Modeling Pharmacokinetics

Abstract

Pharmacokinetics is the study of the fate of xenobiotics in a living organism. Physiologically based pharmacokinetic (PBPK) models provide realistic descriptions of xenobiotics’ absorption, distribution, metabolism, and excretion processes. They model the body as a set of homogeneous compartments representing organs, and their parameters refer to anatomical, physiological, biochemical, and physicochemical entities. They offer a quantitative mechanistic framework to understand and simulate the time-course of the concentration of a substance in various organs and body fluids. These models are well suited for performing extrapolations inherent to toxicology and pharmacology (e.g., between species or doses) and for integrating data obtained from various sources (e.g., in vitro or in vivo experiments, structure–activity models). In this chapter, we describe the practical development and basic use of a PBPK model from model building to model simulations, through implementation with an easily accessible free software.

Appendix

R script for the butadiene PBPK model:

#=================================================================

# Butadiene human PBPK model

# Define and initialize the state variables

y = c("Q_fat" = 0,   # Quantity of butadiene in fat (mg)

      "Q_wp"  = 0,   # ~        in well-perfused (mg)

      "Q_pp"  = 0,   # ~        in poorly-perfused (mg)

      "Q_met" = 0)   # ~        metabolized (mg)

# Define the model parameters

# Units:

# Volumes: liter

# Time:    minute

# Flows:   liter / minute

parameters = c(

  "BDM"    = 73,          # Body mass (kg)

  "Height" = 1.6,         # Body height (m)

  "Age"    = 40,          # in years

  "Sex"    = 1,           # code 1 is male, 2 is female

  "Flow_pul"      = 5,    # Pulmonary ventilation rate (L/min)

  "Pct_Deadspace" = 0.7,  # Fraction of pulmonary deadspace

  "Vent_Perf"     = 1.14, # Ventilation over perfusion ratio

  "Pct_LBDM_wp"   = 0.2,  # wp tissue as fraction of lean mass

  "Pct_Flow_fat"  = 0.1,  # Fraction of cardiac output to fat

  "Pct_Flow_pp"   = 0.35, # ~                          to pp

  "PC_art" = 2,           # Blood/air partition coefficient

  "PC_fat" = 22,          # Fat/blood ~

  "PC_wp"  = 0.8,         # wp/blood  ~

  "PC_pp"  = 0.8,         # pp/blood  ~

  "Kmetwp" = 0.25)        # Rate constant for metabolism (1/min)

# The input air concentration (in parts per million) can vary with time

C_inh = approxfun(x = c(0,120), y = c(10,0), method="constant", f=0, rule=2)

# Check the input concentration profile just defined

plot(C_inh(1:300), xlab = "Time (min)",

     ylab = "Butadiene air concentration (ppm)", type = "l")

# Define the model equations

bd.model = function(t, y, parameters) {

 with (as.list(y), {

  with (as.list(parameters), {

  # Define some useful constants

  MW_bu = 54.0914    # butadiene molecular weight (in grams)

  ppm_per_mM = 24450 # ppm to mM under normal conditions

  # Conversions from/to ppm

  ppm_per_mg_per_l = ppm_per_mM / MW_bu

  mg_per_l_per_ppm = 1 / ppm_per_mg_per_l

  # Calculate Flow_alv from total pulmonary flow

  Flow_alv = Flow_pul * (1 - Pct_Deadspace)

  # Calculate total blood flow from Flow_alv and the V/P ratio

  Flow_tot = Flow_alv / Vent_Perf

  # Calculate fraction of body fat

  Pct_BDM_fat = (1.2 * BDM / (Height * Height) - 10.8 *(2 - Sex) +

                 0.23 * Age - 5.4) * 0.01

  # Actual volumes, 10% of body mass (bones…) get no butadiene

  Eff_V_fat = Pct_BDM_fat * BDM

  Eff_V_wp  = Pct_LBDM_wp  * BDM * (1 - Pct_BDM_fat)

  Eff_V_pp  = 0.9 * BDM - Eff_V_fat - Eff_V_wp

  # Calculate actual blood flows from total flow and percent flows

  Flow_fat = Pct_Flow_fat * Flow_tot

  Flow_pp  = Pct_Flow_pp  * Flow_tot

  Flow_wp  = Flow_tot * (1 - Pct_Flow_pp - Pct_Flow_fat)

  # Calculate the concentrations

  C_fat = Q_fat / Eff_V_fat

  C_wp  = Q_wp  / Eff_V_wp

  C_pp  = Q_pp  / Eff_V_pp

  # Venous blood concentrations at the organ exit

  Cout_fat = C_fat / PC_fat

  Cout_wp  = C_wp  / PC_wp

  Cout_pp  = C_pp  / PC_pp

  # Sum of Flow * Concentration for all compartments

  dQ_ven = Flow_fat * Cout_fat + Flow_wp * Cout_wp + Flow_pp * Cout_pp

  C_inh.current = C_inh(t) # to avoid calling C_inh() twice

  # Arterial blood concentration

  # Convert input given in ppm to mg/l to match other units

  C_art = (Flow_alv * C_inh.current * mg_per_l_per_ppm + dQ_ven) /

          (Flow_tot + Flow_alv / PC_art)

  # Venous blood concentration (mg/L)

  C_ven = dQ_ven / Flow_tot

  # Alveolar air concentration (mg/L)

  C_alv = C_art / PC_art

  # Exhaled air concentration (ppm!)

  if (C_alv <= 0) {

    C_exh = 10E-30 # avoid round off errors

  } else {

    C_exh = (1 - Pct_Deadspace) * C_alv * ppm_per_mg_per_l +

            Pct_Deadspace * C_inh.current

  }

  # Quantity metabolized in liver (included in well-perfused)

  dQmet_wp = Kmetwp * Q_wp

  # Differentials for quantities

  dQ_fat = Flow_fat * (C_art - Cout_fat)

  dQ_wp  = Flow_wp  * (C_art - Cout_wp) - dQmet_wp

  dQ_pp  = Flow_pp  * (C_art - Cout_pp)

  dQ_met = dQmet_wp

  # The function bd.model must return at least the derivatives

  list(c(dQ_fat, dQ_wp, dQ_pp, dQ_met),     # derivatives

       c("C_ven" = C_ven, "C_art" = C_art)) # extra outputs

  }) # end with parameters

 }) # end with y

} # end bd.model

# Define the computation output times

times = seq(from=0, to=1440, by=10)

# Call the ODE solver

library(deSolve)

results = ode(times = times, func = bd.model, y = y, parms = parameters)

# results is basically a table

results

# Plot the results of the simulation

plot(results)

# End

# End Simple Simulation.

#=================================================================

#=================================================================

# Monte Carlo simulations

# We assume that a simple simulation has already been run, so that

# y, parameters, C_inh, and bd.model have all been defined and that

# deSolve has been loaded.

for (iteration in 1:1000) { # 1000 Monte Carlo simulations…

  # Sample randomly some parameters

  parameters["BDM"]      = rnorm(1, 73,   7.3)

  parameters["Flow_pul"] = rnorm(1, 5,    0.5)

  parameters["PC_art"]   = rnorm(1, 2,    0.2)

  parameters["Kmetwp"]   = rnorm(1, 0.25, 0.025)

  # Reduce output times eventually. We only care about time 1440,

  # but time zero still needs to be specified

  times = c(0, 1440)

  # Integrate

  tmp = ode(times = times, func = bd.model, y = y, parms = parameters)

  if (iteration == 1) { # initialize

   results = tmp[2,-1]

   sampled.parms = c(parameters["BDM"],    parameters["Flow_pul"],

                     parameters["PC_art"], parameters["Kmetwp"])

  } else { # accumulate

   results = rbind(results, tmp[2,-1])

   sampled.parms = rbind(sampled.parms,

                    c(parameters["BDM"],    parameters["Flow_pul"],

                      parameters["PC_art"], parameters["Kmetwp"]))

  }

} # end Monte Carlo loop

# Save the results, specially if they took a long time to compute

save(sampled.parms, results, file="MTC.dat.xz", compress = "xz")

# use load(file="MTC.dat.xz") to read them back in

# Plot the results

hist(sampled.parms[,1])

hist(results[,1])

plot(sampled.parms[,1], results[,1])

# End Monte Carlo Simulations.

#=================================================================