The liver is the main site of metabolism for many drugs, leading to the clearance (CL) of these drugs through a variety of oxidative and conjugative metabolic pathways (phase I and II reactions) [1]. Hepatic impairment may lead to changes in the pharmacokinetic (PK) of drugs due to the reduced capacity of the liver to metabolize drugs. As a result, drugs or active metabolites can be accumulated in the body and may reach to toxic levels. Furthermore, pro-drugs needed to be converted to active metabolite(s) for therapeutic benefits, but hepatic impairment may lead to the reduced formation of active metabolite(s), resulting in the reduced efficacy of such products [1].

There can be many liver diseases, such as alcoholic liver disease and chronic infections with hepatitis viruses B and C, acute hepatitis D or E, primary biliary cirrhosis, and primary sclerosing cholangitis [1]. In hepatic disease, there are changes in the activity of metabolizing enzymes and hepatic transporters [2]. Besides liver, reduced duodenal cytochrome P450 3A (CYP3A) activity has also been noted in the hepatic disease [3]. Hepatic disease may also alter kidney function, resulting in the accumulation of a drug and its active or inactive metabolite(s) even though a drug is not primarily metabolized in the liver [1].

The Food and Drug Administration of the United States of America (US FDA) suggests a Child-Pugh score system for the assessment of liver function [1]. The Child-Pugh score system measures several clinical tests such as encephalopathy grade, ascites, serum bilirubin, serum albumin, and prothrombin time. Based on these tests, the FDA has suggested 3 categories of hepatic impairment. Based on Child-Pugh score, these three categories are: category A or mild liver impairment (score 5–6), category B or moderate liver impairment (score 7–9), and category C or severe liver impairment (score 10–15).

Modeling and simulation can be helpful in predicting PK of a drug in subjects with hepatic impairment and finding a suitable dose for the initiation of a dedicated clinical trial [1]. In order to predict PK of drugs in hepatic impairment, whole body physiologically based PK (PBPK) models have been suggested [4–6]. However, there are examples in the literature that suggest that the whole body PBPK model can be simplified (generally known as reduced or lumped PBPK) to predict PK parameters or dose of a drug [7–17].

The objective of this study is to propose a minimal PBPK (mPBPK) model to predict PK parameters such as maximum plasma concentration (C_max) and area under the curve (AUC) in subjects with different degrees of hepatic impairment and compare the predictive performance of the proposed mPBPK model with the predictive performance of the whole body PBPK model as well as with dedicated clinical trials.

Fifty two drugs were used in this study to predict PK parameters such as C_max and AUC in subjects with hepatic impairment by a mPBPK model. The predicted C_max and AUC values from mPBPK model were then compared with the observed C_max and AUC values from dedicated hepatic impairment clinical trials. Out of 52 drugs, 27 drugs were used to compare the predicted C_max and AUC from mPBPK model with the whole body PBPK model. The remaining 25 drugs were used to further strengthen the accuracy of the predictive performance of the proposed mPBPK model.

It should be noted that in this study, the whole body PBPK models were not developed rather C_max and AUC values were taken from the published PBPK studies for the comparison of C_max and AUC values with the predicted values from mPBPK model. Hence, in this manuscript, there are no differential equations and generated C_max-time plots for whole body PBPK. References for whole body PBPK model are provided in this manuscript so that the interested readers can find the whole body PBPK modeling strategy by the respective authors. In order to predict C_max and AUC values for mPBPK, these values were used from healthy subjects from dedicated hepatic impairment clinical trials (Supplementary material). The following is the description of the development of the mPBPK model in this study.

The fold-change (healthy versus liver impairment) for physiological and biochemical parameters used in this study to develop mPBPK model is shown in Table 1. The results of this study are summarized below and in Tables 2, 3, and 4. In supplementary Tables S1–S6, the observed and predicted AUC and C_max values in subjects with different degrees of hepatic impairment by the mPBPK model and whole body PBPK model are shown. In supplementary Tables S7–S12, the predicted and observed AUC and C_max values of 25 drugs with the observed PK values are presented to further demonstrate the robustness of the proposed mPBPK model. In Supplementary example, an example is provided to show how AUC can be predicted from the proposed method (same goes for C_max).

Figures 1, 2, and 3 show the relationship between predicted AUC values by mPBPK and whole body PBPK. Figures 4, 5, and 6 show the relationship between predicted AUC values from mPBPK and the observed AUC values (for all 52 drugs).

Out of 52 drugs, 27 drugs (53 and 33 data points for AUC and C_max, respectively) were predicted by the proposed mPBPK model compared with the whole body PBPK model (Tables 2 and 3). Overall, from mPBPK, 52 drugs with 120 data points for AUC and 93 data points for C_max were predicted in subjects with varying degrees of hepatic impairment (Table 4).

Whole body PBPK models have been suggested for the prediction of PK parameters for special populations such as pediatrics, pregnancy, and renal and hepatic impairment and subsequently for dedicated clinical trials in a specific population. However, over the years, comparative studies between whole body PBPK and minimal or reduced PBPK models have shown that mPBPK models in their predictive performance are as robust and accurate as whole body PBPK models [7–17]. A mPBPK model uses only a few physiological parameters (as few as 3–5) and one or two drug-related physicochemical characteristics and is much simpler to develop than a whole body PBPK model.

The mPBPK model used in this study differs from traditional whole body PBPK models or even other mPBPK models in the sense that no differential equations were developed to generate C_max-time data but retained two basic characteristics of a whole body PBPK or mPBPK models in terms of physiological organs or parameters and product characteristics. Four physiological parameters, two biochemical parameters, and one product characteristic in terms of log P were used to develop physiological factors for different degrees of hepatic impairment. The direct use of C_max and AUC values from healthy subjects along with physiological factors generated in this study made it possible to predict C_max and AUC in subjects with different degrees of hepatic impairment. In other words, the proposed mPBPK model retains the characteristics of a typical whole body or mPBPK model (physiological parameters and product characteristics) but uses a simpler mathematical approach rather than using differential equations to achieve its objective (prediction of C_max and AUC in subjects with hepatic impairment).

The amount of CYP isozymes such as 3A4, 2D6, 1A2, and 2C9 is substantially reduced in subjects with hepatic impairment [4, 5]. Li et al. [18] in their mPBPK model used liver transporter mRNA levels, uptake transporter activity, and efflux transporter activity but not the CYP isozymes. It should be noted that the authors used the same values for all their transporter parameters in all three categories of hepatic impairment. The contents of CYP isozymes, transporter mRNA levels, and transporter activity were not included in this mPBPK model, and yet the prediction of exposure by mPBPK and whole body PBPK were found to be similar.

In a previous study [15], it was shown that the exposure or CL of renally excreted drugs in subjects with renal impairment (from mild to severe) can be predicted with just one single physiological parameter (GFR) with similar accuracy as with a whole body PBPK model. The comparable prediction accuracy of mPBPK model with the whole body PBPK model indicates (in this study and some previous studies) that the extensive number of physiological parameters as well as product characteristics used in a whole body PBPK model are unnecessary.

In a real world, a mPBPK model or other simple empirical models are more attractive than a whole body PBPK model because a mPBPK model is much simpler and cost- and time-effective, yet as robust and accurate as a whole body PBPK model [7–17]. The proposed methods in this study also do not require any special software rather the entire calculation can be done on a spreadsheet in a very short period of time.

There are two limitations of the mPBPK model presented in this study. The application of negative log P value may add more errors in the prediction than the positive log P value. This is of mathematical nature and the magnitude of prediction error will depend on the negative value of log P (for example, –1 or –3). Higher the negative value, higher the prediction error (highly likely). Furthermore, this model can not estimate inter-individual variability.

There is high variability in the observed PK parameters in subjects with hepatic impairment especially, in severe liver impairment. Therefore, the dose adjustment will require a correct estimate of change in the magnitude of exposure to a drug especially, in severe hepatic impairment. A 2-fold prediction error or even 50% prediction error may not be acceptable for the selection of the ‘right dose’ in subjects with hepatic impairment especially, in severe hepatic impairment. A dedicated clinical PK study will provide accurate information regarding the impact of hepatic impairment of a drug. Whole body PBPK or mPBPK may be used to predict exposure to a drug in subjects with hepatic impairment and the predicted exposure (using experience and scientific judgment) can then be used to select a dose to initiate the clinical trial in this population. PK study should be conducted in all three types of hepatic impairment because PK can be different across three groups especially, in subjects with moderate and severe hepatic impairment than healthy subjects.

In conclusion, this study demonstrates that a mPBPK model using 4 physiological parameters, two biochemical parameters, and one parameter as a product characteristic can be developed with reasonable accuracy for the prediction of drug exposure in subjects with varying degrees of hepatic impairment. More accurate prediction was noted in subjects with mild and moderate hepatic impairment as compared to subjects with severe hepatic impairment by both models. In order to evaluate the robustness and accuracy of the proposed model a study with a larger data set particularly, in patients with severe hepatic impairment is needed.

The renowned statistician Box [21] stated that “since all models are wrong the scientist cannot obtain a ‘correct’ one by excessive elaboration.”. On the contrary, following William of Occam should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so over-elaboration and over-parameterization are often the mark of mediocrity.