In[]:=
dtpts1a={{0,8.00},{7,9.83},{14,12.2},{21,13.7},{28,14.2},{35,14.1},{42,13.7},{49,13.2},{56,12.6},{63,12.1},{70,11.6},{77,11.1},{84,10.7},{91,10.4},{98,10.1},{105,9.80},{112,9.56},{119,9.36},{126,9.18}};(*RutherfordE3PointsGeneratedwith:N0=8.0,Pf=10,Pk=1,k1=0.098,k2=0.14,k3=0.02(perfect)*)dtpts1b={{0,9.00},{7,11.1},{14,12.9},{21,13.3},{28,12.8},{35,12.1},{42,11.4},{49,10.9},{56,10.4},{63,10.1},{70,9.79},{77,9.59},{84,9.44},{91,9.33},{98,9.24},{105,9.18},{112,9.14},{119,9.10},{126,9.08}};(*RutherfordE3PointsGeneratedwith:N0=9,Pf=8,Pk=1.19,k1=0.20,k2=0.169,k3=0.050(perfect)*)dtpts1c={{0,7.69},{7,9.02},{14,9.03},{21,8.45},{28,7.68},{35,6.89},{42,6.15},{49,5.49},{56,4.91},{63,4.40},{70,3.96},{77,3.57},{84,3.23},{91,2.94},{98,2.69},{105,2.47},{112,2.28},{119,2.11},{126,1.96}};dtpts1d={{0,7.81},{7,11.00},{14,11.8},{15.38,11.875},{21,11.6},{28,10.9},{35,10.0},{42,9.11},{49,8.22},{56,7.38},{63,6.62},{70,5.92},{77,5.30},{84,4.74},{91,4.24},{98,3.79},{105,3.38},{112,3.02},{119,2.70}};(*RutherfordBtoacure*)(*ExcellentcurvefittingwiththissimplifiedversionofRutherford'sequation:Clear[t,E3Rjthfrd,t,N0,Pf,Pk,k1,k2];E3Rjthfrd[t_,N0_,Pf_,Pk_,k1_,k2_]:=N0+((k1×Pf)/(k2-k1))×((Exp[-(k1×(t-Pk))])-(Exp[-(k2×(t-Pk))]));*)
In[]:=
dtptsX={{60,1},{65,1},{70,1},{75,1},{80,1},{85,1},{90,1},{95,1},{100,1},{105,1},{110,1},{115,1},{120,1},{125,1}};dtptsY={{60,3},{65,3},{70,3},{75,3},{80,3},{85,3},{90,3},{95,3},{100,3},{105,3},{110,3},{115,3},{120,3},{125,3}};(*Benefactors?DonationsarewelcomeatWittenbergUniversityinSpringfieldOhio.PayabletotheSaintAugustineScholarsFoundation.A501(c)(3)charitablescholarshipfoundation.Contributionsare,ofcoursetaxdeductible.Thanks,AlanWhite,President,ArmillaryEquations,LLC*)dtplt1a=ListPlot[dtpts1a,FrameTrue,GridLinesAutomatic,PlotRange{{0,118},{0,15}},ImageSize{575,366},JoinedTrue,MeshAll,PlotStyle{RGBColor["Red"],AbsolutePointSize[10]},AspectRatio2/π,FrameLabel->{Style["Time (days)",Bold,Blue],Style["ProstateAg",Bold,Blue]},LabelStyleDirective[Blue,Large]];dtplt1b=ListPlot[dtpts1b,FrameTrue,GridLinesAutomatic,PlotRange{{0,118},{0,15}},ImageSize{575,366},JoinedTrue,MeshAll,PlotStyle{RGBColor["Blue"],AbsolutePointSize[10]},AspectRatio2/π,FrameLabel->{Style["Time (days)",Bold,Blue],Style["ProstateAg",Bold,Blue]},LabelStyleDirective[Blue,Large]];dtplt1c=ListPlot[dtpts1c,Frame->True,GridLines->Automatic,PlotRange->{{0,118},{0,15}},ImageSize->{575,366},Joined->True,Mesh->All,PlotStyle->{RGBColor["Blue"],AbsolutePointSize[10]},AspectRatio->2/π,FrameLabel->{Style["Time (days)",Bold,Blue],Style["ProstateAg",Bold,Blue]},LabelStyle->Directive[Blue,Large]];dtplt1d=ListPlot[dtpts1d,FrameTrue,GridLinesAutomatic,PlotRange{{0,118},{0,15}},ImageSize{575,366},JoinedTrue,MeshAll,PlotStyle{RGBColor["Purple"],AbsolutePointSize[10]},AspectRatio2/π,FrameLabel->{Style["Time (days)",Bold,Blue],Style["ProstateAg"]},LabelStyleDirective[Blue,Large]];dtpltX=ListPlot[dtptsX,FrameTrue,GridLinesAutomatic,PlotRange{{0,118},{0,15}},ImageSize{575,366},JoinedFalse,MeshAll,PlotStyle{RGBColor["Aqua"],AbsolutePointSize[6]},AspectRatio2/π,FrameLabel->{Style["Time (days)",Bold,Blue],Style["ProstateAg"]},LabelStyleDirective[Blue,Large]];dtpltY=ListPlot[dtptsY,FrameTrue,GridLinesAutomatic,PlotRange{{0,118},{0,15}},ImageSize{575,366},JoinedFalse,MeshAll,PlotStyle{RGBColor["Aqua"],AbsolutePointSize[6]},AspectRatio2/π,FrameLabel->{Style["Time (days)",Bold,Blue],Style["ProstateAg"]},LabelStyleDirective[Blue,Large]];RutherfordEC:=N0+(Pf×k1×k2)×((Exp[-(k1×(t÷Pk))])÷((k2-k1)×(k3-k1))+(Exp[-(k2×(t÷Pk))])÷((k1-k2)×(k3-k2))+(Exp[-(k3×(t÷Pk))])÷((k1-k3)×(k2-k3)));(*RutherfordECfromBatemanH(1918)DifferentialEquationspage:43Seealso:RutherfordE,ChadwickJ,EllisCD(1930)RadiationsfromRadioactiveSubstancespages:10-15equations5and6,Ctrajectory,Figure:2,page:12.(1951Cambridgereprint)*)Clear[t,ECRthfrd,t,N0,Pf,Pk,k3]ECRthfrd[t_,N0_,Pf_,Pk_,k3_]:=N0+(Pf×0.0985×0.0155)×((Exp[-(0.0985×(t-Pk))])÷((0.0155-0.0985)×(k3-0.0985))+(Exp[-(0.0155×(t-Pk))])÷((0.0985-0.0155)×(k3-0.0155))+(Exp[-(k3×(t-Pk))])÷((0.0985-k3)×(0.0155-k3)));
In[]:=
Manipulate[Show[Plot[ECRthfrd[t,N0,Pf,Pk,k3],{t,0,118},FrameTrue,GridLinesAutomatic,AxesOrigin{0,0},PlotRange{{0,118},{0,15}},Prolog{LightBlue,Rectangle[{0,0},{59.4,15}]},(*Note:RadioactiveIodine125halflife=59.4days*)ImageSize{575,366},PlotStyle{RGBColor[0,1,0],Thickness[0.01]},AspectRatio2.0/π,FrameLabel{Style["Time (days)",Bold,Blue],Style["ProstateAg",Bold,Blue]},LabelStyleDirective[Blue,Large]],dtpltX,dtpltY,dtplt1a,dtplt1b,dtplt1c,dtplt1d],{{N0,11.85,Style["Initial Number (N0)",Blue,Bold,Medium]},-10.0,20.0,0.01,ControlTypeSlider,ImageSizeMedium,DefaultBaseStyle{Blue,Bold,Medium},Appearance"Labeled"},{{Pf,-12.02,Style["Power Factor (Pf)",Blue,Bold,Medium]},-25.0,10.0,0.001,ControlTypeSlider,ImageSizeMedium,DefaultBaseStyle{Blue,Bold,Medium},Appearance"Labeled"},{{Pk,15.250,Style["Peak Location (Pk)",Blue,Bold,Medium]},-20.0,30.0,0.0010,ControlTypeSlider,ImageSizeMedium,DefaultBaseStyle{Blue,Bold,Medium},Appearance"Labeled"},{{k1,0.098541,Style["k1",Blue,Bold,Medium]},0.0001,0.750,0.0000010,ControlTypeSlider,ImageSizeMedium,DefaultBaseStyle{Blue,Bold,Medium},Appearance"Labeled"},{{k2,0.0155068,Style["k2",Blue,Bold,Medium]},0.0010,0.020,0.0000010,ControlTypeSlider,ImageSizeMedium,DefaultBaseStyle{Blue,Bold,Medium},Appearance"Labeled"},{{k3,0.0000119,Style["k3",Blue,Bold,Medium]},0.00000010,0.0000250,0.000000005,ControlTypeSlider,ImageSizeMedium,DefaultBaseStyle{Blue,Bold,Medium},Appearance"Labeled"},PaneledTrue,ImageMargins0]
Out[]=
,dtpltX,dtpltY,dtplt1a,dtplt1b,dtplt1c,dtplt1d.
,dtpltX,dtpltY,dtplt1a,dtplt1b,dtplt1c,dtplt1d.
In[]:=
KinetE3R=NonlinearModelFit[dtpts1d,ECRthfrd[t,N0,Pf,Pk,k3],{{N0,11.85},{Pf,-12.02},{Pk,15.25},{k3,0.0000119}},t,Method"Automatic"]
Out[]=
FittedModel
In[]:=
Normal[KinetE3R]
Out[]=
11.8518-0.0183555(122.331-777.901+655.57)
-0.0985(-15.2548+t)
-0.0155(-15.2548+t)
-0.0000119243(-15.2548+t)
In[]:=
KinetE3R["BestFitParameters"]