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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
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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
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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
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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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