# The Application of a Physiologically Based Toxicokinetic Model in Health Risk Assessment

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## Abstract

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## 1. Introduction

## 2. The Construction of the PBTK Model

#### 2.1. Model Characterization

#### 2.1.1. Determination of the Model Structure

#### 2.1.2. Building Mathematical Equations

_{t}/dt is the amount if change of xenobiotics in a compartment per unit time; A

_{t,in}is the amount of entry into the compartment; A

_{t,out}is the quantity leaving the compartment; A

_{t,e}is the amount excreted by the compartment; and A

_{t,m}is the amount of compartment metabolism.

- (1)
- Perfusion-limited model: in this model, xenobiotic concentrations in organs, tissues, and blood reach instant equilibrium without concentration differences. Organs, tissues, and blood are considered as a single compartment with a homogeneous distribution of xenobiotics. The only limiting factor for xenobiotic distribution is blood flow velocity [12,13]. This model is suitable for small lipid-soluble molecules that can readily cross membrane barriers and when organs or tissues are small-sized with high blood flow [7,14,15];
- (2)
- Permeability-limited model: in this model, xenobiotics penetrate organs or tissues through membrane barriers, resulting in a concentration gradient between organs or tissues and blood. Depending on the number of membrane barriers, the permeability-limited model can have two or three sub-compartments. This model is suitable for molecules with polarity or large molecular weight [14,15];
- (3)
- Dispersion model: in this model, xenobiotics are distributed in organs or tissues with a gradient, and the degree of dispersion is evaluated using the dispersion coefficient (D
_{N}). A higher D_{N}indicates a greater dispersion of xenobiotics in organs or tissues. When D_{N}approaches infinity, the dispersion model is similar to the perfusion-limited model [16,17,18,19,20,21]. This model is suitable for xenobiotics with high hepatic clearance [8].

#### 2.2. Definition of Model Parameters

#### 2.3. Model Simulation

#### 2.3.1. Algorithm

^{2}), so a smaller dt

^{2}is needed to reduce this error, although it requires more calculation time. In addition, the Euler algorithm is not applicable to stiff system solutions, for which the Gear algorithm is recommended due to its greater stability [11].

#### 2.3.2. Software

#### 2.4. Model Evaluation

#### 2.4.1. Verification of Measured Data

^{2}, and two-sample Kolmogorov are also not suitable for assessing consistency between predicted and measured values due to the self-correlation of concentration values at different times in pharmacokinetics [11]. To assess the predictive ability of the PBTK model, linear regression analysis can be performed to compare the predicted and measured values. A high prediction ability is indicated when the intercept of the regression equation is close to 0 and the slope and correlation coefficient are close to 1 [10,77]. Another method for assessing the predictive ability of the PBTK model is to calculate the chi-square χ

^{2}as Equation (16). This approach is applicable when there are multiple measured values at the same time point. Peters used this method in an overall PBTK model to evaluate the model fitting [82]. An χ

^{2}value close to 1 indicates better model fitting and prediction performance. The Akaike information criterion (AIC) and Bayesian information criterion (BIC) are commonly used for assessing predictive ability [83,84]. AIC and BIC are calculated using Equations (17) and (18), respectively. Smaller values of AIC and BIC indicate a better model fit. When there is a significant difference between the fitted model and the real model, it is primarily reflected in the likelihood function term −2In(L). When the difference in the likelihood function is not significant, the penalty term 2k, accounting for the number of model parameters, becomes influential. Models with better fit and fewer parameters are preferred as the increase in the number of parameters leads to increased 2k and AIC values. The penalty term for BIC is larger than that for AIC, taking into account the number of observations. This prevents excessive model complexity when there is a large number of observations. The last method is to calculate the fold error (FE), as Equation (19). Firstly, the noncompartmental analysis was performed to obtain the peak concentration, peak time, area under the curve, and other parameters of the observations. Then, the fitting values or predicting values of the above parameters were obtained from corresponding models. Due to the different laboratory conditions, detection methods, and other confounding factors, the measured value and the predicted value cannot be consistent completely, and the model is acceptable when the FE ≤ 2 [80].

#### 2.4.2. Uncertainty Analysis

#### 2.4.3. Variation Analysis

#### 2.4.4. Sensitivity Analysis

#### 2.4.5. Model Optimization

## 3. Applications of Physiologically Based Toxicokinetics (PBTK) in Health Risk Assessment

#### 3.1. Exposure Assessment

#### 3.2. Extrapolation

#### 3.2.1. IVIVE

#### 3.2.2. Dose Extrapolation

#### 3.2.3. Exposure Route Extrapolation

#### 3.2.4. Interspecific Extrapolation

#### 3.3. PBTK of the Special Population

#### 3.4. Metabolic Characteristics and Mechanisms

#### 3.5. Mixture Risk Assessment

## 4. Prospects

- (1)
- The PBTK model is based on the understanding of the in vivo ADME processes of chemicals. However, it is challenging to construct a corresponding PBTK model for certain drugs, such as Class 3 and 4 drugs in the Biopharmaceutics Classification System (BCS), traditional Chinese medicine ingredients, heavy metals, etc., due to limited knowledge about their ADME processes.
- (2)
- The parameters required by the PBTK model are typically obtained through in vivo, in vitro, and in silico assays. However, uncertainties may exist regarding parameters measured using in vitro and in silico assays that require validation with further in vivo experimental data. Moreover, special population groups, such as different races, children, pregnant women, obese individuals, and those who are ill, exhibit distinct physiological and ADME characteristics. Therefore, constructing PBTK models for these special groups necessitates extensive experimental research support.
- (3)
- Validating exposure characteristics among different subjects under various exposure conditions is crucial for establishing a reliable PBTK model. Unfortunately, in toxicological studies, the lack of relevant experimental data poses challenges for verifying the accuracy of PBTK models.
- (4)
- Constructing PBTK models can be relatively complex as it requires researchers to possess fundamental knowledge of toxicokinetics, toxicology, physiology, mathematics, and modeling. This complexity limits accessibility to these models.
- (5)
- For mixtures composed of different types of chemicals, it becomes difficult to construct mixed PBTK models due to variations observed during their respective ADME processes within an organism and also because interaction processes between them can become intricate.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The whole PBTK model (

**A**) and semi-PBTK model (

**B**) of humans. IV, intravenous injection; IA, arterial injection; Q, blood flow; M, metabolites; U, urine; CLint, intrinsic clearance; Ke, excretion rate. The subscripts LU, HT, BR, MU, AD, SK, SP, PA, HA, ST, GU, BO, KI, TH, RP, PP, and LI refer to lung, heart, brain, muscle, fat, skin, spleen, pancreas, hepatic artery, stomach, gut, bone, kidney, thymus, richly perfused tissue, poorly perfused tissue, hepatic vein, respectively.

**Figure 3.**In PBTK models, the left panel (

**A**) is superior to the right panel (

**B**) on the consistency between the predicted value (solid line) and the measured value (hollow circle) by visual inspection.

**Table 1.**The mathematical equations in the PBTK model [15].

Toxicokinetic Process | Equation | |
---|---|---|

Absorption | ||

Respiratory tract | ${C}_{a}=\frac{{Q}_{p}\xb7{C}_{inh}+{Q}_{c}\xb7{C}_{v}}{{Q}_{c}+{Q}_{p}/{P}_{b}}$ | (2) |

Percutaneous | $\frac{{dA}_{sk}}{dt}={K}_{p}\xb7\mathrm{S}\left({C}_{air}-\frac{{C}_{sk}}{{P}_{s:a}}\right)+{Q}_{sk}\xb7\left({C}_{a}-\frac{{C}_{sk}}{{P}_{s:b}}\right)$ | (3) |

Oral | $\frac{d{A}_{o}}{dt}={K}_{o}\xb7\left(A-{A}_{o}\right)$ | (4) |

Intravenous | ${C}_{v}=\frac{{K}_{z}+\left({\sum}_{t}^{n}{Q}_{t}\xb7{C}_{vt}\right)}{{Q}_{c}}$ | (5) |

Distribution | ||

Protein binding | ${C}_{b}=\frac{n\xb7\beta \xb7{K}_{d}\xb7{C}_{f}}{1+{K}_{d}\xb7{C}_{f}}$ | (6) |

Irrigation rate-limiting structure | $\frac{{dA}_{t}}{dt}={Q}_{t}\xb7\left({C}_{a}-{C}_{vt}\right)$ | (7) |

Membrane rate-limiting tissue | $\frac{{dA}_{t}}{dt}={PA}_{t}\xb7\left({C}_{vt}-\frac{{C}_{t}}{{P}_{t}}\right)$ | (8) |

Metabolism | ||

First-order kinetic | $\frac{{dA}_{met}}{dt}={K}_{f}\xb7{C}_{vt}\xb7{V}_{t}=CL\xb7{C}_{vt}$ | (9) |

Second-order kinetic | $\frac{{dA}_{met}}{dt}={K}_{s}\xb7{C}_{vt}\xb7{V}_{t}\xb7{C}_{cf}$ | (10) |

Saturation process | $\frac{{dA}_{met}}{dt}=\frac{{V}_{max}\xb7{C}_{vt}}{{K}_{m}+{C}_{vt}}$ | (11) |

Excretion | ||

Kidney | $\frac{{dA}_{rc}}{dt}=\left(GFR\frac{{T}_{m}}{{K}_{t}+{C}_{p}}\right)\xb7{C}_{p}$ | (12) |

Lung | ${C}_{x}=\left(0.7\xb7\frac{{C}_{a}}{{P}_{b}}\right)+\left(0.3\xb7{C}_{inh}\right)$ | (13) |

_{c}, cardiac output; Q

_{p}, alveolar ventilation; P

_{b}, blood-air distribution coefficient; C

_{a}, xenobiotics concentration in arterial blood; C

_{inh}, xenobiotics concentration in inhaled gas; C

_{v}, xenobiotics concentration in mixed venous blood. Equation (3): A

_{sk}, total amount of skin xenobiotics exposed; t, time; K

_{p}, skin permeability coefficient; S, skin exposure area; C

_{air}, concentration of xenobiotics in the air; C

_{sk}, concentration of exposed skin xenobiotics; P

_{s:a,}skin-air distribution coefficient; Q

_{sk}, skin blood flow; P

_{s:b}, skin-blood partition coefficient. Equation (4): A

_{o}, total amount of xenobiotics absorbed; K

_{o}, oral absorption rate constant; A, oral exposure dose. Equation (5): K

_{z}, intravenous administration rate; Q

_{t}, blood flow in “t” chamber; C

_{vt}, xenobiotics concentration in venous blood of outflow chamber “t”. Equation (6): C

_{b}, binding xenobiotics concentration; n∙β, maximum binding rate; K

_{d}, dissociation constant; C

_{f}, free xenobiotics concentration. Equation (7): A

_{t}, total amount of xenobiotics in “t” chamber. Equation (8): PA

_{t}, mass transfer coefficient; P

_{t}, organ/tissue–blood allocation coefficient. Equation (9): A

_{met}, total amount of xenobiotics metabolism; K

_{f}, first-order metabolic rate constant; V

_{t}, “t” chamber volume; CL, clearance. Equation (10): K

_{s}, second-order me tabolic rate constant; C

_{cf}, cofactor concentration. Equation (11): V

_{max}, maximum velocity of enzyme-catalysis; K

_{m}, Michaelis constant. Equation (12): A

_{rc}, total amount of xenobiotics in kidney; GFR, glomerular filtration rate; T

_{m}, apparent maximum transport rate of the carrier system; K

_{t}, apparent Mieman constant; C

_{p}, xenobiotics concentration in plasma. Equation (13): C

_{x}, xenobiotics concentration in exhaled gas.

Parameter | Data Source | |
---|---|---|

Physiological parameter | Body weight Cardiac output Blood flow to organ or tissue Volume of an organ or tissue Alveolar ventilation | Literature In vivo experiment. |

Physicochemical parameter | Blood-air distribution coefficient Tissue-blood distribution coefficient | Literature In vivo experiment In vitro experiment In silico prediction |

Biochemical parameter | Maximum velocity Michaelis constant First-order rate constant Second-order rate constant | Literature In vivo experiment In vitro experiment In silico prediction |

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**MDPI and ACS Style**

Chen, M.; Du, R.; Zhang, T.; Li, C.; Bao, W.; Xin, F.; Hou, S.; Yang, Q.; Chen, L.; Wang, Q.;
et al. The Application of a Physiologically Based Toxicokinetic Model in Health Risk Assessment. *Toxics* **2023**, *11*, 874.
https://doi.org/10.3390/toxics11100874

**AMA Style**

Chen M, Du R, Zhang T, Li C, Bao W, Xin F, Hou S, Yang Q, Chen L, Wang Q,
et al. The Application of a Physiologically Based Toxicokinetic Model in Health Risk Assessment. *Toxics*. 2023; 11(10):874.
https://doi.org/10.3390/toxics11100874

**Chicago/Turabian Style**

Chen, Mengting, Ruihu Du, Tao Zhang, Chutao Li, Wenqiang Bao, Fan Xin, Shaozhang Hou, Qiaomei Yang, Li Chen, Qi Wang,
and et al. 2023. "The Application of a Physiologically Based Toxicokinetic Model in Health Risk Assessment" *Toxics* 11, no. 10: 874.
https://doi.org/10.3390/toxics11100874