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INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS VOL. 14, NO. 1, MARCH 2009, 33-40 Understanding the Role of Liver Zonation in Toxin Elimination
Shahab SHEIKH-BAHAEI, Sean H. J. KIM, Shahriar SHEIKHBAHAEI and C. Anthony HUNT Abstract- Using game theory and reinforcement learning, we
responsible for eliminating toxins and xenobiotics (includ- created and analyzed generalized agent-based and compartmental
ing drugs) that find their way into blood. The rate of eli- models of hepatic toxin elimination processes to explore plausible
mination, known as causes of hepatic functional zonation. We considered a general situa-
hepatic clearance, is different for each tion in which a group of protective agents (analogous to liver cells)
compound. Histologically, the liver is divided into lobules. cooperate and self-organize their efforts to minimize optimally the
Lobules consist of hepatocytes arranged in a roughly cy- negative effects of toxin intrusions. Following a totally different ap-
lindrical or spherical shape. The central vein (CV), proach, we constructed a physiologically based model of a two-zoned
through which blood exits, is at the center. At the peri- liver to study the physiological consequences of zonation. The results
of the two models support the hypothesis that liver zonation might be

phery are portal vein (PV) triads. A lobule is often de- a consequence of an optimal strategy for toxin clearance.
scribed as being organized into three zones: periportal (up- stream or zone 1), which encircles the portal tracts where Index Terms—liver zonation, multi-agent learning, computa-
blood enters, middle (zone 2), and perivenous (downstream tional biology, self-organizing, Q-learning, game theory
or zone 3), which is poorly oxygenated and located around central vein. Oxygenated blood enters upstream, passes 1. INTRODUCTION
through the mid-zone, and exits downstream. Because of this spatial topology, different liver cells may not have the The liver performs a wide range of functions including de- toxification of blood-borne compounds. So doing protects same exposure to incoming resources and compounds. For the body. A human cannot live more than 24 hours with- example, nutrients (e.g. oxygen) are more available to up- out the liver. Hepatocytes, the parenchymal cells of the stream than to the downstream cells. liver, are among the most complex cells in the body. They Hepatocytes, although genetically identical, exhibit he- cooperate with each other to detoxify xenobiotics by meta- terogeneous enzyme and transporter activities depending bolizing them to less toxic compounds. Over the course of on their location within the lobule. For example, under their evolution they have learned to do so in an effective and optimal way. Hepatocytes express heterogeneous, lo- normal conditions, hepatocytes located downstream ex- cation-dependent enzyme and transporter activities to faci- press more enzymes for xenobiotic metabolism than do up- litate detoxification, apparently following an intrinsic stream hepatocytes. An obvious question is: why? agenda, the principles of which are not fully understood. This phenomenon is known as liver zonation [1]. To gain insight into those processes, we constructed and analyzed a generalized problem of cooperative agents protecting their common wealth from harmful intruders. The agents are as- sumed to have incomplete information about each other and cannot form coalitions). 1.1 Biology Background
The liver is a complex biochemical factory which syn- thesizes, modifies, and metabolizes thousands of sub-stances daily and provides the body with essential sub- stances such as proteins and fats. The liver is also Manuscript received April 1, 2009. This work was supported in part by the CDH Research Foundation and a Graduate Fellowship to S. H. J. Kim from the International Foundation for Ethical Research. This pa- per is extended from "Multi-Agent Based Modeling of Liver Detoxifi- Fig 1. Cross section of a liver lobule. PV: portal triads, CV: central vein, cation" published at Spring Simulation Multiconference 2009, Agent- arrows: direction of blood flow. Oxygenated blood enters the lobule from Directed Simulation Symposium, San Diego, CA, USA, March, 2009. portal veins, and exits from central vein. Usually a liver lobule is de- Shahab Sheikh-Bahaei (e-mail: shahabsb@berkeley.edu) and S. H. scribed as being divided into three functional zones. J. Kim (e-mail: seanhjk@berkeley.edu) are with the UCSF/UC Berke- ley Joint Graduate Group in Bioengineering, University of California, Berkeley, CA 94720, USA. Shahriar Sheikhbahaei (e-mail: sha- To represent hepatic metabolic zonation in physiologi- hriar@berkeley.edu) is with Department of Molecular and Cell Biolo- cally based, pharmacokinetic models, researchers [2][3] gy, University of California, Berkeley, CA 94720, USA usually divide the liver into compartments, each C. A. Hunt is with Department of Bioengineering and Therapeutic representing a different intrahepatic zone. Using a differ- Sciences, University of California, San Francisco, CA 94143, USA (e-mail: a.hunt@ucsf.edu). ent approach, Lamers et al. [4] presents a "mechanistic 34 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL. 14, NO. 1, MARCH 2009 model" which proposes that zonation is induced by porto- ADC = e2·D = (1 – maxE2·p2)·e1·D
central signal gradients. Expected average costs of resource consumption are: 2. METHODS
ARC1 = maxE1·p1·R
ARC2 = maxE2·p2·e1·R
All existing models of liver zonation are top-down models. They fail to hypothesize elementary mechanisms Total expected costs are: that motivate the collective behaviors of liver cells. <Cost1> = ARC1 + ADC = maxE1·p1·R +
(1 – maxE2·p2)(1 – maxE1·p1)·D
(-D, . -D) <Cost2> = ARC2 + ADC = maxE2·p2·(1 – maxE1·p1)·R +
(1 – maxE2·p2)(1 – maxE1·p1)·D
(-R, 0,.0) (0,-R, 0,.0) (0, 0,. -R) Fig 2. The game-theoretic model of the system. Agents either ignore or 1> = maxE1·p1·(R – D) +
eliminate intruders. If Agent i (Ai) eliminates an intruder, it pays the cost {1 – maxE2·p2·(1 – maxE1·p1)}·D
of resource consumption (R). If all agents ignore an intruder, all must pay cost D, which is a consequence of damage caused by the intruder to the <Cost2> = maxE2·p2·(1 – maxE1·p1)(R – D) +
common wealth. p is the elimination strategy of agent i: it is the proba- bility that Ai eliminates an incoming intruder. Ai's ability to eliminate is (1 – maxE1·p1)·D
limited by maxE i (0 maxEi 1). At any given location in the strategy space, agents have a To understand the costs and benefits that may be asso- preferred direction of movement to reduce their costs. For ciated with liver zonation, we began by using the game the two-player game, the direction can be described as a theoretic model as shown in Fig. 2: a sequence of vector field based on the gradients of the above two cost is protecting their common wealth (all extrahepatic tissues) against intruders. Agents are limited in their ability to ∂ < Cost > ∂ < Cost > eliminate intruders. The goal of each is to minimize poten- tial damage caused by the intruders while minimizing re- source consumption. What is the optimal elimination The vector field is shown in Fig. 3 for n = 3 and D/R = 0.6, strategy for agent i? Obviously, an optimal strategy for 1.1, 1.7, 2.3, 3.7 and 10.0. It is easy to find the equilibrium agent i depends on the strategies of other agents who have of each game by inspecting its vector field. The figure shows that the equilibrium changes as D/R increases. Each agent has two options: eliminate or ignore an in- When damage is very small (D/R < 1), all agents ignore truder. The immediate cost for elimination is resource because it is not cost effective to eliminate intruders. consumption, denoted by R. Ignoring an intruder does not When damage is moderate, only downstream agents ex- constitute an immediate cost; however, when all agents ig- pend elimination effort. When damage is large, middle nore an intruder (or it escapes for whatever reason), then agents cooperate with the downstream agents. When dam- all must pay the cost associated with any damage caused to age is large enough, all agents expend elimination effort. the common wealth (denoted by D). It is assumed that a The analysis can be extended to a general case of n signal informs agents of the damage cost at the end of each players as follows: round of play. Agent i cannot eliminate more than maxE i fraction of incoming intruders even if it expends maximum ei = (1 – maxEi·pi)·ei-1 i = 1, 2, … , n eo = 1
elimination effort (0 maxEi 1). The elimination strate- n·D
i, p , is the probability that it eliminates an in- coming intruder. Agents who see intruders earlier, are ARCi = ei-1·maxEi·pi·R
called upstream agents; the others are called downstream i> = ARCi + ADC To analyze the game's equilibrium, we first specified <Costi> = ei-1·maxEi·pi·R + en·D (Eq.3)
that there are only two agents. Their cost functions are where e is the fraction intruders that escape from agent i; calculated as follows: ADC is the average damage cost to each agent; ARC is the average cost of resource consumption to agent i; and 1 = 1 – maxE1·p1
<Costi> is the total expected cost (due to both actions) to e2 = (1 – maxE2·p2)·e1
agent i. The vector field can be calculated the same as for where e and are the fraction of intruders that escape the two-player game, but it is infeasible to visualize and from Agents 1 and 2. Expected average costs of damage to find equilibria. In general, analyzing equilibria of games each agent will be: involving three or more players is hard [5][6]. Sheikh-Bahael et al: Understanding the Role of Liver Zonation in Toxin Elimination 35 The above analysis requires that all agents have a priori ples and [10] for a review of other multi-agent learning knowledge about other agents and the environment. All techniques). the actions available to other agents and all costs with all Q-learning is a primitive form of learning [7] in which combinations of actions are known by all agents. Conse- utility values (Q values) are learned for state-action pairs, quently, it does not offer a mechanism through which au- absent a model of the environment. It provides a simple tonomous agents can reach an optimal strategy. means for agents to learn how to act optimally in an un- We considered a more realistic situation in which known environment. At each step, a Q-learning agent uses agents do not have a priori information about their envi- its new experience to improve its long-term reward esti- ronment (including other agents). We used multi-agent mate by combining new information with prior experience. simulation and enabled agents to learn from experience fol- Each Q-learning strategy is determined by the value lowing a simple reinforcement learning rule. By keeping function, Q, which estimates long-term discounted rewards track of accumulated reward (and penalty), agents could be for each action. General scheme of Q-learning algorithm reinforced to learn an optimal clearance strategy. Their used by each agent is as follows: task was to maximize the long-term average reward per ac-tion. (1) Observe the current state (in this study, there is only Q-learning algorithm [7], a well known rein- forcement learning algorithm, has been shown to converge (2) Choose and execute an action based on the Q-values (available actions are to an optimal decision policy. from a set of available actions, Act Q-learning has a solid foun- = eliminate and = ignore). The agent selects its dation in the theory of Markov decision processes [8]. It is easy to implement and has been used widely in both single- action according to a probability given by the Boltzmann distribution: agent and multi-agent contexts (see [8] and [9] for exam- D/R = 0.6 D/R = 1.1 D/R = 1.7 D/R = 2.3 D/R = 3.7 D/R = 10.0 Fig 3. At any given non-equilibrium point in the strategy space, the agents have a preferred direction of moving in order to decrease their expected costs. The net direction towards which one moves in the strategy space depends on the slope of the cost functions at that point. Cone arrows show the net direction of movement when n = 3 and D/R ratio changes from 0.6 to 10.0. Each game's equilibrium is shown by a star. (A) D/R is small. In this case, all agents ignore the intruders. (B, C, D) D/R is moderately large; the equilibrium is such that Agent 1 ignores (p = 0) but Agent 2 and 3 eliminate. (E, F) When D/R is large enough, Agent 1 starts making elimination effort. 36 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL. 14, NO. 1, MARCH 2009 and Km is the Michaelis-Menten constant. V Q( Act ) / T to be affected by the level of metabolic enzymes and trans- ∑ Q(Act )/T porters expressed by cells. As a result each zone has its max. The two zones are specified to have equal Km T, called "temperature," adjusts the randomness of (3) Observe the new state (for this study, this step is not necessary because there is only one state) and receive an immediate reward. Q value based on the action taken, a, using Eq. Q a) ← 1 ( −α)Q(a) +α(reward + βV ) V = max Q(b) where α is the learning rate (0 ≤ α < 1) and β is the dis- counting factor (0 ≤ β < 1). Here we specified α = 0.1 and = 0.5. V is known as the value of the game and is equal to the maximum Q value. Figure 4. A physiologically-based model to analyze the effects of hepatic 2.1. The Agent-Based Model
zonation on toxicity exposure to the whole body. Boxes are reservoir compartments. Arrows show xenobiotic flow directions. q's are xenobiotic In mammalian livers, an absorbed toxin can be cleared flow rate constants. Xi's are xenobiotic concentration in corresponding re- by any of a sequence of hepatocytes. In order to gain in- servoir compartment shown. pi is the probability that xenobiotic passes sight into that process, we modeled the liver as consisting through path i. of many, parallel sets of toxin eliminating agents arranged Further, we represent the cost to the organism, J, as be- in sequence from PV to CV in Fig. 1. Each agent used a ing proportional to (CL Q-learning algorithm to decide its clearance strategy. An 1)2 + (CL2)2 + (cAUBC)2, where agents' task was to minimize the extrahepatic damage to 1 and CL2 are intrinsic periportal clearance and intrinsic perivenous clearance, respectively; AUBC is the area un- the organism of which they are part. We specified that der the blood concentration curve and c adjusts the relative agents become aware of extrahepatic tissue damage via cost contribution of a fixed dose based on xenobiotic toxic- alarm signals that are quickly released into blood by the ity. damaged tissue. Doing so was based on the fact that hepa-tocytes, like immune cells, express toll-like receptors [15]. They enable cells to detect chemical alarm signals generat- 3. RESULTS
ed by damaged tissues. Figure 5 shows the emergent strategies obtained for 2.2 Physiologically-Based Model
different values of D/R. When D/R is small, agents expend In addition to the agent-based model, we used a tradi- little effort to eliminate intruders. As D/R increases, tional physiologically-based modeling approach to study downstream agents expend more elimination effort than the effects of hepatic zonation on toxicity exposure to the upstream agents. When D/R is large, upstream agents be-whole body (Fig. 4). For simplicity, the liver is gin cooperating and contribute to the elimination process, represented as having two zones: periportal (zone 1) and until all agents are expending maximum effort. Although perivenous (zone 2). Compounds in the liver are assumed the downstream agents always expend an equal or greater to stochastically take one of the following four paths with effort than do upstream agents, it does not mean that probability pi: Path 1: neither of the two zones encounters downstream agents actually eliminate more intruders. For the compound. Path 2: only zone 1 encounters the com- example, when D/R = 5.0, upstream agents eliminate more pound. Path 3: both zones encounter the compound. Path 4: intruders than do downstream agents. only zone 2 encounters the compound. Figure 6 shows the results from the physiologically There is one set of differential equations for each path based model. Figure 6A shows a typical 3D surface of the (Table A1). At each time step, one of the four sets is cho- cost function, J, when c = 0.1. At that toxicity value, the sen according to the probability associated with the corres- minimum cost (Jmin) occurs when CL1 = 0.35 and CL2 = ponding path. In this model, zone 1 and zone 2 eliminate 0.45 (i.e. zone 2 expends more clearance effort than zone compounds independent of each other. We specify that the 1). What happens if toxicity is altered? Figure 6B shows mechanisms of xenobiotic elimination in the liver (includ- how Jmin changes if toxicity varies from 0.05 to 5. It depicts ing uptake transport, biliary efflux and metabolism) follow as toxicity increases, both CL1 and CL2 increase however saturable Michaelis-Menten kinetics. Intrinsic clearance of 2 is greater or equal to CL1 at all toxicity levels. That observation is consistent with the game theoretic and mul- each zone is specified to be CLi = Vmax,i/Km, where Vmax ti-agent models results. is the maximum elimination (metabolic + transport) rate


Sheikh-Bahael et al: Understanding the Role of Liver Zonation in Toxin Elimination 37 communication mechanisms provide appropriate danger signals to agents, informing them about damage caused. Furthermore, agents are assumed to take and use the best Fig 5. Equilibrium of the n-player game changes with D. Upstream is to the left (small numbers) and downstream is to the right. The average eli-mination strategies of 10 agents are shown for different ratios of D/R (shown on each curve) after 10,000 simulation steps (maxE = 0.05). When D/R is small, upstream and downstream agents expend little elimi- nation effort. As D/R ratio increases, downstream agents expend more elimination effort than upstream agents. When D/R is large, upstream agents start to cooperate and contribute to the elimination process. 4. VALIDATION
Liver cells exhibit a similar behavior: they express he- terogeneous, location-dependent enzyme and transporter activities to detoxify compounds. Downstream cells com-mit most to toxin elimination; in other words, xenobiotic metabolism is preferentially located downstream, in the pe-rivenous region [1]. The model also suggests that the loca-tion of hepatotoxicity depends on compound toxicity: less toxic compounds are likely to damage the downstream re-gion, whereas compounds that are more toxic are more Fig 6. Effects of toxicity change in the physiologically-based model. (A) likely to damage the upstream region. Several highly toxic 3D plot of cost function, J, versus CL1 and CL2, when c = 0.1. (B) Dotted compounds [11], such as TCDD (LD50 = 0.034 mg/kg), curve: trajectory of Jmin as c changes from 0.05 to 0.5. For each point on the trajectory the value of corresponding c is reported. Solid line: the uni- cyclochlorotine (LD50 = 2-3 mg/kg), and gossypol (LD50 = 5 mg/kg), selectively damage the upstream zone [12-14]. On the other hand, less toxic compounds such as acetami- Simulations showed that agents adjust their clearance nophen (LD50 = 1295 mg/kg) selectively damage the effort based on the following two factors: the potential downstream zone [11]. damage caused by intruders, and their ranked proximity to the entity being protected. Downstream agents (the ones 5. DISCUSSION
with less proximity to the common wealth) generally ex- pend more elimination effort than do upstream agents, de- We presented a simple, agent-based model of a genera- pending on the threat. lized, hepatic xenobiotic clearance process. The model The emergent, collective behaviors of these agents are consists of a group of agents that, similar to hepatocytes, similar to those of hepatic cells in terms of xenobiotic cooperate to protect a common wealth against toxic intrud- clearance. The model suggests that an underlying mechan- ers. The agents do not have a priori information about ei- ism responsible for liver zonation may be similar to the ther the environment or other agents (e.g., the number of model's simple mechanism. Hepatocytes may possess other agents, actions available to them, costs associated subsystems (e.g. special proteins, signaling pathways, etc.) with their actions, etc.). The agents use Q-learning, a pri- that produce phenomena that have properties that are indis- mitive form of learning, to minimize their long-term dis- tinguishable from those of the Q-learning algorithm. counted costs. Agents are assumed to know the cost of Can hepatocytes learn? At the molecular level, short- their own actions. We also assume that relatively fast term memory (persistent for weeks) within brain forms in


38 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL. 14, NO. 1, MARCH 2009 part by changing the strength of existing synaptic contacts. [13] T. Tatsuno, E. Ito and K. Terao, "Liver injuries induced by cycloch- It can be achieved by altering the amount of neurotransmit- lorotine isolated from Penicillium islandicum." Arch. Toxicol., vol. 55, pp. 39-46, 1984. ter released from presynaptic terminals and/or postsynaptic [14] Y. C. Lin, D. C. Nuber and S. Manabe, "Zone-specific hepatotoxicity receptor concentrations [17]. Simply stated, the molecular of gossypol in perfused rat liver." Toxicon, vol. 29, pp. 787-90, mechanism of learning involves adaptation of synaptic strength based on feedback signals received from other [15] T. R. Billiar, Y. Vodovotz, D. A. Geller, A. N. Salyapongse and S. Liu, "Hepatocyte toll-like receptor 2 expression in vivo and in vi- neurons [18]. Hepatocytes exhibit complex behavior. Re- tro: role of cytokines in induction of rat TLR2 gene expression by cent studies show that toxin-induced hepatocyte injury is lipopolysaccharide." Shock, vol. 14, pp. 361-5, 2000. not a simple passive process regulated by the dose of an [16] M. Hosseini-Yeganeh and A. J. McLachlan, "Physiologically Based inducer compound; rather it is an active process in which Pharmacokinetic Model for Terbinafine in Rats and Humans," An-timicrob. Agents Chemother., vol. 46, pp. 2219, 2002. active signaling plays a crucial role [19]. Hepatocytes [17] E. R. Kandel and J. H. Schwartz, "Molecular biology of learning: change protein expression levels in response to toxic modulation of transmitter release," Science, vol. 218, pp. 433-443, shocks [1] and adjust their sensitivity to signaling mole- cules (for example see [20] and [21]). Upstream hepato- [18] E. R. Kandel, J. H. Schwartz and T. M. Jessell, Essentials of Neural Science and Behavior. Appleton & Lange, 1995, cytes can communicate with downstream hepatocytes via [19] B. E. Jones and M. J. Czaja, "Intracellular signaling in response to blood borne signals [4] and/or intercellular calcium waves toxic liver injury," Am. J. Physiol. Gastrointest. Liver Physiol., vol. [22-23]. On the other hand, downstream cells can commu- 275, pp. 874-878, 1998. nicate back with upstream cells via bile acids [24]. This bi- [20] Y. Xu, B. E. Jones, D. S. Neufeld and M. J. Czaja, "Glutathione modulates rat and mouse hepatocyte sensitivity to tumor necrosis directional communication creates a complex intercellular factor α toxicity," Gastroenterology, vol. 115, pp. 1229-1237, feedback system which might contribute to regulation of adaptation (learning) in the liver. [21] S. Herrmann, M. Seidelin, H. C. Bisgaard and O. Vang, "Indolo [3, 2-b] carbazole inhibits gap junctional intercellular communication in rat primary hepatocytes and acts as a potential tumor promoter," Carcinogenesis, vol. 23, pp. 1861, 2002. We thank Prof. Yoav Shoham and members of the [22] G. DUPONT, T. TORDJMANN, C. CLAIR, S. SWILLENS, M. CLARET and L. COMBETTES, "Mechanism of receptor-oriented UCSF BioSystems Group for helpful suggestions and dis- intercellular calcium wave propagation in hepatocytes," FASEB J., cussions. We gratefully acknowledge research funding vol. 14, pp. 279-289, 2000. provided by the CDH Research Foundation and a Graduate [23] V. Serrière, B. Berthon, S. BOUCHERIE, E. Jacquemin, G. Guillon, Fellowship (SHJK) from the International Foundation for M. Claret and T. Tordjmann, "Vasopressin receptor distribution in the liver controls calcium wave propagation and bile flow 1," FA- Ethical Research. SEB J., vol. 15, pp. 1484-1486, 2001. REFERENCES
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Sheikh-Bahael et al: Understanding the Role of Liver Zonation in Toxin Elimination 39 Shahriar Sheikhbahaei will receive B.A. de-
gree in molecular and cell biology with a con- q42 x4 centration on neurobiology in 2009 from Uni- versity of California, Berkeley, USA. His V max 2 x3 q x1 q3 x3 current research is on plasticity of neuronal synapses. His research interests include synap- tic plasticity, cellular and molecular neurobi- q3 x3 (q41 q42 q43 )x4 ology, developmental neurobiology, neurogenetics and mechan- isms of nervous system structure and function. q43 x4 q5 x5 C. Anthony Hunt received B.S. degrees in
Chemistry and Applied Biology from the Model Parameters. Volumes of distribution: The apparent
Georgia Institute of Technology and the Ph.D. volume of distribution differs from compound to com- degree from the University of Florida. His re-search transitioned from wet-lab to computa- pound and from organ to organ. We assume that the model tional in 1998. He directs the BioSystems compartments are well-stirred and substrates instantly dis-Group. The group's research focuses on de- tribute in the entire tissue volume. The volumes of organs veloping and using advanced modeling and are reported for a 250-g rat in Table A.2. simulation methods to help unravel the complexities of biological systems used in biomedical and pharmaceutical research. Dr. Hunt is a member of several scientific and engineering societies, 1 = Volume of G.I Tract (Vgi stomach + Vsmall intestine + including ACM, IEEE Engineering in Medicine and Biology So- ciety, Society for Computer Simulation International, and Bio- 2 = Volume of zone 1 (Vz1 medical Engineering Society. He is a member and fellow of the ) = (1/4) V American Association for the Advancement of Science and the 3 = Volume of zone 2 (Vz2 American Association of Pharmaceutical Scientists. V4 = Volume of blood (Vb)= Varterial blood V5 = Volume of other organs = V(muscle + skin + adipose Appendix
+ heart + kidney) Table A1. Equations of the model for each of the four paths shown in Figure 4. Table A.2. Physiological parameters of tissues in a 250-g rat [16]. V (ml)
Q (ml/min)
q42 x4 q x1 (q42 q43 )x4 q5 x5 q42 x4 q = blood flow of G.I. Tract (Qgi) =hepatic blood flow (QL) V max1 x2 q x1 q3 x2 q =hepatic arterial flow (QL-Qgi) q =G.I. Tract blood flow (Qgi) q3 x2 (q41 q42 q43)x4 q =effective flow of substrates from blood to other organs which we assume is generally less than the sum total blood q43 x4 q5 x5 flow of the organs. q5=clearance rate of xenobiotics from blood by other or- q42 x4 gans which is primarily done by kidney (renal clearance). The rate constants are listed in Table A3. V max1 x2 q x1 q2 x2 V max 2 x3 q3 x 2 q3 x3 = q x − (q + q + q )x If the liver lobule is roughly approximated as a cylinder (radius r and height h) with two zones, then zone 2 (perivenous) could q43 x4 q5 x5 be thought of as a smaller cylinder with radius r/2. The rest of the volume would represent zone 1 (perportal). The ratio of the two volumes can be calculated as follows: Volume of zone 1 = Vz1 = 2hπr2-hπr2; Volume of zone 2 = Vz2 = hπr2; Vz1/Vtotal= 3πr2/4πr2 = 3/4; and Vz2/Vtotal= πr2/4πr2 = 1/4 40 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL. 14, NO. 1, MARCH 2009 Table A3. Rate constant values. Parameter Value
Metabolic Enzymes (Vmax and Km): We assume that the mechanisms of xenobiotic elimination in the liver (includ-ing uptake transport, biliary efflux and metabolism) are sa-turable. We represent the mechanisms using Michaelis-Menten kinetics.

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