(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 2821734, 112601]*) (*NotebookOutlinePosition[ 2822675, 112633]*) (* CellTagsIndexPosition[ 2822631, 112629]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Exploring Physiologically Based Pharmacokinetic Models", "Title", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->18], Cell[TextData[StyleBox[ "The author presents a Mathematica-based model for assessing the risks of \ styrene inhalation.", FontSlant->"Italic"]], "Subsubtitle"], Cell[CellGroupData[{ Cell["\<\ by Qi Zheng \ \>", "Subtitle", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->18, FontSlant->"Plain", PrivateFontOptions->{"FontPostScriptName"->"Times-MirrorNormal"}], Cell["\<\ Physiologically based pharmacokinetic (PBPK) models are rapidly \ winning acceptance among risk analysts and regulatory agencies (see, \ e.g., [Travis 1989] and [Travis et al. 1989]). PBPK models are in essence \ deterministic compartmental models employing ordinary differential equations \ (ODE's) to explicitly describe the flow, disposition, absorption and \ metabolism within an animal or human body of the chemical under \ consideration. A typical PBPK model consists of from a few to several dozens \ of physiologically meaningful compartments representing the lung, fatty \ tissue, the liver, etc.. Like any mathematical models mimicking complex \ biological systems, PBPK models require that many important physiological and \ biochemical parameters be known a priori or be directly measured in vivo. \ (Statistical estimation of some biochemical parameters are also commonly \ practiced, see, e.g., [Smith and Evans 1995]). Although prototypes of PBPK \ models can be traced many decades back, it was not until the 1980's that PBPK \ models became an important and practical component of risk assessment when \ accurate measurement of the crucially important physiological and biochemical \ parameters ( e.g. alveolar ventilation rates, equilibrium partition \ coefficients and metabolic rates) became feasible. For more information about \ experimental measurement of such parameters, see [Gargas et al. 1986] and \ [Gargas et al. 1989]. Recent years have seen an explosive increase in theoretical research and \ applications of PBPK models (e.g., [Reitz et al. 1988], [Corley et al. \ 1990], [Andersen et al. 1991], [Allen and Fisher 1993] and [Buckley 1995]), \ and consequently an array of slightly different versions of PBPK models burst \ into bloom. The earliest and simplest model that fits the currently accepted \ ``definition\" of a typical PBPK model is perhaps that proposed by Ramsey and Andersen [Ramsey andAndersen 1984] in their pioneering work on styrene \ inhalation pharmacokinetics.Their model (Fig. 1) consists of only 5 \ compartments and yet possesses all the salient features of today's PBPK \ models, and is therefore the focus of the present notebook for the sake of \ simplicity. For more details concerning the interpretation of the model \ depicted in Figure 1, the reader should consult the original paper by Ramsey \ and Andersen.\ \>", "Text", CellMargins->{{Inherited, 41}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ Cell[GraphicsData["PostScript", "\<\ %!PS-Adobe-2.0 EPSF-2.0 %%Creator: dvips 5.51 Copyright 1986, 1993 Radical Eye Software %%Title: Zheng62.gr1.dvi %%CreationDate: Thu Aug 28 14:20:44 1997 %%BoundingBox: 118 276 347 641 %%DocumentFonts: Garamond-Light Garamond-LightItalic Palatino-Roman %%EndComments %DVIPSCommandLine: dvips -p 1 -n 1 -D 100 -E Zheng62.gr1 %DVIPSSource: TeX output 1997.08.28:1419 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S \ N} B /TR{translate}N /isls false N /vsize 11 72 mul N /@rigin{isls{[0 -1 1 0 0 \ 0] concat}if 72 Resolution div 72 VResolution div neg scale isls{Resolution \ hsize -72 div mul 0 TR}if Resolution VResolution 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Cell[CellGroupData[{ Cell[TextData["The Mass Balance Equations"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The starting point of PBPK modeling is the setting up of mass balance \ equations. For each organ or tissue represented by a compartment, the rate \ at which the total amount of chemical in the particular compartment changes \ is the difference between the rate of in-flow (through arterial blood) and \ the rate of out-flow (through venous blood). Take the fatty tissue \ compartment for example. The rate of in-flow of chemical is the velocity of \ blood running through this compartment (", StyleBox["QFat", "Input", FontFamily->"WriCourier", FontWeight->"Light"], ") times the chemical concentration in the arterial blood (", StyleBox["ConcArterical[t]", "Input", FontFamily->"WriCourier", FontWeight->"Light"], "). The rate of out-flow is calculated by analogy. Therefore, the mass \ balance equation for the fat compartment is of the form\n\n", StyleBox["AmtFat'[t]==QFat (ConcArterial[t]-ConcVenousFat[t]).", FontWeight->"Bold"], "\n\nSince the lung compartment serves only as an interface to the outside \ exposure (styrene inhalation in this example), there are only four mass \ balance equations for the four compartments: the fatty tissue, the muscle, \ the richly perfused tissue and the liver.\n" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ MassBalanceEq={ AmtFat'[t]==QFat (ConcArterial[t]-ConcVenousFat[t]), \ \>", "Input", AspectRatioFixed->True], Cell["\<\ AmtMuscle'[t]==QMuscle (ConcArterial[t]- ConcVenousMuscle[t]), \ \>", "Input", AspectRatioFixed->True], Cell["\<\ AmtRichly'[t]==QRichly (ConcArterial[t]- ConcVenousRichly[t]), \ \>", "Input", AspectRatioFixed->True], Cell["\<\ AmtLiver'[t]==QLiver (ConcArterial[t]-ConcVenousLiver[t])-(Vmax \ ConcVenousLiver[t])/(Km+ConcVenousLiver[t])} \ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "Note that the last equation (for the liver) contains an extra term \ describing metabolism of styrene taking place in the liver (to be more \ precise, the word \"liver\" in this context includes the liver per se and \ other tissues with similar metabolic capacity). The metabolism is assumed to \ be saturable, satisfying the well-known Michaelis-Menten equation (", StyleBox["Vmax", "Input"], " is the maximunm reaction rate and ", StyleBox["Km", "Input"], " is the Michaelis constant).\nAt first sight, the above four mass balance \ equations involve nine unknowns. However, concentration of the chemical in a \ particular compartment is related to the total amount of the chemical \ residing in that compartment (that is, Amt/Vol=concentration), and \ concentration in the venous blood running out of a particular compartment is \ linked to the concentration in that particular compartment by the \ tissue:blood partition coefficient. \n\nOf particular note in this regard \ is the assumption that the chemical entering a compartment via arterial blood \ distributes throughout the compartment very rapidly, and hence an \ equilibrium ratio (called ", StyleBox["tissue:blood partition coefficient", FontSlant->"Italic"], ") of the concentration within the compartment to the concentration of \ the venous blood running out of this same compartment is reached almost \ instantaneously. The following rules reflect this important assumption. \ Application of these rules will accomplish the task of expressing the four \ venous blood concentration variables in terms of their corresponding total \ amount variables. " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "ConcToAmtRules={\nConcVenousFat[t]->AmtFat[t]/VolFat/PartCoefFat,\n\ ConcVenousMuscle[t]->AmtMuscle[t]/VolMuscle/PartCoefMuscle,\n\ ConcVenousRichly[t]->AmtRichly[t]/VolRichly/PartCoefRichly,\n\ ConcVenousLiver[t]->AmtLiver[t]/VolLiver/PartCoefLiver};"], "Input", AspectRatioFixed->True], Cell[TextData[ "\nChemical enters the circulation system by the arterial blood's absorbing \ the chemical from alveolar air in the lung. The air:blood partition \ coefficient bridges concentration in arterial blood with that in alveolar \ air, which in turn is directly linked with inhalation exposure. First, the \ simple relation between concentration in alveolar air and that in arterial \ blood. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "AlveolarRule=ConcAlveolar[t]->ConcArterial[t]/PartCoefAirBlood;"], "Input", AspectRatioFixed->True], Cell[TextData[{ "The relation between inhalation exposure level and the total amount of \ chemical absorbed by the body is summarized in the the following ", StyleBox["VenousConcEqn", "Input"], " equation." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ VenousConcEqn= QAlveolar (ConcInhale-ConcAlveolar[t]) == Qt (ConcArterial[t]-ConcVenous[t])/.AlveolarRule;\ \>", "Input", AspectRatioFixed->True], Cell[TextData[ "This equation simply says that the total amount of chemical lost from the \ alveolar air at any moment is the same as the total amount of chemical \ absorbed by all tissues from the blood (measured as the difference between \ arterial blood and venous blood). Note that QAlveolar and Qt are the flow \ speeds of alveolar air and blood circulation through the lung. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ SolnToVenousConcEqn=Solve[{VenousConcEqn},{ConcArterial[t]}][[1]];\ \ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "In the above expression the variable ", StyleBox["ConcVenous[t]", "Input"], " is the concentration in venous blood coming out of the lung, it is \ related to the concentrations in various compartments in the following \ manner." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "ConcVenousRule=ConcVenous[t]->(QFat ConcVenousFat[t]+\n QMuscle \ ConcVenousMuscle[t]\n +QRichly ConcVenousRichly[t]+QLiver \ ConcVenousLiver[t])/Qt;\n "], "Input", AspectRatioFixed->True], Cell[TextData[{ "Now one can express the ", StyleBox["ConcArterial", "Input"], " variable in the four basic variables using the following rule.\n" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["ArterialRule=SolnToVenousConcEqn/.ConcVenousRule;", "Input", AspectRatioFixed->True], Cell["\<\ Physiological parameter values must be incorporated into the mass \ balance equations. Table 2 of [Ramsey and Andersen 1984] provides all these \ parameter values for mice, rats and humans. For illustration purposes, this \ section deals with rat models only. The interested reader can easily \ replace rat parameters with mouse parameters. Human models will be dealt \ with in the subsequent section.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "RatParameters={\nQAlveolar->4.50,Qt->5.64,QLiver->2.11,QFat->0.5,\n\ QMuscle->0.68,QRichly->2.35,VolLiver->0.012,VolFat->0.027,\n\ VolMuscle->0.219,VolRichly->0.015,\n\ PartCoefLiver->2.7,PartCoefFat->50,PartCoefMuscle->1.0,\nPartCoefRichly->5.7,\ \nVmax->3.6,Km->0.36,\nPartCoefAirBlood->40.0};"], "Input", AspectRatioFixed->True], Cell[TextData[ "All the foregoing work now results in an ODE system of mass balance \ equations describing the pharmacokinetics of inhaled styrene within a \ laboratory rat. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ RatODE=MassBalanceEq/.ArterialRule/.ConcToAmtRules/.RatParameters\ \ \>", "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {Derivative[1][AmtFat][t] == 0.5*(-0.7407407407407409*AmtFat[t] + 0.1738374619730552* (4.5*ConcInhale + 0.3703703703703704*AmtFat[t] + 65.12345679012346*AmtLiver[t] + 3.105022831050229*AmtMuscle[t] + 27.48538011695907*AmtRichly[t])), Derivative[1][AmtMuscle][t] == 0.6800000000000001*(-4.566210045662101*AmtMuscle[t] + 0.1738374619730552* (4.5*ConcInhale + 0.3703703703703704*AmtFat[t] + 65.12345679012346*AmtLiver[t] + 3.105022831050229*AmtMuscle[t] + 27.48538011695907*AmtRichly[t])), Derivative[1][AmtRichly][t] == 2.35*(-11.69590643274854*AmtRichly[t] + 0.1738374619730552* (4.5*ConcInhale + 0.3703703703703704*AmtFat[t] + 65.12345679012346*AmtLiver[t] + 3.105022831050229*AmtMuscle[t] + 27.48538011695907*AmtRichly[t])), Derivative[1][AmtLiver][t] == (-111.1111111111111*AmtLiver[t])/ (0.36 + 30.86419753086419*AmtLiver[t]) + 2.11*(-30.86419753086419*AmtLiver[t] + 0.1738374619730552* (4.5*ConcInhale + 0.3703703703703704*AmtFat[t] + 65.12345679012346*AmtLiver[t] + 3.105022831050229*AmtMuscle[t] + 27.48538011695907*AmtRichly[t]))}\ \>", "\<\ {AmtFat'[t] == 0.5 (-0.740741 AmtFat[t] + 0.173837 (4.5 ConcInhale + 0.37037 AmtFat[t] + 65.1235 AmtLiver[t] + 3.10502 AmtMuscle[t] + 27.4854 AmtRichly[t])), AmtMuscle'[t] == 0.68 (-4.56621 AmtMuscle[t] + 0.173837 (4.5 ConcInhale + 0.37037 AmtFat[t] + 65.1235 AmtLiver[t] + 3.10502 AmtMuscle[t] + 27.4854 AmtRichly[t])), AmtRichly'[t] == 2.35 (-11.6959 AmtRichly[t] + 0.173837 (4.5 ConcInhale + 0.37037 AmtFat[t] + 65.1235 AmtLiver[t] + 3.10502 AmtMuscle[t] + 27.4854 AmtRichly[t])), -111.111 AmtLiver[t] AmtLiver'[t] == -------------------------- + 0.36 + 30.8642 AmtLiver[t] 2.11 (-30.8642 AmtLiver[t] + 0.173837 (4.5 ConcInhale + 0.37037 AmtFat[t] + 65.1235 AmtLiver[t] + 3.10502 AmtMuscle[t] + 27.4854 AmtRichly[t]))}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Simulation Using Rat Models"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Now the four mass balance equations contain exactly four unknowns except \ for an indeterminate function ", StyleBox["ConcInhale", "Input"], ", which is in fact the force function of the whole system to be determined \ by the inhalation exposure pattern, viz., how the external exposure changes \ with time. Once ", StyleBox["ConcInhale", "Input"], " is specified, the ODE system of mass balance equations can be numerically \ solved, and various quantities of interest can be readily obtained. \n\nThe \ objective of this section is to simulate the time course of styrene \ concentrations in the four tissue groups and the blood for some prescribed \ external exposure patterns. Following [Ramsey and Andersen 1984], interest is \ focused on eight-hour constant exposures with dose levels ranging from 80ppm \ to 1200ppm. Since the concentration within a compartment is expressed in \ mg/l rather than ppm as for inhalation concentration in the air, a function \ converting ppm to mg/l is handy." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["ppmToMgl[ppm_]:= ppm 104.151/10^3/0.0821/298"], "Input", AspectRatioFixed->True], Cell[TextData[{ "In the above ", StyleBox["ppmToMgl", "Input"], " function, room temperature (25C=298K) is assumed. The molacular weight \ of styrene is taken to be 104.151, and the gas constant is taken to be 0.0821 \ liter*atm/mole/K (c.f. [Masterton and Slowinski 1978, p.99]). For example, \ the following translates a concentration of 1200ppm of styrene into the \ corresponding concentration expressed in mg/l." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["ppmToMgl[1200]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 5.108404384896467\ \>", "\<\ 5.1084\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[ "Assuming that the rat has no previous exposure to styrene, that is, there is \ no styrene residue in the body of the rat, then the following initial \ condition is self-evident."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "initCond={AmtFat[0]==0,AmtMuscle[0]==0,\nAmtRichly[0]==0,AmtLiver[0]==0};"], "Input", AspectRatioFixed->True], Cell["\<\ Now a variety of exposure patterns can be investigated by simply \ specifying appropriate inhalation dose patterns. The dose patterns \ investigated by Ramsey and Andersen (1984) are two-piece step functions, that \ is, the rat is exposed to a constant styrene concentration (say, 1200ppm) in \ the time interval [0,6), and exposure is discontinued after 6 hours of \ exposure. This exposure pattern is now incorporated into the mass balance \ equations. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "ExposurePattern={ConcInhale->If[t<6.0,ppmToMgl[1200.],0.]};\n\ FinalRatODE=Join[RatODE,initCond]/.ExposurePattern//Simplify"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {Derivative[1][AmtFat][t] == -0.3381782477827675*AmtFat[t] + 5.660448221653496*AmtLiver[t] + 0.2698846441590812*AmtMuscle[t] + 2.38899436044842*AmtRichly[t] + 0.3911342894393742*If[t < 6., 5.108404384896467, 0.], Derivative[1][AmtMuscle][t] == 0.04378128671913985*AmtFat[t] + 7.698209581448755*AmtLiver[t] - 2.737979714993878*AmtMuscle[t] + 3.249032330209851*AmtRichly[t] + 0.5319426336375491*If[t < 6., 5.108404384896467, 0.], Derivative[1][AmtRichly][t] == 0.1513029761617333*AmtFat[t] + 26.60410664177143*AmtLiver[t] + 1.268457827547682*AmtMuscle[t] - 16.25710662285149*AmtRichly[t] + 1.838331160365059*If[t < 6., 5.108404384896467, 0.], Derivative[1][AmtLiver][t] == 0.1358507573196839*AmtFat[t] - 41.2363652947457*AmtLiver[t] - (111.1111111111111*AmtLiver[t])/ (0.36 + 30.86419753086419*AmtLiver[t]) + 1.138913198351322*AmtMuscle[t] + 10.08155620109233*AmtRichly[t] + 1.650586701434159*If[t < 6., 5.108404384896467, 0.], AmtFat[0] == 0, AmtMuscle[0] == 0, AmtRichly[0] == 0, AmtLiver[0] == 0}\ \>", "\<\ {AmtFat'[t] == -0.338178 AmtFat[t] + 5.66045 AmtLiver[t] + 0.269885 AmtMuscle[t] + 2.38899 AmtRichly[t] + 0.391134 If[t < 6., 5.1084, 0.], AmtMuscle'[t] == 0.0437813 AmtFat[t] + 7.69821 AmtLiver[t] - 2.73798 AmtMuscle[t] + 3.24903 AmtRichly[t] + 0.531943 If[t < 6., 5.1084, 0.]\\ , AmtRichly'[t] == 0.151303 AmtFat[t] + 26.6041 AmtLiver[t] + 1.26846 AmtMuscle[t] - 16.2571 AmtRichly[t] + 1.83833 If[t < 6., 5.1084, 0.], AmtLiver'[t] == 0.135851 AmtFat[t] - 111.111 AmtLiver[t] 41.2364 AmtLiver[t] - -------------------------- + 0.36 + 30.8642 AmtLiver[t] 1.13891 AmtMuscle[t] + 10.0816 AmtRichly[t] + 1.65059 If[t < 6., 5.1084, 0.], AmtFat[0] == 0, AmtMuscle[0] == 0, AmtRichly[0] == 0, AmtLiver[0] == 0}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell["\<\ Numerical solutions to the above ODE system can now be found to \ yield simulated styrene concentrations in all four tissue groups described by \ the mass balance equations. A 24-hour period is chosen for the simulation \ study, as was done by [Ramsey and Andersen 1984].\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "ODENames={AmtFat,AmtMuscle,AmtLiver,AmtRichly};\n\ RatSoln=NDSolve[FinalRatODE,ODENames,{t,0,24.},MaxSteps->5000]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{AmtFat -> InterpolatingFunction[{0., 24.}, {{0., 0., {0., 0}, {0}}, {0.00002320843591865143, 0., {0.00004640975530133417, 1.999693364257994}, {0}}, {0.00004641687183730287, 0.00002320843591865143, {0.0000928569509574444, 2.001306586058345}, {0}}, {0.0000928337436746057, 0.00004641687183730287, {0.0001858258672692296, 2.004517702905087, 34.58997471864972}, {0, 0}}, {0.0001392506155119086, 0.0000928337436746057, {0.0002789431421086663, 2.007699025780324, 34.26903569016696}, {0, 0}}, {0.0001856674873492115, 0.0001392506155119086, {0.0003722074396441516, 2.010852584607753, 33.9699628885248}, {0, 0}}, {0.0003484769815022197, 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"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Needs[\"Graphics`Graphics`\"];\n"], "Input", AspectRatioFixed->True], Cell[TextData[ "The time course of styrene concentrations in the four tissue groups are now \ presented graphically."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "pFat=Plot[Evaluate[AmtFat[t]/VolFat/.RatParameters/.RatSoln],{t,0,24},\n\ FrameLabel->{\"Time (Hours)\",\"Concentration (mg/l)\"}, \n\ Frame->True,DisplayFunction->Identity,\nPlotLabel->FontForm[\"Fatty \ Tissue\",{\"Helvetica\",12}],\nFrameTicks->{Range[0,24,4],Automatic}]; \n\n\ pLiver=Plot[Evaluate[AmtLiver[t]/VolLiver/.RatParameters/.RatSoln],{t,0,24},\n\ FrameLabel->{\"Time (Hours)\",\"Concentration (mg/l)\"}, \n\ Frame->True,DisplayFunction->Identity,\n\ PlotLabel->FontForm[\"Liver\",{\"Helvetica\",12}],\n\ FrameTicks->{Range[0,24,4],Automatic}];\n\n\ pRichly=Plot[Evaluate[AmtRichly[t]/VolRichly/.RatParameters/.RatSoln],{t,0,24}\ ,\nFrameLabel->{\"Time 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concentration in the four tissue groups. However, in \ pharmacokinetics literature, graphics using semilog coordinates are more \ commonly adopted. In the following, the built-in function ", StyleBox["LogPlot", "Input"], " will be employed to comply with this practice. " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "pLogFat=LogPlot[Evaluate[AmtFat[t]/VolFat/.RatParameters/.RatSoln],{t,0,24},\ \nFrame->True,PlotRange->{.1,3000},PlotStyle->Dashing[{0.01}],\n\ FrameTicks->{Range[0,24,4],Automatic},\nDisplayFunction->Identity]"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "ParametricPlot::pptr: \n {<<2>>} does not evaluate to a pair of real \ numbers at t\n = 0.."], "Message", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData["\<\ Graphics[\"<<>>\"]\ \>", "\<\ -Graphics-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "Note that, since the concentration is zero at time ", StyleBox["t ", FontSlant->"Italic"], "= 0, the error message simply informs that the logarithm of zero is a \ complex number, and hence the message can be safely ignored. Another variable \ of great interest is styrene concentration in (arterial) blood. Although this \ quantity is not directly involved in the mass balance equations, it is \ already defined in the previous section. 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Ramsey \ and Andersen provide parameters for humans solely by rescaling the parameters \ for rats. For simplicity, the parameters for humans in Table 2 of Ramsey and \ Andersen are taken directly without repeating the allometric scaling process.\ \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "HumanParameters={\nQAlveolar->230,Qt->290,QLiver->108,QFat->26,\n\ QMuscle->34,QRichly->121,VolLiver->3.32,VolFat->7.47,\n\ VolMuscle->60.6,VolRichly->4.15,\n\ PartCoefLiver->2.7,PartCoefFat->50,PartCoefMuscle->1.0,\nPartCoefRichly->5.7,\ \nVmax->184,Km->0.36,\nPartCoefAirBlood->52.0};"], "Input", AspectRatioFixed->True], Cell[TextData[ "The mass balance equations for humans are obtained by simply replacing the \ rat parameters with the human parameters in the mass balance equations for \ rats obtained in the second section."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "HumanODE=MassBalanceEq/.ArterialRule/.ConcToAmtRules/.HumanParameters;"], "Input", AspectRatioFixed->True], Cell[TextData[ "To keep the simulation simple, all conditions are 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The \ expression remains the same as that for rats, only some physiological and \ biochemical parameters need changing. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "HumanConcArterialBlood=ConcArterial[t]/.ArterialRule/.\n \ ConcToAmtRules/.HumanExposurePattern/.HumanParameters"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ 0.003396472893533638*(0.06961178045515394*AmtFat[t] + 12.04819277108434*AmtLiver[t] + 0.561056105610561*AmtMuscle[t] + 5.115197632635806*AmtRichly[t] + 230*If[t < 6., ppmToMgl[80.], 0.])\ \>", "\<\ 0.00339647 (0.0696118 AmtFat[t] + 12.0482 AmtLiver[t] + 0.561056 AmtMuscle[t] + 5.1152 AmtRichly[t] + 230 If[t < 6., ppmToMgl[80.], 0.])\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[ "pHuman1=LogPlot[Evaluate[HumanConcArterialBlood/.HumanSoln],{t,0,48.},\n\ PlotRange->{0.00001,10},Frame->True,\nFrameTicks->{Range[0,48,8],Automatic},\n\ DisplayFunction->Identity];"], "Input", AspectRatioFixed->True], Cell["\<\ Another variable of interest in studying the human models is \ styrene concentration of exhaled air. According to the assumption of Ramsey \ and Andersen, alveolar respiration accounts for 70% of total respiration, and \ hence the following definition.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ ConcExhaledAir=(0.7 ConcAlveolar[t]+0.3 \ ConcInhale)/.AlveolarRule/.ArterialRule/. 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This simulation is of some \ practical interest because human exposure to styrene occurs mainly in the \ industrial workplace."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "WorkdayExposure=ConcInhale->If[t<8||\n(t>24)&&(t<32)||(t>48)&&(t<56)||\n \ (t>72)&&(t<80)||(t>96)&&(t<104),ppmToMgl[80],0];\n \n WorkdayODE=\n \ Join[HumanODE,initCond]/.WorkdayExposure//Simplify;\n \n \ WorkdaySoln=NDSolve[WorkdayODE,ODENames,{t,0,168.},\n MaxSteps->5000]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{AmtFat -> InterpolatingFunction[{0., 168.}, {{0., 0., {0., 0}, {0}}, {0.00003106451827921997, 0., {0.000214919514227524, 6.918488556485693}, {0}}, {0.00006212903655843995, 0.00003106451827921997, {0.0004298824570619864, 6.919886569696392}, {0}}, {0.0003740074089765903, 0.00006212903655843995, {0.002590220047334454, 6.933831939363924, 22.35706432512086}, {0, 0}}, {0.0006858857813947407, 0.0003740074089765903, {0.004754879083274919, 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In principle, PBPK models are chemical dependent in that \ both the structure of the model and the biochemical parameters are determined \ by the chemical under investigation. Following this notebook, the reader can \ try many more sophisticated PBPK models alone the same line. The unique value \ of this ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook lies in that it makes all the conceptually important details \ transparent, yet the reader need not worry too much about the purely \ mathematical technicalities. " }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Bibliography "], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Andersen, M.E., Clewell, H.J. III, Gargas, M.L., MacNaughton, M.G., Reitz, \ R.H., Nolan, R.J., and McKenna, M.J. Physiologically based pharmacokinetic \ modeling with dichloromethane, its metabolite, carbon monoxide and blood \ carboxyhemoglobin in rats and humans, ", StyleBox["Toxicology and Applied Pharmacology", FontSlant->"Italic"], " ", StyleBox["108", FontWeight->"Bold"], ":14-27 (1991).\n\nAllen, B.C. and Fisher, J.W. Pharmacokinetic modeling \ of\ntrichloroethylene and trichloroatcetc acid in humans, ", StyleBox["Risk Analysis", FontSlant->"Italic"], " ", StyleBox["13", FontWeight->"Bold"], ":71-86 (1993).\n\n Buckley, L.A., Biologically-based models of dioxin \ pharmacokinetics, ", StyleBox["Toxicology", FontSlant->"Italic"], " ", StyleBox["102", FontWeight->"Bold"], ":125-131 (1995).\n\nGargas, M.L., Andersen, M.E., and Clewell, H.J. III, A \ physiologically based simulation approach for determining metabolic constants \ from gas uptake data. ", StyleBox["Toxicology and Applied Pharmacology", FontSlant->"Italic"], " ", StyleBox["86", FontWeight->"Bold"], ":341-352 (1986).\n\nGargas, M.L., Burgess, R.J., Voisard, D.E., Cason, \ G.H., and Andersen, M.E., Partition coefficients of low-molecular-weight \ volatile chemicals in various liquids and tissues. ", StyleBox["Toxicology and Applied Pharmacology", FontSlant->"Italic"], " ", StyleBox["98", FontWeight->"Bold"], ":87-99 (1989).\n\nCorley, R.A., Mendrala, A.L., Smith, F.A., Staats, D.A., \ Gargas, M.L., Conolly, R.B., Andersen, M.E. and Reitz, R.H., Development of \ a physiologically based pharmacokinetic model for chloroform, ", StyleBox["Toxicology and Applied Pharmacology", FontSlant->"Italic"], " ", StyleBox["103", FontWeight->"Bold"], ":512-527 (1990).\n\nMasterton, W.L. and Slowinski, E.J., ", StyleBox["Chemical Principles with Qualitative Analysis", FontSlant->"Italic"], ", W.B. Saunders Company, Philadelphia, 1978.\n\nRamsey, J.C. and \ Andersen, M.E., A physiologically based\ndescription of the inhalation \ pharmacokinetics of styrene in rats and humans. ", StyleBox["Toxicology and Applied Pharmacology", FontSlant->"Italic"], " ", StyleBox["73", FontWeight->"Bold"], ":159-175 (1984).\n\nRamsey, J.C., Young, J.D., Karbowski, R.J., Chenoweth, \ M.B., McCarty, L.P. and Braun, W.H., Pharmacokinetics of inhaled styrene in \ human volunteers. ", StyleBox["Toxicology and Applied Pharmacology", FontSlant->"Italic"], " ", StyleBox["53", FontWeight->"Bold"], ":54-63(1980).\n\nReitz, R.H., McDougal, J.N., Himmelstein, M.W., Nolan, \ R.J. and Schumann, A.M., Physiologically based pharmacokinetic modeling with \ methylchloroform: Implications for interspecies, high dose/low dose, and dose \ route extrapolations. ", StyleBox["Toxicology and Applied Pharmacology", FontSlant->"Italic"], " ", StyleBox["95", FontWeight->"Bold"], ":185-199 (1988).\n\nSmith, A.E. and Evans, J.S. , Uncertainty in fitted \ estimates of apparent in vivo metabolic constants for chloroform, ", StyleBox["Fundamental and Applied Toxicology", FontSlant->"Italic"], " ", StyleBox["25", FontWeight->"Bold"], ":29-44 (1995).\n\n Travis, C. C. (ed), ", StyleBox["Biologically Based Methods for\nCancer Risk Assessment", FontSlant->"Italic"], ", Plenum Press, 1989.\n \n Travis, C.C., White, R.K., and Arms, A.D., A \ physiologically\nbased pharmacokinetic approach for assessing the cancer risk \ of \ntetrachloroethylene, In Paustenback, D. J. (ed),", StyleBox["The Risk Assessment of Environmental and Human Health Hazards", FontSlant->"Italic"], ", Wiley, 1989." }], "Text", CellMargins->{{Inherited, 69}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["ABOUT THE AUTHOR", "Section"], Cell["\<\ Qi Zheng is a researcher at the National Center for Toxicological \ Research.\ \>", "Text"], Cell[TextData[{ StyleBox["(*", "SmallText"], "address to come*)" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["ELECTRONIC SUBSCRIPTIONS", "Section"], Cell[TextData[{ "Included in the distribution for each electronic subscription is the file \ ", StyleBox["Styrene.nb ", "Input"], "containing ", StyleBox["Mathematica", FontSlant->"Italic"], " code for the material described in this article." }], "Text"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"NeXT 3.0", ScreenRectangle->{{0, 1053}, {0, 832}}, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Automatic, WindowSize->{611, 600}, WindowMargins->{{Automatic, 57}, {73, Automatic}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"NeXTAutomaticEncoding", Magnification->1.25 ] (*********************************************************************** Cached data follows. 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