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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. \ \>", "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. 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The environment is usually represented by \"0\" (zero), so ", Cell[BoxData[ \(TraditionalForm\`k\_i0\)]], " is the fractional excretion coefficient from the ", Cell[BoxData[ \(TraditionalForm\`i\)]], "-th compartment to the outside environment. The input from the \ environment into the ", Cell[BoxData[ \(TraditionalForm\`j\)]], "-th compartment is called ", Cell[BoxData[ \(TraditionalForm\`\(b\_j\)(t)\)]], ". Environment represents the processes that are outside the system. With \ regards to the environment, we only need to know the flow, ", Cell[BoxData[ \(TraditionalForm\`\(b\_j\)(t)\)]], ", into the system from the outside. The ", Cell[BoxData[ \(TraditionalForm\`k\_ij\)]], " are called fractional transfer rate coefficients and they may be a \ function of different variables or constants. " }], "Text"], Cell[TextData[{ "The model of figure 1 consist in two compartments. 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A compartmental model.", "NumberedFigureCaption", CellTags->"fig 2"], Cell[TextData[{ "The compartmental system is represented mathematically in terms of \ quantities such as the concentration or amount of a compound presents in the \ system. In the fig. 2 ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(x\_i\)(t)\)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(x\_j\)(t)\)]], " represent the quantities in compartment ", Cell[BoxData[ \(TraditionalForm\`i\)]], " and ", Cell[BoxData[ \(TraditionalForm\`j\)]], " at time ", Cell[BoxData[ \(TraditionalForm\`t\)]], " since the input", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(b\_i\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\((t)\)\)]], " started. These quantities are called the state variables of the system \ and their evolution in the time are usually differential equations. So the \ system of the figure 2 can be described as it follow:" }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`x\_i' \((t)\) = \(-k\_i0\)\ \(\(x\_i\)(t)\) - k\_ij\ \(\(x\_i\)(t)\) + k\_ji\ \(\(x\_j\)(t)\) + b\ \((t)\)\)], "DisplayFormula"], "\n", Cell[BoxData[ \(TraditionalForm\`x\_j' \((t)\) = k\_ij\ \(\(x\_i\)(t)\) - k\_ji\ \(\(x\_j\)(t)\)\)], "DisplayFormula"] }], "Text"], Cell["or in matrix notation", "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", GridBox[{ {\(x\_i' \((t)\)\)}, {\(x\_j' \((t)\)\)} }], ")"}], " ", "=", RowBox[{ FormBox[ RowBox[{"(", GridBox[{ {\(\(-k\_i0\)\(-\)\(k\_ij\)\(\ \)\), \(k\_ji\)}, {\(k\_ij\), \(-k\_ji\)} }], ")"}], "TraditionalForm"], "+", FormBox[ RowBox[{"(", GridBox[{ {\(b(t)\)}, {"0"} }], ")"}], "TraditionalForm"]}]}], TraditionalForm]], "DisplayFormula"]], "Text"], Cell["\<\ These models are System of Ordinary Differential Equation with Constant \ Coefficients (SODECC). \ \>", "Text"], Cell[TextData[{ "On many ocasions we are interested in models where the inputs ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(b\_i\), "TraditionalForm"], "(", "t", ")"}], TraditionalForm]]], " are acute in ", Cell[BoxData[ \(TraditionalForm\`t\ = \ 0\)]], ", this is ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(b\_i\), "TraditionalForm"], "(", "0", ")"}], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ FormBox[\(b\_i\), "TraditionalForm"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(b\_i\), "TraditionalForm"], "(", "0", ")"}], TraditionalForm]]], " = 0 for ", Cell[BoxData[ \(TraditionalForm\`t \[NotEqual] 0\)]], ". Another frequent situations are the ones where the inputs ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(b\_i\), "TraditionalForm"], "(", "t", ")"}], TraditionalForm]]], " are continuous, this is ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(b\_i\), "TraditionalForm"], "(", "t", ")"}], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ FormBox[\(b\_i\), "TraditionalForm"], TraditionalForm]]], " for ", Cell[BoxData[ \(TraditionalForm\`t\ \[GreaterEqual] 0\)]], " or ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(b\_i\), "TraditionalForm"], "(", "t", ")"}], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ FormBox[\(b\_i\), "TraditionalForm"], TraditionalForm]]], " for ", Cell[BoxData[ \(TraditionalForm\`0 \[LessEqual] t\ < T\)]], ". 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The general multi-input multi-output (MIMO) three-compartment \ model.\ \>", "NumberedFigureCaption", CellTags->"fig 2"], Cell["This model is represented by the following system of ODEs ", "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(\(\(x\& . \_1\)(t)\)\(\ \)\(=\)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(b\_1\)( t)\ - \ \((k\_10\ + \ k\_12\ + \ k\_13)\)\ \(\(x\_1\)( t)\)\ \ + \ k\_21\ \(\(x\_2\)(t)\)\ + \ \(\(k\_31\)\(\ \)\(\(x\_3\)( t)\)\(\ \)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(\(x\& . \_2\)(t)\)\(\ \)\(=\)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(b\_2\)(t)\ + \ k\_12\ \(\(x\_1\)( t)\)\ \ - \ \((k\_20\ + \ k\_21\ + \ k\_23)\)\ \(\(x\_2\)( t)\)\ + \ k\_32\ \(\(x\_3\)(t)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(\(x\& . \_3\)(t)\)\(\ \)\(=\)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`b\_3\ \((t)\)\ + \ k\_13\ \(\(x\_1\)(t)\)\ \ + \ k\_23\ \(\(x\_2\)( t)\)\ - \ \(\((k\_30\ + \ k\_31\ + \ k\_32)\)\(\ \)\(\(x\_3\)( t)\)\(\ \)\)\)]] }], "Text", TextAlignment->Left, TextJustification->0], Cell["in matrix notation ", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ StyleBox[ OverscriptBox[ StyleBox["x", FontWeight->"Bold", FontSlant->"Plain"], "."], FontWeight->"Bold", FontSlant->"Plain"], "(", "t", ")"}], " ", "=", RowBox[{ RowBox[{ StyleBox["A", FontWeight->"Bold", FontSlant->"Plain"], StyleBox[" ", FontWeight->"Bold"], StyleBox["x", FontWeight->"Bold", FontSlant->"Plain"]}], StyleBox[" ", FontWeight->"Bold"], "+", " ", RowBox[{ StyleBox["b", FontWeight->"Bold", FontSlant->"Plain"], "(", "t", ")"}]}]}], ",", " ", \(t \[GreaterEqual] 0\)}], TraditionalForm]], "Text", TextAlignment->Left, TextJustification->0], Cell["with", "Text"], Cell[TextData[{ StyleBox["x", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\((t)\) = \ {\(x\_1\)(t), \ \(x\_2\)(t), \ \ \(x\_3\)(t)}\^T\)]], " \n\n", StyleBox["b", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(\((t)\)\(\ \)\(=\)\(\ \)\({\(b\_1\)(t), \ \(b\_2\)( t), \(b\_3\)(t)}\^T\)\(\ \)\)\)]] }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ "where ", StyleBox["A", FontWeight->"Bold"], " is given by" }], "DisplayFormula", TextAlignment->Left, TextJustification->1], Cell[BoxData[ RowBox[{"A", "=", " ", RowBox[{"(", GridBox[{ {\(a\_11\), \(a\_12\), \(a\_13\)}, {\(a\_21\), \(a\_22\), \(a\_23\)}, {\(a\_31\), \(a\_32\), \(a\_33\)} }], ")"}]}]], "Text", TextAlignment->Left, TextJustification->0], Cell["with", "Text"], Cell[BoxData[{ \(a\_11 = \(-\ \((k\_10\ + \ k\_12\ + \ k\_13)\)\)\ ; a\_12\ = \ k\_21; a\_13 = \ k\_31;\), "\n", \(a\_21 = k\_12; a\_22 = \(-\ \((k\_20\ + \ k\_21\ + \ k\_23)\)\)\ \ ; \ a\_23 = k\_32;\), "\n", \(a\_31 = k\_13; \ a\_32 = \ k\_23; \ a\_33 = \(-\ \((k\_30\ + \ k\_31\ + \ k\_32)\)\);\)}], "Text", TextAlignment->Left, TextJustification->0], Cell["\<\ If we suppose that the substance introduced into the system is a radioactive \ isotope we mush considered the radioactive decay, it is given by a constant \ rate represented by \[Lambda]. This constant is specific for each isotope. \ The decay constant can be interpreted by an equal flow going out of the \ system in each compartment. If we assume a radioactive decay in our \ example:\ \>", "Text"], Cell[BoxData[{ \(a\_11 = \(-\ \((k\_10\ + \ k\_12\ + \ k\_13 + \ \[Lambda])\)\)\ ; a\_12\ = \ k\_21; a\_13 = \ k\_31;\), "\n", \(a\_21 = k\_12; a\_22 = \(-\ \((k\_20\ + \ k\_21\ + \ k\_23 + \ \[Lambda])\)\)\ \ ; \ a\_23 = k\_32;\), "\n", \(a\_31 = k\_13; \ a\_32 = \ k\_23; \ a\_33 = \(-\ \((k\_30\ + \ k\_31\ + \ k\_32 + \ \[Lambda])\)\);\)}], "Text", TextAlignment->Left, TextJustification->0], Cell[TextData[{ "In some circumstances, a physiological model better a compartmental model \ is required, for example, when blood flow or other physiological parameter is \ measured. In those circumstances the transfer rate constants are associated \ with physiologically meaningful values that correspond to the measured \ physiological parameter or may be a function of them. However, if the \ physiological parameters are constants, a compartmental model and a \ physiological model are mathematically equivalent. But the form of the A \ matrix for the compartmental model compared with that for the physiological \ model will be different. For these cases we will directly give the values ", Cell[BoxData[ \(TraditionalForm\`a\_ij\)]], " to the A matrix instead of ", Cell[BoxData[ \(TraditionalForm\`k\_ij\)]], "." }], "Text"], Cell[TextData[{ "The patterns that we have seen can be expanded to systems of ", Cell[BoxData[ \(TraditionalForm\`n\)]], " compartments or ", Cell[BoxData[ \(TraditionalForm\`n\)]], " state variables (in the case of physiological models). 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" }], "Text"], Cell[TextData[{ StyleBox["A", FontWeight->"Bold"], " is a ", Cell[BoxData[ \(TraditionalForm\`n\[Cross]n\)]], " is usually known as the compartmental matrix or system matrix. " }], "Text"], Cell[TextData[{ StyleBox["b", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(\((t)\)\(\ \)\(=\)\(\ \)\({\(b\_1\)(t), \ \(b\_2\)( t), ... , \(b\_n\)(t)}\^T\)\(\ \)\)\)]], "is a column vector where {", Cell[BoxData[ \(TraditionalForm\`\(b\_i\)(t)\)]], "} is the input rate into compartment ", Cell[BoxData[ \(TraditionalForm\`\(\(i\)\(\ \)\)\)]], "from outside system. " }], "Text"], Cell[TextData[{ StyleBox["x", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(\((0)\)\(\ \)\)\)]], " = ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ FormBox[\(x\_1\), "TraditionalForm"], "(", "0", ")"}], ",", " ", RowBox[{ FormBox[\(x\_2\), "TraditionalForm"], "(", "0", ")"}], ",", " ", "...", ",", " ", RowBox[{ FormBox[\(x\_n\), "TraditionalForm"], "(", "0", ")"}]}], "}"}], FormBox[\(\^T\), "TraditionalForm"]}]}], TraditionalForm]]], "are the initial conditions, so ", Cell[BoxData[ \(TraditionalForm\`\(\(\(x\_i\)(0)\)\(,\)\)\)]], " represents the amount or content of species in compartment ", Cell[BoxData[ \(TraditionalForm\`i\)]], " at time ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(t\)\)\)]], " = 0. " }], "Text"], Cell["\<\ SysModel.m provides a few functions to construct and solve Compartmental and \ Physiological models. Most of the examples are made using clearance rates, \ but also has been included some example of non linear models. 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\(TraditionalForm\`x\_0\)]], " the number present at ", Cell[BoxData[ \(TraditionalForm\`t = 0\)]], ". The number of atoms decaying per unit of time is directly proportional \ present at time ", Cell[BoxData[ \(TraditionalForm\`t\)]], ", being ", Cell[BoxData[ \(TraditionalForm\`k\)]], " the constant of proportionality. Then ", Cell[BoxData[ \(TraditionalForm\`x(t)\)]], " is given by the first order linear differential equation" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\[DifferentialD]x\/\[DifferentialD]t = \ \(-\ k\)\ x\ \((t)\)\)], "NumberedEquation"], Cell[TextData[{ "Eq.(2) can be evaluated using ", StyleBox["DSolve ", "DisplayMath", FontFamily->"Courier New"], "which gives the solution in analytical form. Initial conditions x", StyleBox["[0]", "MR"], " ", StyleBox["==", "MR"], " ", StyleBox["x0 ", "MR"], "can have been added." }], "Caption"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eq1", " ", "=", " ", RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "\[Equal]", \(\(-k\)\ x[t]\)}], ",", \(x[0] \[Equal] x0\)}], "}"}], ",", \(x[t]\), ",", "t"}], "]"}]}]], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \({{x[t] \[Rule] \[ExponentialE]\^\(\(-k\)\ t\)\ x0}}\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell[TextData[{ "If we know the values of ", Cell[BoxData[ \(TraditionalForm\`k\)]], " and ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], ", for instant", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(k\ = \ 0.1\)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\(x\)\(\ \)\)\_0 = \ 100\)]], " with apropiated units, we can replace them in the previous expression to \ obtain ", Cell[BoxData[ \(TraditionalForm\`x(t)\)]] }], "Text"], Cell[TextData[{ StyleBox["%", "MR"], " gives the last result generated in the before output and ", StyleBox["expr", FontSlant->"Italic"], " /. ", StyleBox["rules", "TI"], " gives a rule of replacement to be applied on expression ", StyleBox["expr", "TI"], ". " }], "Caption"], Cell[CellGroupData[{ Cell[BoxData[ \(% /. {k \[Rule] 0.1, \ x0 \[Rule] \ 100}\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \({{x[t] \[Rule] 100\ \[ExponentialE]\^\(\(-0.1`\)\ t\)}}\)], "Output", CellLabel->"Out[8]="], Cell[BoxData[ \({{x[t] \[Rule] 100\ \[ExponentialE]\^\(\(-0.1`\)\ t\)}}\)], "Output", CellLabel->"Out[5]=", GeneratedCell->False, CellAutoOverwrite->False] }, Open ]], Cell[TextData[{ "In many occasions it is better build a ", StyleBox["Mathematica", FontSlant->"Italic"], " function where ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`k\)]], " are parameters. They can be replaced with their values when ever it is \ necessary." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x[\ t_, \ x0_, \ k_] = x[t] /. eq1[\([1]\)]\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ \(\[ExponentialE]\^\(\(-k\)\ t\)\ x0\)], "Output", CellLabel->"Out[9]="], Cell[BoxData[ \(\[ExponentialE]\^\(\(-k\)\ t\)\ x0\)], "Output", CellLabel->"Out[6]=", GeneratedCell->False, CellAutoOverwrite->False] }, Open ]], Cell[TextData[{ "Now we can replace k and x", Cell[BoxData[ \(TraditionalForm\`\_0\)]], " with their values or ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(k\ = \ 0.1\)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\(x\)\(\ \)\)\_0 = \ 100\)]], " " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x[t, 0.1, 100]\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \(0.1`\ \[ExponentialE]\^\(\(-100\)\ t\)\)], "Output", CellLabel->"Out[10]="], Cell[BoxData[ \(0.1`\ \[ExponentialE]\^\(\(-100\)\ t\)\)], "Output", CellLabel->"Out[7]=", GeneratedCell->False, CellAutoOverwrite->False] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Examples of exponential decay", "SubsubsectionIcon"], Cell[TextData[{ "Example 1.- Cobalt 60 is a radioactive isotope whose half-life is 5.3 \ year. It is used in medicine as a radiation source to kill cancer cells using \ gamma-emissions from cobalt 60. As it is radioactive its activity decreases \ with the time and the source must be changed periodically according to the \ expression above obtained ", Cell[BoxData[ \(TraditionalForm\`x(t)\ = \ x\_0\ \[ExponentialE]\^\(\(-k\)\ t\)\)]], ", where", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(k\)\)\)]], " is called the disintegration constant or transformation constant." }], "Text"], Cell[TextData[{ "We are interested in evaluate the disintegration constant of ", Cell[BoxData[ \(TraditionalForm\`\^60\)]], "Co. A half-life (", Cell[BoxData[ \(TraditionalForm\`T\)]], ") of 5.3 year means that the radioactivity decrease to the half every 5.3 \ year, this is: if the activity is ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " in ", Cell[BoxData[ \(TraditionalForm\`\(\(t\)\(\ \)\(=\)\(\ \)\(0\)\(\ \)\)\)]], "after a time ", Cell[BoxData[ \(TraditionalForm\`T\)]], " the activity will be ", Cell[BoxData[ \(TraditionalForm\`x\_0/2\)]], ". Therefore we can calculate ", Cell[BoxData[ \(TraditionalForm\`k\)]], " doing ", Cell[BoxData[ \(TraditionalForm\`x(t)\ = \(x\_0/2\ \ and\ t = \ T\)\)]], " in expression ", Cell[BoxData[ \(TraditionalForm\`x(t)\ = \ x\_0\ \[ExponentialE]\^\(\(-k\)\ t\)\)]], ". " }], "Text"], Cell[TextData[{ "We use the function", StyleBox[" x[t, x0, k] ", "DisplayMath"], "before defined" }], "Caption"], Cell[CellGroupData[{ Cell[BoxData[ \(eq = \ Solve[\ x[\ T, \ x0, \ k]\ \[Equal] \ x0/2\ , \ k]\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(Solve::"ifun" \(\(:\)\(\ \)\) "Inverse functions are being used by \!\(Solve\), so some solutions may \ not be found."\)], "Message", CellLabel->"From In[11]:="], Cell[BoxData[ \({{k \[Rule] Log[2]\/T}}\)], "Output", CellLabel->"Out[11]="], Cell[BoxData[ \(Solve::"ifun" \(\(:\)\(\ \)\) "Inverse functions are being used by \!\(Solve\), so some solutions may \ not be found."\)], "Message", CellLabel->"From In[8]:=", GeneratedCell->False, CellAutoOverwrite->False] }, Open ]], Cell[TextData[{ " ", StyleBox["Mathematica", FontSlant->"Italic"], " gives results in terms of inverse functions. In these cases it prints a \ warning message. We can suppress these warning messages" }], "Caption"], Cell[BoxData[ \(Off[Solve::"\"]; Off[InverseFunction::"\"];\)], "Input", CellLabel->"In[12]:=", CellTags->"Solve"], Cell[TextData[{ "This solution is applicable not only to the cobalt 60. It is a general \ relationship between ", Cell[BoxData[ \(TraditionalForm\`k\)]], " and ", Cell[BoxData[ \(TraditionalForm\`T\)]], "." }], "Text"], Cell[TextData[{ "We replace ", Cell[BoxData[ \(TraditionalForm\`k\)]], " with its solution. We call it ", StyleBox["TransConst ", "DisplayMath"], "in order to be used later. 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\(TraditionalForm\`x\ = \ x\_0\ \[ExponentialE]\^\(\(-k\)\ t\)\)]], " where ", Cell[BoxData[ FormBox[ RowBox[{ FormBox["x", "TraditionalForm"], " ", "=", " ", "5"}], TraditionalForm]]], " MBq and ", Cell[BoxData[ \(TraditionalForm\`\(\(t\)\(\ \)\(=\)\(\ \)\(3\)\(\ \)\)\)]], "days, ", Cell[BoxData[ \(TraditionalForm\`k\)]], " is de transformation constant of I-131 and ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " is the variable to be calculated." }], "Text"], Cell["The transformation constant of I-131 is ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(kI131\ = TransConst /. T \[Rule] 8.07\)], "Input", CellLabel->"In[16]:="], Cell[BoxData[ \(0.08589184393555703`\)], "Output", CellLabel->"Out[16]="] }, Open ]], Cell[TextData[{ "We replace", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(t\)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`k\)]], " and ", Cell[BoxData[ \(TraditionalForm\`x\)]], " with their values. " }], "Caption"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(Solve[\ x[\ 3\ , \ x0, kI131\ ]\ \[Equal] \ \ 5\ , \ x0]\)\)\)], "Input", CellLabel->"In[17]:="], Cell[BoxData[ \({{x0 \[Rule] 6.469594574862883`}}\)], "Output", CellLabel->"Out[17]="] }, Open ]], Cell["Therefore should be shipped 6.47 MBq ", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Example of irradiation of a cell population", "SubsubsectionIcon"], Cell[TextData[{ "In a experiment where a cell population is irradiated the survival of \ cells is given by eq.(1) with ", Cell[BoxData[ \(TraditionalForm\`k\)]], " = 1 ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["days", FontSlant->"Plain"], \(-1\)], TraditionalForm]]], ", and we are interested in calculating ", Cell[BoxData[ \(TraditionalForm\`t\)]], " for a specific value ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], " = 20% of initial population ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], ". 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Hence " }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(V\ \ c' \((t)\) = \ \[Nu][ b(t) - c(t)]\)\)\)], "NumberedEquation"], Cell[TextData[{ "Let's consider the case for which the flowing concentration ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(b(t)\), "TraditionalForm"], " ", "=", " ", "b"}], TraditionalForm]]], " constant for and interval ", Cell[BoxData[ FormBox[ RowBox[{"0", "<", FormBox[\(t < T\), "TraditionalForm"]}], TraditionalForm]]], " , and ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(b(t)\), "TraditionalForm"], " ", "=", "0"}], TraditionalForm]]], " since ", Cell[BoxData[ \(TraditionalForm\`t > T\)]], ", " }], "Text"], Cell[BoxData[ \(\(eq2\ = \ V\ \ \(c'\)[t] == \ \[Nu] \((b - c[t])\);\)\)], "Input", CellLabel->"In[19]:="], Cell[TextData[{ "Let ", Cell[BoxData[ \(TraditionalForm\`c(0)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`c\_0\)]], " be the concentration drug in the organ at ", Cell[BoxData[ FormBox[ RowBox[{ FormBox["t", "TraditionalForm"], " ", "=", " ", "0"}], TraditionalForm]]], ". 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Let", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ FormBox[\(x\_1\), "TraditionalForm"], "(", "t", ")"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(x\_2\), "TraditionalForm"], "(", "t", ")"}], TraditionalForm]]], ", where ", Cell[BoxData[ \(TraditionalForm\`t \[GreaterEqual] 0\)]], ", is the mass of the drug in compartment 1 and 2, respectively. Then we \ can describe the process as follow" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[DifferentialD]x\_1\/\[DifferentialD]t = \ b(t)\ - \ drug\ distribution\ rate\ from\ 1\ to\ 2\)\)\)], "NumberedEquation"], Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[DifferentialD]x\_2\/\[DifferentialD]t = \ inflow\ rate\ - \ outflow\ rate\ \((elimination)\)\)\)\)], "NumberedEquation"], Cell[TextData[{ "The previous equations are commonly called ", StyleBox["mass balance equation", FontWeight->"Bold"], ". If it is assumed that rate of transference from each compartment ", Cell[BoxData[ \(TraditionalForm\`i\)]], " is proportional to the mass (or concentration) in this compartment, then \ (3) and (4) become" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\[DifferentialD]x\_1\/\[DifferentialD]t = \ b(t)\ - k\_12\ x\_1\)], "NumberedEquation"], Cell[BoxData[ \(TraditionalForm\`\[DifferentialD]x\_2\/\[DifferentialD]t = \ k\_12\ x\_1\ - \ \(\(k\_20\ x\_1\)\(\[IndentingNewLine]\) \)\)], "NumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`k\_12\)]], " and ", Cell[BoxData[ \(TraditionalForm\`k\_20\)]], " are the constants (>0) of proportionality from 1 to 2 and from 2 to \ environment (elimination). This process is a simple case of first-order \ kinetics. \nBoth ordinary differential equations (ODE) with appropriate \ initial conditions ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], "(0) and ", Cell[BoxData[ \(TraditionalForm\`\(\(x\)\(\ \)\)\_2\)]], "(0) constitute the compartmental metabolic model. In matrix-vector format \ the system of ordinary differential equation (SODE) model is " }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", GridBox[{ {\(x\_1' \((t)\)\)}, {\(x\_2' \((t)\)\)} }], ")"}], " ", "=", RowBox[{ FormBox[ RowBox[{"(", GridBox[{ {\(\(-k\_12\)\(\ \)\), "0"}, {\(k\_12\), \(k\_20\)} }], ")"}], "TraditionalForm"], "+", FormBox[ RowBox[{"(", GridBox[{ {\(b(t)\)}, {"0"} }], ")"}], "TraditionalForm"], "\n"}]}], TraditionalForm]], "NumberedEquation"], Cell[TextData[{ "This system can be solved with ", StyleBox["DSolve", "DisplayMath"], " or using SystemDSolve. Here is assumed k12= 0.3, and k20 = 0.05 and b(t) \ = {0, 0}, an as initial condition is taken {1,0}. 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\(TraditionalForm\`b\_1\)]], " for ", Cell[BoxData[ \(TraditionalForm\`0 < t < T\)]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(b\_1\), "TraditionalForm"], "(", "t", ")"}], " ", "=", " ", "0"}], TraditionalForm]]], " for ", Cell[BoxData[ \(TraditionalForm\`t\ \[GreaterEqual] T\)]] }], "NumberedEquation"], Cell[TextData[{ "Coefficients: ", Cell[BoxData[ \(TraditionalForm\`k\_12\)]], "=2 ", Cell[BoxData[ \(TraditionalForm\`h\^\(-1\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`k\_21\)]], "=1 ", Cell[BoxData[ \(TraditionalForm\`h\^\(-1\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`k\_10\)]], " =0,5 ", Cell[BoxData[ \(TraditionalForm\`h\^\(-1\)\)]], ", \nSingle input: ", Cell[BoxData[ \(TraditionalForm\`b\_0\)]], " = 100 mg\nConstant input: ", Cell[BoxData[ \(TraditionalForm\`b\_1\)]], "=10 mg/h until ", Cell[BoxData[ \(TraditionalForm\`T = 10\)]], " h)" }], "Text"], Cell["The first step is build the compartmental matrix", "Text"], Cell[BoxData[ \(\(model2\ = CompartMatrix[ 2, {{1, 2, 2}, \ {2, 1, 1}, \ {1, 0, 0.5}}];\)\)], "Input", CellLabel->"In[42]:="], Cell[TextData[{ "Here is the retention for ", " ", Cell[BoxData[ \(TraditionalForm\`0 < t < T\)]], " " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ShowODE[model2, \ {0, 0}, {10, 0}, \ t, \ x1] // TableForm\)], "Input", CellLabel->"In[43]:="], Cell[BoxData[ InterpretationBox[GridBox[{ { RowBox[{ RowBox[{ SubsuperscriptBox["x1", "1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(10 - 2.5`\ x1\_1[t] + x1\_2[t]\)}]}, { RowBox[{ RowBox[{ SubsuperscriptBox["x1", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(2\ x1\_1[t] - x1\_2[t]\)}]}, {\(x1\_1[0] == 0\)}, {\(x1\_2[0] == 0\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { Equal[ Derivative[ 1][ Subscript[ x1, 1]][ t], Plus[ 10, Times[ -2.5, Subscript[ x1, 1][ t]], Subscript[ x1, 2][ t]]], Equal[ Derivative[ 1][ Subscript[ x1, 2]][ t], Plus[ 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" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ShowODE[ model2, \ {\*"\"\<\!\(x1\_1\)(T)\>\"", \*"\"\<\!\(x1\_2\)(T)\>\""}, \ {0, 0}, \ t, \ \ x2] // TableForm\)], "Input", CellLabel->"In[45]:="], Cell[BoxData[ InterpretationBox[GridBox[{ { RowBox[{ RowBox[{ SubsuperscriptBox["x2", "1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(\(-2.5`\)\ x2\_1[t] + x2\_2[t]\)}]}, { RowBox[{ RowBox[{ SubsuperscriptBox["x2", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", \(2\ x2\_1[t] - x2\_2[t]\)}]}, {\(x2\_1[0] == "\!\(x1\_1\)(T)"\)}, {\(x2\_2[0] == "\!\(x1\_2\)(T)"\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { Equal[ Derivative[ 1][ Subscript[ x2, 1]][ t], Plus[ Times[ -2.5, Subscript[ x2, 1][ t]], Subscript[ x2, 2][ t]]], Equal[ Derivative[ 1][ Subscript[ x2, 2]][ t], Plus[ Times[ 2, Subscript[ x2, 1][ t]], Times[ -1, Subscript[ x2, 2][ t]]]], Equal[ Subscript[ x2, 1][ 0], "\!\(x1\_1\)(T)"], Equal[ Subscript[ x2, 2][ 0], "\!\(x1\_2\)(T)"]}]]], "Output", CellLabel->"Out[45]//TableForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sol2[t_] = AcuteInput[model2, \ Map[Last, sol1[10]], \ t, \ x2]\)], "Input", CellLabel->"In[46]:="], Cell[BoxData[ \({x2\_1[t] \[Rule] 21.971126807662458`\ \((0.734260642832909`\ \ \[ExponentialE]\^\(\(-3.350781059358212`\)\ t\) + 0.26573935716709096`\ \ \[ExponentialE]\^\(\(-0.1492189406417878`\)\ t\))\) + 44.63368756504546`\ \((\(-0.3123475237772122`\)\ \ \[ExponentialE]\^\(\(-3.350781059358212`\)\ t\) + 0.3123475237772122`\ \ \[ExponentialE]\^\(\(-0.1492189406417878`\)\ t\))\), x2\_2[t] \[Rule] 21.971126807662458`\ \((\(-0.6246950475544242`\)\ \[ExponentialE]\^\(\ \(-3.350781059358212`\)\ t\) + 0.6246950475544242`\ \ \[ExponentialE]\^\(\(-0.1492189406417878`\)\ t\))\) + 44.63368756504546`\ \((0.26573935716709096`\ \ \[ExponentialE]\^\(\(-3.350781059358212`\)\ t\) + 0.734260642832909`\ \[ExponentialE]\^\(\(-0.1492189406417878`\ \)\ t\))\)}\)], "Output", CellLabel->"Out[46]="] }, Open ]], Cell[TextData[{ "We build a compound function that represents the retention divided in two \ intervals: one of them when ", Cell[BoxData[ \(TraditionalForm\`0 < t < T\)]], " and the other one ", Cell[BoxData[ \(TraditionalForm\`t\ \[GreaterEqual] T\)]], ". 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07oo00000goo00@007ooOol000Moo`05001oogooOol00004Ool00`00Oomoo`09Ool00`00Oomoo`02 Ool00`00Oomoo`06Ool00`00Oomoo`3oOolUOol000eoo`<0009oo`D000Moo`05001oogooOol00004 Ool00`00Oomoo`09Ool00`00Oomoo`02Ool00`00Oomoo`06Ool00`00Oomoo`3oOolUOol000eoo`04 001oogoo000=Ool01000Oomoogoo0P000goo00<007ooOol02Goo00<007ooOol00Woo00<007ooOol0 1Woo00<007ooOol0ogoo9Goo000=Ool00`00Ool00003000;Ool01P00Oomoogoo001oo`<000aoo`04 001oogooOol50005Ool00`00Oomoo`3oOolUOol000eoo`03001oogoo00ioo`05001oogooOol0000A Ool01@00Oomoogoo00001goo00<007ooOol0ogoo9Woo000"], ImageRangeCache->{{{127.625, 414.625}, {408.875, 301.875}} -> {-4.48178, \ 7.31202, 0.0294395, 0.0294395}}], Cell["We can represent the above model as ", "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`b\_a' \((t)\)\ = \ \(-\ k\_F\)\)]], " \n", Cell[BoxData[ \(TraditionalForm\`x\_a' \((t)\)\ = k\_F\ - \ k\_a\ \(\(x\_a\)(t)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`x\_1' \((t)\)\ = \ k\_a\ \(\(x\_a\)(t)\)\ - \ \((k\_10 + \ k\_12)\)\ \(\(x\_1\)( t)\)\ \ + \ \(\(k\_21\)\(\ \)\(\(x\_21\)(t)\)\(\ \)\)\)]], " \n", Cell[BoxData[ \(TraditionalForm\`x\_2' \((t)\)\ = \ k\_21\ \(\(x\_1\)(t)\)\ \ - \ \ \(\(k\_12\)\(\ \)\(\(x\_2\)( t)\)\(\ \)\)\)]] }], "NumberedEquation"], Cell[TextData[{ "Zero-order mean that the input is given by a constant rate ", Cell[BoxData[ \(TraditionalForm\`\(\(b\_a\)(t)\)'\)]], " = -", Cell[BoxData[ \(TraditionalForm\`k\_F\)]], ", and therefore a constant input ", Cell[BoxData[ \(TraditionalForm\`k\_F\)]], " to compartment a. The total release amount is called ", Cell[BoxData[ \(TraditionalForm\`d\)]], " that is equivalent to taken as initial condition ", Cell[BoxData[ \(TraditionalForm\`\(b\_a\)(0)\ = \ d\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(b\_a[ t_]\ = \(\(b\_a[ t]\)\(/.\)\(\(DSolve[{\(b\_a'\)[t] + k\_F\ == 0, b\_a[0] \[Equal] d}, b\_a[t], \ t]\)[\([1]\)]\)\(\ \)\)\)], "Input", CellLabel->"In[62]:="], Cell[BoxData[ \(d - t\ k\_F\)], "Output", CellLabel->"Out[62]="], Cell[BoxData[ \(d - t\ k\_F\)], "Output", CellLabel->"Out[58]=", GeneratedCell->False, CellAutoOverwrite->False] }, Open ]], Cell[TextData[{ "We suppose that ", Cell[BoxData[ \(TraditionalForm\`d = 100\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\(k\_F\)\(=\)\(0.1\)\(\ \)\)\)]], " and plot ", Cell[BoxData[ \(b\_a[t]\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Plot[ Evaluate[b\_a[t] /. {d \[Rule] \ 100, k\_F -> 0.1}\ ], \ {t, \ 0, 1200}]\)\(\ \)\)\)], "Input", CellLabel->"In[63]:="], Cell[GraphicsData["PostScript", "\<\ %! 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This means the all ", Cell[BoxData[ \(TraditionalForm\`d\)]], " amount has been already transferred to ", Cell[BoxData[ \(TraditionalForm\`a\)]], " when ", Cell[BoxData[ \(TraditionalForm\`\(b\_a\)(t)\)]], " = 0. We solve this equation to find ", Cell[BoxData[ \(TraditionalForm\`t\)]], " that we will call ", Cell[BoxData[ \(TraditionalForm\`T\)]], ". " }], "Text"], Cell[TextData[{ "We used Solve to find ", Cell[BoxData[ \(TraditionalForm\`T\)]], ". " }], "Caption"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[d - T\ k\_F == \ 0, \ T]\)], "Input", CellLabel->"In[64]:="], Cell[BoxData[ \({{T \[Rule] d\/k\_F}}\)], "Output", CellLabel->"Out[64]="] }, Open ]], Cell["Therefore we should be consider two period: ", "Text"], Cell[TextData[{ "First period: ", Cell[BoxData[ \(TraditionalForm\`\(\(0\)\(<\)\(t\)\(\[LessEqual]\)\(T\)\(\ \)\)\)]], "with ", Cell[BoxData[ \(TraditionalForm\`T\ = \ d/k\_F\)]] }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`x\_a' \((t)\)\ = \ \ k\_F - \ k\_a\ \(\(x\_a\)(t)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`x\_1' \((t)\)\ = \ k\_a\ \(\(x\_a\)(t)\)\ - \ \((k\_10 + \ k\_12)\)\ \(\(x\_1\)( t)\)\ \ + \ \(\(k\_21\)\(\ \)\(\(x\_2\)(t)\)\(\ \)\)\)]], " \n", Cell[BoxData[ \(TraditionalForm\`x\_2' \((t)\)\ = \ k\_12\ \(\(x\_1\)(t)\)\ \ - \ \ \(\(k\_21\)\(\ \)\(\(x\_2\)( t)\)\(\ \)\)\)]] }], "NumberedEquation"], Cell[TextData[{ "\tInitial condition: ", Cell[BoxData[ \(TraditionalForm\`\(x\_a\)(0)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(x\_1\)(0)\)]], " =", Cell[BoxData[ \(TraditionalForm\`\(x\_2\)(0)\)]], " = 0" }], "Text"], Cell[TextData[{ "Second period: ", Cell[BoxData[ \(TraditionalForm\`t > T\)]], ". We call ", Cell[BoxData[ \(TraditionalForm\`t'\)]], " = ", Cell[BoxData[ \(TraditionalForm\`t - T\)]], " " }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`x\_a' \((t')\)\ = \(-\ k\_a\)\ \(\(x\_a\)( t')\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`x\_1' \((t')\)\ = \ k\_a\ \(\(x\_a\)(t')\)\ - \ \((k\_10 + \ k\_12)\)\ \(\(x\_1\)( t')\)\ \ + \ \(\(k\_21\)\(\ \)\(\(x\_2\)(t')\)\(\ \)\)\)]], " \n", Cell[BoxData[ \(TraditionalForm\`x\_2' \((t')\)\ = \ k\_12\ \(\(x\_1\)(t')\)\ \ - \ \ \(\(k\_21\)\(\ \)\(\(x\_2\)( t')\)\(\ \)\)\)]] }], "NumberedEquation"], Cell[TextData[{ "Initial condition: ", Cell[BoxData[ \(TraditionalForm\`\(x\_a\)(0)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(x\_a\)(T)\)]], "; ", Cell[BoxData[ \(TraditionalForm\`\(x\_1\)(0)\)]], " =", Cell[BoxData[ \(TraditionalForm\`\(x\_1\)(T)\)]], "; ", Cell[BoxData[ \(TraditionalForm\`\(x\_2\)(0)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(x\_2\)(T)\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`\(x\_a\)(T)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(x\_1\)(T)\)]], ", y ", Cell[BoxData[ \(TraditionalForm\`\(x\_2\)(T)\)]], " is given by eq. 13 when ", Cell[BoxData[ \(TraditionalForm\`t = T\)]], ". " }], "Text"], Cell[TextData[{ "Now, we give the values of transfer coefficients. In our example they are \ ", Cell[BoxData[ \(TraditionalForm\`k\_F\)]], "=0.1, ", Cell[BoxData[ \(TraditionalForm\`k\_a\)]], "= 2, ", Cell[BoxData[ \(TraditionalForm\`k\_10\)]], "=0.5, ", Cell[BoxData[ \(TraditionalForm\`k\_12\)]], "= 2, ", Cell[BoxData[ \(TraditionalForm\`k\_21\)]], "= 1" }], "Text"], Cell[TextData[{ "We use k with /: (TagSet) to make easier clear all coefficient values \ later. We will can delete its writing ", StyleBox["Clear[k]", "DisplayMath"], " in place of write every ", Cell[BoxData[ \(TraditionalForm\`k\_i\)]], " . Also we write ", Cell[BoxData[ \(k\_F = 1/10\)]], " and ", Cell[BoxData[ \(k\_10 = 1/2\)]], " using rational coefficients better than ", Cell[BoxData[ \(k\_F = 0.1\)]], " and ", Cell[BoxData[ \(k\_10 = 0.5\)]], " because with rational coefficients the computation is made using infinite \ precision. " }], "Caption"], Cell[BoxData[ \(\(\(\(k /: k\_F = 1/10\) \)\(;\)\(\(k /: k\_a = 2\) \)\(;\)\(\(\(k /: k\_10 = 1/2\)\(\ \)\) \)\(;\)\(\(k /: k\_21 = 1\) \)\(;\)\(\ \)\(\(k /: k\_12 = 2\) \)\(;\)\(\ \)\)\)], "Input", CellLabel->"In[65]:="], Cell["Here is the solution to eq13 ", "MathCaption"], Cell[BoxData[ \(\(eq1\ = \ {\(x\_a'\)[t] == \ \ k\_F - \ k\_a\ x\_a[t], \(x1'\)[ t] \[Equal] k\_a\ x\_a[t] - \ \((k\_10 + k\_12)\)\ x1[t] + k\_21\ x2[t], \(x2'\)[t] == k\_12\ x1[t] - k\_21\ x2[t]};\)\)], "Input", CellLabel->"In[66]:="], Cell[CellGroupData[{ Cell[BoxData[ \({x1a[t_], x11[t_], x12[t_]} = {x\_a[t], x1[t], x2[t]} /. \(DSolve[ Join[eq1, {x\_a[0] \[Equal] 0, x1[0] \[Equal] 0, x2[0] \[Equal] 0}], {x\_a[t], x1[t], x2[t]}\ , \ t]\)[\([1]\)] // FullSimplify\)], "Input", CellLabel->"In[67]:="], Cell[BoxData[ \({1\/10\ \[ExponentialE]\^\(-t\)\ Sinh[ t], \(-\(\(\[ExponentialE]\^\(\(-\(1\/4\)\)\ \((9 + \@41)\)\ t\)\ \ \((\((82 - 18\ \@41)\)\ \[ExponentialE]\^\(t/2\) + 41\ 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We have to bear in mind that ", Cell[BoxData[ \(xa[0] = \ x\_a\)]], ", ", Cell[BoxData[ \(x1[0] = \ x\_1\)]], " and ", Cell[BoxData[ \(x2[0] = \ x\_2\)]], " correspond to the retention in the compartment a, 1 and 2 estimated with \ eq13 for ", Cell[BoxData[ \(TraditionalForm\`t\ = \ T\)]], ", that is ", Cell[BoxData[ \(\(\(\ \)\(x\_a\)\)\)]], " = ", StyleBox["x1a[T]", "DisplayMath"], ", ", Cell[BoxData[ \(\(\(\ \)\(x\_1\)\)\)]], " = ", StyleBox["x11[T]", "DisplayMath"], "and", Cell[BoxData[ \(\(\(\ \)\(x\_2\)\)\)]], " = ", StyleBox["x12[T]", "DisplayMath"], ". Also, we know that ", Cell[BoxData[ \(T = d\/k\_F\)]], "for this reason we make the replacement ", Cell[BoxData[ \(T \[Rule] d\/k\_F\)]] }], "Caption"], Cell[BoxData[ \(\(eq2 = DSolve[{\(x\_a'\)[t] == \ \(-\ k\_a\)\ x\_a[t], \(x1'\)[t] \[Equal] k\_a\ x\_a[t] - \ \((k\_10 + k\_12)\)\ x1[t] + k\_21\ x2[t], \(x2'\)[t] == k\_12\ x1[t] - k\_21\ x2[t], x\_a[0] \[Equal] X\_a, x1[0] \[Equal] X1, x2[0] \[Equal] X2}, {x\_a[t], x1[t], x2[t]}\ , \ t];\)\)], "Input",\ CellLabel->"In[69]:="], Cell[BoxData[ \(\({x2a[t_, d_], x21[t_, d_], x22[t_, d_]}\ = \(\({x\_a[t], x1[t], x2[t]} /. eq2[\([1]\)]\)\ /. {X\_a \[Rule] \ x1a[T], X1 \[Rule] \ x11[T], X2 \[Rule] \ x12[T]}\) /. T \[Rule] \ d\/k\_F\ ;\)\)], "Input", CellLabel->"In[70]:="], Cell[TextData[{ "Now we will build a compound function that represent the retention for \ both intervals: one for ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(0 < t \[LessEqual] T\)\)\)]], " where the input is constant and the other for ", Cell[BoxData[ \(TraditionalForm\`t\ > T\)]], " when there isn't any input. We apply ", StyleBox["If", "DisplayMath"], " whose syntax is ", StyleBox["If[", "MR"], StyleBox["condition", "TI"], StyleBox[",", "MR"], " ", StyleBox["A", "TI"], StyleBox[",", "MR"], " ", StyleBox["B", "TI"], StyleBox["]", "MR"], " gives ", StyleBox["A", "TI"], " if ", StyleBox["condition", "TI"], " evaluates to ", StyleBox["True", "MR"], ", and ", StyleBox["B", "TI"], " if it evaluates to ", StyleBox["False", "MR"], ". In our case A correspond to the solutions for the first interval given \ by eq13 (", StyleBox["condition", "TI"], " ", Cell[BoxData[ \(TraditionalForm\`t \[LessEqual] T\)]], "), and B the solutions for the second interval given by eq14 (", StyleBox["condition", "TI"], " ", Cell[BoxData[ \(TraditionalForm\`t > T\)]], ")." }], "Text"], Cell[TextData[{ StyleBox["xa[t, d], x1[t,d]", "DisplayMath"], " and ", StyleBox["x2[t,d]", "DisplayMath"], ", represent the retentions in compartment a, 1 and 2. We live d as \ function variable better than T because T is a function of d. " }], "Caption"], Cell[BoxData[ \(xa[t_, d_] := \ If[\ t \[LessEqual] d\/k\_F, x1a[t], \ x2a[t - d\/k\_F, d]]\)], "Input",\ CellLabel->"In[71]:="], Cell[BoxData[ \(x1[t_, d_] := \ If[\ t \[LessEqual] d\/k\_F, x11[t], \ x21[t - d\/k\_F, d]]\)], "Input",\ CellLabel->"In[72]:="], Cell[BoxData[ \(x2[t_, d_] := \ If[\ t \[LessEqual] d\/k\_F, x12[t], \ x22[t - d\/k\_F, d]]\)], "Input",\ CellLabel->"In[73]:="], Cell["Here is plot the retention. 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59.625}} -> {-3.17454, \ -0.456021, 0.0200746, 0.0200746}}], Cell["This model is represented by the follow ODE System ", "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(\(x\_1' \((t)\)\)\(\ \)\(=\)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(b\_1\)( t)\ - \ \((k\_10\ + \ k\_12\ + \ k\_13)\)\ \(\(x\_1\)( t)\)\ \ + \ k\_21\ \(\(x\_2\)(t)\)\ + \ \(\(k\_31\)\(\ \)\(\(x\_3\)( t)\)\(\ \)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(x\_2' \((t)\)\)\(\ \)\(=\)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(b\_2\)(t)\ + \ k\_12\ \(\(x\_1\)( t)\)\ \ - \ \((k\_20\ + \ k\_21\ + \ k\_23)\)\ \(\(x\_2\)( t)\)\ + \ k\_32\ \(\(x\_3\)(t)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(x\_3' \((t)\)\)\(\ \)\(=\)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`b\_3\ \((t)\)\ + \ k\_13\ \(\(x\_1\)(t)\)\ \ + \ k\_23\ \(\(x\_2\)( t)\)\ - \ \(\((k\_30\ + \ k\_31\ + \ k\_32)\)\(\ \)\(\(x\_3\)( t)\)\(\ \)\)\)]] }], "NumberedEquation"], Cell["in matrix notation ", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ StyleBox["x", FontWeight->"Bold", FontSlant->"Plain"], StyleBox["'", FontWeight->"Plain", FontSlant->"Plain"]}], \((t)\)}], " ", "=", RowBox[{ RowBox[{ StyleBox["A", FontWeight->"Bold", FontSlant->"Plain"], StyleBox[" ", FontWeight->"Bold"], StyleBox["x", FontWeight->"Bold", FontSlant->"Plain"]}], StyleBox[" ", FontWeight->"Bold"], "+", " ", RowBox[{ StyleBox["b", FontWeight->"Bold", FontSlant->"Plain"], "(", "t", ")"}]}]}], ",", " ", \(t \[GreaterEqual] 0\)}], TraditionalForm]], "DisplayFormula", TextAlignment->Left, TextJustification->0], Cell["with", "Text"], Cell[TextData[{ StyleBox["x", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\((t)\) = \ {\(x\_1\)(t), \ \(x\_2\)(t), \ \ \(x\_1\)(t)}\^T\)]], " \n\n", StyleBox["B", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(\((t)\)\(\ \)\(=\)\(\ \)\({\(b\_1\)(t), \ \(b\_2\)( t), \(b\_3\)(t)}\^T\)\(\ \)\)\)]], "\n\nwhere ", StyleBox["A", FontWeight->"Bold"], " is " }], "DisplayFormula", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(\(model\ = CompartMatrix[ 3, \ {{1, 2, k\_12}, {2, 1, k\_21}, {1, 3, k\_13}, {3, 1, k\_31}, {3, 2, k\_32}, {2, 3, k\_23}, {3, 3, k\_33}, {1, 0, \ k\_10}, {2, 0, k\_20}, {3, 0, k\_30}}];\)\)], "Input", CellLabel->"In[83]:="], Cell[CellGroupData[{ Cell[BoxData[ \(model // MatrixForm\)], "Input", CellLabel->"In[84]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-k\_10\) - k\_12 - k\_13\), \(k\_21\), \(k\_31\)}, {\(k\_12\), \(\(-k\_20\) - k\_21 - k\_23\), \(k\_32\)}, {\(k\_13\), \(k\_23\), \(\(-k\_30\) - k\_31 - k\_32\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[84]//MatrixForm="] }, Open ]], Cell["It assumed as initial condition:", "Text"], Cell[BoxData[ \(\(ic\ = {x\_10, \ x\_20, \ x\_30};\)\)], "Input", CellLabel->"In[85]:="], Cell["and as input", "Text"], Cell[BoxData[ \(\(input = {b\_1, \ b\_2, \ b\_3};\)\)], "Input", CellLabel->"In[86]:="], Cell["It can be represented by the system of diferential equation", "Text"], Cell[BoxData[ \(\(eq1 = ShowODE[model, \ ic, input, \ t, \ x];\)\)], "Input", CellLabel->"In[87]:="], Cell[CellGroupData[{ Cell[BoxData[ \(\(eq1 // TableForm\) // TraditionalForm\)], "Input", CellLabel->"In[88]:="], Cell[BoxData[ FormBox[ InterpretationBox[GridBox[{ { RowBox[{ RowBox[{ SubsuperscriptBox["x", "1", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "==", \(b\_1 + \((\(-k\_10\) - k\_12 - k\_13)\)\ \(\(x\_1\)( t)\) + k\_21\ \(\(x\_2\)(t)\) + k\_31\ \(\(x\_3\)(t)\)\)}]}, { RowBox[{ RowBox[{ SubsuperscriptBox["x", "2", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "==", \(b\_2 + k\_12\ \(\(x\_1\)( t)\) + \((\(-k\_20\) - k\_21 - k\_23)\)\ \(\(x\_2\)( t)\) + k\_32\ \(\(x\_3\)(t)\)\)}]}, { RowBox[{ RowBox[{ SubsuperscriptBox["x", "3", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "==", \(b\_3 + k\_13\ \(\(x\_1\)(t)\) + k\_23\ \(\(x\_2\)( t)\) + \((\(-k\_30\) - k\_31 - k\_32)\)\ \(\(x\_3\)( t)\)\)}]}, {\(\(x\_1\)(0) == x\_10\)}, {\(\(x\_2\)(0) == x\_20\)}, {\(\(x\_3\)(0) == x\_30\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { Equal[ Derivative[ 1][ Subscript[ x, 1]][ t], Plus[ Subscript[ b, 1], Times[ Plus[ Times[ -1, Subscript[ k, 10]], Times[ -1, Subscript[ k, 12]], Times[ -1, Subscript[ k, 13]]], Subscript[ x, 1][ t]], Times[ Subscript[ k, 21], Subscript[ x, 2][ t]], Times[ Subscript[ k, 31], Subscript[ x, 3][ t]]]], Equal[ Derivative[ 1][ Subscript[ x, 2]][ t], Plus[ Subscript[ b, 2], Times[ Subscript[ k, 12], Subscript[ x, 1][ t]], Times[ Plus[ Times[ -1, Subscript[ k, 20]], Times[ -1, Subscript[ k, 21]], Times[ -1, Subscript[ k, 23]]], Subscript[ x, 2][ t]], Times[ Subscript[ k, 32], Subscript[ x, 3][ t]]]], Equal[ Derivative[ 1][ Subscript[ x, 3]][ t], Plus[ Subscript[ b, 3], Times[ Subscript[ k, 13], Subscript[ x, 1][ t]], Times[ Subscript[ k, 23], Subscript[ x, 2][ t]], Times[ Plus[ Times[ -1, Subscript[ k, 30]], Times[ -1, Subscript[ k, 31]], Times[ -1, Subscript[ k, 32]]], Subscript[ x, 3][ t]]]], Equal[ Subscript[ x, 1][ 0], Subscript[ x, 10]], Equal[ Subscript[ x, 2][ 0], Subscript[ x, 20]], Equal[ Subscript[ x, 3][ 0], Subscript[ x, 30]]}]], TraditionalForm]], "Output", CellLabel->"Out[88]//TraditionalForm="] }, Open ]], Cell[TextData[{ "We consider ", Cell[BoxData[ \(TraditionalForm\`\(b\_1\)(t)\ = \ 2\ Exp[\(-0.1\)\ t]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(b\_2\)(t) = 2\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(b\_3\)(t)\ = 0.2 Cos[2\ t]\)\)\)]], " that is equivalent to take as initial conditions ", Cell[BoxData[ \(TraditionalForm\`\(\(\(x\_1\)(0)\ = 1\)\(,\)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\(\(x\_2\)(0) = 2\)\(\ \)\(,\)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(x\_3\)(0)\ = \ 3\)]], ". Also we give values for constant coefficients. " }], "Text"], Cell[BoxData[ \(k /: k\_10 = 0.1; k /: k\_20 = 0.2; k /: k\_30 = 0.5; k /: k\_12 = 1; \ k /: k\_13 = 2; \ k /: k\_21 = 1.4; k /: k\_23 = 1.1;; \ k /: k\_31 = 1.1; k /: k\_32 = 0.8;\)], "Input", CellLabel->"In[89]:="], Cell["Here is the solution", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({x1[t1_], x2[t1_], x3[t1_]} = {x\_1[t1], x\_2[t1], x\_3[t1]} /. 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.35534 .21025 L Mistroke .35667 .21108 L .35791 .21173 L .35903 .21215 L .36028 .2124 L .36097 .21242 L .36162 .21237 L .36284 .21205 L .36414 .21142 L .36525 .21067 L .36647 .20965 L .37174 .20431 L .37307 .20316 L .37446 .2022 L .37519 .20182 L .37599 .20151 L .37671 .20133 L .37739 .20125 L .37808 .20126 L .37873 .20135 L .37995 .2017 L .38126 .20232 L .38268 .20321 L .38503 .2049 L .38725 .20637 L .38848 .20699 L .38918 .20724 L .38982 .20741 L .39096 .20754 L .39216 .20742 L .39348 .20698 L .39472 .2063 L .39608 .2053 L .39755 .20398 L .40016 .20138 L .40259 .19917 L .40382 .19828 L .40514 .1976 L .40589 .19734 L .4066 .19719 L .40724 .19714 L .40793 .19717 L .40861 .19728 L .40935 .19749 L .41013 .19781 L .41086 .19818 L .41356 .20003 L .41605 .20192 L .41728 .20273 L .41842 .20333 L .41972 .2038 L Mistroke .42041 .20393 L .42114 .20398 L .42245 .20382 L .42364 .20341 L .44451 .19804 L .44678 .19975 L .4479 .20043 L .44893 .20092 L .45012 .20127 L .45141 .20137 L .4525 .20122 L .45367 .20083 L .45494 .20013 L .45614 .19928 L .45886 .19685 L .46123 .19466 L .46251 .19363 L .46371 .19286 L .4648 .19234 L .46596 .19201 L .46725 .19194 L .46842 .19214 L .46949 .19253 L .47064 .19314 L .47274 .19463 L .4752 .19663 L .47654 .19765 L .4778 .19846 L .47887 .19898 L .48002 .19935 L .4813 .19948 L .48246 .19934 L .48369 .19894 L .48499 .19823 L .48622 .19735 L .48733 .19642 L .48987 .19408 L .49121 .1929 L .49263 .19182 L .49386 .19107 L .49456 .19075 L .49522 .19053 L .49637 .19033 L .4976 .19039 L .49885 .19073 L .50004 .19128 L .50111 .19195 L .50227 .19281 L .5046 .19473 L .50681 .19644 L Mistroke .50803 .1972 L .50868 .19751 L .50936 .19778 L .51057 .19806 L .5117 .19808 L .51239 .19798 L .51315 .19776 L .51386 .19747 L .51451 .19714 L .51706 .19528 L .51966 .19294 L .52093 .19182 L .52209 .19091 L .52321 .19017 L .52444 .18957 L .52573 .18921 L .52696 .18916 L .5276 .18924 L .52831 .18941 L .52904 .18969 L .52974 .19002 L .53231 .19179 L .53484 .19391 L .53622 .19501 L .53751 .19589 L .53824 .1963 L .5389 .19661 L .53964 .19686 L .54043 .19703 L .54109 .19709 L .54181 .19706 L .54311 .19676 L .54792 .19351 L .5503 .19138 L .55249 .18967 L .55371 .18897 L .55501 .18848 L .55576 .18833 L .55646 .18828 L .55776 .18844 L .55897 .18885 L .56008 .18943 L .56258 .19127 L .56536 .19363 L .56672 .19466 L .56796 .19544 L .56927 .19602 L .56999 .19622 L .57067 .19633 L .57141 .19635 L Mistroke .5721 .19627 L .57285 .19609 L .57365 .1958 L .57487 .19514 L .57619 .19421 L .57857 .19217 L .58139 .18975 L .58276 .1888 L .58401 .18815 L .5852 .18778 L .58632 .18766 L .58751 .18778 L .58879 .1882 L .59011 .1889 L .59151 .18988 L .59402 .19198 L .59644 .19395 L .5978 .19484 L .59904 .19542 L .59974 .19564 L .6005 .19578 L .60122 .19581 L .60189 .19576 L .60318 .19543 L .6039 .19512 L .60456 .19476 L .60922 .19102 L .61168 .18899 L .61291 .18819 L .61362 .18782 L .61426 .18755 L .61545 .18725 L .61672 .18722 L .6179 .18745 L .61902 .18789 L .62011 .1885 L .62111 .1892 L .62339 .19108 L .62577 .19309 L .62695 .19395 L .62804 .1946 L .62929 .19513 L .62995 .19531 L .63066 .1954 L .63191 .19535 L .63305 .19506 L .63422 .19453 L .63529 .19387 L .63771 .19193 L .63983 .19005 L Mistroke .64211 .18828 L .6434 .18754 L .64458 .18708 L .64576 .18687 L .64689 .18691 L .64819 .18725 L .6494 .18781 L .65073 .18866 L .65214 .18976 L .65456 .19186 L .65583 .1929 L .65717 .19385 L .65789 .19427 L .65867 .19464 L .65938 .19489 L .66004 .19505 L .66134 .19512 L .66207 .19503 L .66273 .19486 L .68353 .19081 L .68579 .19271 L .6869 .19351 L .68793 .19412 L .68911 .19462 L .6904 .1949 L .6915 .19489 L .69266 .19465 L .69334 .1944 L .69407 .19405 L .69539 .19321 L .69785 .1912 L .70017 .18918 L .7014 .18822 L .70271 .18738 L .70399 .18679 L .70516 .1865 L .70633 .18645 L .70745 .18664 L .70869 .18711 L .70984 .18776 L .71244 .18976 L .71467 .1917 L .71583 .19264 L .71706 .1935 L .71814 .1941 L .71932 .19455 L .72039 .19475 L .72139 .19474 L .72255 .19451 L .7236 .1941 L Mistroke .72481 .19341 L .72595 .19259 L .72857 .19035 L .73099 .18833 L .73222 .18748 L .73336 .18687 L .73456 .18646 L .73588 .18631 L .73656 .18635 L .73721 .18647 L .73843 .1869 L .73974 .18762 L .74117 .18865 L .74568 .19247 L .74692 .19335 L .74822 .19406 L .74933 .19446 L .75055 .19465 L .75125 .19464 L .75189 .19455 L .75262 .19435 L .75332 .19408 L .75455 .19341 L .7559 .19246 L .76092 .18822 L .76213 .18739 L .76344 .18671 L .76419 .18644 L .76488 .18629 L .76556 .18622 L .76622 .18623 L .76689 .18632 L .76761 .18652 L .76891 .18708 L .7697 .18755 L .77044 .18807 L .77184 .18917 L .77419 .19122 L .77668 .19316 L .77798 .1939 L .77872 .19421 L .77941 .19442 L .78062 .19458 L .78192 .19444 L .78314 .19405 L .78383 .19371 L .78448 .19333 L .7872 .19124 L .78959 .18915 L .79089 .18811 L Mistroke .7921 .18729 L .79279 .1869 L .79342 .18661 L .79413 .18637 L .79488 .1862 L .79619 .18616 L .79743 .18641 L .79812 .18667 L .79886 .18703 L .80018 .18787 L .80317 .19033 L .80556 .19235 L .80687 .19327 L .80812 .19394 L .80928 .19435 L .81054 .19452 L .81169 .19442 L .81275 .19411 L .81342 .19382 L .81413 .19343 L .81543 .19256 L .81786 .19054 L .82048 .1883 L .82193 .18729 L .82264 .18689 L .82329 .18658 L .82453 .1862 L .82518 .18611 L .82588 .1861 L .827 .18627 L .82823 .18671 L .82937 .18733 L .8306 .18819 L .83283 .19006 L .83563 .19241 L .83701 .19336 L .83829 .194 L .83944 .19436 L .84067 .19447 L .84138 .19441 L .84204 .19427 L .84328 .19381 L .8445 .19311 L .84584 .19214 L .84855 .18982 L .84986 .1887 L .85109 .18775 L .85229 .18699 L .85341 .18646 L .85471 .18611 L Mistroke .85541 .18605 L .85614 .18609 L .85686 .18621 L .85753 .18642 L .85827 .18673 L .85907 .18716 L .86183 .18922 L .86436 .19143 L .86564 .19245 L .86684 .19327 L .86792 .19385 L .8691 .19427 L .87038 .19444 L .87102 .19442 L .87173 .1943 L .87298 .19388 L .87416 .19325 L .87883 .18947 L .87995 .18853 L .88114 .18764 L .88224 .18695 L .88327 .18646 L .88458 .1861 L .88577 .18603 L .88707 .18627 L .88781 .18653 L .88849 .18685 L .88959 .18752 L .8908 .18841 L .89325 .19051 L .89593 .1927 L .8967 .19321 L .89743 .19363 L .89809 .19393 L .8988 .19418 L .90008 .19441 L .90128 .19435 L .90256 .194 L .90329 .19367 L .90397 .19329 L .90907 .18917 L .91035 .18813 L .91156 .18728 L .91264 .18667 L .91382 .18623 L .91507 .18601 L .91621 .18607 L .91746 .1864 L .91877 .18703 L .9201 .18791 L Mistroke .92135 .1889 L .92418 .19136 L .92533 .19229 L .92657 .19317 L .92775 .1938 L .92885 .1942 L .92983 .19438 L .93091 .19437 L .93204 .19413 L .93311 .1937 L .93428 .19303 L .93551 .19214 L .93772 .19027 L .94027 .18811 L .94153 .18723 L .94269 .1866 L .94371 .18622 L .9448 .18601 L .94588 .18602 L .94708 .18629 L .94807 .18669 L .94914 .18729 L .95107 .18873 L .95327 .19063 L .95567 .19259 L .95676 .19332 L .95793 .1939 L .95904 .19425 L .96004 .19439 L .96112 .19432 L .96212 .19407 L .96322 .1936 L .96438 .1929 L .96847 .18953 L .97025 .18806 L .97123 .18737 L .97215 .18682 L .97324 .18633 L .97424 .18607 L .97517 .18598 L .97619 .18607 L Mfstroke .5 Mabswid .02381 .19325 m .03279 .26251 L .04262 .32533 L .04758 .34768 L .05288 .36952 L .05785 .38733 L .06004 .39359 L .06244 .39902 L .06381 .40138 L .06508 .40313 L .06643 .40452 L .06719 .40512 L .06788 .40556 L .06858 .40592 L .06933 .40621 L .07015 .40645 L .07091 .4066 L .07169 .40672 L .07242 .40679 L .07379 .40691 L .07509 .40705 L .07575 .40715 L .07647 .40729 L .07774 .40764 L .07894 .40809 L .08162 .40953 L .08451 .41145 L .08576 .41226 L .08709 .41302 L .08778 .41335 L .08855 .41364 L .08925 .41384 L .0899 .41396 L .09118 .41397 L .09236 .41372 L .09365 .41314 L .0943 .41272 L .09501 .41218 L .0974 .40974 L .09997 .40638 L .10458 .40003 L .10717 .3972 L .10847 .39611 L .10991 .39518 L .11113 .39459 L .11245 .39414 L .11369 .39383 L .11482 .39361 L .11614 .39335 L .11755 .39301 L Mistroke .11836 .39275 L .11909 .39246 L .12052 .3917 L .12186 .39072 L .1233 .38934 L .12593 .386 L .14608 .36314 L .18713 .32696 L .20759 .31154 L .20879 .31145 L .21005 .31119 L .21071 .31097 L .21144 .31067 L .21273 .30994 L .21393 .30903 L .21523 .30781 L .21758 .30511 L .22193 .29936 L .22439 .29647 L .22558 .29532 L .22667 .29447 L .22784 .29376 L .22893 .29328 L .23009 .29299 L .23135 .29288 L .23259 .29295 L .2339 .29315 L .23504 .2934 L .23626 .29366 L .23739 .29386 L .23842 .29397 L .23946 .29397 L .24044 .29385 L .24158 .29355 L .24264 .2931 L .2438 .29241 L .24504 .29146 L .2647 .27971 L .26584 .28011 L .26706 .28051 L .26821 .28079 L .26925 .28095 L .27043 .28096 L .27154 .2808 L .27275 .28042 L .27405 .27976 L .27538 .27883 L .27679 .27762 L .27931 .27503 L .28164 .27255 L Mistroke .28292 .27129 L .28414 .27024 L .28527 .26944 L .28649 .26876 L .2876 .26836 L .28863 .26815 L .2897 .26812 L .29067 .26822 L .29179 .26849 L .29284 .26885 L .29523 .26989 L .29655 .27047 L .29779 .27094 L .29895 .27125 L .30021 .27142 L .30136 .27139 L .30242 .27118 L .30349 .2708 L .30468 .27017 L .30591 .26932 L .30708 .26836 L .30974 .26582 L .31219 .26347 L .31461 .26159 L .31583 .26092 L .31651 .26063 L .31716 .26044 L .31785 .2603 L .31851 .26024 L .31974 .2603 L .32043 .26043 L .32107 .26061 L .32251 .26115 L .32475 .26228 L .32598 .26293 L .32715 .26349 L .32823 .26393 L .32941 .26427 L .33049 .26443 L .33149 .26443 L .33264 .26425 L .33368 .2639 L .33488 .26331 L .33601 .26257 L .34099 .25824 L .34324 .25637 L .34451 .25555 L .34566 .25497 L .34632 .25474 L .34702 .25456 L Mistroke .34831 .25443 L .3496 .25455 L .35078 .25487 L .35202 .25538 L .35317 .25597 L .3558 .25749 L .35711 .2582 L .35836 .25876 L .35905 .259 L .3598 .2592 L .36112 .25935 L .36237 .25926 L .36355 .25895 L .36461 .2585 L .36576 .25784 L .37071 .25381 L .37296 .25202 L .37419 .25124 L .37534 .25069 L .37632 .25036 L .37738 .25018 L .37854 .25019 L .37961 .25037 L .38079 .25075 L .38189 .25125 L .38441 .25273 L .38571 .25354 L .38707 .25434 L .38834 .25495 L .38952 .25536 L .39053 .25558 L .39162 .25564 L .39283 .25549 L .39393 .25516 L .3951 .25462 L .39637 .25384 L .39868 .25206 L .40139 .24982 L .40272 .24884 L .40393 .24809 L .4052 .24749 L .40591 .24727 L .40657 .24712 L .4073 .24704 L .40796 .24704 L .4087 .24711 L .40948 .24728 L .41078 .24775 L .41217 .24846 L .41471 .25006 L Mistroke .41719 .2516 L .41827 .25215 L .41944 .2526 L .42066 .25287 L .42182 .25292 L .42246 .25286 L .42315 .25273 L .42437 .25231 L .4255 .25175 L .42656 .25109 L .42897 .24926 L .43142 .2473 L .43275 .24635 L .43399 .24563 L .4347 .24531 L .43535 .24507 L .43606 .24488 L .43682 .24476 L .43746 .24474 L .43815 .24478 L .43941 .24504 L .44072 .24554 L .44195 .24619 L .44474 .24801 L .44625 .24901 L .44769 .24985 L .44898 .25044 L .44971 .25068 L .45038 .25083 L .4515 .25095 L .45274 .25085 L .45404 .25049 L .45525 .24994 L .45657 .24915 L .45777 .24829 L .46048 .24616 L .46165 .24528 L .46291 .24444 L .46411 .2438 L .46521 .24337 L .4665 .2431 L .46714 .24306 L .46786 .24309 L .4691 .24333 L .47027 .24375 L .47144 .24434 L .47256 .24502 L .47466 .24645 L .47713 .24808 L .47821 .24865 L Mistroke .47938 .24912 L .48065 .24943 L .48186 .2495 L .48253 .24944 L .48324 .2493 L .48451 .24886 L .48568 .24827 L .48678 .24758 L .48927 .24572 L .49194 .24372 L .49326 .24292 L .49448 .24235 L .49516 .24211 L .49588 .24194 L .4966 .24185 L .4974 .24184 L .49806 .24191 L .49877 .24206 L .50006 .24251 L .50132 .24314 L .50246 .24385 L .50502 .24564 L .50619 .24643 L .50743 .24719 L .50857 .24776 L .50964 .24815 L .51087 .24841 L .51152 .24845 L .51222 .24842 L .51298 .2483 L .51369 .24811 L .51433 .24788 L .51503 .24757 L .51766 .24591 L .52011 .24404 L .52144 .24307 L .52287 .24215 L .52353 .2418 L .52423 .24149 L .52548 .2411 L .52667 .24094 L .52779 .24099 L .52902 .24127 L .53033 .2418 L .53308 .24347 L .53599 .24554 L .53718 .24629 L .53847 .24696 L .53921 .24726 L .53989 .24746 L Mistroke .54057 .2476 L .54121 .24767 L .54244 .24762 L .54374 .24732 L .54497 .24681 L .54609 .24619 L .55142 .24234 L .55275 .2415 L .55416 .24081 L .55489 .24056 L .5557 .24037 L .55643 .24029 L .55712 .24029 L .55782 .24036 L .55848 .2405 L .55971 .24092 L .56041 .24125 L .56105 .24159 L .5625 .24251 L .56492 .24425 L .56619 .24513 L .56753 .24595 L .56871 .24652 L .57 .24694 L .57117 .24711 L .57226 .24708 L .57351 .24682 L .57464 .24639 L .57595 .24569 L .57719 .24488 L .58 .24279 L .58138 .2418 L .58263 .24102 L .58377 .24045 L .58486 .24006 L .58601 .23984 L .58725 .23983 L .58853 .24007 L .58925 .2403 L .58989 .24057 L .59234 .24201 L .59475 .24374 L .59728 .24543 L .59861 .24609 L .59937 .24637 L .60007 .24655 L .6013 .24671 L .602 .24669 L .60263 .24661 L .60384 .24628 L Mistroke .60497 .2458 L .6062 .2451 L .60751 .24421 L .60989 .24243 L .61213 .2409 L .61313 .24035 L .61418 .23989 L .61533 .23957 L .61638 .23945 L .61754 .23952 L .61862 .23977 L .61981 .24023 L .62107 .24091 L .62606 .2444 L .62727 .24515 L .62857 .24579 L .62932 .24607 L .63 .24625 L .63068 .24637 L .63133 .24641 L .63203 .24638 L .63277 .24627 L .63347 .24609 L .6341 .24586 L .63565 .24509 L .63711 .24415 L .63971 .24224 L .64112 .24122 L .64244 .2404 L .64361 .23982 L .64485 .2394 L .64556 .23926 L .64624 .2392 L .64749 .23928 L .64865 .23956 L .64972 .23998 L .65089 .24061 L .65213 .24141 L .65453 .24315 L .65711 .24488 L .65787 .24529 L .65859 .24561 L .65924 .24585 L .65995 .24604 L .66061 .24615 L .66132 .24619 L .6626 .24607 L .66384 .24572 L .66455 .24542 L .6652 .24509 L Mistroke .66795 .24329 L .67049 .24141 L .67173 .24058 L .67288 .23993 L .67352 .23963 L .67423 .23937 L .67494 .23917 L .67569 .23905 L .67701 .23904 L .67825 .23928 L .67896 .23951 L .6797 .23983 L .68102 .24056 L .68401 .24265 L .68642 .24434 L .68775 .2451 L .689 .24563 L .69016 .24593 L .69143 .24603 L .69258 .2459 L .69364 .24561 L .6949 .24506 L .69608 .24438 L .69876 .24248 L .70012 .24147 L .70157 .24049 L .70282 .23978 L .70353 .23946 L .70418 .23922 L .70483 .23904 L .70552 .23892 L .70679 .23889 L 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:@@Q00048Ekg009oo`05:DYJeWooOolQ200200002@@QHaQoogoo9BUJeWooNmhLi`0300001FLiOomk gQcWG_L00Woo00`HaP00124005kgOomkgR48HaQkgWoo8@P3000026LiOomkgR48HaQkgWoo9BT20000 1P@QF]IoogNm9BUJeP9oo`098@P00000001S67ooNmhYBU[F009oo`058@P48@00249JeP02Ool02AcW G_MoogNm9BT48@00125Nm`02Ool01BU:F]Ioogoo8@P00P0000T48FL00`00 00EW>GooNmhLiekg009oo`0<6GooNmhQ26"], ImageRangeCache->{{{0, 719}, {539, 0}} -> {-0.00359504, 0.99648, 1.00001, \ 1.00001}}], Cell["The compartments has been numbered as follow: ", "Text"], Cell[BoxData[ TagBox[GridBox[{ {"1", "\<\"Plasma\"\>"}, {"2", "\<\"RBC\"\>"}, {"3", "\<\"ST0\"\>"}, {"4", "\<\"ST1\"\>"}, {"5", "\<\"ST2\"\>"}, {"6", "\<\"Liver1\"\>"}, {"7", "\<\"Liver2\"\>"}, {"8", "\<\"UrinaryPath\"\>"}, {"9", "\<\"OtherKidney\"\>"}, {"10", "\<\"TrabecularSurface\"\>"}, {"11", "\<\"ExchTrabVol\"\>"}, {"12", "\<\"NonExchTrabVol\"\>"}, {"13", "\<\"CorticalSurf\"\>"}, {"14", "\<\"CorticalVolExch\"\>"}, {"15", "\<\"CorticalVolNonExch\"\>"}, {"16", "\<\"Bladder\"\>"}, {"17", "\<\"Urine\"\>"}, {"18", "\<\"ULI\"\>"}, {"19", "\<\"LLI\"\>"}, {"20", "\<\"FEC\"\>"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Text"], Cell["\<\ Transfer rates are taken from Table A.12.1. (ICRP 78) (For GI compartments \ are applied the trasfer rate included in ICRP 30]).\ \>", "Text"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"Plasma to ST0\"\>", "\<\"k[1,3]->\"\>", "10.5`"}, {"\<\"Plasma to RBC\"\>", "\<\"k[1,2]->\"\>", "0.245`"}, {"\<\"Plasma to urinary bladder\"\>", "\<\"k[1,16]->\"\>", "15.43`"}, {"\<\"Plasma to Kidneys (urinary path)\"\>", "\<\"k[1,8]->\"\>", "2.94`"}, {"\<\"Plasma to other kidney tissue\"\>", "\<\"k[1,9]->\"\>", "0.0122`"}, {"\<\"Plasma to ULI contents\"\>", "\<\"k[1,18]->\"\>", "0.122`"}, {"\<\"Plasma to liver 1\"\>", "\<\"k[1,6]->\"\>", "0.367`"}, {"\<\"Plasma to ST1\"\>", "\<\"k[1,4]->\"\>", "1.63`"}, {"\<\"Plasma to ST2\"\>", "\<\"k[1,5]->\"\>", "0.0735`"}, {"\<\"Plasma to trabecular surface\"\>", "\<\"k[1,10]->\"\>", "2.04`"}, {"\<\"Plasma to cortical surfaces\"\>", "\<\"k[1,13]->\"\>", "1.63`"}, {"\<\"ST0 to plasma\"\>", "\<\"k[3,1]->\"\>", "8.32`"}, {"\<\"RBC to Plasma\"\>", "\<\"k[2,1]->\"\>", "0.347`"}, {"\<\"Kidneys (Urinary path) to urinary bladder\"\>", \ "\<\"k[8,16]->\"\>", "0.099`"}, {"\<\"Other kidney tissue to plasma\"\>", "\<\"k[9,1]->\"\>", "0.00038`"}, {"\<\"Liver 1 to plasma\"\>", "\<\"k[6,1]->\"\>", "0.092`"}, {"\<\"Liver 1 to liver2\"\>", "\<\"k[6,7]->\"\>", "0.00693`"}, {"\<\"ST1 to plasma\"\>", "\<\"k[4,1]->\"\>", "0.0347`"}, {"\<\"ST2 to plasma\"\>", "\<\"k[5,1]->\"\>", "0.000019`"}, {"\<\"Cortical(Bone) surfaces to plasma\"\>", "\<\"k[13,1]->\"\>", "0.0693`"}, {"\<\"Cortical (Bone) surface to exch cortical volume\"\>", \ "\<\"k[13,14]->\"\>", "0.0693`"}, {"\<\"Liver 2 to plasma\"\>", "\<\"k[7,1]->\"\>", "0.00019`"}, {"\<\"Nonexch trab vol to plasma\"\>", "\<\"k[12,1]->\"\>", "0.000493`"}, {"\<\"Nonexch cort vol to plasma\"\>", "\<\"k[15,1]->\"\>", "0.0000821`"}, {"\<\"Cortical(Exch bone) vol to cortical (bone) surfaces\"\>", "\<\ \"k[14,13]->\"\>", "0.0173`"}, {"\<\"Exch trabecular vol to nonexch trabecular vol\"\>", \ "\<\"k[11,12]->\"\>", "0.00578`"}, {"\<\"Trabecular surface to plasma\"\>", "\<\"k[10,1]->\"\>", "0.0693`"}, {"\<\"Exch cortical vol to nonexch cortical vol\"\>", \ "\<\"k[14,15]->\"\>", "0.00578`"}, {"\<\"Exch trabecular vol to trabecular surface\"\>", \ "\<\"k[11,10]->\"\>", "0.0173`"}, {"\<\"Trabecular surface to Exchan. Trabecular vol.\"\>", \ "\<\"k[10,11]->\"\>", "0.0693`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Text"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"Bladder (16) to urine(17)\"\>", "\<\"k[16,17]->\"\>", "12"}, {"\<\"ULI (18) to LLI(19)\"\>", "\<\"k[18,19]->\"\>", "1.8"}, {"\<\"LLI (19) to FEC(20)\"\>", "\<\"k[19,20]->\"\>", "1"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Text"], Cell["The transfer coefficients for uranium are the following ", "Text"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(uraniumModel = {{1, 3, 10.5`}, {1, 2, 0.245`}, {1, 16, 15.43`}, {1, 8, 2.94`}, {1, 9, 0.0122`}, {1, 18, 0.122`}, {1, 6, 0.367`}, {1, 4, 1.63`}, {1, 5, 0.0735`}, {1, 10, 2.04`}, {1, 13, 1.63`}, {3, 1, 8.32`}, {2, 1, 0.347`}, {8, 16, 0.099`}, {9, 1, 0.00038`}, {6, 1, 0.092`}, {6, 7, 0.00693`}, {4, 1, 0.0347`}, {5, 1, 0.000019`}, {13, 1, 0.0693`}, {13, 14, 0.0693`}, {7, 1, 0.00019`}, {11, 12, 0.00578`}, {12, 1, 0.000493`}, {15, 1, 0.0000821`}, {14, 13, 0.0173`}, {14, 15, 0.00578`}, {10, 1, 0.0693`}, {11, 10, 0.0173`}, {10, 11, 0.0693`}, {16, 17, 12}, {18, 19, 1.8}, {19, 20, 1}};\)\)\)], "Input", CellLabel->"In[93]:="], Cell["\<\ The instant urine and faecal rate excrection for a single input 1 in \ compartment 1 can be evaluated as follow \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({qAcuteUrine[t_], \ qAcuteFaecal[t_]} = D[{x\_17[t], \ x\_20[t]} /. AcuteInput[CompartMatrix[20, uraniumModel], PadRight[{1}, 20]\ , \ t, \ \ x], t]\)], "Input", CellLabel->"In[94]:="], Cell[BoxData[ \({\(-6.484613617692136`\)\ \[ExponentialE]\^\(\(-37.94948560700166`\)\ t\ \) + 3.9530844472830466`\ \[ExponentialE]\^\(\(-12.`\)\ t\) + 2.508005291414146`\ \[ExponentialE]\^\(\(-5.3780035102389725`\)\ t\) \ + 0.`\ \[ExponentialE]\^\(\(-1.8`\)\ t\) + 0.`\ \[ExponentialE]\^\(\(-1.`\)\ t\) + 0.001947897735365933`\ \[ExponentialE]\^\(\(-0.3436368331818785`\)\ t\ \) + 8.219485644382199`*^-34\ \[ExponentialE]\^\(\(-0.14818320752681735`\)\ t\ \) + 0.0029994770875515237`\ \[ExponentialE]\^\(\(-0.13884393166609166`\)\ \ t\) + 0.0005840636960559397`\ \[ExponentialE]\^\(\(-0.099`\)\ t\) + 0.014921999311429221`\ \[ExponentialE]\^\(\(-0.09727564424805021`\)\ \ t\) + 0.002184440568345773`\ \[ExponentialE]\^\(\(-0.032193498148806954`\)\ t\ \) + 1.1505115472820008`*^-33\ \[ExponentialE]\^\(\(-0.013496792473182556`\)\ \ t\) + 0.0008737625310680431`\ \[ExponentialE]\^\(\(-0.012583816550724021`\)\ \ t\) + 0.00001023133768930272`\ \[ExponentialE]\^\(\(-0.0004821932901906803`\)\ \ t\) + 2.7619776330188586`*^-7\ \ \[ExponentialE]\^\(\(-0.00037973603767434713`\)\ t\) + 2.525624926900687`*^-7\ \ \[ExponentialE]\^\(\(-0.0001897406391886392`\)\ t\) + 1.4025513903032215`*^-6\ \ \[ExponentialE]\^\(\(-0.00008066471443853646`\)\ t\) + 7.541579132872499`*^-8\ \ \[ExponentialE]\^\(\(-0.000018924282331075318`\)\ t\) + 0.`\ \[ExponentialE]\^\(0.`\ t\), 0.00014948719483443658`\ \[ExponentialE]\^\(\(-37.94948560700166`\)\ \ t\) + 0.`\ \[ExponentialE]\^\(\(-12.`\)\ t\) + 0.0012619446795849`\ \[ExponentialE]\^\(\(-5.3780035102389725`\)\ t\) \ - 0.013687718817788042`\ \[ExponentialE]\^\(\(-1.8`\)\ t\) + 0.012158277773956722`\ \[ExponentialE]\^\(\(-1.`\)\ t\) + 0.00003052454682865503`\ \[ExponentialE]\^\(\(-0.3436368331818785`\)\ \ t\) + 1.0304641599944365`*^-34\ \[ExponentialE]\^\(\(-0.14818320752681735`\)\ \ t\) + 0.00005601548390795479`\ \[ExponentialE]\^\(\(-0.13884393166609166`\)\ \ t\) + 0.`\ \[ExponentialE]\^\(\(-0.099`\)\ t\) + 0.000011478364256597574`\ \ \[ExponentialE]\^\(\(-0.09727564424805021`\)\ t\) + 0.000014132138957412544`\ \[ExponentialE]\^\(\(-0.032193498148806954`\ \)\ t\) + 2.1826003576513065`*^-34\ \[ExponentialE]\^\(\(-0.013496792473182556`\ \)\ t\) + 5.777364473501825`*^-6\ \[ExponentialE]\^\(\(-0.012583816550724021`\)\ \ t\) + 6.794404523969363`*^-8\ \ \[ExponentialE]\^\(\(-0.0004821932901906803`\)\ t\) + 1.8341975864259517`*^-9\ \ \[ExponentialE]\^\(\(-0.00037973603767434713`\)\ t\) + 1.6772871011481807`*^-9\ \[ExponentialE]\^\(\(-0.0001897406391886392`\ \)\ t\) + 9.314603200837`*^-9\ \[ExponentialE]\^\(\(-0.00008066471443853646`\)\ \ t\) + 5.00854728136387`*^-10\ \ \[ExponentialE]\^\(\(-0.000018924282331075318`\)\ t\) + 0.`\ \[ExponentialE]\^\(0.`\ t\)}\)], "Output", CellLabel->"Out[94]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LogPlot[{qAcuteUrine[t], \ qAcuteFaecal[t]}, \ {t, \ 2, 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0.5, k\_g \[Rule] \ 0.1, k\_a \[Rule] \ 2, k\_\(bilis, gi\) \[Rule] \ h[t, 8, 2]\ }\)]] }], "Text"], Cell[BoxData[ \(\(matrixhepatic = CompartMatrix[ 4, \ {{1, 2, \ k12}, \ {1, 0, k10}, {2, 1, \ k21}, {1, 3, \ k\_g}, {\ 3, 4, \ k\_\(bilis, gi\)}, \ {4, 1, \ k\_a}}] /. {k12 \[Rule] 2, k21 \[Rule] \ 1, k10 \[Rule] \ 0.5, k\_g \[Rule] \ 0.1, k\_a \[Rule] \ 2, k\_\(bilis, gi\) \[Rule] \ h[t, 8, 2]\ };\)\)], "Input", CellLabel->"In[101]:="], Cell["Initial condition and input are", "Text"], Cell[BoxData[ \(\(ip\ = {0, 0, 0, 0};\)\)], "Input", CellLabel->"In[102]:="], Cell[BoxData[ \(\(ic2\ = {200, 0, 0, 0};\)\)], "Input", CellLabel->"In[103]:="], Cell[TextData[{ "Then, it can be used ", Cell[BoxData[ ButtonBox[ StyleBox["SystemNDSolve", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], ButtonFunction:>(Internal`PutInformation[ #, LongForm -> False]&), ButtonEvaluator->Automatic, ButtonData:>{"Info3257348699-2143584", "Biokmod`SysModel`SystemNDSolve"}, ButtonFrame->"None", 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Note: This function \ apply the Mathematica function NDSolve for solving the SODE, it should be \ used when MatrixCoefs included some coeffs that are not constants or when it \ was not possible obtain the solution of the model using SystemDSolve."\)], \ "Print", CellTags->"Info3257348699-2143584"], Cell[CellGroupData[{ Cell[BoxData[ \(sol1 = SystemNDSolve[matrixhepatic, \ ic2, ip, {\ t, 0, \ 100}, \ t, \ \ x]\)], "Input", CellLabel->"In[104]:="], Cell[BoxData[ \(NDSolve::"mxst" \(\(:\)\(\ \)\) "Maximum number of \!\(1000\) steps reached at the point \!\(t\) == \ \!\(68.99449950183133`\)."\)], "Message", CellLabel->"From In[104]:="], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{\(x\_1[t]\), "\[Rule]", RowBox[{ TagBox[\(InterpolatingFunction[{{0.`, 68.99449950183133`}}, "<>"]\), False, Editable->False], "[", "t", "]"}]}], ",", RowBox[{\(x\_2[t]\), "\[Rule]", RowBox[{ TagBox[\(InterpolatingFunction[{{0.`, 68.99449950183133`}}, "<>"]\), False, Editable->False], "[", "t", "]"}]}], ",", 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.00165 L .87824 .00164 L .87921 .00162 L .88014 .00161 L .88112 .00159 L .88208 .00158 L .88299 .00156 L .88397 .00155 L .88491 .00154 L .8859 .00152 L .88686 .00151 L .88779 .0015 L .88877 .00148 L .88972 .00147 L .89063 .00146 L .8916 .00144 L .89253 .00143 L .89352 .00142 L .89448 .00141 L .8954 .0014 L .89638 .00138 L .89732 .00137 L .89832 .00136 L .89928 .00135 L .90021 .00133 L .9012 .00132 L .90215 .00131 L .90306 .0013 L .90404 .00129 L .90497 .00128 L .90597 .00127 L .90693 .00125 L .90785 .00124 L .90883 .00123 L .90978 .00122 L .91068 .00121 L .91165 .0012 L .91258 .00119 L .91357 .00118 L .91453 .00117 L .91544 .00116 L .91642 .00115 L .91736 .00114 L .91836 .00113 L .91932 .00112 L .92025 .00111 L .92123 .0011 L .92218 .00109 L Mistroke .92309 .00108 L .92406 .00107 L .925 .00106 L .92599 .00105 L .92695 .00104 L .92787 .00103 L .92885 .00103 L .92979 .00102 L .9307 .00101 L .93166 .001 L .93259 .00099 L .93358 .00098 L .93453 .00097 L .93545 .00097 L .93642 .00096 L .93736 .00095 L .93835 .00094 L .93931 .00093 L .94024 .00092 L .94122 .00092 L .94217 .00091 L .94308 .0009 L .94405 .00089 L .94498 .00088 L .94597 .00088 L .94692 .00087 L .94784 .00086 L .94882 .00085 L .94976 .00085 L .95076 .00084 L .95172 .00083 L .95265 .00082 L .95364 .00082 L .95459 .00081 L .9555 .0008 L .95647 .0008 L .95741 .00079 L .9584 .00078 L .95936 .00077 L .96028 .00077 L .96126 .00076 L .96221 .00075 L .96311 .00075 L .96408 .00074 L .96501 .00074 L .966 .00073 L .96695 .00072 L .96787 .00072 L .96885 .00071 L .96978 .0007 L Mistroke .97078 .0007 L .97174 .00069 L .97267 .00069 L .97365 .00068 L .9746 .00067 L .97551 .00067 L .97619 .00066 L Mfstroke .5 Mabswid .02381 0 m .02474 .11046 L .02575 .21061 L .0267 .28844 L .02761 .35082 L .02859 .40591 L .02952 .44965 L .03052 .48777 L .03148 .51764 L .0324 .54106 L .03338 .56115 L .03433 .57657 L .03524 .58833 L .03621 .59809 L .03714 .60521 L .03761 .60809 L .03813 .61074 L .03908 .61441 L .03956 .61571 L .04006 .61676 L .04031 .61716 L .04057 .61751 L .04083 .61777 L .04106 .61795 L .04117 .61801 L s .04117 .61801 m .04122 .61803 L s .04226 .61803 m .04247 .61792 L .04271 .61775 L .04298 .61752 L .04345 .61697 L .04395 .61623 L .04495 .61435 L .04592 .61208 L .04684 .60956 L .04783 .60658 L .04878 .60347 L .04969 .60029 L .05067 .59676 L .0516 .59324 L .0526 .58939 L .05356 .58561 L .05448 .58192 L .05546 .57794 L .0564 .57409 L .05731 .57037 L .05828 .5664 L .05921 .56257 L .0602 .5585 L .06115 .55459 L .06207 .55084 L .06305 .54687 L .06398 .54304 L .06498 .53897 L .06594 .53504 L .06687 .53125 L .06785 .52721 L .0688 .52331 L .06971 .51955 L .07068 .51554 L .07162 .51167 L .07261 .50755 L .07357 .50357 L .07449 .49975 L .07547 .49568 L .07641 .49177 L .07731 .48801 L .07828 .48402 L .07921 .48018 L .0802 .47611 L .08115 .4722 L .08206 .46845 L .08304 .46448 L .08397 .46068 L .08497 .45665 L .08593 .45279 L Mistroke .08685 .44909 L .08783 .44518 L .08878 .44144 L .08969 .43786 L .09066 .43408 L .09159 .43046 L .09258 .42663 L .09354 .42298 L .09445 .41948 L .09543 .41579 L .09637 .41226 L .09737 .40854 L .09833 .40498 L .09926 .40158 L .10025 .39799 L .10119 .39457 L .10211 .3913 L .10308 .38784 L .10401 .38455 L .10501 .38107 L .10597 .37775 L .10689 .37458 L .10787 .37123 L .10881 .36804 L .10972 .36499 L .11069 .36177 L .11162 .3587 L .1126 .35547 L .11356 .35238 L .11447 .34943 L .11545 .34632 L .11639 .34335 L .11738 .34022 L .11834 .33723 L .11927 .33438 L .12025 .33137 L .1212 .3285 L .12211 .32576 L .12308 .32288 L .12401 .32012 L .125 .31721 L .12596 .31444 L .12688 .3118 L .12786 .309 L .1288 .30634 L .1298 .30354 L .13076 .30086 L .13169 .2983 L .13268 .29561 L .13363 .29304 L Mistroke .13454 .29059 L .13552 .288 L .13645 .28553 L .13745 .28293 L .13841 .28044 L .13933 .27807 L .14031 .27558 L .14126 .27319 L .14217 .27092 L .14313 .26852 L .14406 .26623 L .14505 .26382 L .14601 .26151 L .14693 .25932 L .1479 .257 L .14884 .25479 L .14984 .25246 L .1508 .25023 L .15173 .24811 L .15271 .24587 L .15366 .24374 L .15457 .2417 L .15554 .23955 L .15648 .2375 L .15747 .23534 L .15843 .23328 L .15935 .23132 L .16033 .22924 L .16127 .22726 L .16218 .22537 L .16314 .22338 L .16407 .22148 L .16506 .21948 L .16601 .21756 L .16693 .21574 L .1679 .21382 L .16884 .21198 L .16984 .21005 L .1708 .2082 L .17172 .20644 L .1727 .20458 L .17365 .20281 L .17456 .20112 L .17553 .19933 L .17646 .19763 L .17745 .19584 L .17841 .19412 L .17933 .19249 L .1803 .19077 L .18124 .18912 L Mistroke .18224 .18739 L .18321 .18574 L .18413 .18416 L .18512 .1825 L .18607 .18091 L .18698 .1794 L .18796 .1778 L .18889 .17628 L .18989 .17467 L .19084 .17314 L .19177 .17168 L .19275 .17014 L .19369 .16867 L .1946 .16726 L .19557 .16578 L .1965 .16437 L .19748 .16288 L .19844 .16146 L .19935 .16011 L .20033 .15868 L .20127 .15731 L .20227 .15587 L .20323 .1545 L .20415 .15319 L .20514 .15181 L .20608 .1505 L .207 .14924 L .20797 .14791 L .2089 .14665 L .20989 .14532 L .21085 .14404 L .21177 .14283 L .21275 .14155 L .21369 .14033 L .21459 .13916 L .21556 .13796 L .21649 .13687 L .21748 .13583 L .21843 .13497 L .21934 .13427 L .22031 .13367 L .22125 .13325 L .22176 .13307 L .22225 .13294 L .22248 .13289 L .22273 .13285 L .22296 .13281 L .22317 .13279 L .2234 .13277 L .22352 .13276 L Mistroke .22358 .13276 L .22364 .13276 L .2237 .13275 L .22375 .13275 L .2238 .13275 L .22382 .13275 L .22385 .13275 L .22388 .13275 L .22391 .13275 L .22393 .13275 L .22396 .13275 L .22399 .13275 L .22402 .13275 L .22405 .13275 L .22408 .13275 L .22414 .13275 L .22419 .13275 L .22431 .13276 L .22443 .13276 L .22456 .13277 L .22479 .13279 L .225 .13282 L .22545 .13288 L .22595 .13298 L .22693 .13324 L .22786 .13357 L .22886 .13398 L .22982 .13443 L .23075 .1349 L .23173 .13544 L .23268 .13598 L .23359 .1365 L .23456 .13707 L .23549 .13761 L .23648 .13818 L .23744 .13872 L .23836 .13922 L .23933 .13973 L .24028 .14019 L .24128 .14065 L .24224 .14106 L .24317 .14142 L .24415 .14177 L .2451 .14207 L .24602 .14232 L .24699 .14255 L .24793 .14274 L .24892 .14289 L .24942 .14296 L .24988 .143 L Mistroke .25036 .14305 L .2508 .14307 L .25104 .14309 L .25125 .1431 L .25149 .1431 L .25162 .14311 L .25175 .14311 L .25181 .14311 L .25187 .14311 L .25192 .14311 L .25198 .14311 L .25201 .14311 L .25204 .14311 L .25207 .14311 L .25209 .14311 L .25211 .14311 L .25213 .14311 L .25214 .14311 L .25216 .14311 L .25218 .14311 L .25219 .14311 L .25222 .14311 L .25225 .14311 L .25227 .14311 L .2523 .14311 L .25233 .14311 L .25238 .14311 L .25243 .14311 L .25255 .14311 L .25266 .14311 L .25279 .14311 L .25292 .1431 L .25316 .14309 L .25339 .14308 L .2536 .14307 L .25405 .14303 L .25455 .14298 L .25552 .14284 L .25645 .14265 L .25744 .14238 L .25839 .14206 L .25931 .14171 L .26029 .14127 L .26123 .1408 L .26223 .14024 L .26319 .13965 L .26412 .13904 L .26511 .13836 L .26605 .13766 L .26697 .13695 L Mistroke .26794 .13616 L .26887 .13538 L .26987 .13452 L .27083 .13367 L .27175 .13283 L .27273 .13192 L .27367 .13102 L .27458 .13015 L .27555 .12921 L .27648 .12829 L .27747 .12731 L .27842 .12636 L .27933 .12543 L .28031 .12445 L .28125 .12349 L .28225 .12248 L .28321 .1215 L .28413 .12055 L .28512 .11955 L .28606 .11858 L .28697 .11766 L .28794 .11667 L .28888 .11573 L .28987 .11473 L .29083 .11377 L .29174 .11285 L .29272 .11188 L .29367 .11095 L .29467 .10996 L .29563 .10902 L .29656 .10811 L .29755 .10716 L .2985 .10625 L .29941 .10538 L .30038 .10445 L .30132 .10357 L .30232 .10264 L .30327 .10175 L .3042 .1009 L .30518 .10001 L .30612 .09915 L .30703 .09833 L .308 .09747 L .30893 .09665 L .30992 .09578 L .31088 .09494 L .31179 .09415 L .31277 .09331 L .31371 .09252 L .31471 .09167 L Mistroke .31567 .09087 L .3166 .0901 L .31758 .08929 L .31853 .08852 L .31944 .08778 L .32041 .087 L .32135 .08626 L .32234 .08548 L .3233 .08473 L .32422 .08402 L .3252 .08326 L .32614 .08255 L .32705 .08186 L .32802 .08114 L .32894 .08045 L .32993 .07972 L .33088 .07903 L .3318 .07836 L .33277 .07767 L .33371 .077 L .33471 .0763 L .33567 .07563 L .33659 .07499 L .33758 .07431 L .33852 .07367 L .33943 .07306 L .3404 .07241 L .34133 .07179 L .34233 .07114 L .34328 .07052 L .3442 .06992 L .34518 .0693 L .34612 .0687 L .34712 .06807 L .34808 .06747 L .34901 .0669 L .35 .06629 L .35095 .06572 L .35186 .06517 L .35283 .06459 L .35377 .06403 L .35476 .06345 L .35572 .06289 L .35664 .06236 L .35762 .0618 L .35857 .06127 L .35948 .06076 L .36044 .06022 L .36137 .05971 L .36236 .05917 L Mistroke .36332 .05865 L .36423 .05816 L .36521 .05764 L .36615 .05714 L .36715 .05662 L .36811 .05612 L .36903 .05565 L .37002 .05514 L .37096 .05467 L .37188 .05421 L .37285 .05373 L .37378 .05327 L .37477 .05278 L .37573 .05232 L .37665 .05188 L .37763 .05142 L .37857 .05097 L .37948 .05055 L .38044 .0501 L .38137 .04968 L .38236 .04923 L .38331 .0488 L .38422 .04839 L .3852 .04796 L .38613 .04755 L .38713 .04711 L .38809 .0467 L .38901 .0463 L .38999 .04589 L .39094 .04549 L .39185 .04511 L .39282 .04471 L .39375 .04433 L .39474 .04393 L .3957 .04354 L .39661 .04318 L .39759 .04279 L .39853 .04242 L .39953 .04204 L .40049 .04166 L .40142 .04131 L .4024 .04094 L .40335 .04058 L .40427 .04024 L .40524 .03989 L .40617 .03956 L .40717 .03924 L .40812 .03899 L .40905 .03879 L .40951 .03871 L Mistroke .41003 .03865 L .41026 .03862 L .41051 .0386 L .41075 .03858 L .41097 .03856 L .4112 .03855 L .41132 .03855 L .41145 .03854 L .41152 .03854 L .41159 .03854 L .41165 .03854 L .41169 .03854 L .41172 .03854 L .41175 .03854 L .41177 .03854 L .4118 .03854 L .41182 .03854 L .41183 .03854 L .41186 .03854 L .41189 .03854 L .41192 .03854 L .41194 .03854 L .41196 .03854 L .41201 .03854 L .41205 .03854 L .41208 .03854 L .4122 .03854 L .41231 .03854 L .41242 .03855 L .41268 .03856 L .41292 .03857 L .41338 .03861 L .41388 .03866 L .41486 .0388 L .41581 .03898 L .41672 .03918 L .41768 .03943 L .41861 .0397 L .4196 .04001 L .42056 .04033 L .42148 .04065 L .42245 .041 L .42339 .04135 L .42439 .04171 L .42535 .04207 L .42628 .04241 L .42726 .04276 L .42821 .04309 L .42912 .0434 L .43009 .04372 L Mistroke .43103 .04402 L .43202 .04432 L .43298 .04459 L .4339 .04484 L .43488 .04509 L .43582 .04532 L .43673 .04552 L .43769 .04571 L .43862 .04588 L .43961 .04605 L .44056 .04619 L .44148 .04631 L .44245 .04642 L .44339 .04651 L .44438 .04659 L .44488 .04662 L .44534 .04664 L .44579 .04666 L .44602 .04667 L .44627 .04667 L .44639 .04667 L .44651 .04668 L .44662 .04668 L .44668 .04668 L .44673 .04668 L .4468 .04668 L .44683 .04668 L .44686 .04668 L .44688 .04668 L .44689 .04668 L .44692 .04668 L .44694 .04668 L .44696 .04668 L .44698 .04668 L .44699 .04668 L .44703 .04668 L .44705 .04668 L .44709 .04668 L .44712 .04668 L .44717 .04668 L .44724 .04668 L .44735 .04668 L .44748 .04668 L .4477 .04667 L .44794 .04667 L .4482 .04666 L .44867 .04664 L .44918 .04661 L .45018 .04654 L .45114 .04645 L Mistroke .45206 .04634 L .45305 .0462 L .45399 .04605 L .4549 .04589 L .45587 .0457 L .45681 .0455 L .4578 .04527 L .45876 .04504 L .45968 .04481 L .46065 .04455 L .4616 .04429 L .4625 .04403 L .46347 .04375 L .4644 .04347 L .46538 .04317 L .46633 .04288 L .46725 .04259 L .46822 .04228 L .46916 .04197 L .47015 .04165 L .47111 .04133 L .47204 .04103 L .47302 .0407 L .47396 .04039 L .47487 .04008 L .47584 .03976 L .47677 .03945 L .47776 .03912 L .47872 .0388 L .47963 .03849 L .48061 .03817 L .48155 .03786 L .48255 .03753 L .48351 .03721 L .48444 .03691 L .48542 .03659 L .48637 .03628 L .48728 .03598 L .48826 .03567 L .48919 .03538 L .49018 .03506 L .49114 .03476 L .49206 .03447 L .49304 .03417 L .49399 .03388 L .49489 .0336 L .49586 .03331 L .49679 .03303 L .49778 .03273 L .49873 .03245 L Mistroke .49964 .03218 L .50062 .0319 L .50156 .03162 L .50255 .03134 L .50351 .03106 L .50444 .0308 L .50542 .03053 L .50637 .03026 L .50728 .03001 L .50825 .02975 L .50918 .02949 L .51017 .02923 L .51113 .02897 L .51205 .02873 L .51302 .02847 L .51397 .02823 L .51497 .02797 L .51593 .02772 L .51686 .02749 L .51784 .02724 L .51879 .027 L .51971 .02678 L .52068 .02654 L .52162 .02631 L .52261 .02607 L .52357 .02584 L .52449 .02563 L .52547 .0254 L .52642 .02518 L .52733 .02497 L .52829 .02475 L .52922 .02453 L .53021 .02431 L .53117 .0241 L .53208 .0239 L .53306 .02368 L .534 .02348 L .535 .02327 L .53596 .02306 L .53688 .02287 L .53787 .02266 L .53882 .02246 L .53973 .02228 L .5407 .02208 L .54163 .02189 L .54263 .02169 L .54358 .0215 L .5445 .02132 L .54548 .02113 L .54642 .02095 L Mistroke .54733 .02077 L .54829 .02059 L .54922 .02041 L .55021 .02023 L .55116 .02005 L .55208 .01988 L .55305 .01971 L .55399 .01954 L .55498 .01936 L .55594 .01919 L .55687 .01903 L .55785 .01886 L .5588 .01869 L .55971 .01854 L .56067 .01837 L .56161 .01822 L .5626 .01805 L .56355 .01789 L .56447 .01774 L .56545 .01758 L .56639 .01743 L .56739 .01727 L .56835 .01712 L .56928 .01698 L .57026 .01682 L .57121 .01668 L .57212 .01654 L .5731 .01639 L .57403 .01625 L .57503 .0161 L .57598 .01596 L .5769 .01583 L .57788 .01568 L .57883 .01555 L .57974 .01542 L .5807 .01528 L .58163 .01515 L .58262 .01501 L .58357 .01488 L .58449 .01476 L .58546 .01463 L .5864 .0145 L .5874 .01437 L .58836 .01424 L .58929 .01412 L .59027 .01399 L .59122 .01387 L .59213 .01376 L .5931 .01364 L .59403 .01352 L Mistroke .59502 .0134 L .59598 .01328 L .5969 .01317 L .59788 .01308 L .59882 .013 L .5993 .01297 L .59982 .01294 L .60032 .01292 L .60056 .01291 L .60078 .0129 L .60102 .0129 L .60124 .01289 L .60135 .01289 L .60148 .01289 L .60155 .01289 L .60161 .01289 L .60167 .01289 L .60173 .01289 L .60176 .01289 L .60179 .01289 L .60182 .01289 L .60185 .01289 L .60188 .01289 L .60189 .01289 L .60191 .01289 L .60194 .01289 L .60197 .01289 L .60199 .01289 L .60202 .01289 L .60205 .01289 L .60208 .01289 L .60213 .01289 L .60218 .01289 L .60225 .01289 L .60232 .01289 L .60244 .01289 L .60268 .01289 L .6029 .0129 L .60315 .0129 L .60366 .01292 L .60412 .01294 L .60462 .01296 L .60561 .01303 L .60656 .0131 L .60748 .01319 L .60845 .01329 L .60939 .0134 L .61039 .01352 L .61135 .01365 L .61227 .01377 L Mistroke .61326 .01391 L .6142 .01404 L .61511 .01417 L .61608 .0143 L .61701 .01443 L .618 .01456 L .61896 .01469 L .61988 .01481 L .62086 .01493 L .6218 .01504 L .6228 .01516 L .62376 .01526 L .62469 .01536 L .62567 .01545 L .62662 .01554 L .62754 .01561 L .62851 .01569 L .62944 .01575 L .63044 .01582 L .6314 .01587 L .63232 .01592 L .6333 .01596 L .63424 .016 L .63515 .01603 L .63566 .01604 L .63612 .01605 L .63657 .01606 L .6368 .01606 L .63705 .01606 L .6373 .01606 L .63752 .01607 L .63763 .01607 L .63769 .01607 L .63775 .01607 L .6378 .01607 L .63786 .01607 L .63789 .01607 L .63792 .01607 L .63794 .01607 L .63797 .01607 L .638 .01607 L .63801 .01607 L .63803 .01607 L .63804 .01607 L .63806 .01607 L .63809 .01607 L .63812 .01607 L .63815 .01607 L .63818 .01607 L .63821 .01607 L Mistroke .63828 .01607 L .63835 .01607 L .63848 .01607 L .6386 .01607 L .63871 .01607 L .63896 .01606 L .63919 .01606 L .63941 .01606 L .6399 .01605 L .64039 .01604 L .64084 .01603 L .64179 .016 L .64271 .01597 L .64368 .01592 L .64462 .01587 L .64562 .01581 L .64659 .01575 L .64751 .01568 L .6485 .01561 L .64945 .01553 L .65036 .01545 L .65133 .01536 L .65227 .01527 L .65326 .01518 L .65422 .01508 L .65514 .01499 L .65612 .01488 L .65706 .01478 L .65797 .01468 L .65894 .01458 L .65987 .01447 L .66085 .01436 L .66181 .01425 L .66272 .01415 L .6637 .01404 L .66463 .01393 L .66563 .01382 L .66659 .01371 L .66752 .0136 L .6685 .01349 L .66945 .01338 L .67036 .01327 L .67133 .01316 L .67226 .01306 L .67325 .01294 L .67421 .01283 L .67513 .01273 L .6761 .01262 L .67705 .01252 L .67805 .0124 L Mistroke .67901 .0123 L .67994 .0122 L .68092 .01209 L .68188 .01199 L .68279 .01189 L .68376 .01178 L .6847 .01168 L .68569 .01158 L .68665 .01148 L .68757 .01138 L .68856 .01128 L .6895 .01118 L .69041 .01109 L .69138 .011 L .69231 .0109 L .6933 .0108 L .69425 .01071 L .69517 .01062 L .69614 .01053 L .69708 .01044 L .69808 .01034 L .69904 .01025 L .69997 .01016 L .70095 .01007 L .7019 .00999 L .70281 .0099 L .70378 .00981 L .70472 .00973 L .70571 .00964 L .70667 .00956 L .70759 .00948 L .70857 .00939 L .70951 .00931 L .71042 .00924 L .71138 .00915 L .71231 .00908 L .7133 .00899 L .71425 .00892 L .71517 .00884 L .71614 .00876 L .71708 .00869 L .71807 .00861 L .71903 .00853 L .71996 .00846 L .72094 .00838 L .72189 .00831 L .72279 .00824 L .72376 .00817 L .7247 .0081 L .72569 .00803 L Mistroke .72664 .00796 L .72756 .00789 L .72854 .00782 L .72948 .00775 L .73048 .00768 L .73144 .00761 L .73237 .00755 L .73335 .00748 L .7343 .00741 L .73522 .00735 L .73619 .00729 L .73712 .00722 L .73812 .00716 L .73908 .0071 L .74 .00704 L .74098 .00697 L .74192 .00691 L .74283 .00686 L .7438 .00679 L .74473 .00674 L .74571 .00668 L .74667 .00662 L .74758 .00656 L .74856 .0065 L .7495 .00645 L .7505 .00639 L .75146 .00633 L .75238 .00628 L .75337 .00622 L .75431 .00617 L .75522 .00612 L .75619 .00606 L .75713 .00601 L .75812 .00596 L .75908 .0059 L .75999 .00585 L .76097 .0058 L .76192 .00575 L .76292 .0057 L .76388 .00565 L .76481 .0056 L .7658 .00555 L .76675 .0055 L .76766 .00546 L .76863 .00541 L .76957 .00536 L .77057 .00531 L .77153 .00526 L .77245 .00522 L .77343 .00517 L Mistroke .77438 .00513 L .77528 .00509 L .77625 .00504 L .77718 .005 L .77817 .00495 L .77913 .00491 L .78005 .00487 L .78102 .00482 L .78196 .00478 L .78296 .00474 L .78392 .0047 L .78485 .00466 L .78583 .00462 L .78678 .00458 L .7877 .00454 L .78867 .00451 L .7896 .00449 L .79008 .00448 L .79059 .00447 L .79084 .00446 L .79109 .00446 L .79133 .00446 L .79155 .00446 L .79178 .00446 L .7919 .00446 L .79196 .00446 L .79202 .00446 L .79208 .00446 L .79213 .00446 L .79218 .00446 L .79223 .00446 L .79226 .00446 L .79229 .00446 L .79231 .00446 L .79234 .00446 L .79237 .00446 L .79239 .00446 L .7924 .00446 L .79243 .00446 L .79246 .00446 L .79249 .00446 L .79252 .00446 L .79258 .00446 L .79269 .00446 L .79282 .00446 L .79294 .00446 L .79315 .00446 L .79338 .00446 L .79385 .00446 L .79429 .00447 L Mistroke .79523 .00449 L .79622 .00451 L .79718 .00454 L .7981 .00457 L .79908 .00461 L .80003 .00465 L .80094 .00469 L .8019 .00473 L .80283 .00478 L .80382 .00482 L .80478 .00487 L .80569 .00492 L .80667 .00497 L .80761 .00501 L .80861 .00506 L .80957 .0051 L .81049 .00515 L .81148 .00519 L .81243 .00523 L .81334 .00527 L .81431 .0053 L .81524 .00534 L .81623 .00537 L .81719 .0054 L .81811 .00543 L .81909 .00545 L .82003 .00548 L .82094 .0055 L .8219 .00552 L .82283 .00554 L .82382 .00555 L .82477 .00556 L .82568 .00557 L .82619 .00558 L .82666 .00558 L .82714 .00559 L .8276 .00559 L .82783 .00559 L .82796 .00559 L .82808 .00559 L .82819 .00559 L .82829 .00559 L .8284 .00559 L .82846 .00559 L .82852 .00559 L .82855 .00559 L .82856 .00559 L .82858 .00559 L .82861 .00559 L .82864 .00559 L Mistroke .82867 .00559 L .82869 .00559 L .82872 .00559 L .82875 .00559 L .82878 .00559 L .82882 .00559 L .82888 .00559 L .82894 .00559 L .82899 .00559 L .8291 .00559 L .82921 .00559 L .82944 .00559 L .82968 .00559 L .8299 .00559 L .8304 .00558 L .83086 .00558 L .83136 .00558 L .83233 .00557 L .83327 .00555 L .83427 .00554 L .83523 .00552 L .83615 .0055 L .83713 .00548 L .83808 .00546 L .83899 .00543 L .83996 .0054 L .8409 .00538 L .84189 .00534 L .84284 .00531 L .84376 .00528 L .84474 .00525 L .84568 .00521 L .84668 .00518 L .84765 .00514 L .84857 .00511 L .84956 .00507 L .85051 .00503 L .85143 .005 L .8524 .00496 L .85334 .00492 L .85433 .00488 L .85529 .00484 L .85621 .00481 L .85719 .00477 L .85814 .00473 L .85905 .00469 L .86001 .00465 L .86095 .00462 L .86194 .00458 L .86289 .00454 L Mistroke .86381 .0045 L .86478 .00446 L .86572 .00443 L .86672 .00439 L .86768 .00435 L .86861 .00432 L .86959 .00428 L .87054 .00424 L .87145 .00421 L .87242 .00417 L .87336 .00413 L .87435 .0041 L .87531 .00406 L .87623 .00403 L .87721 .00399 L .87815 .00396 L .87906 .00393 L .88002 .00389 L .88095 .00386 L .88194 .00382 L .88289 .00379 L .88381 .00376 L .88478 .00373 L .88572 .0037 L .88672 .00366 L .88768 .00363 L .8886 .0036 L .88958 .00357 L .89053 .00354 L .89144 .00351 L .89241 .00348 L .89334 .00345 L .89433 .00341 L .89529 .00338 L .89621 .00336 L .89718 .00333 L .89812 .0033 L .89912 .00327 L .90009 .00324 L .90101 .00321 L .902 .00318 L .90295 .00315 L .90386 .00313 L .90483 .0031 L .90577 .00307 L .90677 .00305 L .90772 .00302 L .90864 .00299 L .90963 .00297 L .91057 .00294 L Mistroke .91148 .00292 L .91244 .00289 L .91337 .00287 L .91436 .00284 L .91532 .00282 L .91623 .00279 L .91721 .00277 L .91815 .00274 L .91915 .00272 L .92011 .00269 L .92103 .00267 L .92202 .00265 L .92296 .00262 L .92387 .0026 L .92484 .00258 L .92578 .00256 L .92677 .00253 L .92773 .00251 L .92865 .00249 L .92963 .00247 L .93057 .00245 L .93147 .00243 L .93244 .00241 L .93337 .00239 L .93435 .00236 L .9353 .00234 L .93622 .00232 L .93719 .0023 L .93813 .00228 L .93912 .00226 L .94008 .00224 L .94101 .00222 L .94199 .0022 L .94293 .00218 L .94384 .00217 L .94481 .00215 L .94574 .00213 L .94673 .00211 L .94769 .00209 L .94861 .00207 L .94958 .00205 L .95052 .00204 L .95152 .00202 L .95248 .002 L .95341 .00198 L .9544 .00197 L .95534 .00195 L .95626 .00193 L .95723 .00192 L .95816 .0019 L Mistroke .95916 .00188 L .96011 .00187 L .96104 .00185 L .96202 .00183 L .96296 .00182 L .96387 .0018 L .96483 .00179 L .96576 .00177 L .96675 .00175 L .9677 .00174 L .96862 .00172 L .96959 .00171 L .97053 .00169 L .97153 .00168 L .97249 .00166 L .97341 .00165 L .9744 .00164 L .97534 .00162 L .97619 .00161 L Mfstroke 1.5 Mabswid .02381 0 m .02474 .00552 L .02575 .01053 L .0267 .01442 L .02761 .01754 L .02859 .0203 L .02952 .02248 L .03052 .02439 L .03148 .02588 L .0324 .02705 L .03338 .02806 L .03433 .02883 L .03524 .02942 L .03621 .0299 L .03714 .03026 L .03761 .0304 L .03813 .03054 L .03908 .03072 L .03956 .03079 L .04006 .03084 L .04031 .03086 L .04057 .03088 L .04083 .03089 L .04106 .0309 L .04117 .0309 L .04129 .0309 L .04141 .03091 L .04146 .03091 L .04151 .03091 L .04154 .03091 L .04157 .03091 L .0416 .03091 L .04163 .03091 L .04165 .03091 L .04167 .03091 L .04168 .03091 L .0417 .03091 L .04172 .03091 L .04173 .03091 L .04177 .03091 L .0418 .03091 L .04181 .03091 L .04183 .03091 L .04186 .03091 L .04189 .03091 L .04195 .03091 L .04201 .03091 L .04213 .0309 L .04225 .0309 L .04247 .0309 L Mistroke .04271 .03089 L .04298 .03088 L .04345 .03085 L .04395 .03081 L .04495 .03072 L .04592 .0306 L .04684 .03048 L .04783 .03033 L .04878 .03017 L .04969 .03001 L .05067 .02984 L .0516 .02966 L .0526 .02947 L .05356 .02928 L .05448 .0291 L .05546 .0289 L .0564 .0287 L .05731 .02852 L .05828 .02832 L .05921 .02813 L .0602 .02793 L .06069 .02782 L .06115 .02773 L .0616 .02764 L .06172 .02761 L .06179 .0276 L .06184 .02759 L .06187 .02758 L .0619 .02758 L .06192 .0276 L .06195 .02764 L .06207 .02778 L .063 .02894 L .064 .03015 L .06495 .03132 L .06587 .03243 L .06685 .0336 L .06779 .03471 L .0687 .03578 L .06967 .0369 L .07059 .03797 L .07158 .0391 L .07253 .04017 L .07345 .0412 L .07442 .04228 L .07536 .04331 L .07635 .0444 L .07731 .04543 L .07824 .04642 L .07922 .04746 L Mistroke .08017 .04845 L .08108 .0494 L .08204 .05039 L .08298 .05135 L .08397 .05235 L .08492 .0533 L .08584 .05422 L .08682 .05518 L .08776 .05609 L .08876 .05706 L .08972 .05798 L .09065 .05886 L .09163 .05979 L .09258 .06067 L .09349 .06152 L .09446 .06241 L .0954 .06325 L .09639 .06415 L .09735 .065 L .09827 .06581 L .09925 .06667 L .1002 .06749 L .1011 .06827 L .10207 .0691 L .103 .06988 L .10399 .07071 L .10494 .0715 L .10586 .07226 L .10683 .07305 L .10777 .07381 L .10877 .07461 L .10973 .07537 L .11065 .0761 L .11164 .07687 L .11259 .0776 L .1135 .0783 L .11447 .07904 L .1154 .07974 L .11639 .08048 L .11735 .08119 L .11827 .08186 L .11924 .08258 L .12019 .08326 L .12119 .08397 L .12215 .08465 L .12308 .08531 L .12407 .08599 L .12502 .08665 L .12593 .08727 L .1269 .08793 L Mistroke .12784 .08856 L .12883 .08923 L .12979 .08986 L .13072 .09046 L .1317 .0911 L .13264 .09171 L .13355 .09229 L .13452 .0929 L .13545 .09348 L .13644 .0941 L .13739 .09468 L .13831 .09524 L .13929 .09583 L .14023 .0964 L .14123 .09699 L .14219 .09756 L .14311 .0981 L .1441 .09867 L .14505 .09921 L .14596 .09973 L .14693 .10028 L .14786 .1008 L .14886 .10135 L .14981 .10188 L .15073 .10238 L .15171 .10291 L .15266 .10341 L .15356 .10389 L .15453 .1044 L .15546 .10489 L .15645 .1054 L .1574 .10588 L .15831 .10635 L .15929 .10684 L .16023 .10731 L .16122 .1078 L .16218 .10827 L .16311 .10872 L .16409 .10919 L .16504 .10964 L .16594 .11007 L .16691 .11053 L .16785 .11096 L .16884 .11142 L .16979 .11186 L .17071 .11227 L .17169 .11271 L .17263 .11313 L .17363 .11357 L .17459 .11399 L Mistroke .17552 .11439 L .17651 .11482 L .17746 .11522 L .17837 .11561 L .17934 .11602 L .18028 .1164 L .18127 .11681 L .18223 .1172 L .18315 .11758 L .18413 .11797 L .18508 .11834 L .18598 .1187 L .18695 .11908 L .18788 .11944 L .18887 .11982 L .18982 .12018 L .19074 .12052 L .19172 .12089 L .19265 .12124 L .19365 .1216 L .19461 .12195 L .19554 .12229 L .19652 .12264 L .19747 .12297 L .19838 .12329 L .19935 .12363 L .20029 .12395 L .20128 .12429 L .20223 .12462 L .20315 .12493 L .20413 .12525 L .20507 .12556 L .20598 .12586 L .20695 .12617 L .20787 .12647 L .20886 .12679 L .20981 .12709 L .21073 .12738 L .2117 .12768 L .21264 .12797 L .21311 .12811 L .21363 .12827 L .21387 .12834 L .21401 .12838 L .21413 .12842 L .21419 .12844 L .21424 .12845 L .21427 .12846 L .2143 .1284 L .21432 .12822 L Mistroke .21435 .12805 L .21459 .12651 L .21553 .12069 L .21653 .11482 L .21749 .10945 L .21842 .10453 L .21941 .09957 L .22035 .09503 L .22127 .09087 L .22224 .08667 L .22317 .08283 L .22417 .07896 L .22512 .07541 L .22605 .07218 L .22703 .0689 L .22797 .06591 L .22888 .06318 L .22984 .06041 L .23077 .05787 L .23176 .05531 L .23271 .05297 L .23363 .05083 L .2346 .04867 L .23554 .04669 L .23654 .04468 L .2375 .04285 L .23842 .04118 L .23941 .03948 L .24035 .03793 L .24126 .03651 L .24223 .03506 L .24317 .03374 L .24416 .03241 L .24511 .03118 L .24603 .03006 L .24701 .02893 L .24795 .02788 L .24895 .02683 L .24992 .02586 L .25084 .02498 L .25131 .02454 L .25183 .02408 L .25207 .02387 L .2522 .02376 L .25226 .0237 L .25232 .02365 L .25235 .02362 L .25238 .0236 L .25241 .02361 L .25244 .02362 L Mistroke .25256 .02367 L .25278 .02375 L .25371 .02409 L .2547 .02446 L .25565 .02481 L .25657 .02514 L .25755 .0255 L .25849 .02583 L .25948 .02618 L .26045 .02652 L .26137 .02684 L .26236 .02717 L .2633 .02749 L .26422 .02779 L .26519 .02811 L .26612 .02842 L .26711 .02874 L .26807 .02904 L .26899 .02933 L .26997 .02963 L .27091 .02992 L .27182 .0302 L .27278 .03049 L .27371 .03077 L .2747 .03106 L .27565 .03133 L .27657 .0316 L .27754 .03188 L .27848 .03214 L .27947 .03242 L .28043 .03268 L .28136 .03294 L .28234 .0332 L .28329 .03346 L .2842 .0337 L .28516 .03396 L .2861 .0342 L .28709 .03446 L .28804 .0347 L .28896 .03494 L .28994 .03518 L .29088 .03542 L .29188 .03566 L .29284 .0359 L .29377 .03613 L .29475 .03636 L .2957 .03659 L .29662 .0368 L .29759 .03703 L .29852 .03725 L Mistroke .29952 .03748 L .30048 .0377 L .3014 .0379 L .30238 .03812 L .30332 .03833 L .30423 .03853 L .3052 .03874 L .30613 .03894 L .30711 .03916 L .30807 .03936 L .30898 .03955 L .30996 .03975 L .3109 .03995 L .31189 .04015 L .31286 .04035 L .31378 .04053 L .31476 .04073 L .31571 .04092 L .31662 .0411 L .31759 .04128 L .31853 .04146 L .31952 .04165 L .32047 .04183 L .32139 .04201 L .32237 .04219 L .32331 .04236 L .32431 .04254 L .32528 .04272 L .32621 .04289 L .32719 .04306 L .32814 .04323 L .32906 .04339 L .33003 .04356 L .33097 .04372 L .33196 .04389 L .33292 .04405 L .33385 .0442 L .33483 .04437 L .33577 .04452 L .33668 .04467 L .33765 .04483 L .33858 .04498 L .33957 .04513 L .34053 .04528 L .34144 .04543 L .34242 .04558 L .34336 .04572 L .34436 .04587 L .34532 .04602 L .34625 .04616 L Mistroke .34723 .0463 L .34818 .04644 L .34909 .04657 L .35006 .04671 L .351 .04685 L .35199 .04699 L .35295 .04712 L .35387 .04725 L .35485 .04738 L .35579 .04751 L .3567 .04764 L .35766 .04777 L .35859 .04789 L .35958 .04802 L .36053 .04814 L .36145 .04826 L .36242 .04839 L .36336 .04851 L .36436 .04863 L .36532 .04875 L .36624 .04887 L .36723 .04899 L .36817 .04911 L .36908 .04922 L .37005 .04933 L .37098 .04944 L .37198 .04956 L .37293 .04967 L .37385 .04978 L .37483 .04989 L .37577 .05 L .37677 .05011 L .37773 .05022 L .37866 .05032 L .37965 .05043 L .38059 .05053 L .38151 .05063 L .38248 .05073 L .38342 .05083 L .38441 .05094 L .38537 .05104 L .38629 .05113 L .38727 .05123 L .38822 .05133 L .38912 .05142 L .39009 .05152 L .39102 .05161 L .39201 .05171 L .39297 .0518 L .39388 .05189 L Mistroke .39486 .05198 L .3958 .05207 L .39679 .05216 L .39776 .05225 L .39868 .05234 L .39967 .05243 L .40061 .05251 L .40152 .05259 L .4025 .05268 L .40343 .05276 L .4039 .0528 L .40415 .05283 L .40442 .05285 L .40454 .05286 L .40466 .05287 L .40469 .05287 L .40473 .05288 L .40476 .05288 L .40478 .05284 L .40479 .0528 L .40492 .05246 L .40538 .05125 L .40632 .04887 L .40731 .04646 L .40827 .04426 L .40919 .04225 L .41018 .04022 L .41112 .03836 L .41203 .03666 L .413 .03494 L .41393 .03337 L .41492 .03178 L .41588 .03033 L .41679 .02901 L .41777 .02767 L .41871 .02645 L .41971 .02521 L .42067 .02408 L .4216 .02305 L .42259 .02201 L .42353 .02106 L .42445 .02019 L .42542 .0193 L .42635 .0185 L .42735 .01768 L .42831 .01694 L .42923 .01626 L .43021 .01556 L .43115 .01493 L .43206 .01435 L Mistroke .43302 .01377 L .43395 .01323 L .43494 .01269 L .43589 .01219 L .43681 .01173 L .43779 .01127 L .43872 .01085 L .43972 .01042 L .44068 .01003 L .44161 .00967 L .44208 .00949 L .44232 .0094 L .44259 .0093 L .44272 .00926 L .44279 .00923 L .44282 .00922 L .44285 .00921 L .44287 .00921 L .4429 .00921 L .44296 .00922 L .44308 .00924 L .44354 .00929 L .44446 .00941 L .44545 .00953 L .4464 .00965 L 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(i.e: C in place of c, ", Cell[BoxData[ \(TraditionalForm\`R\_n\)]], " in place of ", Cell[BoxData[ \(TraditionalForm\`r\_n\)]], ", etc.) " }], "Commentary"], Cell["Here is solved the case with \"mixed\" boundary conditions", "Text"], Cell[TextData[{ "Initial condition (IC): t =0, c= 0, 0\[LessEqual]z\[LessEqual]1\nBoundary \ conditions (BC): (The \"mixed\" boundary conditions): For z=0: c \ =\[Delta](z), for z=\[Infinity]: c=0, in other notation: ", Cell[BoxData[ \(TraditionalForm\`c(0) = \[Delta](t)\)]], " y ", Cell[BoxData[ \(TraditionalForm\`c(\[Infinity])\ = 0\)]], "." }], "Text"], Cell[TextData[{ "Eq(17) in ", StyleBox["Mathematica", FontSlant->"Italic"], " notation can be wrote " }], "Text"], Cell[BoxData[ \(D[c[z, t], t] == d\_n\ D[c[z, t], z, z] - \ D[c[z, t], z] - r\_n\ \(d\_n\) c[z, t] \((2)\)\)], "NumberedEquation"], Cell["IC: c[z,0]\[Rule]0", "Text"], Cell["\<\ Here is solved eq.(18) using Laplace Transform. (The right side terms of eq. \ 18 are denoted by eq1 and the left by eq2)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq1 = LaplaceTransform[D[c[z, t], t], \ t, s]\)], "Input", CellLabel->"In[121]:="], Cell[BoxData[ \(\(-c[z, 0]\) + s\ LaplaceTransform[c[z, t], t, s]\)], "Output", CellLabel->"Out[121]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq2 = LaplaceTransform[ d\_n\ D[c[z, t], z, z] - \ D[c[z, t], z] - r\_n\ d\_n\ c[z, t], \ t, s]\)], "Input", CellLabel->"In[122]:="], Cell[BoxData[ RowBox[{ RowBox[{"-", RowBox[{"LaplaceTransform", "[", RowBox[{ RowBox[{ SuperscriptBox["c", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(z, t\), "]"}], ",", "t", ",", "s"}], "]"}]}], "+", RowBox[{ RowBox[{"LaplaceTransform", "[", RowBox[{ RowBox[{ SuperscriptBox["c", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(z, t\), "]"}], ",", "t", ",", "s"}], "]"}], " ", \(d\_n\)}], "-", \(LaplaceTransform[c[z, t], t, s]\ d\_n\ r\_n\)}]], "Output", CellLabel->"Out[122]="] }, Open ]], Cell["\<\ Now we calculate their Laplace Transforms. The are denoted in capital letters \ (i.e: for c[z,0] the LT is C[z,s] ). BC and IC are applied.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq1LT\ = \ eq1 /. {LaplaceTransform[c[z, t], t, s] \[Rule] C[z, s]}\)], "Input", CellLabel->"In[123]:="], Cell[BoxData[ \(\(-c[z, 0]\) + s\ C[z, s]\)], "Output", CellLabel->"Out[123]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eq2LT", " ", "=", " ", RowBox[{"eq2", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"LaplaceTransform", "[", RowBox[{ RowBox[{ SuperscriptBox["c", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(z, t\), "]"}], ",", "t", ",", "s"}], "]"}], "\[Rule]", RowBox[{ SuperscriptBox["C", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(z, s\), "]"}]}], ",", RowBox[{ RowBox[{"LaplaceTransform", "[", RowBox[{ RowBox[{ SuperscriptBox["c", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(z, t\), "]"}], ",", "t", ",", "s"}], "]"}], "\[Rule]", RowBox[{ SuperscriptBox["C", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(z, s\), "]"}]}], ",", \(LaplaceTransform[c[z, t], t, s] \[Rule] C[z, s]\)}], "}"}]}]}]], "Input", CellLabel->"In[124]:="], Cell[BoxData[ RowBox[{\(\(-C[z, s]\)\ d\_n\ r\_n\), "-", RowBox[{ SuperscriptBox["C", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(z, s\), "]"}], "+", RowBox[{\(d\_n\), " ", RowBox[{ SuperscriptBox["C", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(z, s\), "]"}]}]}]], "Output", CellLabel->"Out[124]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(A26 = eq1LT\ == eq2LT /. {c[z, 0] \[Rule] 0}\)], "Input", CellLabel->"In[125]:="], Cell[BoxData[ RowBox[{\(s\ C[z, s]\), "==", RowBox[{\(\(-C[z, s]\)\ d\_n\ r\_n\), "-", RowBox[{ SuperscriptBox["C", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(z, s\), "]"}], "+", RowBox[{\(d\_n\), " ", RowBox[{ SuperscriptBox["C", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(z, s\), "]"}]}]}]}]], "Output",\ CellLabel->"Out[125]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LaplaceTransform[DiracDelta[z], z, s]\)], "Input", CellLabel->"In[126]:="], Cell[BoxData[ \(1\)], "Output", CellLabel->"Out[126]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq26b = DSolve[{s\ C[z] == \(-C[z]\)\ d\_n\ r\_n - \(C'\)[z] + d\_n\ \(C''\)[z], C[0] \[Equal] 1}, C[z], z]\)], "Input", CellLabel->"In[127]:="], Cell[BoxData[ \({{C[ z] \[Rule] \[ExponentialE]\^\(\(z\ \((1 - \@\(1 + 4\ s\ d\_n + 4\ \ d\_n\%2\ r\_n\))\)\)\/\(2\ d\_n\)\) - \[ExponentialE]\^\(\(z\ \((1 - \@\(1 + \ 4\ s\ d\_n + 4\ d\_n\%2\ r\_n\))\)\)\/\(2\ d\_n\)\)\ C[ 2] + \[ExponentialE]\^\(\(z\ \((1 + \@\(1 + 4\ s\ d\_n + 4\ d\ \_n\%2\ r\_n\))\)\)\/\(2\ d\_n\)\)\ C[2]}}\)], "Output", CellLabel->"Out[127]="] }, Open ]], Cell[TextData[{ " The integration constant C[2] can be calculated using the BC: ", Cell[BoxData[ \(TraditionalForm\`\(C\_b\)(\[Infinity])\ = 0\)]], " " }], "Text"], Cell[BoxData[ \(\(Solve[\(eq26b[\([1]\)]\)[\([1, 2]\)] \[Equal] 0, \ C[2]];\)\)], "Input", CellLabel->"In[128]:="], Cell[CellGroupData[{ Cell[BoxData[ \(c = C[2] /. %[\([1]\)] // Simplify\)], "Input", CellLabel->"In[129]:="], Cell[BoxData[ \(1\/\(1 - \[ExponentialE]\^\(\(z\ \@\(1 + 4\ s\ d\_n + 4\ d\_n\%2\ \ r\_n\)\)\/d\_n\)\)\)], "Output", CellLabel->"Out[129]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq26b /. {C[2] \[Rule] 0} // FullSimplify\)], "Input", CellLabel->"In[130]:="], Cell[BoxData[ \({{C[ z] \[Rule] \[ExponentialE]\^\(\(z - z\ \@\(1 + 4\ d\_n\ \((s + \ d\_n\ r\_n)\)\)\)\/\(2\ d\_n\)\)}}\)], "Output", CellLabel->"Out[130]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Cout[s_] = C[z] /. %[\([1]\)]\)], "Input", CellLabel->"In[131]:="], Cell[BoxData[ \(\[ExponentialE]\^\(\(z - z\ \@\(1 + 4\ d\_n\ \((s + d\_n\ r\_n)\)\)\)\/\ \(2\ d\_n\)\)\)], "Output", CellLabel->"Out[131]="] }, Open ]], Cell[TextData[{ "Now it can be evaluated ", Cell[BoxData[ \(\(\(\ \)\(\(C\_out\) \((z, t)\)\(\ \)\)\)\)]], " taken the inverse laplace transform " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cOut[z_, t_] = InverseLaplaceTransform[Cout[s], s, t] // FullSimplify\)], "Input", CellLabel->"In[132]:="], Cell[BoxData[ \(\(\[ExponentialE]\^\(\(-\(\((t - z)\)\^2\/\(4\ t\ d\_n\)\)\) - t\ d\_n\ \ r\_n\)\ z\)\/\(2\ \@\[Pi]\ \@\(t\^3\/d\_n\%3\)\ d\_n\%2\)\)], "Output", CellLabel->"Out[132]="] }, Open ]], Cell["When z=1 the solution is:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cOut[1, t]\)], "Input", CellLabel->"In[133]:="], Cell[BoxData[ \(\[ExponentialE]\^\(\(-\(\((\(-1\) + t)\)\^2\/\(4\ t\ d\_n\)\)\) - t\ \ d\_n\ r\_n\)\/\(2\ \@\[Pi]\ \@\(t\^3\/d\_n\%3\)\ d\_n\%2\)\)], "Output", CellLabel->"Out[133]="] }, Open ]], Cell[TextData[{ "This solution has a little difference with eq.(49) Roberts-Rowland J. \ Pharmacokim.Biopharm 14: 227-260 (1986): ", Cell[BoxData[ \(r\_n\)]], " was removed of the exponent. This mistake has not consequences when ", Cell[BoxData[ \(r\_n\)]], "=0. as happen in Fig 3.(B) of Roberts-Rowland paper" }], "Text"], Cell[TextData[{ "Here is shown this Fig 3.(B) obtain with our method ", Cell[BoxData[ \({d\_n \[Rule] {0.01, 0.1, 0.2, 1, 2}, r\_n \[Rule] 0\)]], " " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[ Evaluate[cOut[1, t] /. {d\_n \[Rule] {0.01, 0.1, 0.2, 1, 2}, r\_n \[Rule] 0}], {t, 0, 4}, \ PlotRange \[Rule] {0, 4}]\)], "Input", CellLabel->"In[134]:="], 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 0.238095 0 0.154508 [ [.2619 -0.0125 -3 -9 ] [.2619 -0.0125 3 0 ] [.5 -0.0125 -3 -9 ] [.5 -0.0125 3 0 ] [.7381 -0.0125 -3 -9 ] [.7381 -0.0125 3 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Metetra .662 .741 .909 r .25184 .40529 .27792 .43027 .31753 .41679 .2918 .39164 Metetra .657 .741 .912 r .2918 .39164 .31753 .41679 .35774 .40312 .33234 .37826 Metetra .645 .743 .921 r .33234 .37826 .35774 .40312 .39856 .38927 .37348 .36594 Metetra .611 .749 .943 r .37348 .36594 .39856 .38927 .44002 .3753 .41521 .3562 Metetra .518 .759 .984 r .41521 .3562 .44002 .3753 .48212 .36144 .45763 .35128 Metetra .209 .686 .935 r .45763 .35128 .48212 .36144 .52489 .34816 .50094 .35327 Metetra .34 0 0 r .50094 .35327 .52489 .34816 .56839 .33642 .54547 .36232 Metetra .515 .038 .06 r .54547 .36232 .56839 .33642 .61277 .32773 .59149 .37468 Metetra .51 .104 .207 r .59149 .37468 .61277 .32773 .65825 .32384 .63894 .38249 Metetra .444 .069 .232 r .63894 .38249 .65825 .32384 .70515 .32584 .6872 .37694 Metetra .29 0 .138 r .6872 .37694 .70515 .32584 .75372 .33277 .73544 .35361 Metetra .73544 .35361 .75372 .33277 .80388 .34064 .7831 .31553 Metetra .838 .837 .845 r .7831 .31553 .80388 .34064 .85507 .34301 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