PBPK (Physiologically Based PharmacoKinetics) Models

Physiologically based pharmacokinetic (PBPK) models consist of a series of mathematical representations of biological tissues and physiological processes in the body of target species aimed at describing the absorption, distribution, metabolism, and excretion of chemicals (Fàbrega et al., 2016). When a chemical substances enter the organism, it is usually distributed to various tissues and organs by blood flow (Nestorov, 2007). Following its distribution in tissues, the substance can bind to various proteins and receptors, undergo metabolism, or can be eliminated unchanged. The concentration vs.

time profiles of the xenobiotic in different tissues, or the amount of metabolites formed, is often used as surrogate markers of its internal dose or biological activity (Andersen et al., 2005). In a sense, PBPK modelling is an integrated systems approach to both understanding the pharmacokinetic behaviour of compounds and predicting concentration vs time profiles in plasma and tissues (Bois et al. 2010).

The biological response results from the interaction between the toxicant and the target tissue. For this reason, models that can predict the target tissue concentration of the toxicologically-active chemical species (parent compound or metabolite) are especially useful and have been applied in the “exposure–dose–response” paradigm. The internal dose metrics (sometime also referred as biological effective dose) replaces the external exposure dose in the derivation of the quantitative dose-response relationship, with the intent of reducing the uncertainty inherent in human health risk assessments based on external exposure dose estimation.

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Fig 1. PBPK model structure and approach, representing the model developmental approach and the required parameters input. BM: Body mass; BSA: Body surface area;

MW: Molecular weight; Ko/w : octanol:water partition coefficient; Fu: Fractional unbound concentrations; B:P: blood to plasma ratio; MSP: microsomal protein.

3.1.1. Approaches to building PBPK models

Building a PBPK model requires gathering a considerable amount of data which can be categorised in three groups, namely: the model structure, which refers to the arrangement of tissues and organs included in the model; the system's data (physiological, anatomical, biochemical data) and chemical-specific data (physicochemical) (see Figure 2.1). The transport of xenobiotics (chemicals) in several tissues is determined by two different approaches: (i) permeability limited (also called as flow limited), and (ii) perfusion limited (also called as diffusion limited), (Bois and Paxman, 1992; Gerlowski and Jain, 1983).

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Permeability-rate –limited Model: This model is also called diffusion limited and could be applied when the distribution of the substance to a tissue is rate-limited by the drug's permeability across the tissue membrane. That condition is more common with polar compounds and large molecular structures. Consequently, the related PBPK models may exhibit different degrees of complexity.

Perfusion-rate-limited Model: This model is also called flow limited kinetics and could be applied when the tissue membrane presents no barrier to distribution. Here, each tissue is considered to be a well-stirred compartment in which the substance distribution is simply limited by blood flow. Thus, the chemical will be delivered to the tissue via the blood, and is assumed to mix throughout the volume of that compartment immediately and completely and normally partition coefficient is used for the distribution of chemical.

Concentrations in the flow limited compartments generally estimated by applying the following equation:

^dC_dtⁱ=^{Qi ∗(Ca−}

Ki:p Ci )

V_i (1) Where Ci is the concentration in the tissue i (nM), Qi is the blood flow in the tissue i (L/h), Ca is the arterial concentration (nM), Ki:p is the partition coefficient of tissue i, and Vi is the volume of the tissue i (L).

3.1.2 Model Parameterization:

There are two approaches of PBPK model building or parameterization: bottom-up and top-down. In bottom-up approach, model parameterization is done based on in- silico prediction or in-vitro understanding of chemical-related ADME mechanisms. It mainly depends on tools for translation of in-vitro data to in-vivo such as IVIVE (in vitro- in vivo extrapolation) and several in-silico tools such as QSAR, in a sense its purely predictive model. In contrast top-down approaches rely on estimation of model parameter by fitting to the observed experimental data. Model parameterization requires two specific parameters namely; System’s and Chemical’s specific input parameters.

System –specific parameters: This comprises of both physiological parameters and biochemical parameters.

Physiological parameters: These parameters are species specific constant. These incudes tissues/organs volume (or weight) and tissues blood flow rate which are specific to the species of interest. These parameters are used to develop species specific PBPK models, the most common being rat, mouse, dog and human. Physiological parameters for developing such models are routinely available in the literatures (Abduljalil et al., 2012; Brown et al., 1997; Sisson et al., 1959; Valentin, 2002).

Biochemical parameters

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Biochemical parameters are the hybrid parameters which depend on both chemical and physiology. Among biochemical parameters, chemicals metabolism is considered to be very important parameters, which are generally derived from in-vitro data using IVIVE methodology. The schema of IVIVE has been provided in the figure 2.

IVIVE generally involves the scaling of in-vitro Vmax parameter was done based on microsomal protein content per gram tissue and weight of tissue per kg body weight.

Vmax was scaled to in-vivo per kg BW from in-vitro cell line studies by using the following equation:

𝑉𝑚𝑎𝑥𝑖𝑛𝑣𝑖𝑣𝑜= (𝑉𝑚𝑎𝑥𝑖𝑛𝑣𝑖𝑡𝑟𝑜∗ 𝑀𝑃𝑃𝐺𝑇 ∗ 𝑉𝑡𝑖𝑠𝑠𝑢𝑒)/𝐵𝑊^.75 (2)

Where, 𝑉𝑚𝑎𝑥 = Maximum metabolic capacity in per gram of microsomal protein, 𝑀𝑃𝑃𝐺𝑇 = the microsomal protein per gram of tissue, Vtissue = the total tissue weight in gram, and BW =is the whole body weight in kg.

Fig. 2: Illustration of Hierarchical structure model approach for metabolic IVIVE scaling.

Chemical-specific parameters

It can be derived by in vivo or in vitro experiment. However, in certain cases when we lack these data, various in-silico approaches can be useful. Among Physicochemical

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parameters partition coefficient is considered one of the most important parameters. It describes the distribution of the chemical between plasma and different organ. There are various tissue composition based algorithm methods to generate partition coefficient data (Poulin and Krishnan, 1996, 1995; Schmitt, 2008; Peyret et al., 2010; Yun et al., 2014).

Tissue composition based algorithm methods for calculating partition coefficients few of them are explained here;

METHOD I (Poulin and Krishnan, 1996, 1995)

𝑃𝑡:𝑃 =(( 𝑃𝑜:𝑤∗ (𝑉𝑛𝑙𝑡+ 0.3 ∗ 𝑉𝑝ℎ𝑡)) + 1 ( 𝑉𝑤𝑡+ 0.7 ∗ 𝑉𝑝ℎ𝑡)) ( 𝑃𝑜:𝑤∗ (𝑉𝑛𝑙𝑝+ 0.3 ∗ 𝑉𝑝ℎ𝑝)) + 1 ( 𝑉𝑤𝑝+ 0.7 ∗ 𝑉𝑝ℎ𝑝) ∗𝑓𝑢𝑝

𝑓𝑢𝑡

𝑃𝑡:𝑃𝑎𝑑𝑖𝑝𝑜𝑠𝑒 =(( 𝐷𝑜:𝑤∗ (𝑉𝑛𝑙𝑡+ 0.3 ∗ 𝑉𝑝ℎ𝑡)) + 1 ( 𝑉𝑤𝑡+ 0.7 ∗ 𝑉𝑝ℎ𝑡)) ( 𝐷_𝑜:𝑤∗ (𝑉_𝑛𝑙𝑝+ 0.3 ∗ 𝑉_𝑝ℎ𝑝)) + 1 ( 𝑉_𝑤𝑝+ 0.7 ∗ 𝑉_𝑝ℎ𝑝) ∗𝑓𝑢𝑝

1

Where,

P_t:P = tissue plasma partition coefficient Po:w = octanol water partition coefficient V_nlt = fractional volume of neutral lipid in tissue Vpht = fractional volume of phospholipid in tissue Vnlp = fractional volume of neutral lipid in plasma Vphp = fractional volume of phospholipid in plasma fup = fractional unbound concentration in plasma fut = fractional unbound concentration in tissue Do:w = olive water partition coefficient

Note: have to take antilog as octanol water partition coefficient always given in log value.

fut value, we can calculate by applying formula of = 1/(1+((1-fup)/fup)*RA)) RA is the ratio of albumin concentration found in tissue over plasma

RA equals 0.15, whereas for nonadipose tissue, RA equal 0.5 (Ellmerer et al., 2000;

Poulin and Theil, 2002).

METHOD II: Schmitt Walter (Schmitt, 2008) 𝑓_𝑢=_𝐶^𝐶𝑢

𝑡𝑜𝑡𝑎𝑙 𝐾𝑡:𝑝= (𝐹_𝑖𝑛𝑡

𝑓𝑢𝑖𝑛𝑡+𝐹_{𝑐𝑒𝑙𝑙} 𝑓𝑢𝑐𝑒𝑙𝑙) ∗ 𝑓𝑢𝑝

Where,

𝐹𝑖𝑛𝑡 = fractional content of interstitial fluid in tissue or volume fraction of interstitial

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𝐹𝑐𝑒𝑙𝑙 = fractional content of cell in tissue or volume fraction of cellular 𝑓𝑢𝑖𝑛𝑡 = unbound fraction in interstitial fluid

𝑓𝑢𝑐𝑒𝑙𝑙 = unbound fraction in cell 𝑓_𝑢^𝑝 = unbound fraction in plasma

Calculation of unbound fraction in interstitial space 𝒇𝒖𝒊𝒏𝒕

Assumption = interstitial fluid is very similar with plasma

Therefore, unbound fraction of interstitium estimated from unbound fraction in plasma

1

𝑓_𝑢^𝑖𝑛𝑡= 𝐹𝑤𝑖𝑛𝑡+^{𝐹𝑝𝑖𝑛𝑡}

𝐹_𝑝^𝑝𝑙 ∗ (¹

𝑓_𝑢^𝑝𝑙− 𝐹𝑤𝑝𝑙

)

Where,

𝐹𝑤𝑖𝑛𝑡 = fractional water content in interstitial 𝐹𝑝𝑖𝑛𝑡

= fractional protein content in interstitial 𝐹_𝑤^𝑝𝑙 = fractional water content in plasma 𝐹_𝑝^𝑝𝑙 = fractional protein content in plasma

𝐹_𝑝^𝑖𝑛𝑡 𝐹_𝑝^𝑝𝑙 = 0.37

Calculation of unbound fraction in cellular space 𝑓𝑢𝑐𝑒𝑙𝑙

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𝑓𝑢𝑐𝑒𝑙𝑙 = 𝐹_𝑤+ 𝐾_𝑛𝑙∗ 𝐹_𝑛𝑙+ 𝐾_𝑛𝑝𝑙∗ 𝐹_𝑛𝑝𝑙+ 𝐾_𝑎𝑝𝑙∗ 𝐹_𝑎𝑝𝑙+ 𝐾_𝑝∗ 𝐹_𝑃 Where,

F_w = fractional content of water in cell Knl = water: neutral lipid partition coefficient Fnl = fractional neutral lipid in cellular space

K_npl = water: neutral phospholipid partition coefficient F_npl = fractional neutral phospholipid in cellular space K_apl = water: acidic phospholipid partition coefficient Fapl = fractional content of acidic phospholipid in cell Kp = water: protein partition coefficient

FP = fractional content of protein in cell Calculation of 𝐊𝐧𝐥

Do:w(pH) = Po:w∗ (₁₊₁₀^1−α_pH−pKa+ α) , for acid

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D_pl:w= 1.294 + .304 ∗ logP (Yun et al., 2014)

If D_pl:w is not there, we can use Po:w value which is close to that Here Dpl:w is equivalent to Knpl

Calculation of 𝐊𝐚𝐩𝐥

Kapl= Knpl∗ (_1+10pH−Pka¹ + 20 ∗ (1 −_1+10pH−Pka¹ )) , for acids

K_apl= K_npl∗ (_1+10Pka−pH¹ + 0.05 ∗ (1 −_1+10Pka−pH¹ )) , for base

Calculation of 𝐊𝐩

Kp= 0.163 + 0.021 ∗ knpl , protein: water partition coefficient