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Edited by: Thomas Hartung, Universität Konstanz, Germany

Reviewed by: Joanna Jaworska, Procter & Gamble, Belgium; Melvin Anderson, The Hamner Institutes for Health Sciences, USA

*Correspondence: George D. Loizou, Mathematical Sciences Unit, Health and Safety Laboratory, Harpur Hill, Buxton, Derbyshire SK17 9JN, UK. e-mail:

This article was submitted to Frontiers in Predictive Toxicity, a aspecialty of Frontiers in Pharmacology.

This is an open-access article subject to a non-exclusive license between the authors and Frontiers Media SA, which permits use, distribution and reproduction in other forums, provided the original authors and source are credited and other Frontiers conditions are complied with.

Physiologically based pharmacokinetic (PBPK) models have a potentially significant role in the development of a reliable predictive toxicity testing strategy. The structure of PBPK models are ideal frameworks into which disparate

Current approaches to testing industrial and agricultural chemicals for potential toxicity in people are inefficient, expensive, and reliant on animal experimentation. As a consequence most chemicals in global commerce today have undergone limited or no safety testing at all (Judson et al.,

Over the last few decades alternative methods of evaluating the toxicological hazard of chemical compounds has focused on the potential of

The need for alternative approaches to toxicity testing is clear. Whether this involves the development of “traditional,” direct, one to one replacement of animal tests with

A PBPK model is an independent, structural model, comprising compartments that correspond directly and realistically to the organs and tissues of the body (e.g., adipose, brain, gut, heart, kidney, liver, lung, muscle, spleen, skin, and bone) and connected by the cardiovascular system. They are mathematical descriptions of biological systems that are translated into computer code and solved computationally. They are frameworks that can capture our understanding of the science underlying the biological processes that lead to disease. A PBPK model is

Sensitivity analysis (SA) allows the model output uncertainty to be ascribed to the source within the model thereby offering a means of evaluating the consistency between internal model structure and the system it tries to emulate (Campolongo and Saltelli,

When trying to establish the contribution of a parameter to model predictions, OAT SA techniques are fairly rapid and simple to implement but can give somewhat misleading results if there are substantial interactions among multiple parameters (Campolongo and Saltelli,

Since we are often interested in concentration-time profiles of xenobiotics in biological systems, here we describe an approach for uncertainty and SA adapted to consider sensitivity indices of PBPK model parameters that vary with time. We propose the elements of a workflow for the application of global SA during PBPK model development and evaluation.

Finally, there is a need to develop increased awareness and confidence in the use of PBPK models. Therefore, an important objective underpinning this work is to remain mindful of the need to develop user-friendly tools that shift the emphasis away from mathematical and programming expertise to the biology underlying RA and to present such information in a manner that is acceptable to toxicologists, risk assessors, and regulators. With the future development of intuitive, user-friendly tools, we believe that the workflow we propose does not require in-depth mathematical expertise and could be undertaken by biological scientists. Mathematical equations are included for the interested reader but may be skipped without diminishing the primary objective of this report and we endeavored to provide examples with clear biological relevance. We hope that this work can make a contribution to the development of good PBPK modeling practice and facilitate the dialog between toxicologists and risk assessors and regulators (Kohn,

A human PBPK model describing a bladder compartment to simulate fluctuations in metabolite concentration in urine caused by micturition (Franks et al.,

Parameter | Abbreviation | Value | Range |
---|---|---|---|

Molecular mass |
MW_{xyl} |
106.17 | – |

Molecular mass MHA (g/mol) | MW_{MHA} |
193.2 | – |

Body mass (kg) | BW | 75 | 49–92 |

Proportion of vascularized tissue (body mass) | VT | 0.91 | – |

Cardiac output (L h^{−1} BW^{0.75}) |
QCC | 12 | 10–20 |

^{−1}) |
_{M} |
11.8 | 9.1–14.6 |

^{−1} mg^{−1} microsomal protein) |
_{max} |
895 | 761–1028 |

Microsomal protein yield per gram wet weight liver (mg g^{−1}) |
MPY | 32 | 18–75 |

Respiratory rate (L h^{−1} BW^{0.75}) |
QPC | 12 | 10–20 |

Respiratory dead space (proportion respiratory rate) | DS | 0.3 | 0.2–0.33 |

Blood:air partition coefficient | Pba | 19 | 12–26 |

Rapidly perfused | Prpda | 117 | 50–150 |

Slowly perfused | Pspda | 53 | 40–80 |

Adipose | Pfaa | 1874 | 1400–2200 |

Liver | Plia | 279 | 150–350 |

Rapidly perfused | QrpdC | 0.48 | |

Slowly perfused | QspdC | 0.22 | 0.2–0.35 |

Adipose | QfaC | 0.05 | 0.09–0.10 |

Liver | QliC | 0.25 | 0.2–0.3 |

Rapidly perfused | VrpdC | 0.09 | |

Slowly perfused | VspdC | 0.604 | |

Adipose | VfaC | 0.19 | 0.07–0.28 |

Liver | VliC | 0.0257 | 0.02–0.031 |

Rate of urine production (L h^{−1}) |
Rurine | 0.07 | 0.06–0.115 |

Urinary creatinine concentration (mmol L^{−1}) |
CRE | 12.5 | 7–15 |

Elimination rate constant (h^{−1}) |
_{1} |
10 | 2–18 |

Where, MRLi is the rate of metabolism of _{MHA} and MW_{xyl} are the molecular weights of MHA and _{B} is the amount of MHA in the blood, _{1}_{B} from the blood to the urine, _{max} is the limiting rate and _{M}_{U} is the amount of MHA in the urine, Vol_{U} is the volume of urine in the bladder and Cre is the concentration of creatinine. The concentration of MHA in the urine was expressed in millimole/mole creatinine. To imitate micturition, the bladder is assumed to fill with urine at a constant (but adjustable) rate and empty at discrete time intervals (when the volume of urine reduces to zero). This enables comparison to be made between model predictions and experimental observations with timed sampling in human volunteer studies (Franks et al.,

The Michaelis–Menten constant _{M}_{max} for hepatic metabolism of _{max} was obtained by multiplying the ^{−1} wet weight liver and the mass of liver (g) (Howgate et al.,

The extended Fourier amplitude sensitivity test (eFAST) method proposed in this paper is one of a suite of methods for a quantitative global SA. Alternative model independent methods for calculating main effect and total effect sensitivity indices include the method of Sobol (

Some alternative methods for a global SA that require many fewer model evaluations are available. These include a class of methods based upon regression analysis (Helton and Davis,

In this study the two-step approach to SA of large models proposed by Campolongo and Saltelli (

The Morris test as a preliminary screening exercise to identify the subset of the most potentially explanatory parameters of model output.

The eFAST for the quantitative analysis of the selected subset of parameters.

The Morris method is global in the sense that it is obtained by taking average values of local measures throughout the input space and produces two sensitivity measures for each parameter, μ and σ. A high μ indicates a factor with an important overall influence on model output; a high σ indicates either a factor interacting with other factors or a factor whose effects are non-linear. The magnitude of μ and σ for each model parameter is relative, i.e., a parameter has a low μ relative to the parameter with the highest μ. For a screening method to be effective, the probability of not identifying a factor that is important must be low. Previous exercises using the Morris method have satisfied this requirement (Campolongo and Saltelli,

The eFAST test is a variance-based global method that is independent of any assumptions regarding model structure (it does not rely on assumptions as to the functional relationship between the model output and its inputs) and is valid for use with non-monotonic models (models that do not give exclusively increasing or decreasing predictions). The method provides a way to estimate the expected value and variance of the dose metric (model output variable) and the contribution of input parameters and their interactions to this variance, given physiologically feasible parameter ranges for inputs. It is important that interactions are identified if the applications of SA are to be fully realized (Campolongo and Saltelli, _{i}_{i}

The workflow comprises the following steps:

Perform the screening exercise using the Morris test

Identify and select the most important parameters

Identify the time period where model output variance is of interest

Perform eFAST on the most potentially explanatory subset of parameters

Present _{i}

In order to test the validity of the screening exercise as a single initial step for parameter sensitivity measures that change over time, the means of μ and σ were calculated over a period of 0–14 h corresponding to the change in venous blood concentration of _{urine}; Figure

_{urine}. In panel

In order to delineate the various steps of a possible workflow the analysis initially was conducted at 40 ppm, the target exposure concentration of the human volunteer studies. For CV eFAST was run at 3–5 h, to investigate parameter sensitivities in the distribution and elimination phases (Figure _{urine} to investigate the early and latter urinary elimination phase (Figure _{M}

Parameter ranges used for both the Morris Screening and eFAST tests were identical and are listed in Table ^{1}

Parameter ranges were set at the 5 and 95th percentiles of the distributions, with the exceptions of the PCs

The mean value and range for _{1}_{B}_{1}

Venous blood ^{3} in volume (Loizou et al.,

The numerical solutions to the model equations were obtained using acslX Libero version 3.0.1.6 (AEgis Technologies^{2}^{3}

Results of the Morris screening exercise for variance in CV and C_{urine} are shown in Figures _{urine}, are annotated. The parameters are ranked in order of importance according to μ for both CV and C_{urine} in Table _{urine}. Unlike the Morris test mean sensitivity indices, it is not straightforward to take averages of _{i}

_{urine}).

Table _{urine}. A tick next to an

The Morris screening test was then conducted at the same time points as the eFAST analysis. Tables _{urine}. The rankings are not exactly the same although more parameters are ranked in the top 10 or bottom 9 by both methods. For CV at 3 h _{urine} the following _{1}_{M}_{1}_{1}_{M}_{1}

_{urine}

The computing time for the Morris screening test for all 19 parameters for both CV and C_{urine} was approximately 8 s.

In addition to calculating _{i}_{urine} did not require scaling to be graphed. The figures also highlight that variance is very sensitive to the units of measurement and may be only weakly related to the uncertainty in the level of the substance (in blood, urine, or any body compartment of interest) that arises due to uncertainty in the parameters of the model. This does not impact on the functionality of variance-based methods for SA, however the standard deviation (the square root of the variance) is a more appropriate measure of the underlying uncertainty in the model output that results from parameter value uncertainty. Some outputs in the PBPK model may be much more sensitive to the inputs than others. The total variance can be used to compare the sensitivity of model outputs (to the model inputs) provided those outputs have the same units of measurement. For example, if the variance of the substance in the liver was found to be much larger than the variance of the substance in the kidney, one could conclude that the liver parameter value (e.g., mass, perfusion rate, PC) was a much more sensitive parameter than the corresponding kidney parameter.

Figures _{i}_{i}

_{i} (black bar) and any interactions with other parameters (grey bar) given as a proportion of variance. The ribbon, representing variance due to parameter interactions, is bounded by the cumulative sum of main effects (lower bold line) and the minimum of the cumulative sum of the total effects (upper bold line), _{urine} at 5 h and _{urine} at 8 h.

The amount of variance that is accounted for by including all parameters up to a specified point is bounded below by the cumulative sum of the ordered main effects (the base of the ribbon) and is bounded above by the cumulative sum of the total effects. Note that there is multiple accounting of the interactions associated with each parameter, which causes the total to exceed 100% eventually. In fact, we have an even stricter limit for the upper bound: it cannot exceed 100% minus the sum of the main effects that are not included in the cumulative sum up to that point (the top of the ribbon).

More formally, the lower bound of the included variance for the first k parameters is given by

and the upper bound is given by

where

Thus, the amount of variance that is accounted for by including all parameters up to a specified point can be determined by taking a cross section of the ribbon above that parameter, i.e., the range from _{k}_{k}

The contribution of different parameters to variance of CV changes with time. The most notable being _{max}_{max}_{M}_{max}

Figures _{urine} at 5 and 8 h. In this case _{urine}. Beyond _{i}_{i}

Figure

The computing time for the eFAST analysis of 19-model parameters for CV and C_{urine} was approximately 13 h each.

The probability of non-identification of important parameters by the Morris test has been reported to be low (Campolongo et al., _{urine} disagreement with the Morris screening occurred at each time point. For both CV and C_{urine} the disagreements in rankings were small and occurred with parameters that accounted for a very small percentage of dose metric variance. It is recommended that the screening phase be performed at the same time points as the eFAST analysis.

The computational cost of performing global SA may be an important criterion for determining whether a screening exercise is required. In this study the eFAST conducted on a PBPK model with 19 parameters took approximately 13 h to complete on a dual core desktop PC. However, the run time increases substantially as the number of parameters is increased. In a preliminary investigation the number of model parameters was increased to 25 (each compartment requires a mass, perfusion rate and PC) by adding a kidney and brain compartment. The run time for eFAST analysis of the 25-parameter model increased to 72 h. Therefore, in the case of PBPK models with modest numbers of parameters, e.g., 19–25, the screening should be used to reduce the number of parameters for quantitative analysis by de-selecting the 10

The eFAST results are consistent with other studies showing that the influence of model parameters on model output follow a Pareto-like distribution (Campolongo et al., _{urine}. At 3 and 5 h for CV and 3 and 8 h for C_{urine}, the computation of sensitivity measures occurs in a region of parameter space that corresponds to the time period where the concentration of

The keys points are that it is important to conduct the SA at a dose within the range of any experimental data as the results from the SA may be sensitive to the dose. However, an advantage of conducting the SA at multiple doses is that the sensitivity of results, in effect the degree of interaction between the parameters of the PBPK model and the dose, can be assessed. Perhaps most importantly, whilst there were some important differences in the most sensitive parameters, there was little change in the ordering of the least important parameters. There was a consistency in the least important parameters at all three doses, therefore the approach suggested in this workflow, of eliminating the least sensitive parameters, was independent of the dose.

The primary objective in the paper was to introduce the elements of a workflow for SA and some of the technical details have been omitted from the methodology for clarity of presentation. One important issue is on the assumed probability distribution for the parameters. In the examples presented in the paper uniform distributions were assumed in all cases. Oakley and O’Hagan (

If little information is known about the model parameters it may be reasonable to assume uniform distributions on the inputs (although a robust SA might also examine the results under a variety of assumed probability distributions). However, if it is feasible to obtain reliable probability distributions from sources such as PopGen these should be used in sensitivity analyses. After appropriate transformations of the parameters a variety of distributions can be used in eFAST. Some minor differences in the SA results in this work would have resulted from different probability distributions on the inputs.

In practice there is a balancing act between the availability of a suite of probability distributions and ease of use. The authors consider uniform distributions as a default setting, coupled with a choice of uniform, log-uniform, normal, and log-normal probability distributions for model parameters to be a reasonable compromise.

Despite greater computational cost a number of factors shift the balance in favor of eFAST over OAT techniques. These are: (i) different parameters are affected by different ranges of variation and uncertainty in different regions of parameter space, i.e., different patterns of parameter sensitivity predominate in different regions of parameter space, (ii) the presence of significant interactions between parameters should never be discounted, (iii) presentation of quantitative information on SA in the form of a bar chart is intuitive. Specifically, the presentation of the main effects, _{i}

We have defined the basis of a workflow for SA of PBPK models that is computationally feasible, accounts for interactions between parameters, and can be displayed in an intuitive manner. When used to analyze PBPK models containing up to 25 parameters, Morris screening should be used to identify the 10

This work was supported by CEFIC-LRI (Grant No: LRI-B3.7.2-HSLC-081010). The authors thank Conrad Housand and Robin McDougall of The Aegis Technologies Group, Inc for their help in configuring the global sensitivity analysis scripts and for also reviewing the manuscript.

The plots were created using R (R Development Core Team,

The data are assumed to have three columns:

• “Parameter,” containing the name of the parameter

• “Main Effect,” containing the size of the main effects

• “Interaction,” containing the size of the interactions

In practise, you are likely to read in the data from a file, but for simplicity, here is some code to create sample data.

Sometimes it is easier to use the data in wide format, other times in long format. This data fortification process returns both, plus some extra columns.

The plot is essentially a bar chart plus a cumulative frequency ribbon, with some minor adjustments to improve appearance of the plot.

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