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Enterohepatic Circulation Effect in Physiologically Based Pharmacokinetic Models: The Sorafenib Case Roberto Andrea Abbiati and Davide Manca* PSE-Lab, Process Systems Engineering Laboratory, Dipartimento di Chimica, Materiali e Ingegneria Chimica “Giulio Natta”, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy ABSTRACT: Sorafenib was recently approved by both the FDA and EMA for the treatment of unresectable hepatocellular carcinoma (HCC), renal cell carcinoma (RCC), and thyroid cancer (DTC). This paper proposes a physiologically based model to describe its pharmacokinetic properties. Special attention is devoted to the enterohepatic circulation, which causes the characteristic double-peak in the drug’s concentration−time curves. This physiologically based pharmacokinetic (PBPK) model has a structure suitable for different mammals. The paper focuses on the anatomy and physiology of mice aiming to develop a physiologically consistent model to produce accurate simulation of sorafenib pharmacokinetics, in agreement with literature data. The in silico description of sorafenib pharmacokinetics allows understanding further its properties, by complementing the experimental data with numerical simulations based on a mechanistic approach. Modeling and simulation is a rapidly growing branch of quantitative systems pharmacology and supports new drug development by shortening analysis time, reducing costs, and delivering reliable pharmacokinetic knowledge.

1. INTRODUCTION Sorafenib is a recently approved molecule for the treatment of advanced-stage renal cell carcinoma (RCC, FDA 2005 and EMA 2006), hepatocellular carcinoma (HCC, FDA 2007 and EMA 2006), and differentiated thyroid cancer (DTC, FDA 2013).1,2 RCC is the major malignancy generated in the kidneys and accounts for 90−95% of total incidence,3 i.e., 2% of all adult tumors.4 Among liver malignancies, HCC occurs in 90% of cases,5 although Faruque et al.6 report more conservative statistics, i.e., 70−85% of total liver cancer occurrence. Worldwide, HCC is identified as the third cause of death due to cancer.7 Altogether, kidneys and liver malignancies in the U.S. account for about 7% of cancer mortality (4% liver and 3% kidneys in males, while occurrence is slightly reduced in females).8 These data both motivate and justify detailed studies on sorafenib pharmacokinetics and the implementation of advanced tools to assist pharmacists’ and clinicians’ activity. Sorafenib (Nexavar, Bayer) is a small molecule that acts as a multikinase inhibitor. Specifically, sorafenib acts upstream against B-raf, C-raf and downstream against VEGFR2, PDGFRβ, and FGFR1. These elements allow controlling both growth-factor pathways and formation of tumor vascular vessels.9,10 There are a limited number of literature publications regarding pharmacokinetic (PK) properties. Pawaskar et al.11 ascribe this scarcity to the fact that most preclinical data were submitted as reports to FDA and thus were not published. In © 2017 American Chemical Society

particular, little is known about sorafenib distribution in different body organs/tissues. This is also due to difficulties in carrying out in-human studies and is related to the high costs of both sorafenib and in-animal tests. The weak knowledge of PK properties is a relevant issue when cancer-treatment drugs are concerned, because on one hand medicines ought to act locally in the affected organs/tissues, while on the other hand it is advisable to limit side effects, which are particularly unpleasant and may bring on suffering states to debilitated patients. According to the biopharmaceutical classification system (BCS), in vitro studies catalog sorafenib as a class II or possibly class IV drug.12 Sorafenib shows a good permeability through biological membranes, despite a poor solubility. Actually, sorafenib has a strong permeability across Caco-2, while it is a weak substrate with respect to P-glycoprotein. This allows a good absorption rate for the dissolved drug but a limited capability to cross blood-brain barrier.3,13 A direct consequence is the limited bioavailability of sorafenib due to low solubility in the gastrointestinal lumen, while the dissolved drug fraction can easily enter the blood circulation and spread in organs and tissues. Pharmacokinetic models, preferably based on both anatomical and physiological elements, can play an important role in Received: Revised: Accepted: Published: 3156

September 22, 2016 March 7, 2017 March 7, 2017 March 7, 2017 DOI: 10.1021/acs.iecr.6b03686 Ind. Eng. Chem. Res. 2017, 56, 3156−3166

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Figure 1. Sorafenib concentration in mice organs/tissues as measured by Pawaskar et al.11 For the sake of readability, only central values of the distribution are shown without standard deviation bars (whose values can be found in Figure 4). Blood concentration is lower with respect to the concentration measured in most organs at the same time. It is worth observing the presence of a rise in concentration values about 6 h after administration, which can be ascribed to the role played by EHC. Plasma to blood ratio is assumed 0.55.

determining drug absorption, distribution, metabolism, and elimination (i.e., ADME) paths and dynamically assessing the drug presence in specific organs/tissues. These bits of information can supplement experimental data and provide experts with a more extensive view. Interestingly, past the oral administration of sorafenib, there is evidence of a double peak in the concentration profile of both organs and tissues. This is commonly justified in the literature as a consequence of enterohepatic circulation (EHC) of bile and confirmed by experiments in bile duct cannulated rats.3 The recirculation of absorbed drugs induced by the bile occurs in a few steps (in mammals featuring the gallbladder): (i) intestinal absorption of drug; (ii) blood distribution to the liver via the hepatic portal vein; (iii) drug diffusion from blood to hepatocytes and subsequent excretion of a fraction of drug into the bile; (iv) delivery of bile to the gallbladder, where it accumulates and concentrates, with loss of water; (v) the bile is periodically transported from gallbladder into the duodenum through the Oddi’s sphincter; (vi) a fraction of recirculated drug goes through a second intestinal absorption. In mammals without the gallbladder (e.g., rats, horses) this process is continuous and step (iv) is missing. Conversely, mammals equipped with the gallbladder (e.g., men and mice14) accumulate the bile, during fasting, and periodically deliver it back into the intestine as a bolus. This periodic phenomenon yields a prompt increase of drug concentration in the intestinal lumen and brings on cyclical concentration peaks. EHC may have serious consequences on PK behavior of drugs and has to be taken into account, in particular for drugs with narrow therapeutic index and tendency to accumulate in tissues, since an erroneous assessment of peak concentration may lead to undesired consequences. Modeling tools in pharmacokinetic analysis are becoming progressively popular, and their use is more frequent and appreciated by regulatory agencies, although submitted reports still have to rely on experimental evidence.15−18

It is worth observing how model predictions can be important when cancer-treatment drugs are involved, because of the suffering caused to patients in human trials and the extreme difficulty to assess drug concentration in organs/ tissues. An ambitious, but concrete, goal of PK modeling is the achievement of a physiologically based pharmacokinetic model (PBPK) capable of determining the predominant ADME path of a drug and evaluating the correlations that extrapolate in silico predictions from small mammals to humans. Mechanistically based systems, as PBPK models are, have found extensive application in pharmacometrics. A number of recent papers, realizing the burdensome, complexity, and critical activity of full PBPK model parametrization focused on reducing model intricacy by devising either lumped compartment models19 or so-called minimal-PBPK models.20 Other papers defined the EHC characteristics and proposed conceptually similar but differently formulated models.21−23 Purpose of this article is the implementation of a PBPK model to include and describe accurately the process of enterohepatic circulation. In addition, we focus on a specific anticancer drug (i.e., sorafenib). Anticancer drugs development can profit from the use of modeling tools due to the limitation imposed to the experimental practice. A major driver of this research is the fidelity in the description of physiological processes and the consistency with known drug PK properties. With reference to previously published data by Pawaskar et al.,11 a reduced-complexity PBPK model, accounting for EHC, transit-like intestinal absorption, and blood-protein binding properties of sorafenib, is developed and simulated. A fundamental advantage of PBPK use is the possibility to simulate drug course in experimentally unobservable body regions, so that pharmacologist can have more instruments to make inference on drugs PK properties. One important goal consists in providing clinicians and researchers with a simple, effective, and flexible tool to evaluate sorafenib PK. 1.1. Sorafenib Pharmacokinetics. The 2006 EMA report on sorafenib3 provides some reference information on the 3157

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Figure 2. Compartmental structure of the physiologically based model for the description of sorafenib pharmacokinetics. Drug administration is oral (i.e., PO, left dashed violet arrow) while elimination (right dashed red arrows) occurs via hepatic metabolism (CLH), and through faeces (FLIL E ). The classical characterization of the intestinal absorption as a transit-like process is further improved with the introduction of the gallbladder compartment. This allows considering the enterohepatic circulation phenomenon, and quantifying its role in the double peak of some drugs concentration profiles. Parameter R accounts for the binding action of plasma proteins to drug molecules.

allows verifying the model’s capability to describe drug concentrations in the whole body. The proposed PBPK model is a reduced-complexity compartmental model, with perfusion-limited distribution characteristics. This hypothesis sounds reasonable as far as sorafenib (likewise for most BCS class II drugs) is concerned. A simplified version of that model was applied to study the PK profiles of patients after an intravenous administration of remifentanil (an analgesic drug).25 Earlier applications of a similar model are available in Di Muria et al.,26 where authors introduced for the first time the gastrointestinal circulatory system compartment. This compartment is particularly important in the field of PBPK modeling because it allows the implementation of a lumped model with reduced complexity, in contrast to previous works where highly complex, full body PBPK models were necessary to correctly model the gastrointestinal absorption and the first pass liver metabolism.27 Figure 2 shows the PBPK model that accounts for nine compartments. In general, the compartment volumes are modeled as well-stirred and perfectly mixed vessels, so both radial and longitudinal concentration gradients are neglected. We recognize that this hypothesis is an intrinsic limitation of current state-of-the-art PBPK models. For instance, such a hypothesis may result in an inappropriate model of liver physiology, as observed by Chu et al.28 Nonetheless, this work aims at detailing hepatic properties with a good level of accuracy as extensively discussed in the following section. In case of oral administration, gastric gavage is the typical technique adopted in mice treatments. For this reason, we consider three major absorption sites: the gastric lumen (GL), the small (SIL), and large (LIL) intestinal lumina. Once

comprehension of PK characteristics. We highlight some extracts from that report with respect to the ADME process for CD-1 mice, given a sorafenib-tosylate-salt oral administration. Drug intestinal absorption is almost complete (measured as 78.6%) and produces a plasma concentration peak 1.5−2 h after administration. The measured terminal halflife is 6.1 h, while the total plasma clearance is 0.65 L/kg, mainly due to hepatic activity. Sorafenib is highly bound to plasma proteins, with EMA reporting a value of 99.5%. With respect to distribution, sorafenib reaches high concentrations in various organs and tissues in reason for a good permeability. Despite the high permeability, sorafenib shows low penetration across blood/brain barrier. Final excretion of unabsorbed sorafenib and metabolites occurs quite completely through the faeces (>90%) with renal contribution being minimal.24 Figure 1 rationalizes the PK data of Pawaskar et al.11 with the purpose of clarifying both distribution extent and lifetime of sorafenib in mice.

2. MATERIALS AND METHODS The main purpose of this paper is the attainment of a modelassisted pharmacokinetic analysis of sorafenib behavior. This is achieved by applying a novel pharmacokinetic model, accurately based on mammalian anatomical and physiological characteristics. The proposed model is tailored to mice, despite it can also be applied to humans via a suitable parameter reidentification as its structural (i.e., equation-oriented) organization does not change. The well-known advantage of performing experiments on mice consists in the possibility to measure concentration profiles in both organs and tissues. This 3158

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case of sorafenib. The portal vein conveys the drug absorbed by the intestinal-lumen membrane to the liver, while the hepatic artery is a further inlet mean for that organ. Once in the liver, the drug moves into the hepatocytes and undergoes a metabolic process, is partially excreted into the bile and in part enters the central circulatory system.31,32 The gallbladder stores the drug fraction that is excreted into the bile. That small organ accumulates and concentrates the bile, resulting in an increase of drug concentration. Eventually, after a given accumulation time, during digestion, the whole content of gallbladder is expelled via the Oddi’s sphincter into the duodenum tract of SIL, where the drug may be partially absorbed again. Although the processes responsible for the drug recirculation are rather complex and interconnected (e.g., hepatic conjugation and intestinal deconjugation,33 bile concentration in the gallbladder34), we adopted a reasonable simplification in the PK modeling, which is based on the key steps that were detailed in the Introduction. The periodicity of the EHC process is also a further issue that should be considered. Indeed, the steps of bile accumulation, concentration, and release in the duodenum occur cyclically during digestion. Because of very limited information, reduced sampling data (for instance, Pawaskar et al.11 measured just six samples of drug concentration over 24 h, with three samples in the 6−24 h interval) and the fact that oral administration is intrinsically discontinuous, we decided to take into account only the first EHC contribution, as it is the most relevant to the PK characterization. Under these circumstances, we deem this decision reasonable, as literature studies on mice (and even more on humans) are few and usually do not provide quantitative measures of bile flows, gallbladder accumulation, and contraction intervals. Finally, the variability of this process is high among individuals and over its periodic reproduction. 2.2. Model. This section details the constitutive equations of the PBPK model and defines the meaning of adaptive parameters and their values. The proposed model consists of a system of 8 + NR ordinary differential equations (ODE) with suitable initial conditions, where NR is the number of subcompartments that discretize SIL. Indeed, SIL can be modeled by a series of well-stirred compartments (see also Pawaskar et al.,11 Jain et al.12). The higher the number, the more detailed the numerical evaluation of the bolus transit through the intestinal tract. The optimal value NR = 7 is determined as a compromise between results precision and numerical efficiency. Each equation of the ODE system models the mass-balance of a compartment, where the differential term on the left-hand side quantifies the drug accumulation in that compartment. Remaining elements (on the right-hand side of each equation) are drug inlet, outlet, and depletion terms. The GL comes with an initial condition of drug concentration that is equal to the administered dose divided by lumen volume. Gastric Lumen, GL

absorbed through the intestinal wall, the drug enters the gastrointestinal circulatory system (GICS), which lumps in a dedicated compartment the blood vessels of the mesenteric artery, the portal vein, and the related microcirculatory vessels. As far as the description level of the model is concerned, we refer to blood flows, while, at the computational level, we consider the plasma fraction in blood (see section 2.2 for further details). The drug enters the liver via the portal vein, where hepatic metabolism, bile excretion, and introduction into the systemic circulation occur (see the following section). The fraction of drug that survives the hepatic metabolism reaches the central circulatory system and distributes to the various organs and tissues of the mammalian body. With the rationale of preserving a simplified approach, consistent with the limited experimental information available, the remaining organs and tissues are lumped into just two compartments according to blood perfusion. Richly perfused organs are assumed to account for a higher drug concentration and are assigned to the highly perfused organs compartment (HO). Similarly, the less perfused sites show lower concentrations of drug and are lumped into the poorly perfused tissues compartment (PT). For the sake of clarity, the HO compartment lumps the kidneys, the spleen, the heart, the bladder, the lungs, and the sexual organs. Conversely, the PT compartment is more heterogeneous and lumps the adipose tissues, the bones, the skin, the muscles, and the brain. Actually, the brain deserves a specific comment, as it is well-known for being intensely perfused. Despite this characteristic, the distribution of drugs in the brain is hampered by the blood-brain barrier, which protects this organ from potentially harmful substances. Consequently, we assume that, with respect to sorafenib, the brain can be assimilated to a less perfused organ. The muscles are another peculiar tissue as in case of intense physical activity their blood perfusion is significant. Despite this, in the case of either analgesic administration29 or antitumor treatment,30 the average physical activity of muscles is reduced, which ascribes them to the PT compartment. Finally, other minor distribution sites are neglected to preserve the model simplicity and avoid possible risks of overparameterization (see also sections 2.2 and 2.3). This model formulation is specifically appropriate for sorafenib in the case of HCC treatment, as it considers the liver compartment explicitly. Nonetheless, it is worth observing that the modular structure of the model allows adding further compartments as a function of the specific interest/feature to be studied/forecast. 2.1. Liver and Enterohepatic Circulation Modeling. An accurate description of both the gastrointestinal tract and the liver is necessary to correctly model and predict the PK of drugs. In fact, either inconsistent physiology or oversimplified hypotheses may lead to poor understanding of important phenomena. In addition, it is worth highlighting that PBPK modeling was conceived as a tool to assist in the PK translation from animals to humans. Under this perspective, the physiological consistency of models is a necessary requirement, although not sufficient, to implement reliable PK simulators. Sorafenib PK is particularly sensitive to this issue due to its poor solubility, delayed absorption, and large enterohepatic circulation. To achieve a reasonable compromise between precision and simplicity we made the following assumptions: (i) GL does not contribute to drug absorption but introduces a delay time; (ii) SIL is the only compartment responsible for drug absorption; (iii) the LIL absorption is neglected in the

dC GL(t ) = −F GL(t ) dt

(1)

Small Intestinal Lumen, inlet, SIL1 3159

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Industrial & Engineering Chemistry Research Table 1. Parameter Values for Sorafenib PBPK Modela symbols

values

units

VGL VSIL VLIL VGB VP VL VHO VPT QHV QHA QPV QK QGB out

1.01 0.762 0.3 0.04 1.285 1.51 1.484 18.216 1.245 0.153 1.092 0.7 0.001

cm3 cm3 cm3 cm3 mL cm3 cm3 cm3 mL/min mL/min mL/min mL/min mL/min

VGICS tGL tSIL tLIL tGB jPL Rapp jP−HO jP−PT

0.005 80 180 210 260 1 0.75 1 1

cm3 min min min min

jSIL A jHO−P jPT−P jLP EffH QGB in

0.043 0.033 0.146 3.6 0.026 0.036

min−1 min−1 min−1

jSIL CA jLIL A jLIL CA

0 0 0

min−1 min−1 min−1

min−1 min−1

mL/min

description

type

Individualized Parameters GL compartment volume SIL compartment volume LIL compartment volume GB compartment volume Plasma compartment volume (55% of blood) Liver compartment volume Highly perfused Organs compartment volume Poorly perfused Tissues compartment volume Hepatic Vein volumetric flow Hepatic Artery volumetric flow Portal Vein volumetric flow Plasma volumetric flow to kidneys Bile volumetric flow leaving GallBladder Assigned Parameters GICS compartment volume GL residence characteristic time SIL residence characteristic time LIL residence characteristic time Time corresponding to GallBladder emptying into the SIL Plasma-Liver partition coefficient Drug fraction bound to plasma proteins (Apparent value) Plasma to Highly perfused Organs mass transfer coefficient Plasma to Poorly perfused Tissues mass transfer coefficient Unknown Parameters (Whose Values Are Regressed) SIL to GICS mass transfer coefficient Highly perfused Organs to Plasma mass transfer coefficient Poorly perfused Tissues to Plasma mass transfer coefficient Liver-Plasma partition coefficient Hepatic efficiency of elimination Bile volumetric flow to GallBladder (diluted bile) Neglected Parameters GICS to SIL counter mass transfer coefficient LIL to GICS mass transfer coefficient GICS to LIL counter mass transfer coefficient

refs

individualized individualized individualized individualized individualized individualized individualized individualized individualized individualized individualized individualized individualized

35 36 36 37 35 38 38 38 38 38 38 38 35

assigned assigned assigned assigned assigned assigned assigned assigned assigned

29 39 26,39 39 11 b 11 b b

unknown/regressed unknown/regressed unknown/regressed unknown/regressed unknown/regressed unknown/regressed neglected neglected neglected

This is the full parameter set for the model eqs 1−11. bAssignment based on the “a posteriori” model identifiability analysis (see section 2.3). Couples of unknown parameters showing a strong correlation index were assigned a unit value to one parameter and the other parameter was determined by the nonlinear regression routine. a

dC1SIL(t ) V GL = −C1SIL(t )jASIL + F GL(t ) SIL − F1SIL(t ) dt V1

Large Intestinal Lumen, LIL V SIL dC LIL(t ) SIL = −C LIL(t )jALIL + FNR (t ) NR − FELIL(t ) dt V LIL V GICS LIL + C GICS(t )jCA (1 − R app) LIL V

GB Q out C GICS V GICS SIL (t )jCA (1 − R app) SIL + C GB(t ) SIL + ActBile nSIL V1 V1

(2)

Small Intestinal Lumen intermediate, SILn, n = 2, ..., NR − 1

GastroIntestinal Circulatory System, GICS

V nSIL dCnSIL(t ) −1 t = −CnSIL(t )jASIL + FnSIL − FnSIL(t ) ( ) −1 dt V nSIL C GICS V GICS SIL + (t )jCA (1 − R app) SIL nSIL Vn

⎛ Q PV dC GICS(t ) SIL = −C GICS(t )⎜⎜ + jCA (1 − R app) GICS dt j V ⎝ PL

(3)

⎞ LIL + jCA (1 − R app)⎟⎟ + ⎠

Small Intestinal Lumen outlet, SILNR SIL SIL V NR dC NR (t ) SIl SIL SIL −1 = −C NR − FNR t (t )jASIL + FNR ( ) (t ) −1 SIL dt V NR

C GICS V GICS SIL + (t )jCA (1 − R app) SIL nSIL V NR

(5)

+ C LIL(t )jALIL (4)

NR

∑ CnSIL(t )jASIL n=1

Q PV V LIL P + C t ( ) V GICS V GICS

V nSIL V GICS (6)

Liver 3160

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Figure 3. Drug concentration profile as a function of time and space along SIL. SIL spatial discretization moves from 1 (entrance) to 7 (exit). It is possible to observe two important features: (i) the double peak of concentration due to enterohepatic circulation of bile, (ii) the rapid decrease of drug concentration along the intestine, which is related to the rather rapid absorption of sorafenib by GICS.

⎛ Q HV ⎞ Q inGB dC L(t ) CLH ⎟ L = −C L(t )⎜⎜ + − C t ( ) ⎟ L dt VL ⎠ VL ⎝ jLP V + C P(t )

Q HA Q PV GICS + C t ( ) jPL V L jPL V L

Table 1 reports the meaning and value of parameters of the PBPK equations with superscripts referring to the organ/ tissue/compartment where the parameter is applied and subscripts providing further information. For the sake of clarity, CL stands for clearance and is the product of QPV · EffH. EffH quantifies the liver capability to purify the blood flow and assumes values in the [0−1] range, where 0 means no drug removal while 1 means complete removal/elimination. Fi is a flow term that describes the drug transit along the intestinal compartments and is calculated as Ci/ti where i is a specific gastrointestinal compartment. The residence time of every subcompartment of SIL is assumed to be tSIL/NR. This simple but effective formulation assures two advantages: (i) it models a longitudinal profile of drug concentration in the small intestinal lumen, as SIL is the principal absorption site (see Figure 3); (ii) the higher the concentration in a given compartment (Ci) the higher the corresponding Fi as these terms are directly proportional. By doing so, we indirectly acknowledge the fact that absorption through the intestinal wall has a saturation effect because of its nonlinear nature. To account for this issue, instead of making the absorption term jSIL A depend on the drug’s local concentration, we opted to model the drug transport to the following compartments by Fi terms. Finally, ActBile is a Boolean variable responsible for opening/ closing the bile flow between the gallbladder and the intestine. Until the gallbladder accumulates the bile, the flow leaving that organ is null and ActBile = 0. Conversely, when the gallbladder contracts and pours bile into the duodenum ActBile = 1. Under general operating conditions, the gallbladder fills and empties periodically and ActBile should assume 0/1 values accordingly. However, as discussed above, due to the lack of experimental data we modeled just one bile accumulation/release step. We assumed that ActBile becomes 1 at tGB = 260 min based on experimental data.11

(7)

Plasma ⎛ dC P(t ) = −C P(t )⎜⎜jP−PT (1 − R app) + jP−HO (1 − R app) dt ⎝ +

Q HA Q PV ⎞⎟ Q HV V PT PT L + + + C ( t ) j C ( t ) PT−P jPL V P V P ⎟⎠ VP jLP V P

+ C HO(t )jHO−P

V HO VP

(8)

Highly perfused Organs, HO dC HO(t ) VP = −C HO(t )jHO−P + C P(t )jP−HO (1 − R app) HO dt V (9)

Poorly perfused Tissues, PT dC PT(t ) VP = −C PT(t )jPT−P + C P(t )jP−PT (1 − R app) PT dt V (10)

Gall Bladder, GB GB Q out Q inGB dC GB(t ) ActBile + C L(t ) GB = −C GB(t ) GB dt V V

(11)

Equations 2−4 describe the NR discrete sections of SIL and specifically define the inlet zone (SIL1), the intermediate segments (SILn), and the outlet zone (SILNR). 3161

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Once the whole set of parameters is respectively calculated, assigned, neglected, and identified, one can use their values for future applications of the PBPK mice model respect to the administration of sorafenib tosylate in solution. Furthermore, most parameters can remain unchanged also in case of different administration routes, for instance in the case of sorafenib tissue-distribution properties (e.g., jPT−P), as once sorafenib is absorbed into blood, it behaves all the same (i.e., independently of the administration route/formulation). For the sake of clarity, in the case of different race (e.g., dogs, monkeys) but same active principle (e.g., sorafenib), the model parameters would/might change and should be, respectively, found in the literature, assigned as individualized features, and identified according to the same procedure described above. The set of “unknown parameters” found for the mice could play the role of initial values for the nonlinear regression procedure applied to that different race. 2.3. Numerical Methods. In this paper we introduced a dynamic model based on the definition of a system of ordinary differential equations (ODE, eqs 1−11). These equations include certain model parameters (see Table 1), and some of those (i.e., Unknown Parameters) need to be determined via a regression procedure respect to the experimental data. A prerequisite to the application of such models is the assessment of model structural identifiability. In other terms it is necessary to verify that the model formulation is such that, given the availability of a specific data set, it is possible to determine a unique set of model parameters. If this condition is not respected, the regressed parameters value becomes meaningless as the parameter set may not be unique and assume random values.41 Model identifiability test is also important in the design of experiments, as the estimation of certain parameters may be possible only in presence of specific experimental data, while those parameters may be inaccessible in case of inappropriate data collection.42 To verify model structural identifiability, we applied the freeware software “DAISY”.41,43 Structural identifiability is a strict condition because a model that does not respect it has to be reformulated. The most common methods to verify the structural identifiably are the Taylor series and the differential algebra approaches. DAISY implements the second method and it was capable of proving the structural identifiability of our PBPK model. As far as the numerical features of the PBPK model are concerned, it was initially implemented in Matlab and then ported to Fortran 90 programming language to increase both the numerical robustness and CPU efficiency. Indeed, the ODE system showed to be stiff and its numerical solution was significantly improved by introducing the DVODE solver.44 That routine founds its efficiency on the Nordsieck array formulation and a multivalue approach to solution where the integration step is continuously adapted and varied according to the critical features of the ODE system. The identification of “unknown parameters” was carried out by BURENL,45 which is a nonlinear regression routine with statistical analysis capabilities. DVODE, BURENL and the Fortran 90 numerical implementation produced a 40−50 CPU-time fold acceleration respect to the original Matlab version. After the nonlinear regression of the “unknown parameters”, an a posteriori test of model parameters identifiability (or practical identifiability) was run to verify the goodness of the regressed values. This test accounts also for the quality of the available experimental data (e.g., quantity of experimental data and accuracy). Parameters estimation quality was assessed via a

It is worth detailing an important aspect of the model definition: the model considers the plasma fraction of blood, so that the hematocrit fraction of blood is neglected. This decision depends on the fact that blood is a liquid connective tissue and plasma fraction is the transportation vector that acts as a solvent and suspending medium.40 The parameters of eqs 1−11 are categorized in four different classes (see “type” column in Table 1). There are two main reasons for that classification. The former depends on a numerical principle that suggests keeping things as simple as possible to avoid possible overparameterizations. This can be achieved by reducing, as far as possible, the number of unknown parameters to be identified. The latter refers to the physiological approach adopted to model the PK of mammals. Actually, it is suitable to exploit the organ, tissue, and drug features based on either the physical data of treated animals/ patients or the available literature data on active principles. According to this approach, whenever it is possible, parameters are either determined from literature or assigned from animal/ patient data. For instance, the parameters related to anatomical properties (e.g., organ volumes, blood flow rates) are determined via literature correlations with respect to body mass. We name these parameters “Individualized”. A second class of parameters (i.e., “Assigned”) are assumed to be constant and are either taken or adapted from available literature data (e.g., publications, reports, measurements), since it is not viable to determine more specific individual values. This is the case of gastrointestinal transit times, which are assumed constant and animal/patient independent. Within the “assigned” category, special attention must be devoted to R (i.e., drug fraction bound to plasma proteins). We assumed that only the unbound fraction of drugs can leave blood flow via a passive mechanism and diffuse into organs/tissues. Accordingly, we did not apply any specific modeling to acknowledge further transport phenomena (e.g., active transport). As a consequence, it was not possible to obtain a realistic simulation of sorafenib PK by simply applying the available literature value (i.e., 99.5%) for plasma protein binding,3 as the drug would result almost completely confined to the plasma compartment. To overcome this limitation, without introducing further complexities in the model, we considered an apparent binding parameter, Rapp, which is assumed equal to 75% (see also Table 1) and is assigned arbitrarily. The “Unknown Parameters” category refers to unknownvalue parameters that must be determined with a regression procedure. A constrained nonlinear regression identifies the “unknown parameters” by minimizing an objective function that measures the distance between experimental drug concentrations (i.e., data in plasma and selected tissues (Pawaskar et al.11)) and model predicted values. An example of “Unknown Parameters” is jPT−P, which accounts for the mass transfer coefficient from PT (i.e., lumped compartment of Poorly perfused Tissues) to Plasma compartments. An in vitro assessment of that parameter is practically unfeasible, as one should quantify the drug permeability in a large number of different tissues and then determine an averaged value. Remaining parameters in Table 1 are “Neglected” as they assume a zero value. “Neglected Parameters” play a role in the general definition of the PBPK model. They should be accounted for some drugs with specific intestinal absorption features. For our purposes, and with respect to sorafenib, they can be neglected and set to 0. 3162

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Industrial & Engineering Chemistry Research Table 2. A Posteriori Statistical Analysis of Parameters Assessmenta parameter jHO−P jPT−P jSIL A jLP EffH QGB in

value 3.31 1.46 4.32 3.59 2.61 3.58

× × × × × ×

10−2 10−1 10−2 100 10−2 10−2

t-value

95% confidence interval 1.68 3.33 1.40 6.57 4.76 1.14

× × × × × ×

10−2 10−2 10−2 10−1 10−3 10−2

4.13 9.16 6.46 11.49 11.50 6.58

standard deviation 8.02 1.59 6.68 3.13 2.27 5.44

× × × × × ×

relative (local) SI

10−3 10−2 10−3 10−1 10−3 10−3

0.815 1.035 0.047 0.458 0.925 0.809

a “Unknown parameters” were regressed with the numerical methods described in section 2.3. This table reports the values with 95% confidence interval, t-values to be compared with a t reference value of 2.101, standard deviation, and relative Sensitivity Index.

Figure 4. PBPK model compared to Pawaskar et al.11 experimental data with standard deviation bars. Sorafenib dose is 20 mg/kg, with mice body weight of 27.5 g. Simulated curves are shown on a semilog scale and are compared to experimental data, showing a good consistency with the experimental data as confirmed by the correlation indexes (Plasma, 0.99; Liver, 0.94; HO, 0.96; PT, 0.86). Note that the PT correlation index is 0.86 after a significant experimental data uncertainty (expressed by the standard deviation bars).

t test, where t values for the estimated parameters (based on eq 12) were compared to a reference value based on a t-Student distribution with 18 (i.e., NExpData − NUnknownParameters) degrees of freedom. As a general rule, parameters with a t value higher than the reference t are considered acceptable (a more comprehensive description of model-based design of experiments can be found in ref 42. ti =

Pi vi

SI =

dOutput |Output − Output′| = dPi Pi ·macheps

(13)

SI values are compared in terms of relative (local) SI, as from eq 14.46 SI relative =

∂AUC Pi · ∂Pi AUC

(14)

Here AUC represents the model output chosen as a reference for the model perturbation, i.e., the sum of the Areas Under the Curve of Plasma, Liver, Highly perfused Organs, and Poorly perfused Tissues concentration vs time curves. Table 2 reports the results of the sensitivity analysis and the statistical data on the model “Unknown Parameters”.

(12)

Here P refers to the regressed parameter value, v is the variance of the parameter that is obtained from the parameters variance/ covariance matrix. Index i represents the ith parameter of the regressed set. Additionally, a local sensitivity analysis was performed to determine the relative importance of the regressed parameters on the model outcome. By definition, the sensitivity index can be determined assigning a small perturbation to each parameter while keeping the other constants. Sensitivity index (SI) is defined as the ratio of the model output variation over the parameter perturbation, as shown in eq 13.

3. RESULTS This section shows the main results of the model compared to experimental data. Given a rather good consistency, the numerical simulations can be used to figure out an in-depth understanding of sorafenib PK, by complementing the information available from experimental studies. Figure 4 3163

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Figure 5. Sorafenib elimination pathways: (A) concentration−time profiles of the eliminated drug and refers to the administered dose of 20 mg/kg with a mice body weight of 27.5 g. The residual difference between eliminated mass and total administered dose, at the final time of simulation, is the drug fraction still present in the body. (B) Proposes the same data at times of 5, 10, 15, and 20 h after administration. Data are normalized with respect to the total administered dose. Histogram bars highlight that the major contribution to drug elimination is the hepatic metabolism while fecal elimination is minimal. This simulation ignored the renal excretion contribution because literature confirms it to be negligible when compared to hepatic metabolism.24 Note that products of metabolism are still collected in faeces during an in vivo study, and here feces refer just to the unmodified sorafenib. This is also consistent with the literature that assigns to the hepatic metabolism, the major role in sorafenib elimination.11

very tight confidence intervals. There is not any significant correlation among the parameters, as it is confirmed by the variance/covariance and correlation matrixes. Finally, the sensitivity analysis suggests that in terms of model output variation (considered as a concentration−time curve AUC), the intestinal absorption coefficient (jSIL A ) plays a minor role, while all other parameters (apart from jLP) are comparable in terms of influence on the simulation results.

shows the model simulation in different organs and tissues with respect to experimental concentrations. The trend of PK model is consistent with experimental data. In particular, the initial period (until the eighth hour, where most of the experimental values are measured) shows a reasonable behavior and a good consistency with the EHC hypothesis, which is responsible for the double peak of drug concentration. In general, the bile recirculation prolongs the drug lifetime at a range of high concentrations, and this is a critical point that should be considered both in drug development stages and for administration assessment. Since the main goal of PBPK models is to reproduce the complete ADME process of drugs, these models can provide further information beyond the classic PK profiles. For instance, Figure 5 shows the different compartmental contributions to sorafenib metabolism and excretion. This is an important and sometimes overlooked strength of modeling in pharmacokinetics. In section 1.1, sorafenib PK was commented on with respect to EMA statements, and the most important contribution to plasma clearance was recognized to be the hepatic metabolism. Fecal excretion in Figure 5 is the unmodified sorafenib fraction, which is expelled from the organism and does not consider the metabolites, which are accounted for by the hepatic metabolism. These data are particularly interesting because they show the advantage of a PBPK model with respect to a standard compartmental model missing any physiological meaning. Indeed, it is possible to simulate drugs PK to investigate tissue concentrations that were not experimentally studied. The quality of the model and the goodness of the regressed parameters were tested via a priori and a posteriori identifiability analysis. The model structural (a priori) identifiability confirmed that the model was globally identifiable. Then, knowing the experimental data value and standard deviation from the Pawaskar et al.11 study, a practical (a posteriori) identifiability test was performed, and Table 2 reports the results of the statistical analysis. Regressed model parameters confirmed to be robust estimates. Indeed, all parameters passed the t-test and showed

4. DISCUSSION Sorafenib is a rather recently commercialized drug for the treatment of specific malignancies. Being a novel molecule, its pharmacokinetic properties are still not completely understood. By recognizing the potentiality of modeling tools in new drug discovery, we developed and applied a reduced PBPK model to describe the PK of sorafenib. This model is capable of maintaining a close similarity with mammalian physiology, even though it preserves a rather good level of simplicity. A peculiar feature of this physiological model is that it accounts for EHC, a phenomenon that was recognized to strongly characterize concentration-vs-time profiles of sorafenib and other orally administered drugs. With the purpose of verifying the model capability of determining drug concentration in tissues, we referred to the Pawaskar et al.11 PK study in mice. As a further source of uncertainty in sorafenib PK, clinical studies revealed important insights and showed wide interindividual variability among patients that was quantified as 40−80% of the standard administered dose.47,48 Investigating the origin of this variability, Jain et al. reported “baseline bodyweight... accounts for 4% of inter-individual variability”.12 To improve the model capability and address this complexity, we proposed a formulation which determines some model parameters as a function of body mass by exploiting literature correlations.38 These parameters are named “individualized” and can play a significant role when the PBPK model is applied to simulate sorafenib ADME routes in humans. Concerning other model parameters, the PBPK model has to undergo a parameter identification procedure (i.e., nonlinear regression) 3164

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Industrial & Engineering Chemistry Research to determine some adaptive parameters (i.e., “unknown parameters”), which can be neither found in the literature nor measured. However, once identified respect to some reference experimental data, these parameters can be assumed constant for future applications. Globally, the model is able to reproduce concentrations in multiple distribution regions of mammals as confirmed by the results shown in Figure 4. This allows moving a step forward and analyzing both qualitatively and quantitatively PK predictions with respect to previous literature reports. Sorafenib experimental measures show significant distribution of drug in the organism, with a large presence in the liver and other organs, which are richly perfused by blood. This occurs despite a very high binding fraction to plasma proteins (>99%). We suppose that this wide distribution can be justified via active transport mechanisms that make possible the migration of bound drugs across blood vessel walls. We attempted to simulate drug distribution by directly adopting the literature value for drug binding to blood proteins and by considering the passive transport only, but this approach resulted in the impossibility to achieve concentrations comparable with organ/tissue experimental values. Consequently, the R parameter, which determines the drug to plasma proteins binding, was arbitrarily set equal to a lower value (i.e., 75%), which still accounts for a high binding. Concerning drug metabolism, the hepatic activity is rather intense as confirmed by the simulation (Figure 5). Hepatocytes account for large presence of sorafenib, but only a smaller portion is available for metabolism as a fraction is conjugated in the bile and another fraction returns to blood. Enterohepatic circulation was quantified by considering a specific compartment (i.e., the gallbladder) which acts as a purely accumulating vessel until the Oddi’s sphincter releases the bile to the duodenum, thereby producing the characteristic double peak in the PK curves. In this work, the sorafenib circulation through the gallbladder depends on the drug concentration in the bile and consequently the accurate assessment of the bile volumetric flow is critical. By recognizing the importance of this value, we specified two different volumetric flows, contrary to the literature where it is common to find a single value.35 This depends on the fact that the bile leaving the liver is diluted, and one of the functions of gallbladder is to absorb water and concentrate the bile. This causes the bile flow leaving the gallbladder (QGB out) to be smaller than the entering one (QinGB), and this influences the concentration of the drug which is carried along. We believe that understanding PK via modeling tools, in the direction of more detailed qualitative/quantitative information, will get further improvement and validation from additional knowledge provided that in vitro and preclinical studies become available. This article can also play the role of foundation to further specific studies such as pharmacodynamic analysis, PK translation from animals to humans, and multiscale modeling toward tumor sites.49 In this last case, PBPK modeling would provide the input data of drug concentration for the neoplasm analysis.

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Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the financial support from the Italian Ministry of Education through the PRIN 2010-2011 (Grant 20109PLMH2) fund. Francesco Trotta and Rina Cavallaro of the PRIN project are also acknowledged for their fruitful discussions and contributions to unraveling the important details behind the anatomical and physiological features at the basis of the PK model.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*Phone +39 02 23993271. E-mail: [email protected]. ORCID

Davide Manca: 0000-0003-2055-9752 3165

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