Support:Documents:Examples:Estimate Input Parameters Simultaneously with Flow

Estimating Input Function Parameters Simultaneously with Flow

This example demonstrates an approach to simultaneously estimating input functions parameters along with blood flow in the context of the PET blood flow model.

Tissue Uptake

The flow model has one tissue compartment. Radioactive water (or another freely diffusible molecule) in the blood is assumed to rapidly exchange with the water in the tissue. The tissue model has two rate constants: K₁ and k₂ and can be depicted in a diagram:

It can also be described by the differential equation:

dC_T/dt = K₁ C_p - k₂ C_T

where C_T is the tissue concentration and C_p is the plasma concentration of radioactive water. C_T and C_p are interpreted as either molar concentrations or decay-corrected activity concentration (Salinas 2007).

Instead of estimating K₁ and k₂, it sometimes is advantageous to estimate a different set of parameters defined as

lambda = K₁/k₂ and F = K₁.

F is blood flow (perfusion) per unit volume of tissue (for simplicity we're assuming 100% extraction) and lambda is the ratio of tissue to blood concentrations at equilibrium.

To use this parameterization in COMKAT, one needs to express the rate constants in terms of lambda and F: K₁ = F, k₂ = K₁/lambda

This example demonstrates how to estimate F (simultaneously with input function parameters). The example could be easily modified to estimate lambda as well.

Input Function

This example also demonstrates how to use a parameterized input function. Specifically, the plasma concentration vs. time t is modeled as

C_p = (p₁t - p₂ - p₃) exp(p₄ t) + p₂ exp(p₅ t) + p₃ exp(p₆t) [aka Feng input]

This input function has been implemented in fin.m in the COMKAT examples folder.

Pixel Values

In a PET image, pixel values represent the radioactivity concentration. There are contributions from radioactivity in blood and tissue. The arterial blood concentration includes radioactive water in plasma and the blood cells and is designated C_a. Here we assume C_a can be described using the Feng input equation but with different values for p₁, p₂, ... p₆ than were used for C_p. The tissue activity concentration is C_T x S where S is the exponentially-decaying specific activity and C_T is given above. The pixel activity concentration is calculated as the activity concentration in the tissue plus the pixel's fractional blood space F_v times the arterial blood activity concentration: PET = C_TS + F_vC_a.

COMKAT Implementation

PET indicates a tissue

Text

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Code

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Figures

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figure
plot(1:10)
title('Example Figure')
ylabel('y-axis label')
xlabel('x-axis label')
set(1,'PaperPosition',[0.25 2.5 3 2])
print -dpng ExampleFigure.png

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That is it. It is so simple that even a caveman (or professor) can do it.