Modeling the Collective Behavior of Hepatocytes in Clearing Toxins

Shahab Sheikh-Bahaei1 and C. Anthony Hunt 1,2

1 The UCSF/UCB Joint Graduate Group in Bioengineering, University of California, Berkeley, CA,
2 The Department of Biopharmaceutical Sciences, BioSystems Group, University of California, San Francisco, CA


I. Methods





Figure 1. The schematic of the simple game (A) and its mapping to the liver (B).


  • Filled circles (A1 and A2): Agents 1 and 2 (also referred to as players).
  • Unfilled circles (N1 and N2):  Nature players (map to extracellular spaces, such as sinusoidal vessels etc, in periportal and perivenous regions, respectively)
  • Arrows: possible routs a compound can take.
  • (-c1,-c2) at terminal nodes indicates the costs of c1 and c2 imposed to players 1 and 2, respectively. ci can be one of the following:
    • 0: no cost,
    • e: cost of elimination (e.g. energy/resource consumption),
    • T: toxicity costs (e.g. the damage caused by the compound to the external tissue).
  • Li: percentage of incoming compounds that player i does not encounter.
  • Pi: percentage of encountered compounds that player i eliminates.

Description of the game:

  • At each time step one simulated compound enters the game:

1. Nature forces at N1 pass the compound to N2 with probability L1; or alternatively, pass it to player 1 with probability 1-L1.

2. Player 1 either eliminates the compound with probability P1 and pays the elimination cost (e), or takes the risk and pass the compound to N2 with probability 1-P1.

3. Nature forces at N2 pass the compound to the terminal node with probability L2, in which case both players have to pay the toxicity cost T. Conversely, the forces might pass the compound to player 2, with probability 1-L2.

4. Player 2 either eliminates the compound with probability P2 and pays the elimination cost (e), or takes the risk and passes the compound to the terminal node and makes both players pay the toxicity cost T.

Mappings to the liver:

  • Players 1 and 2: xenobiotic eliminating agents in the periportal and perivenous regions of the liver, respectively.
  • Pi: Clearance Strategy of player i -- the probability that the xenobiotic eliminating agents in zone i eliminate an encountered compound. In other words, it maps to the clearance probability as a consequence of all complex stochastic events that eventually result in elimination of the compound seen by zone i. The events include: binding to transporters, membrane proteins, or enzymes, metabolism, excretion to bile, etc.
  • Li: Percentage of incoming compounds that cannot be seen by zone i due to physical constraints outside cells’ control. Examples include: remaining out of hepatocytes’ reach by staying inside the sinusoidal vessels due to saturated or damaged transporters, etc.  Li is the percentage of compounds players i cannot process even with maximum effort.





Text Box:

Figure 2. A physiologically-based model to analyze the effects of hepatic zonation on toxicity exposure to the whole body.


  • Boxes: reservoir compartments
  • Arrows: xenobiotic flow directions
  • q's: xenobiotic flow rate constants
  • Xi’s: xenobiotic concentration in corresponding reservoir compartment shown
  • pi: probability that a xenobiotic particle passes through path i.


Description of the model:

The model uses a traditional physiologically-based PKPD approach.  Liver is assumed to have 2 zones: periportal (zone 1) and pervenous (zone 2). Compounds in the liver are assumed to stochastically take one of the four paths with probability pi:

  • Path 1: neither of the two zones encounters the compound.
  • Path 2: only zone 1 encounters the compound.
  • Path 3: both zones encounter the compound.
  • Path 4: only zone 2 encounters the compound.

There is one set of differential equations for each path. At each time step, one of the four sets is chosen according to the probability associated with the corresponding path. In this model, Zone 1 and Zone 2 eliminate compounds independent of each other. We assume that the mechanisms of xenobiotic elimination in the liver (including uptake transport, biliary efflux and metabolism) follow saturable Michaelis-Menten kinetics. Intrinsic clearance of each zone is assumed to be CLi = Vmax,i/Km, where Vmax  is maximum elimination (metabolic + transport) rate and Km Michaelis-Menten constant. Vmax is assumed to be affected by the level of metabolic enzymes and transporters expressed by cells. As a result each zone has its own Vmax­. The two zones are assumed to have equal Km values.

Further, we assume that the cost to the organism, J, is proportional to (CL1) 2 + (CL2) 2 + (cAUBC)2, where CL1 and CL2 are intrinsic periportal clearance and intrinsic perivenous clearance, respectively; AUBC is the area under the xenobiotic-concentration-in-the-blood curve and c adjusts the relative cost contribution of a fixed dose based on xenobiotic toxicity.



Table 1.  Equations of the model for each of the four path shown in Figure 2.

Path 1:

Path 2:

(Equation set 1)


Path 3:


Path 4:






Figure 3. n-player expansion of the simple game presented in Figure 1.


  • Filled circles (A1 … An): Agents (also referred to as players).
  • Unfilled circles:  nature forces out of agent’s control
  • Arrows: possible routs a compound can take due to natural forces and agents’ actions
  • (-c1,-c2, … ,cn) at terminal nodes indicates the costs of c1, c2,…,c­n  imposed to players 1,2,…n, respectively. ci can be one of the following:
    • 0: no cost.
    • e: cost of elimination (e.g. energy/resource consumption),
    • T: toxicity costs (e.g. the damage caused by the compound to the external tissue).


Description of the game:

  • At each time step one simulated compound enters the game. As it goes through the agents, it either bypasses agent i with probability Li or otherwise gets eliminated by player i with probability (1-Li)Pi. The player that eliminates the compound pays the elimination cost (e); others pay no cost. If no player eliminate the compound all players pay the toxicity cost T.
  • In this version of the game, each player has the ability to adjust its clearance strategy.  The objective of each player is to adjust its strategy to minimize its own average costs. By keeping track of accumulated rewards (and penalties), players can be reinforced to learn an optimal clearance strategy. In this work each player independently uses the Q-learning algorithm (4).  Their goal is to maximize the long-term average reward per action.  A simplified explanation of the Q-learning algorithm follows:
    • Given a set of available actions, Actions= {Act1= process, Act2=ignore}), a Q-learning agent selects its action according to the probabilities P(Act1), P(Act2)=1- P(Act1).
    • After the game is finished (i.e. all agents performed their actions), players receive their reward or penalty (payoff), according to the outcome of their actions. (obviously the payoff they receive depends on the actions of all other agents too).
    • Agents take note of the action and the resulting payoff: For each action they keep a Q value: Q(Act1), Q(Act2). Q(Acti) represents the average payoff agents have received so far by performing action i.
    • Using the updated Q’s after each step, agents update their action probabilities P(Act1) and P(Act2) employing the Boltzmann distribution (Eq 2).


Mappings to the liver:

  • Players 1,…,n: xenobiotic eliminating subsystems along the liver lobule. Players 1,…,n/2 represent the ones closer to the portal zone. Players n/2+1,…,n represent the ones closer to the central zone of the liver. One player does not necessarily map to one hepatocyte.
  • Pi: Clearance Strategy of player i -- the probability that the xenobiotic eliminating subsystem i eliminate an encountered compound. In other words, all complex stochastic events that eventually result in elimination of a compound by the subsystem. The events include: binding to transporters, membrane proteins, or enzymes, metabolism, excretion to bile, etc.
  • Li: Percentage of incoming compounds that cannot be seen by subsystem i due to physical constraints outside cells control. Examples include: remaining out of hepatocytes’ reach by staying inside the sinusoidal vessels due to saturated or damaged transporters, etc. 
  • Q-learning algorithm: hypothesized learning/adapting capabilities of hepatocytes.
    • Q’s: collection of proteins, endogenous compounds, motifs or pathways that can adapt according to signals received by cells. Collectively they may be called memory of the cell, which might include genetically conserved information.
    • Translation of P’s from Q’s: collection of motifs or signaling pathways which can affect the enzyme/transporter expression (and consequently the rate of elimination) according to the cell’s memory of the past.


II. Results


A                                                                                                                     B                                                                                                        



Figure 4.  Effects of toxicity change in the 2-player game.


Surface with solid marker lines: cost surface of player 1

Surface with dashed marker lines: cost surface of player 2

Lighter green corresponds to lower costs; purple corresponds to higher costs, as shown by the color-bar.

Arrows on the surfaces: players’ preferred moving direction in order to lower their costs

Star: the equilibrium of the corresponding game



Players’ cost functions and accordingly the equilibrium of the game are highly influenced by the toxicity level. Three examples of cost functions are shown.  The equilibrium of each game is shown by a small white star. The arrows are the directions towards which players are inclined to move in their strategy space to reduce costs. The direction of arrows eventually takes them to the equilibrium from any given point in the strategy space.

(A) When toxicity is low (0.5), the equilibrium of the game is at (p1=0, p2=0). The cost surface of player 1 is steeper than that of player 2. As a result, player 2’s contribution is less costly.

(B) When toxicity is higher than the energy cost (1.2 > 1), the equilibrium of the game moves to (p1=0, p2=1), which means player 1 makes none while player 2 makes maximum clearance effort.

(C) Another change of equilibrium happens when toxicity is high enough (in this particular example, at least %64 higher than energy cost). In this case, both players make maximum effort to clear the toxins (p1=1, p2=1). Calculations of cost surfaces are presented in Appendix B.



Figure 5.  Effects of toxicity change in the physiologically-based model.

Legend: A:

3D surface: plot of cost function, J, versus CL1 and CL2, when c=0.1. (CL1, Cl2, c and J are defined in Figure 2 legend). The units of the two intrinsic clearances are mL/h.

2D curves: contours of the cost function. Blue colors correspond to lower costs. Red colors correspond to higher costs.


Dotted 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 unity line



(A): Typical surface of the cost function (J) versus intrinsic periportal clearance (CL1) and intrinsic perivenous clearance (CL2) when c=0.1. If CL1 and CL2 are both small, the energy cost is low; but the toxicity cost is high since the compound is not cleared fast enough. On the other hand, high values of CL1 and CL2 reduce toxicity cost but increase energy consumption.  Consequently, for a given c, there exists an optimal pair of CL1 and CL2 at which J is at its minimum (Jmin).

(B): The trajectory of optimal J (Jmin) as c (toxicity) increases. The values of c are shown on the trajectory. The solid line is the unity line. Note that the trajectory is above the unity line which means CL2,optimal>=CL1,optimal. It demonstrates that in order to minimize the exposure of toxic compounds to the body, it’s less expensive if zone 2 makes more clearance effort than zone 1.



Clearance effort

A-- Toxicity=1.0

B -- -- Toxicity=2.0

Toxicity Processed

Clearance effort

C -- Toxicity=3.0

upstream                                               downstream

D -- Toxicity=5.0

upstream                                               downstream

Toxicity Processed


Distance to periportal region

     Distance to periportal region


Figure 6.  Effects of toxicity change in the n-player game.


  • ith blue bar: average (n=1000) clearance strategy (also referred to as clearance effort) of the ith player (after 10,000 steps)
  • ith red bar: total toxicity amount processed by the ith player (after 10,000 steps)


  • Emerged in silico collective clearance strategies of the n-player game for different toxicity values are shown. Each graph depicts the strategies of 20 agents playing the expanded n-player game after 10,000 steps.  Also the total toxicity amounts each player has processed are shown (red bars).  Using Q-learning algorithm, each agent independently learned its own strategy to minimize its energy and toxicity costs. The converged collective strategies are such that agents in the upstream (periportal) region don’t make as much clearance effort as agents in downstream region unless the compound is highly toxic.


  • The distribution of Total toxicity processed amounts (red bars) differs from that of clearance efforts. At low toxicity levels (A), downstream players process more compounds than what upstream players do. And, at high toxicity values (C and D), upstream players process more compounds, although their clearance effort is less or equal to downstream players. At some moderate toxicity levels, however, players in the middle process the most although they don’t make as mush clearance effort as the downstream players (B). As a result: downstream, middle and upstream players are more susceptible to low, medium and high toxicity levels, respectively. (This observation cannot be made in a 2-zoned model.)





Figure 7A. 2D expansion of the n-player game. Players are organized in a circular 2D shape similar to a liver lobule. At each step of the game, particles are placed in the portal area (shown in pink). They stochastically and simultaneously move towards the central region according the flow shown. Each agent on the grid plays one of the two actions (ignore or process) upon encountering a particle (similar to the game shown in Figure 3). Agents use Q-learning algorithm to minimize their long-term average cost. Particles that reach the exit area (light blue area) are removed. All players are penalized for exited particles according to their assigned toxicity.



toxicity :                   25



Figure 7B.  Effects of toxicity change on 2D gradients of clearance strategies (P’s).


  • Each point on the graphs shows the average (n=500) clearance strategy of the agent located on that point.
  • Color mapping to clearance effort:
  • The evolution of zonation is shown and compared for 3 different toxicity levels: 25 (left column), 10 (middle column) and 5 (right column).
  • Players were pre-exposed to toxicity level of 50 for 500 steps then 500 steps of zero toxicity.



Figure compares the emergence of 2D distribution of clearance strategies for the 3 different toxicity levels. In all three conditions, downstream agents make more clearance effort than upstream agents. Upstream agents make more contribution for higher toxicity levels. Other important parameters are as follows: L=0.8, T=0.01, dose=100, space_size=31x31.



Video 1. Shows how the players respond and self-organize their strategies when toxicity is changed.


Video 2. Shows how the players respond and self-organize their strategies when other agents die due to toxicity. Green bars show the amount of toxicity processed by the corresponding agent. Black solid curve shows the average strategies of the agents (+/- standard deviation). (A) Toxicity level is low: downstream players die first. Upstream players increase their clearance effort to compensate. (B) Toxicity level is moderate: mid-zonal players die first. (C) Toxicity level is high: upstream players die first.





III. Validation



Table 2. LD50 and zone activity of toxic compounds found in the literature. pp: periportal,  pv: perivenous The table summarizes the LD50 and zone activity of compounds found in the literature. We intended to report all such data found in the literature, however LD50 values are not available for all compounds, due to ethical implications of measuring LD50. Although the zone activity data are from different papers and may not be directly comparable, they provide initial support for our hypothesized model: highly toxic compounds affect periportal region, while low toxic compounds affect perivenous region, and compounds with medium toxicity affect middle regions of the liver. It is consistent with the model presented in this work (see figures 6 and 7).

Chemical Name

LD50 Rat

Oral (mg/kg) {{23 Anonymous 2008}}







{{22 Birnbaum,L S. 1999}}




“Animals treated with CC developed degeneration and necrosis of the hepatocytes almost exclusively in the periportal region of the liver lobule.” {{24 Tatsuno,T. 1984}}


3.03(f), 3.76(m)


“…phosphorus damage periportal

tissue.” {{33 Thurman,R G. 1984}}




“Gossypol selectively damages the periportal region of lobules in perfused rat liver” {{25 Lin,Y C. 1991}}


23-28 (mice)

pv &


“necrosis was detected in midzonal and pericentral regions of the liver lobule accompanied by fatty changes.” {{26 Thurman,R G. 1997}}




“Hepatic and renal lesions consisted of periportal intracanalicular cholestasis but no hepatic and renal tubular hemosiderosis.” {{23 Anonymous 2008}}




{{21 Katz,N. 1989}}




& pp

“In rats pretreated with saline, corn oil or PB, coumarin produced centrilobular hepatic necrosis, whereas in rats pretreated with 20 MC or AR0, coumarin produced periportal hepatic necrosis”{{29 Evans,J G. 1993}}




“Acute Exposure/ Experiments .../have/ shown that large doses of ferrous sulfate produce severe liver damage ...At post mortem in naturally occurring cases only organ consistently affected is liver, characteristic finding being periportal necrosis” {{23 Anonymous 2008}}




“Lipid accumulations occurred & appeared in distinctive periportal zonation pattern” {{27 Kendall,M W. 1979}}




“the extent of cell death was higher in perivenous cells than in periportal cells” {{28 Lindros,K O. 1992}}




“Statistically significant treatment related differences were limited to the liver in which an increase in hepatic lesions was observed ... which usually had a minimal amount of hepatocellular fatty change and periportal hepatocellular hypertrophy” {{23 Anonymous 2008}}




“…histologic examination of liver tissues 24 hours after treatment with 120 mg/kg APAP showed areas of perivenous hemorrhage and necrosis in APAPtreated IL-10 KO mice” {{30 Pohl,Lance R. 2002}}




“…diallyl phthalate is metabolized to allyl alcohol, a potent periportal hepatotoxicant…” {{31 Sipes,I G. 1986}}




{{21 Katz,N. 1989}}

Most drugs






(image removed due to copyright restrictions)

Figure 8. In vivo effects of toxicity change on perfused rat liver, example 1.

The figure shows the dose-dependent zonal distribution of Cysteine uptake in a perfused rat liver.  X axis is the percent distance to periportal region. Y axis is percent peak radioactivity measured from autographs taken 30s after administration of tracer [35S]Cysteine in a perfused rat liver (6).  As concentration of Cysteine increases, it becomes more toxic (LD50=1890mg/kg), and as a result the clearance strategies of hepatocytes change: The hepatocytes close to the periportal region do not make as much clearance effort as those closer to the perivenous region, unless there is a high toxicity threat. This behavior is consistent with the behavior of agents in the proposed game trying to simultaneously minimize their energy and toxicity costs (thus improving their own fitness as well as the whole organism fitness). The data is taken from {{11 Kaplowitz,N. 1992}}.


Figure 9.  In vivo effects of toxicity change on rat liver, example 2.

Dose dependent localization of TCDD (LD50=0.034 mg/kg) and effect of CYP1A2 mRNA expression in perivenous and periportal hepatocytes obtained from rats treated with 0.01, 0.3 and 10.0 ug TCDD/kg.  X-axis is the concentration of [3H]TCDD in each liver-cell population expresses as attograms (ag) [3H]TCDD/viable hepatocyte.

The data is consistent with the model: at low toxicity values perivenous hepatocytes express more CYP1A2 than periportal hepatocytes. As toxicity increases both perivenous and periportal CYP1A2 expression increases. At the highest toxicity value (0.3LD50) periportal and perivenous hepatocytes express almost equal amounts of CYP1A2. Data from {{22 Birnbaum,L S. 1999}}.




1. Billiar,T R.; Vodovotz,Y.; Geller,D A.; Salyapongse,A N.; Liu,S.Hepatocyte toll-like receptor 2 expression in vivo and in vitro: role of cytokines in induction of rat TLR2 gene expression by lipopolysaccharide.Shock, 2000, 14, 3, 361-5, BioMedical Press, Augusta, Ga.

2. Matzinger,The Danger Model: A Renewed Sense of SelfScience, 2002, 296, 5566, 301, American Association for the Advancement of Scienc

3. Lucier,G W.; Clark,G C.; McCoy,Z.; Portier,C J.; Goldstein,J A.; Tritscher,A M.Dose-response relationships for chronic exposure to 2,3,7,8-tetrachlorodibenzo-p-dioxin in a rat tumor promotion model: quantification and immunolocalization of CYP1A1 and CYP1A2 in the liver.Cancer research, 1992, 52, 12, 3436-42, Waverly Press, Baltimore, Md.

4. Tomita,Masaru; Nakajima,Hiromu; Naito,Yasuhiro; Ohno,HiroshiConstruction of a biological tissue model based on a single-cell model: a computer simulation of metabolic heterogeneity in the liver lobule.Artificial life, 2008, 14, 1, 3-28, MIT Press, Cambridge, MA

5. Zacharewski,Timothy R.; Jump,Donald B.; Harkema,Jack R.; Chittim,Brock; Tashiro,Colleen; Burgoon,Lyle D.; Boverhof,Darrell R.Temporal and dose-dependent hepatic gene expression patterns in mice provide new insights into TCDD-Mediated hepatotoxicity.Toxicological Sciences, 2005, 85, 2, 1048-63, Academic Press, San Diego, CA

6. Jungermann,K.Zonation of metabolism and gene expression in liver.Histochemistry and Cell Biology, 1995, 103, 2, 81-91, Springer-Verlag, Heidelberg

7. Gebhardt,R.Metabolic zonation of the liver: regulation and implications for liver function.Pharmacology therapeutics, 1992, 53, 3, 275-354, Pergamon, Oxford

8. Walsh,P J.; Mommsen,T P.Metabolic and enzymatic heterogeneity in the liver of the ureogenic teleost Opsanus beta.The journal of experimental biology, 1991, 156, 407-18, Published for the Company of Biologists Ltd. by the Cambridge University Press, London

9. Sawada,Yasufumi; Ohtani,Hisakazu; Tsujimoto,Masayuki; Kogetsu,Hirokuni; Kumagai,Yoichi; Nasu,RisaPhysiologically based pharmacokinetic model for pralmorelin hydrochloride in rats.Drug metabolism and disposition, 2005, 33, 10, 1488-94, American Society for Pharmacology and Experimental Therapeutics, etc.], [Bethesda, Md., etc.,

10. Lamers,W H.; Grange,T.; Moorman,A F.; Ruijter,J M.; Sassi,H.; Christoffels,V M.A mechanistic model for the development and maintenance of portocentral gradients in gene expression in the liver.Hepatology, 1999, 29, 4, 1180-92, No longer published by Elsevier, Philadelphia , PA

11. Kaplowitz,N.; Ookhtens,M.; Yamamuro,W.; Wong,E.; Chan,E T.; Saiki,H.Zonal distribution of cysteine uptake in the perfused rat liver.The Journal of biological chemistry, 1992, 267, 1, 192-6, American Society for Biochemistry and Molecular Biology [etc.], [Baltimore, etc.]

12. Kietzmann,T.; Jungermann,K.Oxygen: modulator of metabolic zonation and disease of the liver.Hepatology, 2000, 31, 2, 255-60, No longer published by Elsevier, Philadelphia, PA

13. Roze,D.; Michod,R E.Cooperation and conflict in the evolution of multicellularity.Heredity, 2001, 86, Pt 1, 1-7, Oliver and Boyd, London

14. Turchin,Peter; Burtsev,MikhailEvolution of cooperative strategies from first principles.Nature, 2006, 440, 7087, 1041-4

15. Axelrod, Robert; Axelrod,David E.; Pienta, Kenneth J.Evolution of cooperation among tumor cellsProceedings of the National Academy of Sciences of the United States of America, 2006, 103, 36, 13474, National Academy of Sciences

16. Fardel,Olivier; Lagadic-Gossmann,Dominique; Guillouzo,André; Payen,Léa; Huc,Laurence; Loewert,Maud; Sparfel,LydieAcute cytotoxicity of the chemical carcinogen 2-acetylaminofluorene in cultured rat liver epithelial cells.Toxicology Letters, 2002, 129, 3, 245-54, Elsevier, Amsterdam,

17. Walum, ErikAcute oral toxicityEnvironmental health perspectives, 1998, 106, Suppl 2, 497, National Institute of Environmental Health Science, [Research Triangle Park, N.C.]

19. Watkins,Technical Note: Q-LearningMachine learning, 1992, 8, 3/4, 279, Kluwer Academic Publishers, Boston/U.S.A.

20. Vincent,Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics2005, Cambridge University Press, Cambridge

21. Katz,N.; Jungermann,K.Functional specialization of different hepatocyte populations.Physiological reviews, 1989, 69, 3, 708-64, American Physiological Society

22. Birnbaum,L S.; Alcasey,S K.; Lucier,G W.; Lindros,K O.; Blanton,J.; Walker,N J.; Richardson,V M.; Santostefano,M J.Dose-dependent localization of TCDD in isolated centrilobular and periportal hepatocytes.Toxicological Sciences, 1999, 52, 1, 9-19, Academic Press, San Diego, CA

23. HSDB Hazardous Substances Data Bank2008, 2008, TOXNET United States National Library of Medicine,

24. Tatsuno,T.; Ito,E.; Terao,K.Liver injuries induced by cyclochlorotine isolated from Penicillium islandicum.Archives of toxicology, 1984, 55, 1, 39-46, Springer-Verlag, Berlin

25. Lin,Y C.; Nuber,D C.; Manabe,S.Zone-specific hepatotoxicity of gossypol in perfused rat liver.Toxicon, 1991, 29, 6, 787-90, Pergamon, Amsterdam,

26. Thurman,R G.; Mason,R P.; Connor,H D.; Raleigh,J A.; Arteel,G E.; Stachlewitz,R F.Development and characterization of a new model of tacrine-induced hepatotoxicity: role of the sympathetic nervous system and hypoxia-reoxygenation.The Journal of pharmacology and experimental therapeutics, 1997, 282, 3, 1591-9, American Society for Pharmacology and Experimental Therapeutics, [Bethesda, Md.]

27. Kendall,M W.Light and electron microscopic observations of the acute sublethal hepatotoxic effects of Mirex in the rat.Archives of environmental contamination and toxicology, 1979, 8, 1, 25-41, Springer, New York

28. Lindros,K O.; Anundi,I.Evidence for cytochrome P450 2E1-mediated toxicity of N-nitrosodimethylamine in cultured perivenous hepatocytes from ethanol treated rats.Pharmacology toxicology, 1992, 70, 6 Pt 1, 453-8, Munksgaard International Publishers], [Copenhagen

29. Evans,J G.; Lake,B G.Effect of pretreatment with some mixed-function oxidase enzyme inducers on the acute hepatotoxicity of coumarin in the rat.Food and Chemical Toxicology, 1993, 31, 12, 963-70, Pergamon, Oxford

30. Pohl,Lance R.; Shah,Anjali G.; George,John W.; Martin,Jackie L.; Amouzadeh,Hamid R.; Reilly,Timothy P.; Masubuchi,Yasuhiro; Bourdi,MohammedProtection against acetaminophen-induced liver injury and lethality by interleukin 10: role of inducible nitric oxide synthase.Hepatology, 2002, 35, 2, 289-98, No longer published by Elsevier, Philadelphia, PA

31. Sipes,I G.; Schram,K H.; Carter,D E.; Eigenberg,D A.Examination of the differential hepatotoxicity of diallyl phthalate in rats and mice.Toxicology and Applied Pharmacology, 1986, 86, 1, 12-21, Academic Press, San Diego [etc.]

33. Thurman,R G.; Kauffman,F C.; Matsumura,T.; Belinsky,S A.Rates of allyl alcohol metabolism in periportal and pericentral regions of the liver lobule.Molecular pharmacology, 1984, 25, 1, 158-64, American Society for Pharmacology and Experimental Therapeutics, [Bethesda, Md.]

35. El Samad, Hana; Khammash, M.; Homescu, Chris; Petzold, LindaOptimal Performance of the Heat-Shock Gene Regulatory NetworkIFAC World Congress, 2005, 16, Prague, Czech Republic




Table A1. Physiological parameters of tissues in a 250-g rat {{9 Sawada,Yasufumi 2005}}.


Volume (ml)

Flow rate (ml/min)



Qgi = 9.8



QL = 11.8

Blood vessels


QB = 43





Table A2. Rate constant values.