Maths -- model for drug absorption. Simple, but gives rather realistic graphs.

[VelocideX]

After a this thread I was spurred to create a simple differential equation model of absorption of stuff in the body.

Here's the assumptions I've made:
1) Pill reaches stomach as a whole. No "slow absorption" into the stomach. Stomach starts the problem with highest amount of substance
2) Stomach clears exponentially. Seems pretty reasonable to me, at least to a first approximation. I don't think its linear, and its got to tail off to 0 sometime, so you'd need some interpolation between a linear model and an exponential one. In any case, exponential IS linear in a small enough region.
3) Intestines receive stomach contents immediately. No time is allowed for "traversing the length of the intestines".
4) Intestines clear at a rate proportional to the amount of stuff in them. This assumes that no saturation is reached for whatever transporter mechanism exists. It would be accurate for low amounts of stuff, I guess
5) Amount of stuff entering blood is proportional to the amount in the intestines. This has to follow from the last one, it's just a conservation of stuff really
6) Blood clears proportional to how much is in it. Again, exponential decline. Seems reasonable to me; the concept of a blood half-life has no meaning without exponential decline.

You end up getting nice graphs like this

Amount of stuff in the intestines:

Amount of stuff in blood:

Admittedly I've just inserted random constants, but they're based on proportions I thought were sensible.

If anyone wants I can post the differential equation and the solutions. The solutions have a closed analytic form, though in its generality (no constant substitution) its a little hideous. Not too bad though, much better than I was expecting.

 

[ld50 vs ssri]

Nice one VelocideX, thats a rather good eqution showing the differences in absorption rates. Wish i was better at the mathematics side of things.. Be interesting to see the differential equation and the solutions.

 

[VelocideX]

Here's the differential equation representing the intestines. Note i've used B for amount in blood, Z for amount in intestines (Maple doesn't like using I... it thinks its the square root of minus one heh)

First term is input from stomach, last is output from intestines

Next equation is the blood. First term is input from intestines, last is exponential removal from blood:

Here's solutions for B and Z

For the constants for those graphs I've chosen k1 = 1, k2 = 0.5, k3 = 0.1

 

[Pippin]

wow baby. you're supadupa clever *beams*

(sings - my boyfriend is so smart S.M.A.T)


:D

 

[1bRAVEheart1]

This is probably completely unrelated, but I found it interesting anyway.

I put two aspirin in water the other day. The first one I dropped the whole one in and it floated on the top and dissolved. The second one I broke up into smaller bits and dropped them in and they sank and dissolved a little bit, but not very much.

I was expecting the second one I dropped in to dissolve faster as it had greater surface area, but it didn't. I'm assuming it didn't like dissolving after sinking for some reason, but I don't know why.

It definitely didn't have anything to do with the amount of water, as there was heaps in the glass.

bRAVEheart

 

[Pippin]

btw if anyone's wondering...my song is to the tune of that simpsons one.... (no im not a simpsons geek, that one was just about the only episode i ever saw)

 

[thomo1981m]

is there anywhere that can actually quote (reference) this as its simalar to what a mate of mine will need for his court case

 

[VelocideX]

Why would he need this for his court case? I hardly think that a court case would turn on pharmacokinetic particulars... especially not some rought approximation to absorption, based on zero knowledge of absorption coefficients...

 

[VelocideX]

I'll also note I've never incorporated normalisation into those... you need extra constants to do it, though its certainly possible.

(Normalisation = area under curve being consistent)

 

[BigTrancer]

Hehe, now just replace the stomach contents exponential with a cosine function to simulate someone taking a pill every two hours and watch what happens to the graphs for B and Z...

BigTrancer

 

[thomo1981m]

i posted a while back he was in a car accident and high amount of mdma in his system the police say he is a high regular user but he rarley pops a pill

 

[thomo1981m]

i think it was due to the amount of mdma in his blood

 

[KemicalBurn]

(Maple doesn't like using I... it thinks its the square root of minus one heh)

lol! :D that might screw the graph up abit

but nevertheless, impressive work VelocideX have you thought about cross posting in Advanced DD?

 

[VelocideX]

It doesn't matter if he was a regular user or not. They can only prosecute on the evidence before him. He had MDMA in his blood and therefore he's guilty of driving under the influence of drugs. It doesn't matter whether they accuse him of regularly or not -- thats irrelevant, and any good lawyer would argue that.

 

[VelocideX]

Hehe, now just replace the stomach contents exponential with a cosine function to simulate someone taking a pill every two hours and watch what happens to the graphs for B and Z...

BigTrancer

You'd need something like cos^2 because otherwise the stomach contents start to drain backwards (negative flow)

ahahah i just triedi t. THe intestines reach equilibrium and blood levels go on this massive oscillating function upwards for a while, then they also reach a semi-equilibrium state, with blood levels rising and falling in a ~cos type function

 

[VelocideX]

lol! :D that might screw the graph up abit

but nevertheless, impressive work VelocideX have you thought about cross posting in Advanced DD?

Sure, but people don't generally like cross-posting

 

[KemicalBurn]

it was merely a suggestion to spread the word 8) as im sure there are others who dont look in Aust. DD (like the rest of the world) who would be interested in this

 

[phase_dancer]

Fantastic piece of work VelocideX. And thanks for thinking of us by posting here in Aus DD, as I'm sure those in Advanced DD would also be impressed. Now, apart from individual variations such as gastrointestinal motility and other variables, as I see it, apart from what you've purposely excluded you would also need to account for :


1) Any possible ionisation with the drug in question, as the pKa values (3-8 ) of some drugs (weak acids and bases) means they will essentially be at equilibrium in the stomach or duodenum-small intestine:

For a weak base;

B + H3O+ <--->( BH+) + H2O and pH = pKa log ([BH+]/ )


The non-protonated form will be much more lipid soluble and thus permeate easily into cell walls.


2) Differences in compartmental pH as these will also alter absortion rates, as rate of ionisation will alter steady state distribution between aqueous compartments. Being a weak base, amphetamine or MDMA will also reabsorb from plasma to the stomach, the rate of which will also vary with concentration.

3) Carrier mediated transport (edit: I know you mentioned ignoring this but it's probably relevant to a small degree). Is this facillitated diffusion or active transport? i.e. is their some energy involved -ATP hydrolysis or cationic migration etc. With this, is the need to account for carrier availability and saturation from competitive inhibition (preferential binding by another species)


Like you said, a basic model VelocideX, but a damn fine representation

.

 

[fastandbulbous]

Small world

The graphs are remarkably similar to the charging and discharging of a capacitor, though a resistor (const voltage source). It's also governed by exponential equations

 

[BigTrancer]

There's quite a massive literature on drug pharmacokinetics out there, too, in which people try to forumlate their specific drugs and delivery methods to optimise graphs like the above so that the right amount of drug stays in a person's system for the right amount of time... before ever going to clinical trial.

BigTrancer