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= Example 1: Estimate Input Parameters, Rate Constants Known =
+
== Official Examples ==
−
Here we assume that the input function has the form
 
  
−
Cp = (p1t - p2 - p3) exp(-p4t) + p2 exp(-p5t) + p3 exp(-p6t)
+
* [http://comkat.case.edu/index.php?title=Support:Documents:Examples:Estimate_physiological_parameters_using_a_physiologically_based_model A physiologically based model]
  
−
where Cp is the plasma concentration of 18FDG, pi are the input function parameters to be estimated, and t is time. (This form comes from Feng, Huang, Wang. "Models for Computer simulation Studies of Input Functions for Tracer Kinetic Modeling with Positron Emission Tomography", International Journal of Biomedical Computing, 32(2):95-110, March 1993.)
+
* [http://comkat.case.edu/index.php?title=Support:Documents:Examples:Estimate_Parametric_Image_with_Matlab_Distributed_Computing_Server Estimate Parametric Image with Matlab Distributed Computing Server]
  
−
'''Step 1''' To set this up in MATLAB for simulation, create the following function and put it in a file called refCp.m:
+
* [http://comkat.case.edu/index.php?title=Support:Documents:Examples:Model_Corrected_Input_Function Model Corrected Input Function]
  
−
function [fval, jac] = refCp(p, t)
+
* [http://comkat.case.edu/index.php?title=Support:Documents:Examples:Estimate_Change_of_Neurotransmitter Estimate Changes of Neurotransmitter after a Stimulus (ntPET)]
−
  if (nargout > 0),
 
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        t = t(:);  % make it a column vector
 
−
        fval = (p(1)*t - p(2) - p(3)) .* exp(-p(4)*t) + p(2) * exp(-p(5)*t) + p(3) * exp(-p(6)*t);
 
−
        if (nargout > 1),
 
−
            jac = [t .* exp(-p(4)*t) ...
 
−
                  -exp(-p(4)*t) + exp(-p(5)*t) ...
 
−
                  -exp(-p(4)*t) + exp(-p(6)*t) ...
 
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                  -t .* (p(1)*t - p(2) - p(3)) .* exp(-p(4)*t) ...
 
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                  -t * p(2) .* exp(-p(5)*t) ...
 
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                  -t * p(3) .* exp(-p(6)*t)];
 
−
        end
 
−
    end
 
−
return
 
  
−
This function can be used to plot an example input curve using these commands
+
* [http://comkat.case.edu/index.php?title=Support:Documents:Examples:Estimate_Input_Delay_and_Rate_Constants Estimate Input Delay and Rate Constants]
  
−
    t=0:.1:60;
+
* [http://comkat.case.edu/index.php?title=Support:Documents:Examples:Estimate_Input_Parameters%2C_Rate_Constants_Known Estimate Input Parameters, Rate Constants Known]
−
    pin = [28 0.75 0.70 4.134 0.1191 0.01043];
 
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    plot(t,refCp(pin,t))
 
−
    set(gca,'FontSize',6);
 
−
    axis([0 60 0 5]);
 
−
    xlabel('Minutes');
 
−
    ylabel('pmol/ml');
 
  
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[[Image:inputExampleFigure.png]]
+
* [http://comkat.case.edu/index.php?title=Support:Documents:Examples:Estimate_Input_Parameters_Simultaneously_with_Flow Estimate Input Parameters Simultaneously with Flow]
  
−
Note that he above function, refCp.m not only calculates values Cp, it also optionally calculates values for the derivative of Cp with respect to parameter vector p. These will be needed later since to fit the input function, it is important to quantify how changes in values of p will effects Cp values which, in turn, effect model-predicted tissue concentration reflected in voxels or regions of interest.
+
* [http://comkat.case.edu/index.php?title=Support:Documents:Examples:Simplified_reference_tissue_method Simplified reference tissue method]
  
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'''Step 2''' Create an FDG model
+
* [http://comkat.case.edu/index.php?title=Support:Documents:Examples:Glucose_Minimal_Model Glucose Minimal Model]
  
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    cm = compartmentModel;
+
* [http://comkat.case.edu/index.php?title=Support:Documents:Examples:Enzyme-substrate_%28Michaelis-Menten%29_kinetics Enzyme-substrate (Michaelis-Menten) kinetics]
−
 
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    %        k1    k2      k3    k4
 
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    ktrue=[0.1020 0.1300 0.0620 0.0068];
 
−
 
−
    % define the parameters
 
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    cm = addParameter(cm, 'k1',    ktrue(1));    % 1/min
 
−
    cm = addParameter(cm, 'k2',    ktrue(2));    % 1/min
 
−
    cm = addParameter(cm, 'k3',    ktrue(3));    % ml/(pmol*min)
 
−
    cm = addParameter(cm, 'k4',    ktrue(4));    % 1/min
 
−
    cm = addParameter(cm, 'sa',    75);          % specific activity of injection, kBq/pmol
 
−
    cm = addParameter(cm, 'dk',    log(2)/109.8); % radioactive decay
 
−
    cm = addParameter(cm, 'PV',    1);            % (none)
 
−
 
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    % define input function parameter vector
 
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    cm = addParameter(cm, 'pin', [28; 0.75; 0.70; 4.134; 0.1191; 0.01043]);
 
−
 
−
    % define compartments
 
−
    cm = addCompartment(cm, 'Ce', '');
 
−
    cm = addCompartment(cm, 'Cm', '');
 
−
    cm = addCompartment(cm, 'Junk', '');
 
−
 
−
    % define plasma input function
 
−
    % specifying function as refCp with parameters pin
 
−
    cm = addInput(cm, 'Cp', 'sa', 'dk', 'refCp', 'pin'); % plamsa pmol/ml
 
−
 
−
    % connect inputs and compartments
 
−
    cm = addLink(cm, 'L', 'Cp', 'Ce', 'k1');
 
−
    cm = addLink(cm, 'K', 'Ce', 'Junk', 'k2');
 
−
    cm = addLink(cm, 'K', 'Ce', 'Cm', 'k3');
 
−
    cm = addLink(cm, 'K', 'Cm', 'Ce', 'k4');
 
−
 
−
    % specify scan begin and end times
 
−
    ttt=[ ones(6,1)*5/60; ...   %  6 frames x  5  sec
 
−
          ones(2,1)*15/60; ...  %  2 frames x 15  sec
 
−
          ones(6,1)*0.5;...      %  6 frames x  0.5 min
 
−
          ones(3,1)*2;...        %  3 frames x  2  min
 
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          ones(2,1)*5;...        %  2 frames x  5  min
 
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          ones(10,1)*10];        % 10 frames x 10  min
 
−
    scant = [[0;cumsum(ttt(1:(length(ttt)-1)))] cumsum(ttt)];
 
−
    cm = set(cm, 'ScanTime', scant);
 
−
 
 
−
    % define an output which is sum of Ce and Cm adjusted to account
 
−
    % for partial volume (PV).  Ignore vascular activity.
 
−
    Wlist = {...
 
−
        'Ce', 'PV';
 
−
        'Cm', 'PV'};
 
−
    Xlist = {};
 
−
    cm = addOutput(cm, 'PET', Wlist, Xlist);
 
−
 
−
    % solve model and generate example output
 
−
    [PET, PETindex]=solve(cm);
 
−
    plot(0.5*(PET(:,1)+PET(:,2)),PET(:,3), '.');
 
−
    set(gca,'FontSize',6);
 
−
    axis([0 60 0 100])
 
−
    xlabel('Minutes');
 
−
    ylabel('KBq/ml');
 
  
−
[[Image:modelOutput_1.png]]
+
* [http://comkat.case.edu/index.php?title=Support:Documents:Examples:FDG_with_Time-varying_Rate_Constants FDG with Time-varying Rate Constants]
−
 
−
'''Step 3''' Use model output as perfect "data"
 
  
−
    data = PET(:,3);
+
* [http://comkat.case.edu/index.php?title=3D_Reslicing_using_COMKAT_image_tool_(basic) 3D Slicing using COMKAT Image Tool (basic)]
  
−
'''Step 4''' Fit the "data"
+
* [http://comkat.case.edu/index.php?title=Support%3ADocuments%3AExamples%3A_Reslicing_using_COMKAT_image_tool_(advanced) 3D Slicing using COMKAT Image Tool (advanced)]
  
−
    % tell model abbout data to be fit
 
−
    cm = set(cm, 'ExperimentalData', data);
 
−
 
−
    % specify parameters to be adjusted in fitting
 
−
    cm = addSensitivity(cm, 'pin');
 
−
 
−
    % set parameter values initial guess, lower and upper bounds
 
−
    pinit = [ 10; 0.4;  0.4;  3;  0.05; 0.01];
 
−
    plb =  [ 10; 0.1;  0.1;  1;  0.05; 0.001];
 
−
    pub =  [100; 2. ;  2. ; 10;  1.  ; 0.05];
 
−
 
−
    % actually do the fitting
 
−
    pfit = fit(cm, pinit, plb, pub);
 
−
 
−
Values of parameter pfit estimates are [28.0167; 0.7652; 0.7059; 4.1586; 0.1257; 0.0105] and are reasonably close to the true values (pxEval(cm, 'pin')) [28; 0.75; 0.70; 4.134; 0.1191; 0.01043]
 
  
−
'''Step 5''' Examine model outputs for "data", initial guess, and fit
+
== User-contributed Examples ==
  
−
    PETfit = solve(set(cm, 'ParameterValue', 'pin', pfit));
+
* [http://comkat.case.edu/index.php?title=Support:Documents:Examples:Instructions_for_Adding_Your_Own_Example Instructions for Adding Your Own Example]
−
    PETinit = solve(set(cm, 'ParameterValue', 'pin', pinit));
 
−
    t = 0.5*(PET(:,1)+PET(:,2));
 
−
    plot(t,data,'.', t, PETinit(:,3), t, PETfit(:,3));
 
−
    set(gca,'FontSize',8);
 
−
    axis([0 60 0 75])
 
−
    xlabel('Minutes');
 
−
    ylabel('KBq/ml');
 
−
    ah=legend('Data', 'Initial Guess', 'Fit','Location','SouthEast');
 
−
    set(ah,'FontSize',6,'Box','off');
 
−
 
 
−
[[Image:inputFitExample.png]]
 
−
 
 
−
= Simplified reference tissue method =
 
−
 
 
−
This example demonstrates a COMKAT implementation of the simplified reference tissue model (SRTM) described by Lammertsma and Hume [Neuroimage. 4(3 Pt 1), 153-158 (1996)]. By simultaneously analyzing time-activity curves from areas with and without receptors, the SRTM provides a means to estimate the binding potential which is an index of receptor concentration and ligand affinity.
 
−
 
 
−
==Background==
 
−
 
 
−
Input functions are a fundamental component of pharmacokinetic modeling. They describe the time-course of (radio)pharmaceutical which is accessible to the extravascular space. In PET receptor studies the arterial plasma concentration of a ligand is taken as the input function. As there is desire to obviate blood sampling, the SRTM is a popular method used to analyze ligand-receptor data.
 
−
 
 
−
The typical model equations used for a region containing receptors include state equations for free (F) and bound (B) ligand:
 
−
<tr>
 
−
    <td width="507">dF/dt = k1 Cp &#8211; (k2+k3)F + k4 B </td>
 
−
    <td width="105">Eq 1 </td>
 
−
  </tr>
 
−
  <tr>
 
−
    <td width="507">dB/dt = k3 F &#8211; k4 B</td>
 
−
    <td width="105">Eq 2 </td>
 
−
  </tr>
 
−
where Cp is the arterial plasma input function and tracer conditions are assumed (B << Bmax) so that k3 = kon (Bmax – B) ≈ kon Bmax.
 
−
 
 
−
Model Figure
 
−
<br>[[Image:ModelFigure.PNG]]
 
−
 
 
−
In a region devoid of receptors, k3 (and therefore B) is zero so Eq 1 may be simplified. Denoting this region as the reference region, or rr, the equation is
 
−
<table width="622" border="0">
 
−
  <tr>
 
−
    <td width="507" height="52">dF<sub>rr</sub>/dt = k1 Cp &#8211; k2<sub>rr</sub> F<sub>rr</sub>  </td>
 
−
    <td width="105">Eq 3 </td>
 
−
  </tr>
 
−
</table>
 
−
 
 
−
This may be rearranged to obtain an expression for the plasma input
 
−
<table width="622" border="0">
 
−
  <tr>
 
−
    <td width="507" height="52"> Cp = Cpn / k1<sub>rr</sub></td>
 
−
    <td width="105">Eq 4 </td>
 
−
  </tr>
 
−
</table>
 
−
 
 
−
where
 
−
<table width="622" height="66" border="0">
 
−
  <tr>
 
−
    <td width="507" height="52"> Cpn = (dF<sub>rr</sub>/dt + k2<sub>rr</sub> F<sub>rr</sub>)</td>
 
−
    <td width="105">Eq 5 </td>
 
−
  </tr>
 
−
</table>
 
−
 
 
−
is the normalized plasma input function.
 
−
 
 
−
This expression (Eq 4) for Cp may be substituted into the equations for regions that have receptors, aka, target tissue which we denote as tt
 
−
<table width="622" border="0">
 
−
  <tr>
 
−
    <td width="507" height="52">dF<sub>tt</sub>/dt = (k1<sub>tt</sub>/k1<sub>rr</sub>) Cpn &#8211; (k2<sub>tt</sub>+k3<sub>tt</sub>)F<sub>tt</sub> + k4<sub>tt</sub> B<sub>tt</sub> </td>
 
−
    <td width="105">Eq 6 </td>
 
−
  </tr>
 
−
  <tr>
 
−
    <td width="507" height="47">dB<sub>tt</sub>/dt = k3<sub>tt</sub> F<sub>tt</sub> &#8211; k4<sub>tt</sub> B<sub>tt</sub></td>
 
−
    <td width="105">Eq 7 </td>
 
−
  </tr>
 
−
</table>
 
−
 
 
−
Data from target tissue and from reference regions could be, in theory, simultaneously fit using the model embodied in Eq. 4 to 6. In practice there is often an issue of identifiability. Hence, assumptions are made to reduce the number of parameters to estimate. One assumption is that the equilibrium concentration ratio between plasma and tissue free space is the same in the reference region and target tissue:
 
−
<table width="622" border="0">
 
−
  <tr>
 
−
    <td width="507" height="52">k1<sub>tt</sub>/  k2<sub>tt</sub>= k1<sub>rr</sub> /k2<sub>rr</sub></td>
 
−
    <td width="105">Eq 8 </td>
 
−
  </tr>
 
−
</table>
 
−
 
 
−
this implies
 
−
<table width="622" border="0">
 
−
  <tr>
 
−
    <td width="507" height="52">k1<sub>tt</sub>/k1<sub>rr</sub> =  k2<sub>tt</sub>/k2<sub>rr</sub></td>
 
−
    <td width="105">Eq 9 </td>
 
−
  </tr>
 
−
</table>
 
−
 
 
−
and this ratio may be denoted by R1
 
−
<table width="622" border="0">
 
−
  <tr>
 
−
    <td width="507" height="52">R1 = k1<sub>tt</sub>/k1<sub>rr</sub> =  k2<sub>tt</sub>/k2<sub>rr</sub></td>
 
−
    <td width="105">Eq 10 </td>
 
−
  </tr>
 
−
</table>
 
−
 
 
−
Making the substitution into Eq 6 we obtain
 
−
<table width="622" border="0">
 
−
  <tr>
 
−
    <td width="507" height="52">dF<sub>tt</sub>/dt = R1 Cpn &#8211; (k2<sub>tt</sub>+k3<sub>tt</sub>)F<sub>tt</sub> + k4<sub>tt</sub> B<sub>tt</sub> </td>
 
−
    <td width="105">Eq 11 </td>
 
−
  </tr>
 
−
</table>
 
−
 
 
−
This equation could be solved simultaneously with Eq. 7 to obtain time-courses of free and bound radioligand. Analytic solutions derived for more general conditions (Eq 1 and 2) are given in the appendix in the document thad describes validation of the COMKAT implementation of the FDG model. Solutions for Ftt and B tt would then be summed to obtain the total concentration in the target tissue which is the quantity measured in an imaging study
 
−
<table width="622" border="0">
 
−
  <tr>
 
−
    <td width="507" height="52">C<sub>t</sub><sub></sub> = F<sub>tt</sub>+B<sub>tt</sub> </td>
 
−
    <td width="105">Eq 12 </td>
 
−
  </tr>
 
−
</table>
 
−
 
 
−
In general, the solution Ctt is a weighted sum of two exponential terms. For ligands in which the association and dissociation of ligand from the receptor is much faster than the transport of ligand between the vascular and extravascular space, two exponentials are not evident from the data. That is, a single exponential model adequaltey describes the data because the two compartments exchange so quickly that they look like a single compartment. (Mathematically one of the eigenvalues or exponential coefficients is very large in magnitude.) The differential equation for such a single-exponential model is
 
−
<table width="622" border="0">
 
−
  <tr>
 
−
    <td width="507" height="45">dC<sub>tt</sub> /dt = k1<sub>tt </sub>Cp &#8211;k2<sup>a</sup><sub>tt</sub> C <sub>tt</sub> </td>
 
−
    <td width="105">Eq 13 </td>
 
−
  </tr>
 
−
</table>
 
−
 
 
−
where k2att is the the apparent k2tt and is related to the the other parameters as
 
−
<table width="622" border="0">
 
−
  <tr>
 
−
    <td width="507" height="45">k2<sup>a</sup><sub>tt</sub> = k2<sub>tt </sub>/(1+BP)<sub></sub> </td>
 
−
    <td width="105">Eq 14 </td>
 
−
  </tr>
 
−
</table>
 
−
 
 
−
where BP = k3tt/k4tt is the binding potential. Expressing the input in terms of Cpn, this is equivalent to the one-tissue compartment model depicted below
 
−
 
 
−
Simplified Model Figure
 
−
[[Image:SimplifiedModelFigure.png]]
 
−
 
 
−
== COMKAT Implementation ==
 
−
 
 
−
comkat\examples\dasbSRTM.m
 
−
 
 
−
Step 1 To set this up in MATLAB, create an input function that implements Eq. 5 [Cpn = (dFrr/dt + k2rr Frr)]. In this k2rr is a parameter to estimate amd Frris taken as experimentally measured reference region concentration. To obtain Frr at arbitrary times requires interpolation. For this a smoothing spline is used. Its piecewise-polynomial coefficients are easily obtained
 
−
 
 
−
ppCref = csaps(tref, Cref, 0.01);
 
−
 
 
−
where tref and Cref are the sample times and concentrations from the reference region. With Frr expressed in this form, the polynomials may be analytically differentiated once to obtain the piecewise polynomial coefficients of dFrr/dt
 
−
 
 
−
ppdCrefdt = fnder(ppCref, 1);
 
−
 
 
−
The input Cpn may be evaluated as Cpn(t) = k2rr x ppval(ppCref. t) + ppval(ppdCrefdt.t) with the complete function given in comkat\examples\refInput.m
 
−
 
 
−
function [Cpn, dCpndk2ref] = refInput(k2ref, t, pxtra)
 
−
  ppCref = pxtra{1};
 
−
  ppdCrefdt = pxtra{2};
 
−
  if (nargout > 0),
 
−
    Cpn = k2ref * ppval(ppCref, t) + ppval(ppdCrefdt,t);
 
−
    idx = find(Cpn < 0);
 
−
    Cpn(idx) = 0;
 
−
    if (nargout > 1),
 
−
      dCpndk2ref = ppval(ppCref, t);
 
−
      dCpndk2ref(idx) = 0;
 
−
    end
 
−
  end
 
−
return
 
−
 
 
−
For the sake of estimating the value of k2rr, this function must provide the derivative of Cpn with respect to k2rr when two output arguments are requested.
 
−
 
 
−
This input function is included in the compartment model by using these commands
 
−
 
 
−
sa = 1; % specific activity
 
−
dk = 0; % decay constant (=0 since data are decay corrected)
 
−
xparm = { ppCref, ppdCrefdt };
 
−
cm = addInput(cm, 'Cpn', sa, dk, 'refInput', 'k2ref', xparm);
 
−
 
 
−
Step 2 Create a model. To simultaneously fit three regions – midrbain, amygdala, and thalamus – create a model that has the compartments and links depicted as shown. Three Region Model
 
−
 
 
−
[[Image:ThreeROIFigureSRTM.PNG]]
 
−
 
 
−
The commands are
 
−
 
−
% specify compartments
 
−
cm = addCompartment(cm, 'Cmid');
 
−
cm = addCompartment(cm, 'Camy');
 
−
cm = addCompartment(cm, 'Ctha');
 
−
cm = addCompartment(cm, 'Sink'); % comkat requires closed system
 
−
 
 
−
% specify links
 
−
cm = addLink(cm, 'L', 'Cpn',  'Cmid',  'R1mid');  % connect input to
 
−
cm = addLink(cm, 'K', 'Cmid', 'Sink',  'k2ppmid');
 
−
 
−
cm = addLink(cm, 'L', 'Cpn',  'Camy',  'R1amy');
 
−
cm = addLink(cm, 'K', 'Camy', 'Sink',  'k2ppamy');
 
−
 
−
cm = addLink(cm, 'L', 'Cpn',  'Ctha',  'R1tha');
 
−
cm = addLink(cm, 'K', 'Ctha', 'Sink',  'k2pptha');
 
−
 
−
% specify outputs (neglect vascular contrib) 
 
−
% for simultaneously fitting target tissues
 
−
cm = addOutput(cm, 'Mid', {'Cmid', 'PV'}, { } );
 
−
cm = addOutput(cm, 'Amy', {'Camy', 'PV'}, { } );
 
−
cm = addOutput(cm, 'Tha', {'Ctha', 'PV'}, { } );
 
−
 
−
% specify parameters and values
 
−
cm = addParameter(cm, 'PV',    1);
 
−
cm = addParameter(cm, 'k2ref', 0.05);
 
−
 
−
cm = addParameter(cm, 'R1mid', 0.8);
 
−
cm = addParameter(cm, 'R1amy', 1.0);
 
−
cm = addParameter(cm, 'R1tha', 1.2);
 
−
 
−
cm = addParameter(cm, 'BPmid', 0.5);
 
−
cm = addParameter(cm, 'BPamy', 1.0);
 
−
cm = addParameter(cm, 'BPtha', 1.5);
 
−
 
−
cm = addParameter(cm, 'k2ppmid', 'k2ref*R1mid/(1+BPmid)');
 
−
cm = addParameter(cm, 'k2ppamy', 'k2ref*R1amy/(1+BPamy)');
 
−
cm = addParameter(cm, 'k2pptha', 'k2ref*R1tha/(1+BPtha)');
 
−
 
 
−
The rest of the details are quite straightforward and are all contained in the example file comkat\examples\dasbSRTM.m. The only thing slightly unusual is that multiple time-activity curves must be specified so they can be simultaneously fit. To do this, just use
 
−
 
 
−
cm = set(cm, 'ExperimentalData', PETdata(:, [1 2 3]));
 
−
where columns [1 2 3] contain 'Mid', 'Amy', and 'Tha' data [the same order as the adddOutput() commands] and the rows of PETdata correspond to different frames.
 
−
 
 
−
= Glucose minimal model =
 
−
= Enzyme-substrate (Michaelis-Menten) kinetics =
 

Latest revision as of 19:48, 7 August 2012

Official Examples


User-contributed Examples

Retrieved from "https://comkat.cwru.edu/index.php?title=Support:Documents:COMKAT_Examples&oldid=2169"