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The two-stage carcinogenesis theory was \ then succinctly couched in the birth-and-death process language by Kendall's \ pioneering work, resulting in three types of stochastic carcinogenesis \ models. Moolgavkar and Venzon's (1979) subsequent seminal work recast the \ two-stage models as important tools for connecting cancer survival and \ hazard rate functions with the biologically most important cellular kinetic \ parameters, i.e., cell division rate, cell death rate and cell mutation rate. \ The survival and hazard rate functions are sometimes justifiably called two \ fundamental quantities, as these two quantities were inextricably woven into \ the majority of carcinogenesis modeling efforts during the last two decades \ or so. \n\nHowever, compelling biological evidence suggests that two \ mutations are not sufficient to cause certain cancers. For example, \ Kopp-Schneider and Portier (1995) analyzed mouse skin painting experiment \ data that indicate the need for more than two stages in cancer models. \ Regarding colorectal tumor, Fearon and Vogelstein (1990) commented: \ \"mutation in at least four or five genes are required for the formation of a \ malignant tumor.\" Using radiation effects data, Little (1995) also argued \ that some extra stages should be appended to the two-stage cancer models. At \ the instigation of those authors, Zheng (1997a) offered systematic \ computational recipes for a rather broad class of multistage carcinogenesis \ models. The package CarcinoMod is a ", StyleBox["Mathematica", FontSlant->"Italic"], " implementation of all those recipes. Specifically, CarcinoMod provides \ functions for computing the survival and hazard rate functions for all eight \ types of models presented in Zheng (1997a). CarcinoMod can handle models \ with both constant parameters and time-dependent parameters. Moreover, for \ the extensively studied ", Cell[BoxData[ \(TraditionalForm\`\(B\_2/0\)/0\)]], " model, all existing closed form expressions for the survival and hazard \ rate functions are separately implemented, enhancing computational accuracy \ and efficiency for this particularly simple model.\n\nThe goal of this \ notebook is to demonstrate the various functionalities that CarcinoMod \ provides. For mathematical details, the reader is referred to the technical \ papers listed in the reference section. We begin with the simpler cases --- \ models with constant parameters." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(SetDirectory["\<~/CarcinoMod\>"]\)], "Input"], Cell[BoxData[ \("/home/zheng/CarcinoMod"\)], "Output"] }, Open ]], Cell[BoxData[ \(<< CarcinoMod.m\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["General k-stage model with constant parameters", "Section"], Cell[TextData[{ "For models with constant parameters (also called homogeneous models), the \ name of a CarcinoMod function is composed of two parts: the model name and \ the quantity to be computed. For example, to compute the survival function \ for a ", Cell[BoxData[ \(TraditionalForm\`\(B\_k/N\)/0\)]], " model, we invoke the function BN0Survival. The first part of the function \ name \"BN0\" indicates that this function is for ", Cell[BoxData[ \(TraditionalForm\`\(B\_k/N\)/0\)]], " models where k is an arbitrary positive integer. The function name itself \ does not contain any information about the exact number of stages in the \ model, because k can be inferred by checking the number of cellular \ parameters supplied by the user. The second part \"Survival\" is simply an \ English word with its first letter capitalize, the meaning of which is \ self-explanatory. By analogy, the function name A00Hazard suggests that this \ function is for computing the hazard rate function for ", Cell[BoxData[ \(TraditionalForm\`\(A\_k/0\)/0\)]], " models. It is particularly worth noting that, for any constant parameter \ model considered in Zheng (1997a), the solution of one ordinary differential \ equation (ODE) system yields all the values for both the survival and the \ hazard rate functions. Therefore, these functions are so designed that, by \ specifying a positive real number T, the user get the survival (or hazard \ rate) function as an interpolating function object that are valid on the \ whole interval [0,T]." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?BN0Survival\)\)], "Input"], Cell[BoxData[ \("BN0Survial[T,\!\({...,{\[Beta]\_i,\[Delta]\_i,\[Mu]\_i},...},N\_0\)] \ returns the survival function for a \!\(B\_k/N/0\) model as an interpolating \ function defined on the time interval [0,T]. Note k is automatically \ determined by the number of triplets in the second argument. The optional \ parameter \!\(N\_0\) (default = 1) represents the total number of normal \ cells at t=0. The birth, death and mutation rates for stage i cells are {\!\(\ \[Beta]\_i, \[Delta]\_i,\[Mu]\_i}\). Use this function when all parameters \ are constant quantities, otherwise BN0NonhomoSurvival should be invoked."\)], "Print"] }, Open ]], Cell[TextData[{ "The following code draws the survival function for a ", Cell[BoxData[ \(TraditionalForm\`\(B\_3/N\)/0\)]], " model on the time interval [0,2000]. 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Because closed -form formulae for the ", Cell[BoxData[ \(TraditionalForm\`\(B\_2/0\)/0\)]], " model are available (Kopp-Schneider et al. 1994 and Zheng 1994) and was \ separately built into CarcinoMod, we can use the ", Cell[BoxData[ \(TraditionalForm\`\(B\_2/0\)/0\)]], " model as a benchmark." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?B200Survival\)\)], "Input"], Cell[BoxData[ \("B200Survival[t,\!\(\[Nu],{\[Beta],\[Delta],\[Mu]}\)] returns the \ survival function evaluated at time t for a \!\(B\_2/0/0\) model. Other \ parameters are: \!\(\[Nu]\): normal cell initiation rate, \!\(\[Beta]\): \ birth rate of initiated cells, \!\(\[Delta]\): death rate of initiated cells, \ \!\(\[Mu]\): mutation rate of initiated cells. This function uses an exact \ analytic formula and is also valid for both \!\(A\_2/0/0\) and \!\(C\_2/0/0\) \ models."\)], "Print"] }, Open ]], Cell[BoxData[ \(survExact[t_] := B200Survival[t, 5.5, {23.7, \ 23.58, 5.5*10^\((\(-7\))\)}]\)], "Input"], Cell[BoxData[ \(\(g1 = Plot[survExact[t], {t, 0, 250}, PlotStyle -> RGBColor[0, 1, 0], DisplayFunction -> Identity]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(survODE = B00Survival[250, 5.5, {{23.7, 23.58, 5.5*10^\((\(-7\))\)}}]\)], "Input"], Cell[BoxData[ \("-SurvivalFunction-"\)], "Output"] }, Open ]], Cell[BoxData[ \(\(g2 = Plot[survODE[t], {t, 0, 250}, PlotStyle -> RGBColor[1, 0, 1], DisplayFunction -> Identity]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Show[g1, g2, Frame -> True, DisplayFunction -> $DisplayFunction]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 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different computational approaches are almost \ indistinguishable. It indicates that the ODE based general approach performs \ satisfactorily, at least in this example. The two corresponding hazard rate \ curves also match excellently." }], "Text"], Cell[BoxData[ \(hazExact[t_] := B200Hazard[t, 5.5, {23.7, \ 23.58, 5.5*10^\((\(-7\))\)}]\)], "Input"], Cell[BoxData[ \(\(g1 = Plot[hazExact[t], {t, 0, 120}, PlotStyle -> RGBColor[0, 1, 0], DisplayFunction -> Identity]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(hazODE = B00Hazard[120, 5.5, {{23.7, 23.58, 5.5*10^\((\(-7\))\)}}]\)], "Input"], Cell[BoxData[ \("-HazardRateFunction-"\)], "Output"] }, Open ]], Cell[BoxData[ \(\(g2 = Plot[hazODE[t], {t, 0, 120}, PlotStyle -> RGBColor[1, 0, 1], DisplayFunction -> Identity]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Show[g1, g2, Frame -> True, DisplayFunction -> $DisplayFunction]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: 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considerably more computational resources. \ Specifically, for each given ", Cell[BoxData[ \(TraditionalForm\`t\_0 > 0\)]], ", one needs to solve an ODE system to obtain the survival probability ", Cell[BoxData[ \(TraditionalForm\`S(t\_0)\)]], ". For this reason, each CarcinoMod function for computing survival \ probabilities for nonhomogeneous models returns only numerical quantities, \ instead of interpolating function objects. Time dependency of a particular \ parameter is specified by a pure function. For example, if a cell birth rate \ varies with the time t as ", Cell[BoxData[ \(TraditionalForm\`\[Beta](t) = 0.028\ \((1 + 0.00001\ t\^2)\)\)]], ", it is supplied to a CarcinoMod function either as 0.028 (1+0.00001 #)& \ or as Function[t, 0.028 (1+0.00001 t^2)]. The dummy variable t can be \ replaces with any legitimate symbols. The name of a CarcinoMod function for \ a nonhomogeneous model has the same structure as the name for its \ corresponding homogeneous model, except that the second part of the name \ always begins with \"Nonhomo\". 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For example, the constant parameter 0.001 can be either 0.01& or \ Function[t,0.001]. If all cellular parameters are constant quantities, \ BN0Survival is more efficient."\)], "Print"] }, Open ]], Cell[TextData[{ "Consider a ", Cell[BoxData[ \(TraditionalForm\`\(B\_2/N\)/0\)]], " model. Assume that the cellular birth, death and mutation rates for \ normal (stage 0) cells are ", Cell[BoxData[ \(TraditionalForm \`\(\[Beta]\_0\)(t) = 0.03\ \((1 + 0.00000001\ t\^2)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\[Delta]\_0\)(t) = 0.028 \((1 + 0.00001\ t)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\[Mu]\_0\)(t) = 0.00001\)]], ", and for initiated (stage 1) cells are ", Cell[BoxData[ \(TraditionalForm\`\(\[Beta]\_1\)(t) = 0.03\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Delta]\_1\)(t\)\)]], ")=0.028, and ", Cell[BoxData[ \(TraditionalForm\`\(\[Mu]\_1\)(t) = 0.00001\)]], ". The survival probability at t=12345 can be found by calling \ BN0NonhomoSurvival." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(BN0NonhomoSurvival[ 12345, {{0.030 \((1 + 0.00000001 #^2)\)&, 0.028\ \((1 + 0.00001\ #)\)&, 0.00001&}, {0.030&, 0.028&, 0.00001&}}, 5]\)], "Input"], Cell[BoxData[ \(0.708819965002409624`\)], "Output"] }, Open ]], Cell["\<\ Although pure functions provide a very flexible way of supplying \ time-dependent parameters to CarcinoMod functions, it is sometimes desirable \ that parameters be given in traditional form. For example, if all but only \ one or two parameters are constant quantities, it is tedious to type a long \ list of parameters as pure functions. The function MakeParameterFunctional \ allows the user to type parameters in traditional form and then transform \ them into pure function.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?MakeParameterFunctional\)\)], "Input"], Cell[BoxData[ \("MakeParameterFunctional[para,t] translates a list of triplets of \ time-dependent parameters into the form of pure functions. 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Because the hazard rate at a given point ", Cell[BoxData[ \(TraditionalForm\`t\_0 \[GreaterEqual] 0\)]], " reflects the behavior of the corresponding survival function in a \ neighborhood of that point, it is necessary to compute a number of survival \ probabilities around ", Cell[BoxData[ \(TraditionalForm\`t\_0\)]], " in order to obtain the hazard rate at ", Cell[BoxData[ \(TraditionalForm\`t\_0\)]], ". Consequently, with nonhomogeneous models, several ODE systems must be \ solved to get one value of the hazard rate function. It is therefore natural \ and economical that, with time-dependent parameter models, CarcinoMod \ functions return interpolating function objects for hazard rate functions, as \ in the case of homogeneous models. Numerical differentiation of the survival \ function ", Cell[BoxData[ \(TraditionalForm\`S(t)\)]], " is simple in principle but tricky in practice, if the survival function \ is not sufficiently smooth. In order to compute ", Cell[BoxData[ \(TraditionalForm\`S' \((t)\)\)]], ", CarcinoMod first samples the survival function ", Cell[BoxData[ \(TraditionalForm\`S(t)\)]], ", creates an interpolation function object for ", Cell[BoxData[ \(TraditionalForm\`S(t)\)]], ", and then differentiates the interpolating function. Three options are \ available that allow the user to control how these interpolating function \ objects are constructed. The option HazardInterpolationPoints specifies how \ many sampling points should be used to compute the corresponding survival \ probabilities. The second option, HazardInterpolationOrder, specifies the \ order of the interpolating function that will eventually be returned to the \ user. The third option, HazardBoundaryStretch, if set to some ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon] > 0\)]], ", will enlarge the prescribed interval from [0,T] to ", Cell[BoxData[ \(TraditionalForm\`\([0, \((1 + \[Epsilon])\) T]\)\)]], ". 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Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Functions for the ", Cell[BoxData[ \(TraditionalForm\`\(B\_2/0\)/0\)]], " model" }], "Section"], Cell[TextData[{ "Due to its structural simplicity, the ", Cell[BoxData[ \(TraditionalForm\`\(B\_2/0\)/0\)]], " model possesses several closed-from analytic formulae for the survival \ and hazard rate functions that were published during the past few years. In \ fact, all the analytic formulae derived for the survival and hazard rate \ functions for the ", Cell[BoxData[ \(TraditionalForm\`\(B\_2/0\)/0\)]], " models are also valid for the ", Cell[BoxData[ \(TraditionalForm\`\(A\_2/0\)/0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(C\_2/0\)/0\)]], " models (see Zheng 1998). To retain the historical flavors of the field, \ CarcinoMod functions using these analytic formulae all begin with the prefix \ \"B200\" as if they were solely for the ", Cell[BoxData[ \(TraditionalForm\`\(B\_2/0\)/0\)]], " models. We have already encountered two of these functions: B200Survival \ and B200Hazard. Another function in this group is B200PiecewiseLinearHazard \ which computes the hazard rate function for a ", Cell[BoxData[ \(TraditionalForm\`\(B\_2/0\)/0\)]], " model that allows the first mutation rate (the intensity of a Poisson \ process) to change with the time in a piecewise linear manner. The formula \ was derived by this author in an unpublished manuscript in 1995 and is useful \ when the tissue under consideration is still growing. Because in a ", Cell[BoxData[ \(TraditionalForm\`\(B\_2/0\)/0\)]], " model, normal cell growth is assumed to follow some deterministic law \ (e.g., the logistic law) and the rate at which normal cells mutate to \ initiated cells is proportional to the size of the normal cell population, it \ is not reasonable to assume constant initiation rates when dealing with \ growing tissues. Piecewise linear functions allow the user to approximate the \ normal cell growth pattern with piecewise linear functions." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?B200PiecewiseLinearHazard\)\)], "Input"], Cell[BoxData[ \("B200PiecewiseLinearHazard[t,\!\({...,{t\_i,a\_i, \ b\_i},...},{\[Beta],\[Delta],\[Mu]}\)] returns the hazard rate at time t for \ a \!\(B\_2/0/0\) model whose normal cell initiation rate varies with the time \ in a piecewise linear fashion. Specifically, normal cells mutate to initiated \ cells at rate \!\(a\_i +b\_i t\) at the time interval \ \!\([t\_\(i-1\),t\_i)\). The cellular birth, death and mutation rates for \ initiated cells, \[Beta], \[Delta] and \[Mu] must all be constant quantities. \ This function uses an exact analytic formula and is also valid for \ \!\(A\_2/0/0\) and \!\(C\_2/0/0\) models."\)], "Print"] }, Open ]], Cell["\<\ In the following example, we assume that a growth function is \ composed of three pieces of linear functions.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(normalGrowth = {{200, 0.01, 0.00195}, {400, 0.25, 0.00075}, {Infinity, 0.51, 0.0001}}\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"200", ",", StyleBox["0.01`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", StyleBox["0.00195000000000000017`", StyleBoxAutoDelete->True, PrintPrecision->3]}], "}"}], ",", RowBox[{"{", RowBox[{"400", ",", StyleBox["0.25`", StyleBoxAutoDelete->True, PrintPrecision->2], ",", StyleBox["0.00075`", StyleBoxAutoDelete->True, PrintPrecision->2]}], "}"}], ",", RowBox[{"{", RowBox[{ InterpretationBox["\[Infinity]", DirectedInfinity[ 1]], ",", StyleBox["0.509999999999999964`", StyleBoxAutoDelete->True, PrintPrecision->2], ",", StyleBox["0.0001`", StyleBoxAutoDelete->True, PrintPrecision->1]}], "}"}]}], "}"}]], "Output"] }, Open ]], Cell["\<\ For comparison purposes, we first draw the graph of this piecewise \ linear function. 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Cell[TextData[{ "The last two functions in this group deal with the ", Cell[BoxData[ \(TraditionalForm\`\(B\_2/0\)/0\)]], " model with all four parameters piecewise constant. The algorithm for \ computing the survival function for this case can be found in Zheng (1995a). \ It was first implemented with ", StyleBox["Mathematica", FontSlant->"Italic"], " as a user function for ", StyleBox["MathSource", FontSlant->"Italic"], " item 0207-931 (Zheng 1995b). The algorithm for computing the hazard \ rate function for the same scenario was recently published by Heidenreich et \ al. (1997), and is also a user function of CarcinoMod. We here present the \ same five-piece example from ", StyleBox["MathSource", FontSlant->"Italic"], " item 0207-931." }], "Text"], Cell[BoxData[ \(\(fivePieces = {{4000, 0.0017, 0.0015, 0.03, 0.0000003}, \n \ \ \ \ \ \ \ \ \ \ \ \ {8000, 0.0015, 0.0014, 0.02, 0.0000001}, \n \ \ \ \ \ \ \ \ \ \ \ {12000, 0.0016, 0.0016, 0.03, 0.0000002}, \n \ \ \ \ \ \ \ \ \ \ \ {16000, 0.0014, 0.0012, 0.02, 0.0000005}, \n \ \ \ \ \ \ \ \ {Infinity, 0.0015, 0.0013, 0.01, 0.0000004}}; \)\)], "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(?B200PiecewiseConstantSurvival\)\)], "Input"], Cell[BoxData[ \("B200PiecewiseConstantSurvival[t,\!\({...,{t\_i,\[Beta]\_i, \ \[Delta]\_i,\[Nu]\_i,\[Mu]\_i},...}\)] returns the survival function \ evaluated at time t for a \!\(B\_2/0/0\) model whose cellular parameters are \ piecewise constant. Specifically, on the ith interval\!\([t\_\(i-1\),t\_i)\), \ the birth, death and mutation rates for initiated cells are \!\(\[Beta]\_i, \ \[Delta]\_i\) and \!\(\[Mu]\_i\), respectively, and normal cell initiation \ rate is \!\(\[Nu]\_i\). The last partition point is always \!\(\[Infinity]\). \ This function uses an exact analytic formula and is also valid for \ \!\(A\_2/0/0\) and \!\(C\_2/0/0\) models."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(surv = B200PiecewiseConstantSurvival[#, fivePieces]&\)], "Input"], Cell[BoxData[ \(B200PiecewiseConstantSurvival[#1, fivePieces]&\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[surv[t], {t, 0, 20000}, Frame -> True, GridLines -> Automatic]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 4.7619e-05 0.0147151 0.588604 [ [.02381 -0.0125 -3 -9 ] [.02381 -0.0125 3 0 ] [.2619 -0.0125 -12 -9 ] [.2619 -0.0125 12 0 ] [.5 -0.0125 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Even if the \ exposure dose is perfectly piecewise constant, the internal dose is almost \ always continuous, as already demonstrated by many physiologically based \ pharmacokinetic (PBPK) models. See Zheng (1997b) for a PBPK model implemented \ with ", StyleBox["Mathematica", FontSlant->"Italic"], ". Caution therefore must be used before adopting carcinogenesis models \ with piecewise constant parameters." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Summary", "Section"], Cell[TextData[{ "CarcinoMod provides the user with 32 functions to compute the survival and \ hazard rate functions for the eight types of carcinogenesis models given in \ Zheng (1997a). For each type of models, there are four user functions whose \ names have a common 3-letter prefix to indicate the models that the functions \ address. For example, the prefix \"BN0\" signifies functions for the ", Cell[BoxData[ \(TraditionalForm\`\(B\_k/N\)/0\)]], " models. This prefix is followed by either Survival, NonhomoSurvival, \ Hazard or NonhomoHazard. Such function names are self-explanatory and easy to \ remember. All these 32 functions adopt the ODE-based computational approach \ as described in Zheng (1997a). In addition, there are five functions whose \ names have the prefix \"B200\". These five function names no doubt remind \ the user of the ", Cell[BoxData[ \(TraditionalForm\`\(B\_2/0\)/0\)]], " model, although they are in fact valid for the ", Cell[BoxData[ \(TraditionalForm\`\(A\_2/0\)/0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(C\_2/0\)/0\)]], " models as well. These five functions use exact formulae to compute the \ survival and hazard rate functions. Many of the stochastic carcinogenesis \ models developed during the last two decades or so are sequential, \ unidirectional compartmental models, and all such models can be handled by \ CarcinoMod. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Acknowledgments", "Section"], Cell[TextData[{ "A major part of this work was performed while the author was at Wolfram \ Research, Inc. as a ", StyleBox["Mathematica", FontSlant->"Italic"], " visiting scholar in December, 1997." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ "Fearon, E.R. and Vogelstein, B. (1990). A genetic model for colorectal \ tumorigenesis, Cell, 61:759-767.\n\nHeidenreich, W.F., Luebeck, E.G. and \ Moolgavkar, S.H. (1997). Some properties of the hazard function of the \ two-mutation clonal expansion model, Risk Analysis, 17:391-399.\n\nKendall, \ D.G. (1960). Birth-and-death processes, and the theory of carcinogenesis, \ Biometrika, 47:13-21.\n\nKopp-Schneider, A., Portier, C.J. and Sherman, C. D. \ (1994). The exact formula for tumor incidence in the two-stage model, Risk \ Analysis, 14:1079-1080.\n\nKopp-Schneider, A. and Portier, C.J. (1995). \ Carcinoma formation in NMRI mouse skin painting studies is a process \ suggesting greater than two stages, Carcinogenesis, 16:53-59.\n\nLittle, M.P. \ (1995). Are two mutations sufficient to cause cancer? Some generalizations of \ the two-mutation model of carcinogenesis of Moolgavkar, Venzon, and Knudson, \ and of the multistage model of Armitage and Doll, Biometrics, 51:1278-1291.\n\ \nMoolgavkar, S.H. and Venzon, D.J. (1979). Two-event models for \ carcinogenesis: Incidence curves for childhood and adult tumors, Mathematical \ Biosciences, 47:55-77.\n\nZheng, Q. (1994). On the exact hazard and survival \ functions of the MVK stochastic carcinogenesis models, Risk Analysis, \ 14:1081-1084.\n\nZheng, Q. (1995a). On the MVK stochastic carcinogenesis \ model with Erlang distributed cell life lengths, Risk Analysis, 15:495-502.\n\ \nZheng, Q. (1995b). ", StyleBox["MathSource", FontSlant->"Italic"], " item 0207-931, abstracted in The ", StyleBox["Mathematica", FontSlant->"Italic"], " Journal Volume 6 (1996), Issue 1, p.17 .\n\nZheng, Q. (1997a). A unified \ approach to a class of stochastic carcinogenesis models, Risk Analysis, \ 17:617-624.\n\nZheng, Q. (1997b). Exploring physiologically based \ pharmacokinetic models, ", StyleBox["Mathematica", FontSlant->"Italic"], " in Education and Research, 6:22-28.\n\nZheng, Q. (1998). On a trinity \ role of the hazard and survival functions of the two-stage carcinogenesis \ models, Communications in statistics -- Computation and Simulation, to \ appear.\n\n" }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 800}, {0, 580}}, WindowToolbars->"EditBar", WindowSize->{732, 556}, WindowMargins->{{4, Automatic}, {Automatic, 1}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "Magnification"->1}, StyleDefinitions -> "Default.nb", MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@0000000000000006P801T1T00000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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