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Model of Neurotransmitter PET (ntPET)

Overview

The changes of endogenous neurotransmitter by PET scanning after a stimulus have been proved with different approaches. In stead of detecting the increase or decrease of a neurotransmitter, it is important to characterize the temporal change of a neurotransmitter after a stimulus. Morris proposed the new technique (called ntPET) for capturing the dynamic changes of a neurotransmitter after a stimulus and demonstrated the ability of this method to reconstruct the temporal characteristics of an enhance in neurotransmitter concentration.

Generally, there are two separate injections of tracer for ntPET: one for the rest condition (without any stimuli) and the other for the activation condition (after a stimulus). Therefore, the model can be described as:

This model can be described by the differential equations:

dF/dt = K₁C_p - K₂F - K_on[B_max - B - B2 - B^en]F + k_offB

dB/dt = K_on[B_max - B - B2 - B^en]F - k_offB

dF2/dt = K₁C_p2 - K₂F2 - K_on[B_max - B - B2 - B^en]F2 + k_offB2

dB2/dt = K_on[B_max - B - B2 - B^en]F2 - k_offB2

dB^en/dt = K^en_on[B_max - B - B2 - B^en]F^en -k^en_offB^en

C_p is the plasma concentration of tracer at the first injection (the "rest" condition).

F and B are free (unbound) and bound molar concentrations of tracer after the first injection

C_p2 is the plasma concentration of tracer at the second injection (the "activation" condition).

F2 and B2 are free (unbound) and bound molar concentrations of tracer after the second injection.

F^en and B^en are free and bound molar concentrations of a neurotransmitter released by endogenous ligand after a stimulus.

B_max is the maximum number of receptors that can be bound by neurotransmitter or tracer.

B_max - B - B2 - B^en is the concentration of available receptors. This also indicates that binding is saturable.

K₁, K₂, K_on, K_off, K^en_on and K^en_off are rate constants.

Implementing the Compartment Model

Create a new model for ntPET

Step 1. Create the compartemt model

    cm = compartmentModel;

    %Set scan time of the whole scan
    inj2Delay = 240; % min between 1st and 2nd inj
    disp('Define one long study: inj 1 = baseline, inj 2 = activation');
    scantInj1 = [[0:0.1:0.9 1:0.25:1.75 2:0.5:4.5 5:1:59]' [0.1:0.1:1 1.25:0.25:2 2.5:0.5:5 6:1:60]']; 
    scantInj2 = scantInj1 + inj2Delay;
    scant = [scantInj1; scantInj2]; 
    deltaT = scant(:,2) - scant(:,1);
    tmid   = 0.5*(scant(:,1) + scant(:,2));

    %Set parameters of the model
    cm = addParameter(cm, 'PV', 1 );    
    cm = addParameter(cm, 'Fv', 0);     
    cm = addParameter(cm, 'sa', 1);    % uCi/pmol
    cm = addParameter(cm, 'dk', 0);    % assume data are decay-corrected        
    cm = addParameter(cm, 'k1',      0.9);   
    cm = addParameter(cm, 'k2',      0.4);   
    cm = addParameter(cm, 'kon' ,   0.01);   
    cm = addParameter(cm, 'koff',    0.14);  
    cm = addParameter(cm, 'k2ref',  0.3);
    cm = addParameter(cm, 'Bmax',  80);
    cm = addParameter(cm, 'konEN', 0.25);
    cm = addParameter(cm, 'koffEN', 25);     

    %model for baseline condition:
    cm = addCompartment(cm, 'F');
    cm = addCompartment(cm, 'B',      'Bmax');
    cm = addCompartment(cm, 'Junk');

    %model for activation condition:
    cm = addCompartment(cm, 'F2');
    cm = addCompartment(cm, 'B2',      'Bmax');
    cm = addCompartment(cm, 'B^{EN}', 'Bmax');

Step 2. Set the input function for two injections

    % Create input functions for two injections
    t = 0:0.05:305; % coverage out to 6+ h
    Cp1Data = fengInput([2 0.5 8.5 0.22 0.2 -4. -0.1 -0.02], t);
    Cp2Data = fengInput([2 inj2Delay+0.5 8.5 0.22 0.2 -4. -0.1 -0.02], t);
    ppCp1 = spline(t, Cp1Data);
    ppCp2 = spline(t, Cp2Data);
    
    figure; plot(t, ppval(ppCp1, t), 'r', t, ppval(ppCp2,t), 'b', 'LineWidth', 2); title('Exogenous Input Functions'); legend('Cp1', 'Cp2')

    cm = addInput(cm, 'C_p', 'sa', 'dk', 'ppval', ppCp1);
    cm = addInput(cm, 'C_p2', 'sa', 'dk', 'ppval', ppCp2);

Step 3. Create a function described the change of concentrations of the endogenous neurotransmitter. To detect the change, the parameters of the function will be estimated.

The change of concentrations of the endogenous neurotransmitter after a stimulus is described by a gamma variate function. F^en = Basal + Gamma[t-Delay]^Alpha exp(-Beta[t-Delay]). Note: Basal is the concentration of the endogenous neurotransmitter without any stimulus. Gamma, Alpha and Beta control the change of concentrations of the endogenous neurotransmitter. Delay is the delay time.

    DApar = [50 27 1 0.1 240+10]; % = [Basal Gamma Alpha Beta Delay] 
    cm = addParameter(cm, 'DA', DApar);
    cm = addInput(cm, 'F^{EN}', 0, 0, 'gammavariate', 'DA'); %DA defined by gamma variate function at activation condition
    figure; plot(t, gamma(DApar, t), 'b', 'LineWidth', 2); title('Endogenous Input Function'); legend('F^{en}'); 
    ax = axis; axis([ax(1) ax(2) 0 ax(4)]); % ensure minimum y-axis value is zero

The endogenous input function is implemented as (create the below function and put in in a file called gammavariate.m):

function [y, grad] = gammavariate(p,t) 
% parameters [p(1) p(2) p(3) p(4) p(5)] = [Basal G alpha beta delay]
    y = p(1).*ones(size(t)); 
    idx = find(t >= p(5));
    dt  = t(idx) - p(5);
    
    y(idx) = p(1) + p(2)*(dt.^p(3)).*exp(-p(4).*dt);
    
    if nargout > 1,
        grad = zeros(length(t), 5); 
        grad(:,1)=ones(size(t));

        if ~isempty(idx), 
            grad(idx,2) = (dt).^p(3).*exp(-p(4).*(dt));
            grad(idx,3) = p(2).*(dt).^p(3).*log(dt).*exp(-p(4).*(dt));
            grad(idx,4) = -p(2).*(dt).^(p(3)+1).*exp(-p(4).*(dt));
            grad(idx,5) = -p(2)*p(3)*(dt).^(p(3)-1).*exp(-p(4).*(dt)) + p(2)*p(4)*(dt).^p(3).*exp(-p(4).*(dt)); %dy/ddelay
        end
    end
return

Step 4. Set link between each compartment and add output.

    %links/output for baseline condition:
    cm = addLink(cm, 'L',  'C_p',     'F',      'k1');
    cm = addLink(cm, 'K',  'F',       'Junk',   'k2');
    cm = addLink(cm, 'K2', 'F',       'B',      'kon');
    cm = addLink(cm, 'K',  'B',       'F',      'koff');

    disp('Output is sum of radioactivity in free and bound compartments')
    WList = {...
          'F', 'PV'; 
          'F2', 'PV';
          'B', 'PV';
          'B2', 'PV'};
    XList = {}; 
    cm = addOutput(cm, 'PET Scan', WList, XList);

    %links/output for activation condition:
    cm = addLink(cm, 'L',  'C_p2',     'F2',      'k1');
    cm = addLink(cm, 'K',  'F2',       'Junk',    'k2');
    cm = addLink(cm, 'K2', 'F2',       'B2',      'kon');
    cm = addLink(cm, 'L2', 'F^{EN}',  'B^{EN}', 'konEN');   
    cm = addLink(cm, 'K',  'B2',       'F2',      'koff');
    cm = addLink(cm, 'K',  'B^{EN}',  'Junk',    'koffEN');