(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 212567, 4244] NotebookOptionsPosition[ 208539, 4124] NotebookOutlinePosition[ 209438, 4153] CellTagsIndexPosition[ 209395, 4150] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Two Compartment Model Solution with ", FontSize->20], StyleBox["Mathematica ", FontSize->20, FontSlant->"Italic"], StyleBox["and Illustration of Central Volume Principle\n", FontSize->20], StyleBox["Author: Michael Jerosch-Herold", FontSize->10] }], "Title", CellGroupingRules->{"TitleGrouping", Inherited}, CellChangeTimes->{{3.480082439095111*^9, 3.480082468044767*^9}, { 3.4800825859495106`*^9, 3.480082687132298*^9}, {3.480082719654158*^9, 3.4800827506009283`*^9}, 3.4800951737424912`*^9, {3.480096428777275*^9, 3.480096465961402*^9}, 3.4801018449670634`*^9, {3.4801588892196846`*^9, 3.4801588900464864`*^9}, {3.480158942119378*^9, 3.480159021476717*^9}, { 3.4801590723796062`*^9, 3.480159083190426*^9}, {3.4801603522572527`*^9, 3.4801604353585987`*^9}, {3.4801607264551096`*^9, 3.480160746547945*^9}, { 3.4801611243806086`*^9, 3.4801611253634105`*^9}, {3.480250369439494*^9, 3.480250400341262*^9}}, TextAlignment->Center, Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[{ StyleBox["Abstract", FontSize->16], StyleBox["\nThis didactic example starts with a two-compartment model as \ illustrated in Figure 1 of this ", FontSize->12], StyleBox["Mathematica", FontSize->12, FontSlant->"Italic"], StyleBox[" notebook. The two compartment model could be used to analyze the \ contrast enhancement in myocardial tissue. The analytical solution of the \ two-compartment model is combined with appropriate initial conditions to \ obtain the impulse response. An impulse response is calculated for specific \ parameter values. Next the use of the impulse response is demonstrated by \ calculating the tissue response for a arterial input with the shape of a \ gamma-variate function. 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Alternatively one can normalize the same curve to unit \ area. In this latter case the resulting graph gives the probability for the \ residence times in the two-compartment region.", FontFamily->"Arial", FontSize->14]], "Text", CellChangeTimes->{{3.4801817560026937`*^9, 3.480181845671651*^9}, { 3.4802712971523027`*^9, 3.4802713856023617`*^9}, {3.480793414719355*^9, 3.4807934604639716`*^9}, {3.4807934939308863`*^9, 3.480793670462983*^9}, { 3.4807937843584976`*^9, 3.480793797205232*^9}, 3.494094971834979*^9}, FontColor->RGBColor[1, 0, 0], Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[StyleBox["Plot the solution for the specified parameter values. \ Show the tracer concentration in the vascular and interstitial compartments, \ and the weighted sum of the two, which corresponds to the \ contrast-enhancement one would observe by external detection, using an \ imaging device.", FontFamily->"Arial", FontSize->14]], "Text", CellChangeTimes->{{3.480159536287219*^9, 3.4801595903569145`*^9}}, Background->RGBColor[0.87, 0.94, 1]], Cell[BoxData[ RowBox[{ RowBox[{"TotalRes", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"v1", " ", RowBox[{"c1", "[", "t", "]"}]}], "+", " ", RowBox[{"v2", " ", RowBox[{"c2", "[", "t", "]"}]}]}], ")"}], "/", "v1"}], "/.", "ImpulseResp"}], " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"v1", "\[Rule]", "0.1"}], ",", " ", RowBox[{"v2", "\[Rule]", "0.22"}]}], "}"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.480267079892089*^9, 3.4802670851613903`*^9}, 3.4802672465016184`*^9, {3.4802673144915075`*^9, 3.480267343814184*^9}, { 3.4802673805512857`*^9, 3.4802673886027465`*^9}, {3.480267425999885*^9, 3.480267463020003*^9}, {3.4802675562253337`*^9, 3.480267556824368*^9}, { 3.480267610771454*^9, 3.4802676152257085`*^9}, {3.4802676613863487`*^9, 3.4802676642585125`*^9}, {3.480267732024389*^9, 3.480267742482987*^9}, { 3.4802679376261487`*^9, 3.480267957511286*^9}, {3.480268064267392*^9, 3.480268066752534*^9}, {3.48026833768303*^9, 3.480268342532308*^9}, { 3.4802694116674585`*^9, 3.4802694140765967`*^9}, {3.480269695376686*^9, 3.480269716860915*^9}, {3.480269855562848*^9, 3.4802698690516195`*^9}, 3.480270058431452*^9, {3.4802700890372024`*^9, 3.4802700906322937`*^9}, { 3.480271074805585*^9, 3.480271078236781*^9}, {3.480330143590085*^9, 3.480330159174513*^9}, {3.4803303627704706`*^9, 3.480330363597272*^9}, { 3.4804240159290533`*^9, 3.4804240557303305`*^9}, {3.480424116541808*^9, 3.4804241507737665`*^9}, {3.480424206695965*^9, 3.4804242139553804`*^9}, { 3.4807910655377645`*^9, 3.480791065709365*^9}, 3.480791350831066*^9, 3.480791439064821*^9, {3.4807915615572357`*^9, 3.4807915817592716`*^9}, { 3.4807916384029713`*^9, 3.480791641164176*^9}, {3.4807916881514587`*^9, 3.480791688479059*^9}, {3.480791846242136*^9, 3.48079185430735*^9}, { 3.480791909422247*^9, 3.4807919172222605`*^9}, {3.4807920019928093`*^9, 3.4807920189032393`*^9}}], Cell[BoxData[ RowBox[{"Bfunc", " ", ":=", " ", RowBox[{"Function", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", " ", "y", ",", " ", "z"}], "}"}], ",", " ", RowBox[{"TotalRes", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"t", "\[Rule]", " ", "x"}], ",", " ", RowBox[{"PS", "\[Rule]", " ", "y"}], ",", " ", RowBox[{"F", "\[Rule]", "z"}]}], "}"}]}], ",", " ", "Listable"}], "]"}]}]], "Input", CellChangeTimes->{{3.480268195601904*^9, 3.4802682268776927`*^9}, { 3.4802683616604023`*^9, 3.4802683787353783`*^9}, {3.4802684348595886`*^9, 3.4802684876576085`*^9}, {3.480268535540347*^9, 3.4802685412606745`*^9}, { 3.4802686474387474`*^9, 3.480268648793825*^9}, {3.4802687460143857`*^9, 3.4802687734919577`*^9}, {3.4802688243558664`*^9, 3.4802688725756245`*^9}, { 3.480268928253809*^9, 3.480268962425764*^9}, {3.4802690472896175`*^9, 3.480269053368965*^9}, {3.480269126835168*^9, 3.4802691270901823`*^9}, { 3.4802691595460386`*^9, 3.480269162135186*^9}, {3.480269196091129*^9, 3.4802692517163105`*^9}, {3.4802693198012047`*^9, 3.4802693385612774`*^9}, { 3.48026938230678*^9, 3.48026938494493*^9}, {3.480271087204294*^9, 3.4802710913855333`*^9}, {3.4807914434172287`*^9, 3.480791451685243*^9}, { 3.4807915674072466`*^9, 3.480791567656847*^9}}], Cell["\<\ In the next graph you can interactively change the blood flow, and the \ permeability-surface area product through the sliders with the \"PS\" and \"F\ \" labels at the top. This allows to see how the changes of these parameters \ affect the probability for the tracer. Note that an increase of PS causes a \ prolonged washout of contrast, while at the smallest PS value the tracer is \ confined to the vascular compartment and the traversal of the tracer through \ the tissue unit is relatively quicker.\ \>", "Text", CellChangeTimes->{{3.4802702684784656`*^9, 3.480270387184255*^9}, { 3.4802704919192457`*^9, 3.4802705157436085`*^9}, {3.4807917138759036`*^9, 3.4807917843568277`*^9}}, Background->RGBColor[0.87, 0.94, 1]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Bfunc", "[", RowBox[{"t", ",", " ", "PS", ",", " ", "F"}], "]"}], ",", " ", RowBox[{"{", " ", RowBox[{"t", ",", " ", "0.0", ",", " ", "0.1", ",", " ", "2"}], "}"}], ",", " ", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\