Talk:Basic Physics of Nuclear Medicine/Dynamic Studies in Nuclear Medicine

I am quite sure that you miscalculated something in the following section.

http://en.wikibooks.org/wiki/Basic_Physics_of_Nuclear_Medicine/Dynamic_Studies_in_Nuclear_Medicine#Models_with_Three_Compartments

The argument is like this:

You wrote (mind the brackets):

q_3=-A_5*[exp(-l_1*t)-A_2*exp(-l_2*t)]

The model is such that

q_3(t=0)=0

This obviously means:

0=-A_5*[1-A_2]

The case A_5=0 is not interesting because it implies q_3(t)=0 for all times, which certainly does not describe the system we are interested in, so we can dived by A_5 yielding:

1=A_2

But you wrote

A_2=0.35

thus we got a contradiction.

Yours Dirk Hünniger from German Wikibooks

I think the correct equation is.

q_3=-A_5*[exp(-l_1*t)-exp(-l_2*t)]

So just one A_2 removed. I tried to calculate from scratch. I put up the differential equation and initial condition from the image showing the compartments and there paths of exchange:

$$ \begin{array}{ccc} q_1(t=0)&=&1 \\ q_2(t=0)&=&0 \\ q_3(t=0)&=&0 \end{array} $$

$$ \begin{array}{ccc} \overset{\text{ }}{\frac{\mathrm{d} q_1}{\mathrm{d} t}}&=&-\left( k_{13} + k_{12} \right) \cdot q_1 + k_{31} q_3 \\ \overset{\text{ }}{\frac{\mathrm{d} q_2}{\mathrm{d} t}}&=& k_{12} \cdot q_1 \\ \overset{\text{ }}{\frac{\mathrm{d} q_3}{\mathrm{d} t}}&=& k_{13} \cdot q_1 - k_{31} q_3 \end{array} $$

This is differential equation of the form:

\frac{dq}{dt}=B q

where q is a vector and B is Matrix. Wikipedia says that the solution is easy to get if you know the eigenvalues and eigenvectors of B.

So I put that into wolfram:

http://www.wolframalpha.com/input/?i=eigenvalues({{(-b-a)%2C0%2Cc}%2C{a%2C0%2C0}%2C{b%2C0%2C-c}})

and had a close look at the result.

Renography
I think the Section http://en.wikibooks.org/wiki/Basic_Physics_of_Nuclear_Medicine/Dynamic_Studies_in_Nuclear_Medicine#Renography

is wrong.

I was trying to figure out the equations and I don't see that they are right. There are a few problems.

First of all I can understand it for tt0 it turns wired. q4 decreases in a monotonous manner. This ok. But the rate of decrease is governed by l1 and l2. But l1 and l2 are function of k12, k13 and k31 an not other parameters. But the decrease of the concentration of the tracer must depend on exchange rates between the compartment 4, 5 and 6 too. So this is why I don't believe in this equation. I propose to use a model of five compartments, specifically 1,3,4,5 and 6 and calculate that with the methods I have given here:

http://de.wikibooks.org/wiki/Physikalische_Grundlagen_der_Nuklearmedizin/_Mathematischer_Anhang

The differential equation I propose is given by the following image:



Since we got five compartments, this might lead to a polynomial of degree 5. So we might not be able to solve for the eigenvalues. But if we are lucky it might work. Otherwise we can still calculate things in a numeric manner using the same approach.

Independent of the exact method of solution we are going to choose it seems very important to me to write down the differential equations we are going to solve, which is not yet done for this case.

Dirk Hünniger (talk) 08:26, 10 April 2010 (UTC)

Results
I did the proposed calculation. It is written down here:

http://de.wikibooks.org/wiki/Physikalische_Grundlagen_der_Nuklearmedizin/_Mathematischer_Anhang#Renographie

it is lengthy and not very interesting. The resulting Plots are here:

http://de.wikibooks.org/wiki/Physikalische_Grundlagen_der_Nuklearmedizin/_Dynamische_Studien_in_Nuklearmedizin#Renographie

There is one interesting point that is different between the two models.

Kierans models says:

"There is a door from the renal parenchyma to the renal pelvis. In the beginning this door is closed, and the tracer is located in the intravascular compartment. Thus it accumulates in the renal parenchyma by flow from the intravascular compartment. After two minutes the door opens. Now there is a big flow from the renal parenchyma to the renal pelvis and from there to the bladder, this flow is bigger than the flow from the intravascular compartment to the renal parenchyma. So the concentration of the tracer in the kidney decreases"

--Kieran complained that I misquoted him here. And he is very angry with me because of what I wrote here. But this is how I understand his model.

My Model says:

"There is no door. The tracer accumulates in the kidney by flow from the intravascular compartment. There is also a flow from the intravascular compartment to extravascular compartment and this is really large. So the tracer leaves the intravascular compartment to the extravascular compartment and thus the concentration of the tracer in the intravascular compartment deceases rapidly. Thus there is no input to the kidney from the intravascular compartment anymore and so the concentration of the tracer in the kidney decreases"

So to put is simply:
 * Kieran says the flow leaving the kidney increases.
 * I say the flow into the kindey decreases

--Dirk Hünniger (talk) 06:03, 12 April 2010 (UTC)