plotting function can create the result of time-dependent sensitivity measurement to determine the parameter impact on model output over time.
In addition, through using the check, the parameter with
sensitivity and convergence indices over the given condition can be
preliminary detected for all output variables. Based on our previous
study, we proposed the heatmap visualization approach
heat_check to distinguish “influential” and
“non-influential” parameters with a “cut-off” point. Through the given
argument order, we can select the specific order of
sensitivity measurement that we’re interested in.
In the default setting, the heat_check can only show the
influential parameters. The argument show.all is used to
show all results. Adding the index = "CI" in the function
can further investigate the convergence index. Based on the current
setting of sampling size, most parameters cannot reach the acceptable
criteria of convergence. Therefore, a higher number of sampling is
necessary. The sample size of convergence in the current PBPK model is
8,192 (Hsieh et al. 2018). However, based
on the current sample size we still can find 6 parameters that can be an
important parameter for the plasma APAP concentration.