Simple mathematical model for drug absorption -- nice graphs!

I'm not going to repost the entire thread, but take a wander over to this thread in Aus Drug Discussion.

It's a simple mathematical model I've created, and the graphs it produces are rather convincing.

The stupid thing doesn't actually conserve the amount of stuff floating around as it moves, but that just requires a multiplicative constant (I think...)

I'd love to hear people's opinions on it, or how it could be modified.

There are two things I think you need to try and do, make is so pharmacokinetic constants can be chucked in (half life and dose especially)....

What you also need to distribution.. A single compartment model should work, so that MDMA distibutes into a comparment (independent of metabolism) in a way dependent on plasma concentration... it's just kinda the opposite of absorbtion from intestines...

You should look up a paper on pharmacokinetic modeling...

half-life is easy. it's ln(2)/k3.

umm so you're saying material would go from bloodstream --> some compartment (e.g. brain) then back to the bloodstream?

What sort of model is realistic? (I don't know enough about pharmacokinetics).

Rate at which things enter the brain (say) is proportional to the difference between the amount in the brain and the amount in the blood? That would imply some sort of diffusive model wouldn't it?

Actually I suppose not. After all the compartment would equilibriate when the difference between the concentration in it and the plasma was zero... it can't really keep filling, can it?

Actually that could have just been guessed anyway
dF/dt = k4*B - F

its just another term. Ill stick it in any see what happens

Ok I've attempted a new model... same as before

Stomach --> Intestines --> Blood* --> outflow (excretion)

*Now blood has an two-way connection beteen a compartment where the flow is proportional to the difference between them. e.g if compartment has nothing in it, and blood level is high, there will be a strong flow towards the compartment, and vice versa.

MAPLE gives an exact solution to this after several minutes of working (!), I'm still trying to have a think if this works correctly... or if it even makes sense. Something bothers me about the fact that I haven't modelled a separate flow into the compartment and out of it, though all that's going to do is add more terms to the differential equation, which sucks... more computing time.

HAHAHAH IT WORKS :D

There's a general formula, but its rediculously long...

Here's a graph... the first curve = blood, last one is brain... ive presumed that the equilbrium happens rather quickly, as blood flows quickly

What about if you've got some membrane / barrier (e.g. BBB) that's a really poor absorber? Then you get this


Which looks kinda like what you'd expect. First (on the left) is blood, second (the smaller one) is brain/compartment.

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I've just realised with the compartment model you can no longer talk about amounts... it doesn't matter that the amount in the blood is higher than the amount in the compartment, only that the concentration is higher... hmmm

so yeah these graphs have to be in concentrations... like mol/L or some other equivalent units.

I think so anyway... in any case I've got fuck all idea about how you'd do normalisation... it would be a fucking bitch of a problem to solve.

I'm not entirely sure my problem makes sense anymore.

If you really wanted to do it properly, you'd have to redesign the problem.

And treat each container as being of volume Vi, for some i (e.g. Vb = volume of blood). Then you'd need to define the concentration as B/Vb. And then you'd need to define the problem in those terms...

e.g. using such things as d/dt (B/Vb)... I think that's more likely to give you a correct model, as then concentrations will equilibriate

e.g. dF/dt = k(B/vB - F/Vf)

I suppose that's not really all that much of a change though, you've just gotta go around and put in volumes etc as appropriate in the differences. cool

Yup, you can only talk about concentrations... unless you get into something like intranasal, where you'll have non-disolved stuff, and a huge majority of the substance wont be available for diffusion.

One thing I think you could do to simplefy your model is to drop the stomach, as absorption from there will be extremely limited..

Here is a review on pharmacokinetic modeling-
Whole body pharmacokinetic models.
Nestorov I.
Clin Pharmacokinet. 2003;42(10):883-908.

There's no absorption from the stomach, only movement from the stomach to the intestines.... it takes account of the fact that intestinal delivery isn't instananeous due to breakdown in the stomach

you can always tune the parameter to make it very fast

Incidentally would you have any idea what the volume of blood the brain holds is? and the body?

I've introduced a term for concentration and it seems to work much better now

I'm still trying to work out normalisation...

The body holds... in a dude, about 60% water.. so about 42L and 7.5% of that is plasma... so 3.15L

Heres a website with some of that stuff.

As far as how much plasma is in the brain, I would say a very very small percentage, but I have no idea.

I was saying that you could get rid of the stomach, because if you took a pill on an emptry stomach, then the the stomach contents would be transfered to the intestines basically as a bolus.

Really? Fair enough. I assumed there must have been some sort of emptying...

I haven't put much thought to it, but atm I can't really see how I can construct the model without the brain volume. At least in any sort of reasonable attempt to conserve material, which is kinda necessary if the model is going to be realistic. It only has to be approximate

Ok I've ditched the stomach, which made things somewhat easier, but at the same time this is getting beyond the realm of analytical solutions. The analytical solution is like pages and pages and pages now... I've ditched it for a polynomial. Bad news is that the polynomial has to be of at least order 40 to accurately represent the function in the relevant range
its got massive coefficients. Some of them are 100 digits long.

The good news is maple actually calculates this all faster than the analytic form heh

OK I've converted the polynomials to something reasonable e.g. non-rational form (evaluated decimal coefficients) and its much tidier. This seems to be the way to go. MAPLE seems to calculate this much faster anyway. Joys of polynomials... can make the polynomial order arbitrarily large in any case.

Actually fuck that. I was trying polynomials of order 150 (!) and keeping things to 20 sig figs and the calculations were going wacko... not surprised. And it was taking longer.

Analytical is better. I was wrong too, its faster. Heaps. And the graphs don't go bezerk.

AND i seem to have solved the normalisation problem too.

In the above graph, the green line definately looks like an XR brand drug. The only ones I have encountered are tramadol and effexor. Whereas the red line just indicates a standard pill formulation. Except that the time scale would need modifying since no drug is gonna kick in that quickly.

What you have got there is sort of the two extremes of drug formulation; a spike and a flat curve. Do you think you could draw a line that diplays intermediacy between the two?

The green line is indicative of the amount in the brain, not in the bloodstream. It is derived by assuming that the drug transfers relatively slowly through some barrier (e.g. the blood brain barrier).

I haven't included an XR formulation in here, though that could quite easily be done with a little modification.

This model is good in that it shows a very good approximation (visually, at least) to the standard pill formation. A few simple differential equations go a long way...

You can get anything you want out of it... this is the power of it. All you need to do is adjust the proportionality constants to determine the half-life of clearing, the half-life between the intestines and the blood, and the effectiveness of transferring between the blood and the brain, and then recompute the solution. The general form is a bitch, but MAPLE only takes a few seconds to find the solution if you give numerical values for the constants.

I love these sorts of thoughts about drug research. Model the shit out of everything

Hahaha yeah sometimes I go through old threads, in this forum especially. There’s some cool stuff to be had


and yeah def don’t think the OP‘s model was right, it relies on a lot of assumptions. But nearly all rigorous science tends to develop better mathematical models as the discipline grows. neuropharmacology doesn’t have as much of that yet as I think is possible and I’m curious to see where that goes