(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 9.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. For additional information concerning CDF *) (* licensing and redistribution see: *) (* *) (* www.wolfram.com/cdf/adopting-cdf/licensing-options.html *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 1063, 20] NotebookDataLength[ 182166, 6175] NotebookOptionsPosition[ 171296, 5835] NotebookOutlinePosition[ 172750, 5892] CellTagsIndexPosition[ 172512, 5880] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Optimality for models given by a system of ordinary differential equation \ \>", "Title"], Cell["Last update: 2013-08-27", "Reference"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Author", FontWeight->"Bold"], ": Juan M. Rodr\[IAcute]guez-D\[IAcute]az and Guillermo S\[AAcute]nchez-Le\ \[OAcute]n " }], "Author"], 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"], 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"], Cell[CellGroupData[{ Cell[BoxData["$Version"], "Input"], Cell[BoxData["\<\"9.0 for Microsoft Windows (64-bit) (January 25, \ 2013)\"\>"], "Output"] }, Open ]] }, Open ]], Cell["", "Section"], Cell[CellGroupData[{ Cell["\<\ Overview of compartmental and biokinectic equivalents models\ \>", "Section"], 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"], 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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(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", "on"], TraditionalForm], ",", FormBox[ SubscriptBox["k", "off"], TraditionalForm]}], ")"}], TraditionalForm]]], "]" }], "Text"], 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]]], ". 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