PBPK Modeling to Simulate the Fate of Compounds in Living Organisms

Abstract

Pharmacokinetics study 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 1

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 fonction 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 Carldo 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 Cardo 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. #=================================================================