- On 10 Sep 1999 at 22:44:58, Sarawut Oo-puthinan (sarawuto.at.nu.ac.th) sent the message

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Dear pharmacokineticists

I am studying the release of cloxacillin from hydroxyapatite

(bone matrix

substitute, which used in bone injury) into bone. Which can the model be

used to simulate the profile of release of antibiotic in bone?

At now, we are determining the diffusion of the drug from the bone

substitute in the buffered solution.

The preliminary result shows that the release may be complete

within 1-2

days.

Thank you in advance

Sarawut Oo-puthinan

Faculty of Pharmaceutical Sciences, Naresuan University, Thailand - On 12 Sep 1999 at 12:52:12, ml11439.at.goodnet.com (Michael J. Leibold) sent the message

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Hello Sarawut,

Fick's Law (dc/dt= K(cs-ct)) suggests that the dissolution

process could be modeled pharmacokinetically as the first order

absorption equation used to model the dissolution and absorption

of oral medications (L(ft)= KaFD/(s+ka) or ft= KaFDe-kat).

However, the sustained release of medications can also be

modeled as the Ko equation for continuous, constant release. This

is a simplification but may be adequate particularly if you find

the release from bone cement occurs at a constant rate.

My personal studies have indicated that sustained release

systems such as transdermal systems, depot injections or SR oral

preparations can be modeled using the first order absorption, one-

comparment model:

Cp= [KaFD/(Ka-Ke)Vd][e-Ket - e-Kat]

Although there are more complex pharmacokinetic equations

which have been developed [particularly in the case of the

transdermal systems], this equation can predict plasma concen-

trations.

In the case of predicting plasma concentrations from the

release of antibiotic from bone cement, I think the above

equation could be used. A more sophisticated approach

would be to model the system as three comparment model with

release of the antibiotic from the third compartment. However,

I think that the bone cement would comprise a drug-delivery

system, and not a functional pharmacokinetic compartment with

mutidirectional microconstants.

I don't have the literature readily available, but there

have been pharmacokinetic studies on the release of aminoglycosides

from bone cement. The pharmacokinetic model used in these studies

should be applicable.

However, it occurs to me that tissue concentrations would be

a big issue in this study. In that case, you may want to model

this system as a two compartment model with a continuous infusion

into the peripheral compartment. The peripheral compartment would

represent the infusion of antibiotic directly into the site of

infection. The plasma concentrations would be modeled as resulting

from the diffusion of antibiotic from the peripheral compartment.

[SI-A][Xs]= [Us]

[(s+k10+k12) -k21][X1s]= [ 0 ]

[ -k12 (s+k21)][X2s] [ Ko/s(1-e-Ts)]

That is, the matrix system resulting from infusion into the

peripheral compartment.

The equation for the central (plasma) compartment would be:

Cp= Ko(1-e-at)k21/[Vc(a)(b-a)] + Ko(1-e-bt)k21/[Vc(b)(a-b)]

The equation for the peripheral (tissue) compartment would be:

Cp= Ko(1-e-at)(k10+k12-a)/[Vp(a)(b-a)] +

Ko(1-e-bt)(k10+k12-b)/[Vp(b)(a-b)]

This later equation could be used to predict concentrations at

the site of action, the tissue or peripheral compartment.

Alternatively, the same matrix system could be used but with

a first order input into the peripheral compartment like the initial

one-compartment equation I suggested. That is, the usual first order

absorption into a two compartment model, but absorption would

occur into the tissue compartment where the drug would then diffuse

into the plasma compartment.

[(s+k10+k12) -k21][X1s]= [ 0 ]

[ -k12 (s+k21)][X2s] [ KaFD/(S+Ka) ]

The point being that you could place an emphasis on predicting

tissue concentrations with the later two models. However, the above

perpheral is not homogeneous with the rest of the peripheral physio-

logic compartment since the released antibiotic will mix with plasma

before it mixes with the rest of the peripheral compartment.

Therefore, a more accurate pharmacokinetic model might be:

k13 k12

Ko-> Cpt3 <-> Cpt1 <-> Cpt2

k31 \ k21

\

k10->

In this case, Cpt3 represents the homogeneous cement tissue compartment

from the which the antibiotic immediately mixes, and diffuses into the plasma

from where it distributes to the rest of the body. For the above three

compartment model, the Laplace-tranformed system matrix is:

[SI-A][Xs]= [Us]

[(s+k13+k12+k10) -k21 -k31 ] [X1s] = [ 0 ]

[ -k12 (s+k21) 0 ] [X2s] = [ 0 ]

[ -k13 0 (s+k31)] [X3s] = [Ko/s(1-e-ts)]

The Laplace-transformed compartmental quantities are:

X1s = Ko/s(1-e-Ts)(s+k21)k31/[(s+a)(s+b)(s+g)]

X2s = Ko/s(1-e-Ts)k31k12/[(s+a)(s+b)(s+g)]

X3s= Ko/s(1-e-Ts)[(s+k12+k12+k13)(s+k21)-k12k21]/[(s+a)(s+b)(s+g)]

However, the above three compartment model could be simplified by

combining the tissue compartment with central compartment and just modeling

cement tissue compartment as the only other compartment, but with the

understanding that it is not homogeneous with the rest of the tissue

space usually considered as part of the peripheral compartment.

Good luck with your study!

Mike Leibold, PharmD, RPh

ML11439.-a-.goodnet.com - On 13 Sep 1999 at 14:09:49, ml11439.at.goodnet.com (Michael J. Leibold) sent the message

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Hello Sarawut,

In short, the 2c model might be the most useful if you specify

that the cement-tissue is the only peripheral compartment in the

model, and the rest of the peripheral tissue compartment is part

of the central compartment. This would mean that this two compartment

model is not the usual two compartment model in which the distribution

phase represents the distibution of the antibiotic into the second

compartment. This model does not have a distribution phase from the

central comparment into the second compartment,and the second compartment

only represent the cement-tissue compartment and not the rest of the

tissue space.

k12

[KaFDe-kat]->Cpt2 <-> Cpt1->k10

k21

[SI-A][Xs]= [Us]

[(s+k10+k12) -k21][X1s]= [ 0 ]

[ -k12 (s+k21)][X2s] [ KaFD/(S+Ka) ]

The Laplace-transformed compartmental quantities are:

X1s= KaFD(k21)/(s+ka)(s+a)(s+b)

X2s= KaFD(S+k10+k12)/(s+ka)(s+a)(s+b)

The equations for the concentrations in the central in perheral

compartments are:

Central compartment:

Cp= KaFD(k21)e-kat/Vc(a-ka)(b-ka) +

KaFD(k21)e-at/Vc(ka-a)(b-a) +

KaFD(k21)e-bt/Vc(ka-b)(a-b)

Cement-tissue comparment:

Ct= KaFD(k10+k12-ka)e-kat/Vt(a-ka)(b-ka) +

KaFD(k10+k12-a)e-at/Vt(ka-a)(b-a) +

KaFD(k10+K12-b)e-bt/Vt(ka-b)(a-b)

I think that this model could adequately model the physiological

situation, unless there is something I don't know about the release

of antibiotic from the bone cement you describe. It is actually a

one compartment model with a second compartment added on representing

the drug delivery system and the surrounding tissue.

I hope this was of some use!! I just thought I saw a mathematical

solution to the pharmacokinetic problem you describe.

Mike Leibold, PharmD, RPh

ML11439.-a-.goodnet.com - On 14 Sep 1999 at 11:25:33, David_Bourne (david.-at-.boomer.org) sent the message

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>Date: Tue, 14 Sep 1999 01:36:56 -0700 (MST)

>X-Sender: ml11439.-at-.pop.goodnet.com

>Mime-Version: 1.0

>To: postmaster.at.boomer.org

>From: ml11439.at.goodnet.com (Michael J. Leibold)

>Subject: PharmPK Re: Model for determining release of drug in bone

>

>Hello,

>

> I just wanted to report a slight error in one of my equations

>in the email of Sept 12, 1999. The following equation is incorrect:

>

>X3s= Ko/s(1-e-Ts)[(s+k12+k12+k13)(s+k21)-k12k21]/[(s+a)(s+b)(s+g)]

>

>The correct equation is:

>

>X3s= Ko/s(1-e-Ts)[(s+k10+k12+k13)(s+k21)-k12k21]/[(s+a)(s+b)(s+g)]

>

>This is regarding Laplace-transformed quantities of a three compartment

>model representing a bone cement infusion system.

>

> Mike Leibold, PharmD, RPh

> ML11439.-a-.goodnet.com

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