- On 19 Jan 2004 at 16:35:19, Toufigh Gordi (tgordi.-at-.buffalo.edu) sent the message

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Dear all,

I have constructed a PK/PD model similar to what follows:

dA1/dt= -A1*C*K12

dA2/dt= A1*C*K12 - A2*K20

I need to compare the time course of the system with respect to the

transfer from compartment 1 to 2 and the elimination from compartment

2. The main objective is to determine which process is faster: transfer

from A1 to A2, or elimination from A2. My problem is that K12 is a

second order rate constant, dependent on A1 and C, whereas K20 is a

first order rate constant, dependent on A2. Thus K12 has the units of

1/(C*time), while K20 has the unit of 1/time. Can anybody suggest a

general approach in comparing the two rates? References to published

material are highly welcome.

Regards,

Toufigh Gordi

[What is the differential equation for 'C'? Does C vary with time

within a particular experiment? - db] - On 20 Jan 2004 at 08:48:05, kim.travis.aaa.syngenta.com sent the message

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I would have thought that a simple and functionally-relevant way of

doing

this would be to see how your model behaves. I assume that the chemical

starts in compartment 1 - if not then what follows may not apply. If

your

model predicts only low levels in compartment 2 at any timepoint then

tranfer out of 2 is fast compared to transfer from 1 to 2. If chemical

builds up in compartment 2 then the opposite is true. You could do

this by

simulation, or as the model is so very simple you could analytically

derive

the maximum concentration in compartment 2 over time, given known

initial

conditions.

There is a danger in overinterpretting the parameters of such models and

forgetting that it is how the whole model and data behaves that is

important,

Kim - On 20 Jan 2004 at 11:15:32, "J.H.Proost" (J.H.Proost.aaa.farm.rug.nl) sent the message

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Dear Dr. Gordi,

With respect to your question about the model:

> I have constructed a PK/PD model similar to what follows:

>

> dA1/dt= -A1*C*K12

> dA2/dt= A1*C*K12 - A2*K20

I have three comments:

1. I do not understand the rationale of this model. As was asked by

David Bourne: what is C?

It seems that your equations refer to an absorption model, in which the

rate of absorption is dependent also on the concentration (or whatever)

of something else?

2. The question how to compare the rate constants is simple. The

absorption rate constant (or 'drug entry constant') is C*K12, which has

the unit of reciprocal time.

3. The question why one would compare the rate constants is not clear

to me. Perhaps there may be a valid reason for this; if so, please

explain. If C*K12 can be regarded as an absorption rate constant, and

K20 as an elimination rate constant, a comparison of the numeric values

is meaningless. Both constants refer to completely different processes.

'C*K12' is the rate of removal of drug from comp. 1 divided by the

amount in comp. 1, and 'K20 is the rate of removal of drug from comp. 2

divided by the amount in comp. 2. Both rate constants are dependent on

the volumes of the corresponding compartments. What would we learn from

the knowledge which parameter, C*K12 or K20, is largest?

Best regards,

Hans Proost

Johannes H. Proost

Dept. of Pharmacokinetics and Drug Delivery

University Centre for Pharmacy

Antonius Deusinglaan 1

9713 AV Groningen, The Netherlands

tel. 31-50 363 3292

fax 31-50 363 3247

Email: j.h.proost.aaa.farm.rug.nl - On 20 Jan 2004 at 14:13:13, Toufigh Gordi (tgordi.-at-.buffalo.edu) sent the message

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It seems that the problem is more complex than I thought. I had

originally given these equations:

"

dA1/dt= -A1*C*K12

dA2/dt= A1*C*K12 - A2*K20

"

Here comes additional information about the model. The full model

describes the effect of a drug on the number of organisms. C is

time-variable drug concentrations, which could be simplified as

following a mono-exponential elimination rate (although in reality it

could be much more complex, but still follow a mono-exponential decay).

A1 is the number of viable organisms, which due to a direct drug effect

are transferred to an "injured" form (A2). Thus, A2 is the number of

injured organisms, which are then cleared from the body. The full model

is again more complex than the one presented here, but I don't find it

of relevance to my aim, i.e. comparing the rate of transfer from A1 to

A2 and A2 out. The observed number of organisms are the sum of A1 and

A2 since there is no practical way of separating the viable and injured

organism. To complicate the situation even more, there is another

transfer to A2: from another viable, but not visible, organism

compartment, with another second-order rate constant.

During the model-building, I tried a model without any "injured"

compartment but the fits gets much better when it is added to the

system, which would imply the A2 to 0 is slower than A1 to A2. However,

I would like to have a more solid evidence of this by a possible

comparison of the two rates. So the question is: is there any way of

comparing a first-order (in this case K20) rate constant and a

second-order (K12) rate constant?

I think it is a good idea, as proposed by Kim Travis, to simulate the

model and see how the numbers change.

Thank you!

T. Gordi - On 20 Jan 2004 at 12:40:41, Walt Woltosz (walt.-at-.simulations-plus.com) sent the message

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The following message was posted to: PharmPK At 02:15 AM 1/20/2004,

Hans Proost wrote:

"1. I do not understand the rationale of this model. As was asked by

David Bourne: what is C?

It seems that your equations refer to an absorption model, in which the

rate of absorption is dependent also on the concentration (or whatever)

of something else?"

I also wonder what C refers to in the equation.

For passive diffusion, which governs the absorption for most drugs, the

absorption rate is dependent on two concentrations - one on each side

of the membrane. Let's remember that the modern definition of

absorption is crossing the apical membrane into the enterocytes, not

reaching the portal vein or systemic circulation (this was emphasized

repeatedly at the Oral Drug Delivery Course in Vail, Colorado last

week).

For passive diffusion, Ficks' Law says that molecules cross a membrane

at a rate that is determined by the difference in concentration across

the membrane:

J = D * (C1 - C2)

2. The question how to compare the rate constants is simple. The

absorption rate constant (or 'drug entry constant') is C*K12, which has

the unit of reciprocal time.

The practice of assuming an absorption rate "constant" is a simplifying

assumption that, in our experience, rarely holds true. In fact, Ka is

never a constant, although in some situations, it can be treated as one

with acceptable error.

Consider the equation for Fick's Law above and what can cause the flux

(J) to change.

D: for absorption, this is the product of the local effective

permeability (Peff) and the surface/volume ratio. The local effective

permeability, Peff, changes with position in the gastrointestinal

tract. It is different in duodenum, jejunum, ileum, and colon for most

drugs. The differences are caused by different surface area, different

tight junction gap (for paracellular transported drugs), different pH

(ionization effects), and different transporter expression (for influx

or efflux). Sometimes the differences are small. Usually the colon Peff

is significantly different from the small intestine values.

C1 and C2: assume C1 is the lumen concentration, and C2 is the

concentration in the enterocytes. When C1 >> C2, you can treat C2 as

zero. But how long does that last? As the drug is absorbed, C1 drops

and C2 increases. When they become equal, absorption stops. When the

enterocyte concentration is higher than lumen, the flux is reversed,

and drug is exsorbed/secreted into the lumen (a phenomenon that has

been observed for iv doses). The old-fashioned approach of assuming an

absorption rate "constant" ignores this, and it ignores the fact that

the absorption rate steadily decreases as C1 decreases and C2

increases. In our experience, using a concentration gradient (Fick's

Law) approach provides far more accurate and consistent results -

enabling a single model to predict plasma concentration-time results

for different doses without a change in model parameters.

Walt Woltosz

Chairman & CEO

Simulations Plus, Inc. (SIMU)

1220 W. Avenue J

Lancaster, CA 93534-2902

U.S.A.

http://www.simulations-plus.com

Phone: (661) 723-7723

FAX: (661) 723-5524

E-mail: walt.aaa.simulations-plus.com

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