(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 9.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 514330, 13443] NotebookOptionsPosition[ 493950, 12858] NotebookOutlinePosition[ 495309, 12910] CellTagsIndexPosition[ 495063, 12898] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["", "Abstract", CellChangeTimes->{{3.583237028160064*^9, 3.5832370614197173`*^9}, { 3.5866102531893806`*^9, 3.586610306346696*^9}, 3.5867799314218817`*^9}], Cell["\<\ This document describe some of the computation included in the paper:\ \>", "AbstractSection", CellChangeTimes->{{3.5406259859024515`*^9, 3.540625993702465*^9}, { 3.5406261278471007`*^9, 3.5406261377375183`*^9}, 3.583235560791264*^9, { 3.58323691967708*^9, 3.5832369567275105`*^9}, {3.586779934276726*^9, 3.586779938254772*^9}}], Cell[CellGroupData[{ Cell["\<\ Optimality for models given by a system of ordinary differential equation \ \>", "Title", CellChangeTimes->{{3.5406259859024515`*^9, 3.540625993702465*^9}, { 3.5406261278471007`*^9, 3.5406261377375183`*^9}, 3.583235560791264*^9, { 3.58323691967708*^9, 3.5832369567275105`*^9}, {3.586779934276726*^9, 3.586779938254772*^9}}], Cell["Last update: 2014-01-10", "Reference", CellChangeTimes->{{3.39928025384801*^9, 3.399280259603493*^9}, { 3.4228525171714554`*^9, 3.4228525382666454`*^9}, {3.422936332120079*^9, 3.422936340401223*^9}, {3.442218589593443*^9, 3.4422185944372244`*^9}, { 3.479816247027495*^9, 3.4798162579804096`*^9}, {3.513395822423673*^9, 3.513395880159526*^9}, {3.5185321133491745`*^9, 3.518532117927387*^9}, 3.5194440623039017`*^9, {3.5406260031092815`*^9, 3.5406260316573315`*^9}, { 3.543642073116476*^9, 3.5436420765328827`*^9}, {3.54497113674302*^9, 3.54497114245263*^9}, {3.5451310961930685`*^9, 3.545131097675071*^9}, { 3.54557490726515*^9, 3.545574912444359*^9}, {3.569391362912204*^9, 3.5693913662662096`*^9}, {3.5706017596182275`*^9, 3.570601761521469*^9}, { 3.583235676872295*^9, 3.5832356809439416`*^9}, {3.58617370467741*^9, 3.5861737085306435`*^9}, {3.586583761258679*^9, 3.5865837619294243`*^9}, { 3.5866988864290733`*^9, 3.586698887614686*^9}, {3.586779974150753*^9, 3.586779977223981*^9}, {3.5992284722048283`*^9, 3.5992284775798388`*^9}}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Authors", FontWeight->"Bold"], ": Juan M. Rodr\[IAcute]guez-D\[IAcute]az and Guillermo S\[AAcute]nchez-Le\ \[OAcute]n " }], "Author", CellChangeTimes->{{3.3992809464453416`*^9, 3.3992809874687996`*^9}, { 3.583235271650755*^9, 3.5832353363135557`*^9}, {3.583241161906254*^9, 3.583241187630952*^9}, 3.586610337781839*^9}], Cell[TextData[{ "Department of Statistics, Faculty of Science, Pl. de los Ca\[IAcute]dos \ s/n, 37008 Salamanca, Spain\n", StyleBox["ENUSA Industrias Avanzadas S.A. Salamanca, Spain", FontFamily->"NimbusRomNo9L-Regu"] }], "Institution", CellChangeTimes->{{3.5832353644019766`*^9, 3.5832354080669155`*^9}, { 3.583235607498228*^9, 3.58323565233321*^9}}], Cell["\<\ Many processes are given by a system of ordinary differential equations, very \ often without an analytical solution. When there are unknown parameters, that \ need to be estimated, optimum experimental design approach offers quality \ estimators for the different objectives of the practitioners. But almost \ every optimality criteria needs to deal with the linearized model for \ computing optimal designs, and this can be a great problem when it is not \ possible to obtain the analytical form of the model. In this work, a \ procedure for findingoptimal designs for models given as solutions to a \ system of ordinary differential equations is described. Some important models \ like the compartmental one, are studied through actual case studies, \ obtaining the corresponding optimal designs.\ \>", "Abstract", CellChangeTimes->{{3.583237028160064*^9, 3.5832370614197173`*^9}, { 3.5866102531893806`*^9, 3.5866102564607077`*^9}}], Cell[CellGroupData[{ Cell[BoxData["$Version"], "Input"], Cell[BoxData["\<\"9.0 for Microsoft Windows (64-bit) (January 25, \ 2013)\"\>"], "Output", CellChangeTimes->{3.3992752647824697`*^9, 3.422852547121887*^9, 3.4422186266093054`*^9, 3.4798162427619514`*^9, 3.479969368478038*^9, 3.5194436028419547`*^9, 3.5398472812103806`*^9, 3.5407338621176844`*^9, 3.5694804164916124`*^9, 3.570540055670437*^9, 3.5832368306624255`*^9, 3.599228482126673*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Overview of compartmental and biokinectic equivalents models\ \>", "Section", CellChangeTimes->{{3.5447940797892375`*^9, 3.544794111753694*^9}, 3.5453251975932136`*^9}], Cell["\<\ Compartmental analysis has applications in clinical medicine, \ pharmacokinetics, internal dosimetry, nuclear medicine, ecosystem studies and \ chemical reaction kinetics. It can be described as the analysis of a system \ in terms of compartments which separate the system into a finite number of \ component parts which are called compartments. Compartments interact through \ the exchange of species. Species may be a chemical substance, hormone, \ individuals in a population and so on. A compartmental system is usually \ represented by a flow diagram or a block diagram. A general introduction to \ this theory can be found in Anderson (1983), Godfrey (1983) and Jazquez \ (1985).\ \>", "Text", CellChangeTimes->{{3.544762612104905*^9, 3.544762616177314*^9}, { 3.545279847977592*^9, 3.5452798753281565`*^9}, 3.5453251975932136`*^9}], Cell[TextData[{ "We adopt the convention of representing compartments with circles or \ rectangles. The flow into or out of the compartments is represented by \ arrows. The ", Cell[BoxData[ FormBox[ SuperscriptBox["i", "th"], TraditionalForm]]], " compartment of a system of ", StyleBox["n ", FontSlant->"Italic"], "compartments is labelled ", Cell[BoxData[ FormBox["i", TraditionalForm]]], " and the size (amount or content) of the component in compartment ", Cell[BoxData[ FormBox["i", TraditionalForm]]], " as ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ SubscriptBox["x", "i"], TraditionalForm], FormBox[ RowBox[{"(", FormBox["t", TraditionalForm], ")"}], TraditionalForm]}], TraditionalForm]]], ". The exchange between compartments, or between a compartment and the \ environment is labeled ", Cell[BoxData[ FormBox[ SubscriptBox["k", "ij"], TraditionalForm]]], ", where ", Cell[BoxData[ FormBox["i", TraditionalForm]]], " represents the flow from ", Cell[BoxData[ FormBox["i", TraditionalForm]]], " to ", Cell[BoxData[ FormBox["j", TraditionalForm]]], ". The environment is usually represented by \"0\" (zero), so ", Cell[BoxData[ FormBox[ SubscriptBox["k", "i0"], TraditionalForm]]], " is the fractional excretion coefficient from the ", Cell[BoxData[ FormBox["i", TraditionalForm]]], "-th compartment to the outside environment. The input from the \ environment into the ", Cell[BoxData[ FormBox["j", TraditionalForm]]], "-th compartment is called ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["b", "j"], "(", "t", ")"}], TraditionalForm]]], ". Environment represents the processes that are outside the system. With \ regards to the environment, we only need to know the flow, ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["b", "j"], "(", "t", ")"}], TraditionalForm]]], ", into the system from the outside. The ", Cell[BoxData[ FormBox[ SubscriptBox["k", "ij"], TraditionalForm]]], " are called fractional transfer rate coefficients and they may be a \ function of different variables or constants. 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As\[IAcute] se indica en Bates, en el parrafo que sigue a la ec \ (5.14). \nPara calcular X(p)(0) lo que hace es d(Xo)/dp, es decir deriva \ X(0)= Xo respecto de a. Lo anterior lo corrobora en el ejemplo de la pag. 179 \ Tetracycline 5.La conclusi\[OAcute]n es que siempre se debe tomar X(p)(0)=0, \ excepto cuando las condiciones iniciales son funcion de p, esto es X(0)= \ Xo(p).\nPara mi tiene sentido fisiol\[OAcute]gico, si p es un factor de \ trasferencia no experimentara ningun efecto sobre las condiciones iniciales y \ por tanto suderivada sera cero. Sin embargo si p es un parametro que afecta a \ las condiciones iniciales su derivada sera distinto de cero. Por ejemplo: en \ el modelo pulmonar la deposicion en el pulmon en t = 0 de pendende del amad \ p. En este caso x(0) si depende del AMAD p y por tanto su valor es distinto \ de cero y su derivada en t=0 tambien lo es. 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In the \ follow we won`t consider compartment 4 that is not relevant in our case. We \ will assume a flow from compartment 1 to outside given by a transfer \ coefficients ", Cell[BoxData[ FormBox[ SubscriptBox["k", "10"], TraditionalForm]]], ". The coefficient transfer values, in ", Cell[BoxData[ FormBox[ SuperscriptBox["days", RowBox[{"-", "1"}]], TraditionalForm]]], ", taken from ICRP 78 are ", Cell[BoxData[ FormBox[ SubscriptBox["k", "10"], TraditionalForm]]], "= 1.9404, ", Cell[BoxData[ FormBox[ SubscriptBox["k", "30"], TraditionalForm]]], "= 0.01155 and ", Cell[BoxData[ FormBox[ SubscriptBox["k", "31"], TraditionalForm]]], "= 0.0462. We will suppose that ", Cell[BoxData[ FormBox[ SubscriptBox["k", "12"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["k", "23"], TraditionalForm]]], " are unknown, although we know that their values will be about ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["k", "12"], "=", "0.8"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["k", "23"], "=", "0.0078"}], TraditionalForm]]], ". We wish estimate them taken experiment data from compartment 1 . The \ problem consist on decide by DOE the best moment to taken the sample. 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this \ interval (about 3 days) is happening a input b1(t) to compartment 1 and \ negligible flow from other compartments reach the compartment 1. In the \ second interval the opposite happens.\ \>", "Text", CellChangeTimes->{{3.5865824498813343`*^9, 3.5865825271041713`*^9}, { 3.5865826787865176`*^9, 3.586582750100932*^9}, {3.5865848856627235`*^9, 3.5865848868015385`*^9}, {3.5866081867683487`*^9, 3.58660819383533*^9}, { 3.586608274255392*^9, 3.586608275940235*^9}, {3.5866099309751625`*^9, 3.586609999276992*^9}, {3.586610381149175*^9, 3.586610523665346*^9}, 3.5866105883134036`*^9}], Cell[TextData[{ "We wish obtain ", Cell[BoxData[ FormBox[ SubscriptBox[ SubscriptBox["x", "i"], RowBox[{"(", "\[Beta]", ")"}]], TraditionalForm]]], " = {\[PartialD]", Cell[BoxData[ FormBox[ SubscriptBox["x", "i"], TraditionalForm]]], "/\[PartialD]", Cell[BoxData[ FormBox[ SubscriptBox["k", "12"], TraditionalForm]]], ", \[PartialD]", Cell[BoxData[ FormBox[ SubscriptBox["x", "i"], TraditionalForm]]], "/\[PartialD]", Cell[BoxData[ FormBox[ SubscriptBox["k", "23"], TraditionalForm]]], "} , ", StyleBox["i", FontSlant->"Italic"], " = {1,2,3} to be used later for computing the Optimal Design." }], "Text", CellChangeTimes->{{3.586171738477047*^9, 3.58617178220424*^9}, { 3.586172027298059*^9, 3.58617222568505*^9}, {3.5861737326796594`*^9, 3.5861737565634775`*^9}, {3.586173788730969*^9, 3.586173796718299*^9}}], Cell[TextData[{ "We are going to apply different methods to obtain ", Cell[BoxData[ FormBox[ SubscriptBox["x", RowBox[{"(", "\[Beta]", ")"}]], TraditionalForm]]], " in order to make a comparison. Each method is computated in a new \ Mathematica session in order to compare the computation time. 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We discart this method\ \>", "Text", CellChangeTimes->{{3.570537848836604*^9, 3.5705378809885235`*^9}, { 3.5861748227100716`*^9, 3.5861748579039335`*^9}, {3.586441210291201*^9, 3.5864412603365355`*^9}, {3.5865237180612264`*^9, 3.5865237217988086`*^9}, { 3.586613603828793*^9, 3.5866136068488836`*^9}, {3.5866144743566704`*^9, 3.58661456792787*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"func", "[", RowBox[{"30", ",", " ", "0.80", ",", " ", "0.0078", ",", " ", "k12"}], "]"}], ",", RowBox[{"func", "[", RowBox[{"30", ",", " ", "0.80", ",", " ", "0.0078", ",", " ", "k23"}], "]"}]}], "}"}], "//", "AbsoluteTiming"}]], "Input", CellChangeTimes->{{3.5455748074717746`*^9, 3.545574865909477*^9}, 3.5705373225552135`*^9, {3.570537691399985*^9, 3.570537699777274*^9}, { 3.5865235894623556`*^9, 3.5865235911939325`*^9}, {3.5865236527478046`*^9, 3.5865236761636467`*^9}, {3.586523816599084*^9, 3.586523825662788*^9}, { 3.586612261416685*^9, 3.586612268764474*^9}, {3.5866125651252728`*^9, 3.5866125887286777`*^9}, {3.5866132122370453`*^9, 3.586613222457352*^9}, { 3.586613546437071*^9, 3.5866135497471704`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"86.93372530000000608652044320479035377502`7.959788203770649", ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "0.0015973937795900459`"}], ",", RowBox[{"-", "0.41548634021496195`"}], ",", "0.01670002391366156`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.035059909631866816`", ",", RowBox[{ "-", "3.97846435172047356773830826287168016435`12.783652321549557"}], ",", "2.08295562826884982858823459622921321473`12.818983758183833"}], "}"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.545574861369869*^9, 3.5455748904015203`*^9}, 3.5705378767920575`*^9, 3.57053998947898*^9, 3.583236879678142*^9, 3.5864439062693024`*^9, 3.586523642607644*^9, 3.586523769724348*^9, 3.586523926652135*^9, 3.586612415299031*^9, 3.58661271908562*^9, 3.5866135569873877`*^9, 3.5866137003616886`*^9, 3.5992292817777214`*^9}] }, Open ]], Cell[TextData[{ "Note that this function is equivalente to the previous function: ", StyleBox["X1[a_,b_,ti_]:={fa[a,b,ti][[1]], fb[a,b,ti][[1]]}", "Input"] }], "Text", CellChangeTimes->{ 3.5705193803855424`*^9, {3.5861748625683575`*^9, 3.586174908308015*^9}, 3.5861749467779493`*^9, {3.5864593597707157`*^9, 3.586459373140071*^9}, { 3.5865200534379187`*^9, 3.5865201112138877`*^9}, 3.5865239560353003`*^9, 3.5866140593443236`*^9, {3.58669262331178*^9, 3.586692628459832*^9}, 3.5866928761565886`*^9, {3.5866929549374013`*^9, 3.586692976918025*^9}}], Cell["\<\ Conclusion: Method 1 and 3 are very fast and they are also the easiest for \ programming. We will compare both methods in a OED \ \>", "Conjecture", CellChangeTimes->{ 3.5705193803855424`*^9, {3.5861748625683575`*^9, 3.586174908308015*^9}, 3.5861749467779493`*^9, {3.5864593597707157`*^9, 3.586459373140071*^9}, { 3.5865200534379187`*^9, 3.5865201112138877`*^9}, 3.5865239560353003`*^9, 3.5866140593443236`*^9, {3.58669262331178*^9, 3.586692628459832*^9}, 3.5866928761565886`*^9, {3.5866929549374013`*^9, 3.586693003469426*^9}, { 3.586693038491833*^9, 3.5866930463855286`*^9}, {3.5866950043149385`*^9, 3.586695024064742*^9}, {3.5866950604287176`*^9, 3.586695142828767*^9}, { 3.5992309290275507`*^9, 3.5992309465432076`*^9}, 3.599232366975646*^9}, Background->RGBColor[0.87, 0.94, 1]], Cell[BoxData[ RowBox[{"Quit", "[", "]"}]], "Input", CellChangeTimes->{{3.586495697678523*^9, 3.58649570484824*^9}}, CellID->15373341] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[" Optimal experiment design", "Subsubsection", CellChangeTimes->{{3.5447648936261168`*^9, 3.5447649346752415`*^9}, { 3.544851717596448*^9, 3.544851722245256*^9}, {3.545063235162685*^9, 3.545063237112688*^9}, 3.545131135848338*^9, {3.586520121119978*^9, 3.586520131369278*^9}}], Cell[TextData[{ "We will suppose that ", Cell[BoxData[ FormBox[ SubscriptBox["k", "12"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["k", "23"], TraditionalForm]]], " are unknown, although we know that their values will be about ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["k", "12"], "=", "0.8"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["k", "23"], "=", "0.0078"}], TraditionalForm]]], ". We wish estimate them taken experiment data from compartment 1 . The \ problem consist on decide by DOE the best moment to taken the sample. We will \ use ", StyleBox["D", FontSlant->"Italic"], "-optimal design." }], "Text", CellChangeTimes->{{3.4803390881888547`*^9, 3.480339089904855*^9}, 3.5693319909815617`*^9, {3.5694774897970414`*^9, 3.5694775111222787`*^9}, { 3.5694775872036123`*^9, 3.5694775939740243`*^9}, 3.569477646046916*^9, { 3.569477703548617*^9, 3.5694777203498464`*^9}, {3.5694779145409875`*^9, 3.5694779344466224`*^9}, {3.5865788478350096`*^9, 3.586579268368416*^9}, { 3.586579835801256*^9, 3.586579910735496*^9}, {3.586580612352733*^9, 3.58658067392659*^9}, {3.586580762644667*^9, 3.586580768557188*^9}, { 3.5865815580073614`*^9, 3.5865816119839334`*^9}, 3.5865849416049004`*^9}], Cell[CellGroupData[{ Cell["Method 1", "Subsubsubsection", CellChangeTimes->{{3.5865243287431936`*^9, 3.5865243310864925`*^9}, { 3.586693439135195*^9, 3.5866934404612093`*^9}}], Cell["\<\ Here we will the optimal design experiment computing the derivatives using \ the method 1 that we have yet described\ \>", "Text", CellChangeTimes->{{3.586518077402085*^9, 3.5865181364643106`*^9}, { 3.5866934963877335`*^9, 3.586693497542195*^9}, {3.586694338437674*^9, 3.5866943419945073`*^9}}], Cell[TextData[{ "We wish find ", StyleBox["t ", FontSlant->"Italic"], ":{", Cell[BoxData[ FormBox[ SubscriptBox["t", "0"], TraditionalForm]]], ",.., ", Cell[BoxData[ FormBox[ SubscriptBox["t", "i"], TraditionalForm]]], ", ... ", Cell[BoxData[ FormBox[ SubscriptBox["t", "n"], TraditionalForm]]], "} of the model given by eq. (4). (or (5)) using ", StyleBox["D", FontSlant->"Italic"], "-optimal design when the analitycal expression of ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "1"], "(", RowBox[{"t", ",", " ", "a", ",", "b"}], ")"}], TraditionalForm]]], " can not be found. [f(t, \[Beta]) =", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "1"], "(", RowBox[{"t", ",", " ", FormBox[ SubscriptBox["k", "12"], TraditionalForm], ",", FormBox[ SubscriptBox["k", "23"], TraditionalForm]}], ")"}], TraditionalForm]]], "]" }], "Text", CellChangeTimes->{{3.5448512851792884`*^9, 3.544851305552924*^9}, { 3.5448513409337864`*^9, 3.5448514828784356`*^9}, {3.5448515275101137`*^9, 3.5448517021212206`*^9}, {3.544851749826104*^9, 3.5448518977455645`*^9}, { 3.5448519286492186`*^9, 3.5448519318628244`*^9}, {3.54485204897223*^9, 3.5448520868646965`*^9}, {3.544959592609887*^9, 3.5449596156043277`*^9}, { 3.586517955890416*^9, 3.586517959962119*^9}, 3.586584952306582*^9, { 3.5866943968757896`*^9, 3.586694402507449*^9}}], Cell[TextData[{ "1.- It is defined a model ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", RowBox[{"t", ",", " ", "\[Beta]"}], ")"}], " "}], TraditionalForm]]], "where the unknown parameters are ", Cell[BoxData[ FormBox[ RowBox[{"\[Beta]", " ", "=", " ", RowBox[{"{", RowBox[{"a", ",", "b"}], "}"}]}], TraditionalForm]]], ". In our case we call ", Cell[BoxData[ FormBox[ RowBox[{"\[Beta]", " ", "=", " ", RowBox[{"{", RowBox[{ SubscriptBox["k", "12"], ",", SubscriptBox["k", "23"]}], "}"}]}], TraditionalForm]]], ". 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Because the sample will be taken in compartment 1 , we extract of fa \ and fb the derivatives corresponding to ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "1"], "(", "t", ")"}], TraditionalForm]]], " " }], "Text", CellChangeTimes->{{3.5398482299416466`*^9, 3.5398482465244756`*^9}, { 3.540126025609722*^9, 3.540126025609722*^9}, {3.540624755880291*^9, 3.5406247594214973`*^9}, {3.5406253882274017`*^9, 3.540625389023003*^9}, { 3.586518397049366*^9, 3.586518441806262*^9}, {3.5865185312575493`*^9, 3.5865185638307433`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"X1", "[", RowBox[{"a_", ",", "b_", ",", " ", "ti_"}], "]"}], ":=", RowBox[{"{", " ", RowBox[{ RowBox[{"fa", "[", RowBox[{"a", ",", "b", ",", " ", "ti"}], "]"}], ",", " ", RowBox[{"fb", "[", RowBox[{"a", ",", "b", ",", " ", "ti"}], "]"}]}], "}"}]}]], "Input", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmViYGAQBWIQHedY9tsh4ZVjqcrXPyD6yhtVRkcg/WGvEheIDtPT 4gfR82xtBUA0QyyXGIj+9GiJJIhu0NRRBdF8RZvB9JuQw4kgukfBOQlEJyWa zgHRGoFT54Lovo4ZS0G00DSFlSCajeX6RhB9r+YFmLZbNXm/SeIrx2UWXUdB dFPQq3sgepP9wUcg+stO3b4Dma8cTXn4JoJotV1vzx8C0mZ5hjdB9K5Hc1IP A2mlE3fB9IR2rcUus4DmcsUuAdGrl5rsBdExeXvA9Ftz9scg+ljwXzCduuSr kweQftO20RVER7zRbtu/A6i+62QXiFZhzVwCogW8c5aD6CVswufeA2n/ihlg GgDQ2qhO "]], Cell[TextData[{ "5.- A typical election for compute the covariance matrix is assumed that \ that the relationship between samples decays exponentially with increasing \ time-distance between them, that is ", Cell[BoxData[ FormBox["\[CapitalGamma]", TraditionalForm]]], " = {", Cell[BoxData[ FormBox[ SubscriptBox["l", "ij"], TraditionalForm]]], "} with ", Cell[BoxData[ FormBox[ SubscriptBox["l", "ij"], TraditionalForm]]], "= exp {\[Rho]|", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["t", "j"], " ", "-"}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["t", "j"], TraditionalForm]]], "|}.", StyleBox["For computational purpose we have found more appropriate to use \ the distance ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["d", "i"], TraditionalForm]]], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], "= ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["t", "i"], "-", SubscriptBox["t", RowBox[{"i", "-", "1"}]]}], TraditionalForm]]], ", ", StyleBox["instead of ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["t", "i"], TraditionalForm]]], ", then ", Cell[BoxData[ FormBox[ SubscriptBox["t", "i"], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["\[Sum]", "i"], SubscriptBox["d", "i"]}], TraditionalForm]]], " being ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["d", "0"], " ", "=", SubscriptBox["t", "0"], " "}], TraditionalForm]]], ". That is for a two points design . We suppose a 3-points design. 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We also assumed k12= 0.80, k23=0.0078\ \>", "Text", CellChangeTimes->{{3.540625487303176*^9, 3.540625491327983*^9}, 3.5436447801117687`*^9, 3.5436448765019035`*^9, {3.543644996322071*^9, 3.5436450167921*^9}, {3.5449611717455993`*^9, 3.5449611776499367`*^9}, { 3.5449612360022745`*^9, 3.5449612482409744`*^9}, {3.586518589992215*^9, 3.5865186073863325`*^9}, {3.5865242025153203`*^9, 3.586524231254264*^9}}], Cell["7.- Then we can obtain the information matrix", "Text", CellChangeTimes->{{3.5406256750183053`*^9, 3.540625677093109*^9}, 3.5436441346508646`*^9}], Cell[TextData[{ "M = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["X", "T"], " ", SuperscriptBox["\[CapitalSigma]", RowBox[{"-", "1"}]], " ", "X", " "}], TraditionalForm]]] }], "Text"], Cell["\<\ m := X . Inverse[\[CapitalSigma]]. 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Because the sample will be taken in compartment 1 , we extract of fa \ and fb the derivatives corresponding to ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "1"], "(", "t", ")"}], TraditionalForm]]], " " }], "Text", CellChangeTimes->{{3.5398482299416466`*^9, 3.5398482465244756`*^9}, { 3.540126025609722*^9, 3.540126025609722*^9}, {3.540624755880291*^9, 3.5406247594214973`*^9}, {3.5406253882274017`*^9, 3.540625389023003*^9}, { 3.586518397049366*^9, 3.586518441806262*^9}, {3.5865185312575493`*^9, 3.5865185638307433`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"X1", "[", RowBox[{"a_", ",", "b_", ",", " ", "ti_"}], "]"}], ":=", RowBox[{"{", " ", RowBox[{ RowBox[{"fa", "[", RowBox[{"a", ",", "b", ",", " ", "ti"}], "]"}], ",", " ", RowBox[{"fb", "[", RowBox[{"a", ",", "b", ",", " ", "ti"}], "]"}]}], "}"}]}]], "Input", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmViYGAQBWIQHedY9tsh4ZVjqcrXPyD6yhtVRkcg/WGvEheIDtPT 4gfR82xtBUA0QyyXGIj+9GiJJIhu0NRRBdF8RZvB9JuQw4kgukfBOQlEJyWa zgHRGoFT54Lovo4ZS0G00DSFlSCajeX6RhB9r+YFmLZbNXm/SeIrx2UWXUdB dFPQq3sgepP9wUcg+stO3b4Dma8cTXn4JoJotV1vzx8C0mZ5hjdB9K5Hc1IP A2mlE3fB9IR2rcUus4DmcsUuAdGrl5rsBdExeXvA9Ftz9scg+ljwXzCduuSr kweQftO20RVER7zRbtu/A6i+62QXiFZhzVwCogW8c5aD6CVswufeA2n/ihlg GgDQ2qhO "]], Cell[TextData[{ "5.- A typical election for compute the covariance matrix is assumed that \ that the relationship between samples decays exponentially with increasing \ time-distance between them, that is ", Cell[BoxData[ FormBox["\[CapitalGamma]", TraditionalForm]]], " = {", Cell[BoxData[ FormBox[ SubscriptBox["l", "ij"], TraditionalForm]]], "} with ", Cell[BoxData[ FormBox[ SubscriptBox["l", "ij"], TraditionalForm]]], "= exp {\[Rho]|", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["t", "j"], " ", "-"}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["t", "j"], TraditionalForm]]], "|}.", StyleBox["For computational purpose we have found more appropriate to use \ the distance ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["d", "i"], TraditionalForm]]], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], "= ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["t", "i"], "-", SubscriptBox["t", RowBox[{"i", "-", "1"}]]}], TraditionalForm]]], ", ", StyleBox["instead of ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["t", "i"], TraditionalForm]]], ", then ", Cell[BoxData[ FormBox[ SubscriptBox["t", "i"], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["\[Sum]", "i"], SubscriptBox["d", "i"]}], TraditionalForm]]], " being ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["d", "0"], " ", "=", SubscriptBox["t", "0"], " "}], TraditionalForm]]], ". That is for a two points design . We suppose a 3-points design. The first \ is defined by the user \n\n ", Cell[BoxData[ FormBox["\[CapitalGamma]", TraditionalForm]]], " where " }], "Text", CellChangeTimes->{{3.540625406183033*^9, 3.5406254109722414`*^9}, { 3.5451304552347426`*^9, 3.5451304744383764`*^9}, {3.5866948349907103`*^9, 3.586694835879989*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"\[CapitalGamma]", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "\[Rho]"}], " ", "d1"}]], ",", SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "\[Rho]"}], " ", RowBox[{"(", RowBox[{"d1", "+", "d2"}], ")"}]}]]}], "}"}], ",", RowBox[{"{", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "\[Rho]"}], " ", "d1"}]], ",", "1", ",", SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "\[Rho]"}], " ", "d2"}]]}], "}"}], ",", RowBox[{"{", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "\[Rho]"}], " ", RowBox[{"(", RowBox[{"d1", "+", "d2"}], ")"}]}]], ",", SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "\[Rho]"}], " ", "d2"}]], ",", "1"}], "}"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{ 3.540625252787964*^9, {3.5406254244350653`*^9, 3.540625431891878*^9}, 3.5451111354347157`*^9, 3.5451304391355143`*^9, 3.5451305206924577`*^9, 3.570973146430005*^9, 3.5866948469092965`*^9}], Cell[TextData[{ "6.- Now it is computed the covariance matrix \[CapitalSigma] = ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Sigma]", "2"], TraditionalForm]]], " ", Cell[BoxData[ FormBox["\[CapitalGamma]", TraditionalForm]]] }], "Text", CellChangeTimes->{3.58658496043428*^9}], Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{"\[CapitalSigma]", " ", "=", " ", RowBox[{ FormBox[ SuperscriptBox["\[Sigma]", "2"], TraditionalForm], "*", "\[CapitalGamma]"}]}], ";"}]}]], "Input"], Cell["We assume ", "Text", CellChangeTimes->{{3.543644917601961*^9, 3.543644926561973*^9}, { 3.5865838294645925`*^9, 3.5865838849387655`*^9}, {3.5866938663143682`*^9, 3.586693873693274*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"\[Rho]", "=", "1"}], ";", RowBox[{"\[Sigma]", " ", "=", "1"}], ";"}], " "}]], "Input", CellChangeTimes->{{3.5401260979494495`*^9, 3.5401261152186794`*^9}, { 3.540625474339553*^9, 3.5406254961483912`*^9}, {3.54062563813984*^9, 3.5406256421646476`*^9}, {3.5406256729591017`*^9, 3.540625686000725*^9}, { 3.544972054071789*^9, 3.544972054539789*^9}, {3.5865838911008244`*^9, 3.58658389197444*^9}, 3.586693878420108*^9}], Cell["\<\ We will also need give the initial values of \[Beta] the standard \ deviation of the measures. We also assumed k12= 0.80, k23=0.0078\ \>", "Text", CellChangeTimes->{{3.540625487303176*^9, 3.540625491327983*^9}, 3.5436447801117687`*^9, 3.5436448765019035`*^9, {3.543644996322071*^9, 3.5436450167921*^9}, {3.5449611717455993`*^9, 3.5449611776499367`*^9}, { 3.5449612360022745`*^9, 3.5449612482409744`*^9}, {3.586518589992215*^9, 3.5865186073863325`*^9}, {3.5865242025153203`*^9, 3.586524231254264*^9}}], Cell["7.- Then we can obtain the information matrix", "Text", CellChangeTimes->{{3.5406256750183053`*^9, 3.540625677093109*^9}, 3.5436441346508646`*^9}], Cell[TextData[{ "M = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["X", "T"], " ", SuperscriptBox["\[CapitalSigma]", RowBox[{"-", "1"}]], " ", "X", " "}], TraditionalForm]]] }], "Text"], Cell["\<\ m := X . Inverse[\[CapitalSigma]]. Transpose[X];\ \>", "Text", CellChangeTimes->{3.5401265957751226`*^9, 3.540127525209155*^9}], Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{"m1", "[", "ti_", "]"}], " ", ":=", RowBox[{ RowBox[{"Transpose", "[", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"X1", "[", RowBox[{"0.80", ",", " ", "0.0078", ",", " ", "#"}], "]"}], "&"}], ",", "ti"}], "]"}], " ", "]"}], ".", " ", RowBox[{"Inverse", "[", "\[CapitalSigma]", "]"}], ".", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"X1", "[", RowBox[{"0.80", ",", " ", "0.0078", ",", " ", "#"}], "]"}], "&"}], ",", "ti"}], "]"}]}]}]}]], "Input", CellChangeTimes->{ 3.5401265957751226`*^9, {3.540127179340948*^9, 3.5401272427706594`*^9}, { 3.5401273167771893`*^9, 3.540127317697591*^9}, {3.540393197485012*^9, 3.5403932003554173`*^9}, {3.5403935426668186`*^9, 3.5403935450848227`*^9}, {3.5403935904809027`*^9, 3.5403937458727756`*^9}, 3.5407341091509185`*^9, 3.5436450896722016`*^9, {3.543645189872342*^9, 3.5436451932623467`*^9}, {3.5436458896933217`*^9, 3.543645893233327*^9}, { 3.5436459248133707`*^9, 3.54364593122338*^9}, {3.543646082853592*^9, 3.5436460897736015`*^9}, {3.545110745902032*^9, 3.545110752906444*^9}, 3.5451118269839306`*^9, {3.545119548714531*^9, 3.545119574844577*^9}, { 3.5864963768445544`*^9, 3.586496382211092*^9}, {3.586501939572788*^9, 3.5865019626060905`*^9}}], Cell["\<\ 8.- Finally the determinant of the information matrix is maximized as \ function of d0, d1 and d2. We constrain the d values to a maximun of t=50 \ becouse to longer time the concentration will be very low (lower than the \ detection limit)\ \>", "Text", CellChangeTimes->{{3.5406256750183053`*^9, 3.540625677093109*^9}, 3.5436441346508646`*^9, {3.5865191232799416`*^9, 3.5865193094676056`*^9}, { 3.5866939867944436`*^9, 3.5866939886508627`*^9}, {3.5866940817058077`*^9, 3.586694264321303*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"obj", "[", RowBox[{ RowBox[{"d0_", "?", "NumericQ"}], ",", RowBox[{"d1_", "?", "NumericQ"}], ",", RowBox[{"d2_", "?", "NumericQ"}]}], "]"}], ":=", RowBox[{"Det", "[", RowBox[{"m1", "[", RowBox[{"{", RowBox[{"d0", ",", RowBox[{"d1", "+", "d0"}], ",", RowBox[{"d0", "+", "d1", "+", "d2"}]}], "}"}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.570890730462919*^9, 3.570890749312152*^9}, { 3.5708909540580053`*^9, 3.570890954633597*^9}, {3.5708911881635303`*^9, 3.5708912130578814`*^9}, 3.578373348839544*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sol1", "=", 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That is for a two points design", " . We suppose a 3-points design. 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