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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

   % Set scan time
   inj2Delay = 240; % min between 1st and 2nd inj (6 hours delay)

   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));

   %MODEL CONSTRUCTION   
   cm = compartmentModel;

   %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');