(*********************************************************************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 182894, 10114]*) (*NotebookOutlinePosition[ 183949, 10150]*) (* CellTagsIndexPosition[ 183905, 10146]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["A Carcinogenesis Model \nat the Nucleotide Level "], "Title", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell[TextData[ "Qi Zheng \nNational Center for Toxicological Research \nU.S. Food and Drug \ Administration "], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell[TextData["Introduction "], "Section", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell[TextData[ "Rapid and accelerating advances in biomedical cancer research have led a \ dramatic increase in the amount of attention given to mathematical modeling \ of carcinogenesis. The advent of comprehensive symbolic computation systems \ like Mathematica has facilitated research in the area. A cancer model that \ describes mutational events at the nucleotide level has been proposed \ recently by Q. Zheng, D. Gaylor and W. Lutz (American Statistical Association \ 1995 Proceedings of the Biometrics Section, pp. 349-354). The computational \ aspects of this model (both symbolic and numerical) which were appreciably \ more complex than those of the traditional two-stage model were \ investigated using Mathematica and reported by this author at the Tenth \ International Conference on Mathematical and Computer Modelling and \ Scientific Computing in Boston on July 6, 1995 (see \"Combining Symbolic and \ Numerical computing in Carcinogenesis Modeling\" in the Proceedings of the \ Tenth International Conference) . The present notebook is based on the \ results presented at the Conference and focuses on computational aspects. For \ biological and modeling aspects, the interested reader should consult the \ two aforementioned references. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[" \nSetOptions[\"stdout\",PageWidth->70]; "], "Input", AspectRatioFixed->True], Cell[TextData["Cumulants and the Cumulant Generating Function "], "Section", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell[TextData["How to Derive Differential Equations "], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell[TextData[ "The model is essentially a four-compartment stochastic process. The joint \ cumulants of the process provide valuable statistical information. To solve \ for these cumulants, first derive a system of ordinary differential \ equations (ODE) that the cumulants of up to a given order must satisfy. Such \ ODEs may be derived from a partial differential equation (PDE) that the \ cumulant generating function (cgf) satisfies. The K[ ] function defined \ below produces symbolically the cgf for the model truncated at a specified \ order ord. Each term k[i1,i2,i3,i4][t] represents a cumulant function of \ order i1+i2+i3+i4. \n "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "\nK[ord_]:=Select[ \nSum[k[i1,i2,i3,i4][t] x1^i1 x2^i2 x3^i3 x4^i4 /(i1! i2! \ i3! i4!), \n \ {i1,0,ord},{i2,0,ord},{i3,0,ord},{i4,0,ord}]/.k[0,0,0,0][t]->0, \n \ ((Exponent[#,x1]+Exponent[#,x2]+Exponent[#,x3]+ \n Exponent[#,x4])<=ord)&]; \ \n "], "Input", AspectRatioFixed->True], Cell[TextData[ "The right hand side of the aforementioned PDE should also be expanded to \ the same given order. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "RHS[ord_]:=Select[ Expand[(b Sum[x1^i/i!,{i,1,ord}]+ \n a \ Sum[(-x1+x2)^i/i!,{i,1,ord}]+ \n d1 Sum[(-x1)^i/i!,{i,1,ord}]) \ D[K[ord],x1]+ \n (p1 Sum[(x1-x2)^i/i!,{i,1,ord}]+p2 \ Sum[(x1-x2+x3)^i/i!,{i,1,ord}]+ \n d2 Sum[(-x2)^i/i!,{i,1,ord}]) \ D[K[ord],x2]+ \n (p3 Sum[(-x3+x4)^i/i!,{i,1,ord}]+ \n u \ Sum[x4^i/i!,{i,1,ord}]+ \n d3 Sum[(-x3)^i/i!,{i,1,ord}]) D[K[ord],x3]+ \n \ v Sum[x1^i/i!,{i,1,ord}] ] , \n \ ((Exponent[#,x1]+Exponent[#,x2]+Exponent[#,x3]+Exponent[#,x4])<=ord)&] \n "], "Input", AspectRatioFixed->True], Cell[TextData[ "By equating coefficients of the left hand side of the PDE (which is simply \ D[K,t]) with those of the right hand side, a system of ODEs results. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "odeForCumulants[ord_]:=List@@LogicalExpand[ \n \ Series[D[K[ord],t],{x1,0,ord},{x2,0,ord},{x3,0,ord}, \n \ {x4,0,ord}]==Series[RHS[ord], \n \ {x1,0,ord},{x2,0,ord},{x3,0,ord},{x4,0,ord}]]; \n "], "Input", AspectRatioFixed->True], Cell[TextData["The Mean and Variance-Covariance Functions "], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell[TextData[ "The first and second order cumulants are the best known and the most widely \ used. The first order cumulants are referred to as means or expectations, \ while the second order cumulants are called variances (or covariances). One \ can easily obtain an ODE system for these cumulants. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "odeOrderTwo=odeForCumulants[2]; \nodeOrderTwo[[5]] \n "], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ -(d3*k[0, 0, 1, 0][t])/2 - (p3*k[0, 0, 1, 0][t])/2 + d3*k[0, 0, 2, 0][t] + p3*k[0, 0, 2, 0][t] - (p2*k[0, 1, 0, 0][t])/2 - p2*k[0, 1, 1, 0][t] + Derivative[1][k[0, 0, 2, 0]][t]/2 == 0\ \>", "\<\ -(d3 k[0, 0, 1, 0][t]) p3 k[0, 0, 1, 0][t] ---------------------- - ------------------- + 2 2 d3 k[0, 0, 2, 0][t] + p3 k[0, 0, 2, 0][t] - p2 k[0, 1, 0, 0][t] ------------------- - p2 k[0, 1, 1, 0][t] + 2 (k[0, 0, 2, 0])'[t] ------------------- == 0 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ " \nAs one may readily see from the above output, the resulting ODEs are \ not yet in a form pleasing to the human eye. Further symbolic processing is \ needed to render the equations to a \"canonical\" form. First, fractions \ should be eliminated. Because the equations are in the form lhs==0, one \ only needs to process the left hand sides of these equations. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "odeOrderTwo=(Numerator[Together[#]])&/@(First/@odeOrderTwo); \n\ odeOrderTwo[[5]] \n "], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ -(d3*k[0, 0, 1, 0][t]) - p3*k[0, 0, 1, 0][t] + 2*d3*k[0, 0, 2, 0][t] + 2*p3*k[0, 0, 2, 0][t] - p2*k[0, 1, 0, 0][t] - 2*p2*k[0, 1, 1, 0][t] + Derivative[1][k[0, 0, 2, 0]][t]\ \>", "\<\ -(d3 k[0, 0, 1, 0][t]) - p3 k[0, 0, 1, 0][t] + 2 d3 k[0, 0, 2, 0][t] + 2 p3 k[0, 0, 2, 0][t] - p2 k[0, 1, 0, 0][t] - 2 p2 k[0, 1, 1, 0][t] + (k[0, 0, 2, 0])'[t]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ " \nSecondly, collect coefficients for each cumulant . "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "rule1=(a_ k[i__][t_]+ b_ k[i__][t_]):>(a+b) k[i][t]; \n\ odeOrderTwo=odeOrderTwo//.rule1; \nodeOrderTwo[[5]] "], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ (-d3 - p3)*k[0, 0, 1, 0][t] + (2*d3 + 2*p3)*k[0, 0, 2, 0][t] - p2*k[0, 1, 0, 0][t] - 2*p2*k[0, 1, 1, 0][t] + Derivative[1][k[0, 0, 2, 0]][t]\ \>", "\<\ (-d3 - p3) k[0, 0, 1, 0][t] + (2 d3 + 2 p3) k[0, 0, 2, 0][t] - p2 k[0, 1, 0, 0][t] - 2 p2 k[0, 1, 1, 0][t] + (k[0, 0, 2, 0])'[t]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ " \nThirdly, after locating the derivative term in each expression, translate \ these expressions back to ODEs. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "posOfDerivative=Flatten[(Position[#,_. k[__]'[_]])&/@odeOrderTwo]; \n\ odeOrderTwo=MapThread[(#[[#2]]==-Drop[#,{#2}])&, \n \ {odeOrderTwo,posOfDerivative}]; \nodeOrderTwo[[5]] \n \ "], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ Derivative[1][k[0, 0, 2, 0]][t] == -((-d3 - p3)*k[0, 0, 1, 0][t]) - (2*d3 + 2*p3)*k[0, 0, 2, 0][t] + p2*k[0, 1, 0, 0][t] + 2*p2*k[0, 1, 1, 0][t]\ \>", "\<\ (k[0, 0, 2, 0])'[t] == -((-d3 - p3) k[0, 0, 1, 0][t]) - (2 d3 + 2 p3) k[0, 0, 2, 0][t] + p2 k[0, 1, 0, 0][t] + 2 p2 k[0, 1, 1, 0][t]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ " \nFurther simplification and sorting transform the ODEs into textbook \ format. At this point, one may rename these cumulants with more common \ notation (e.g., m stands for mean, etc.). "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "rule2={k[1,0,0,0]->m1,k[0,1,0,0]->m2, \n\ k[0,0,1,0]->m3,k[0,0,0,1]->m4,k[2,0,0,0]->v11, \n\ k[1,1,0,0]->v12,k[1,0,1,0]->v13,k[1,0,0,1]->v14, \n\ k[0,2,0,0]->v22,k[0,1,1,0]->v23,k[0,1,0,1]->v24, \n\ k[0,0,2,0]->v33,k[0,0,1,1]->v34,k[0,0,0,2]->v44}; \n \n\ TableForm[Sort[Simplify[odeOrderTwo//.rule2]]] \n "], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ TableForm[{Derivative[1][m1][t] == v - (a - b + d1)*m1[t] + (p1 + p2)*m2[t], Derivative[1][m2][t] == a*m1[t] - (d2 + p1 + p2)*m2[t], Derivative[1][m3][t] == p2*m2[t] - (d3 + p3)*m3[t], Derivative[1][m4][t] == (p3 + u)*m3[t], Derivative[1][v11][t] == v + (a + b + d1)*m1[t] + (p1 + p2)*m2[t] - 2*(a - b + d1)*v11[t] + 2*(p1 + p2)*v12[t], Derivative[1][v12][t] == -(a*m1[t]) - (p1 + p2)*m2[t] + a*v11[t] - (a - b + d1 + d2 + p1 + p2)*v12[t] + (p1 + p2)*v22[t], Derivative[1][v13][t] == p2*m2[t] + p2*v12[t] - (a - b + d1 + d3 + p3)*v13[t] + (p1 + p2)*v23[t], Derivative[1][v14][t] == (p3 + u)*v13[t] - (a - b + d1)*v14[t] + (p1 + p2)*v24[t]\\ , Derivative[1][v22][t] == a*m1[t] + (d2 + p1 + p2)*m2[t] + 2*a*v12[t] - 2*(d2 + p1 + p2)*v22[t], Derivative[1][v23][t] == -(p2*m2[t]) + a*v13[t] + p2*v22[t] - (d2 + d3 + p1 + p2 + p3)*v23[t], Derivative[1][v24][t] == a*v14[t] + (p3 + u)*v23[t] - (d2 + p1 + p2)*v24[t], Derivative[1][v33][t] == p2*m2[t] + (d3 + p3)*m3[t] + 2*p2*v23[t] - 2*(d3 + p3)*v33[t], Derivative[1][v34][t] == -(p3*m3[t]) + p2*v24[t] + (p3 + u)*v33[t] - (d3 + p3)*v34[t], Derivative[1][v44][t] == (p3 + u)*(m3[t] + 2*v34[t])}]\ \>", "\<\ m1'[t] == v - (a - b + d1) m1[t] + (p1 + p2) m2[t] m2'[t] == a m1[t] - (d2 + p1 + p2) m2[t] m3'[t] == p2 m2[t] - (d3 + p3) m3[t] m4'[t] == (p3 + u) m3[t] v11'[t] == v + (a + b + d1) m1[t] + (p1 + p2) m2[t] - 2 (a - b + d1) v11[t] + 2 (p1 + p2) v12[t] v12'[t] == -(a m1[t]) - (p1 + p2) m2[t] + a v11[t] - (a - b + d1 + d2 + p1 + p2) v12[t] + (p1 + p2) v22[t] v13'[t] == p2 m2[t] + p2 v12[t] - (a - b + d1 + d3 + p3) v13[t] + (p1 + p2) v23[t] v14'[t] == (p3 + u) v13[t] - (a - b + d1) v14[t] + (p1 + p2) v24[t] v22'[t] == a m1[t] + (d2 + p1 + p2) m2[t] + 2 a v12[t] - 2 (d2 + p1 + p2) v22[t] v23'[t] == -(p2 m2[t]) + a v13[t] + p2 v22[t] - (d2 + d3 + p1 + p2 + p3) v23[t] v24'[t] == a v14[t] + (p3 + u) v23[t] - (d2 + p1 + p2) v24[t] v33'[t] == p2 m2[t] + (d3 + p3) m3[t] + 2 p2 v23[t] - 2 (d3 + p3) v33[t] v34'[t] == -(p3 m3[t]) + p2 v24[t] + (p3 + u) v33[t] - (d3 + p3) v34[t] v44'[t] == (p3 + u) (m3[t] + 2 v34[t])\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[TextData["How to Obtain Numerical Solutions \n "], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell[TextData[ "There are 34 cumulants of order up to 3. To numerically solve and plot them \ require additional symbolic manipulation. For the present purpose, one \ needs these ODEs in their \"raw\" form. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["odeOrderThree=odeForCumulants[3]//.rule1; \n "], "Input", AspectRatioFixed->True], Cell[TextData[ " \nNow specify numerical values for the cell kinetic parameters. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "odeOrderThree=odeOrderThree//.{v->0.03,b->0.0019,d1->0.0015, \n \ d2->0.0016,d3->0.0017,a->0.0005, \n \ p1->0.0005,p2->0.0005,p3->0.0005,u->0.0000003}; \n "], "Input", AspectRatioFixed->True], Cell[TextData[ " \nIn addition to creating a list of equation names, determine and attach \ the initial conditions to the original ODE list. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " \nodeNames=Table[If[i1+i2+i3+i4<=3,k[i1,i2,i3,i4],{}], \n \ {i1,0,3},{i2,0,3},{i3,0,3},{i4,0,3}]//Flatten//Rest; \n\ initial=Table[If[i1+i2+i3+i4<=3,k[i1,i2,i3,i4][0]==0,{}], \n \ {i1,0,3},{i2,0,3},{i3,0,3},{i4,0,3}]//Flatten//Rest; \n\ odeWithInit=Join[odeOrderThree,initial]; \n "], "Input", AspectRatioFixed->True], Cell[TextData[ "Note that the NDSolve function in Mathematica Version 2.2 does not always \ behave correctly when the functions have indexed names, like k[2,0,0,1]. \ When computing numerical values, NDSolve sometimes also computes numerical \ values for the indices, after which it is no longer able to recognize the \ functions. This limitation of NDSolve has been corrected for Version 3 of \ Mathematica. However, there are several simple ways to deal with this \ problem in Version 2.2. One of the easiest ways is to use the NProtectedAll \ attribute to protect the function indices from numerical computation. The \ NProtectedAll attribute was added to Mathematica after the manual was \ printed, and so is not included in the manual. The name NProtectedAll has \ been changed to NHoldAll for Version 3. \n "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "SetAttributes[k, NProtectedAll]; \n\ odeSoln=NDSolve[odeWithInit,odeNames,{t,0,5000}]; \n "], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "To have a feeling of the shapes of these cumulant functions, one can plot \ a few of them . 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They help answer a variety of questions arising from quantitative \ cancer research. This section briefly demonstrates computing and plotting of \ these two important functions. \n \nThe survival function is computed \ using the characteristic method from standard partial differential equation \ \[RawEscape] theory. 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During this visit, Tom Issaevitch and David Withoff of \ Wolfram Research made several valuable and specific coding suggestions, which \ led to more elegant and efficient code in the present notebook. I am \ indebted to S.R. Sipcic at Boston University College of Engineering for \ kindly inviting me to present part of this work to the Tenth International \ Conference on Mathematical and Computer Modeling and Scientific Computing. \ Partial support provided by Wolfram Research during the visit is gratefully \ acknowledged. This work was also supported by an appointment to the \ Postgraduate Research Program at the National Center for Toxicological \ Research administered by the Oak Ridge Institute for Science and Education \ through an interagency agreement between the U.S. Department of Energy and \ the Food and Drug Administration. 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