- On 3 Apr 1999 at 22:09:41, "Bonate, Peter, Quintiles" (pbonate.-at-.qkcm.quintiles.com) sent the message
I have never had occassion to model a drug with flip-flop kinetics before.
I understand that if you model a one-compartment system as
C=A*exp(-alpha*t)+B*exp(-beta*t) then you have to assume that Ka>Kel. But
what if you model a system as a series of differential equations:
dX1/dt = -Ka*X1
dX2/dt = Ka*X1 - Kel*X2
with scaling parameter C=X2/V? Are the output rate constants from such a
fit really the right values? The standard errors are quite precise and the
fit appears excellent.
Unfortunately, I do not have iv data. Any comments on this would be
appreciated. Thanks.
PETER L. BONATE, PhD.
Clinical Pharmacokinetics
Quintiles
POB 9708 (L4-M2828)
Kansas City, MO 64134
phone: 816-767-6084
fax: 816-767-3602
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- On 3 Apr 1999 at 22:10:20, "Nick Holford" (n.holford.at.auckland.ac.nz) sent the message
Yes (well at least within the usual caveat of all models being wrong). The
closed form solution parameterised in A,alpha,B,beta is obtained by
integrating the DEs and hidiing the DE parameterisation. The hiding of the
parameters is part of the legacy of having mathematicians rather than
biologists think about how to express the model. It does not change the
solution. However, reparameterisation can change the final solution if the
obj function minimum is not well defined because the search depends on the
parameterisation. Any differences in the model prediction are usually small.
If they are big something is wrong with the way the model is coded or the
data is very poor for identifying the model parameters.
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, Private Bag 92019, Auckland, New Zealand
email:n.holford.-a-.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556
http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.html
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- On 5 Apr 1999 at 22:46:20, David_Bourne (david.-a-.pharm.cpb.uokhsc.edu) sent the message
[Three replies - db]
X-Sender: st005899.at.brandywine.otago.ac.nz
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Date: Sun, 4 Apr 1999 21:54:58 +1200
To: PharmPK.at.pharm.cpb.uokhsc.edu
From: Robert Purves
Subject: Re: PharmPK flip-flop kinetics
A one-compartment system with absorption kinetics can be modelled in at
least three ways. As an expression:
C=FD/V*ka*{exp[-ke*t] - exp[-ka*t]}/(ka-ke)
where FD is the available dose, and V, ka, ke are parameters to be estimated.
As a single differential equation:
dC/dt=FD/Vd*ka*exp[-ka*t] - ke*C; C=0, t=0
As two differential equations:
dC/dt=ka*Depot/Vd - ke*C; C=0, t=0
dDepot/dt=-ka*Depot; Depot=FD, t=0
Taking for demonstration purposes FD=10 (fixed) and a small data set:
t C
0.25 34
0.5 48
1 47
3 10
parameter estimates for an unweighted fit in all three versions are
"normal" "flip-flop"
V 0.099 0.05
ke 0.99 1.95
ka 1.95 0.99
Sum of squares = 0.132
The three versions of the model equations behave identically, and thus
cannot help to disambiguate the normal/flip-flop choice; there is no magic
associated with the differential equations. The usual methods are (1) by an
arbitrary and indefensible decision, for example that ka MUST be > ke, (2)
from prior information, or (3) with IV data fitted simultaneously.
Lastly, I would recommend against the practice of modelling this kinetic
system with A*exp(-alpha*t)+B*exp(-beta*t), since the appearance of 4
parameters is illusory. In such a kinetic model we require that C(0)=0,
from which it follows that we must take B=-A, and there are only three
parameters for estimation. If we (wrongly) choose to estimate all 4
parameters of A*exp(-alpha*t) + B*exp(-beta*t) from the example data-set,
we get A=-176.9 alpha=2.14 B=174.3 beta=0.94, results which have no
discernible relation to the correct values for the one-compartment model
with absorption. If, on the other hand, we replace B by -A, we can recover
the correct values of the two rate constants (corresponding to ka and ke
above, and with, alas, the same flip-flop ambiguity).
Robert Purves
Pharmacology Department
University of Otago
P.O. Box 913
Dunedin, New Zealand
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Date: Sun, 04 Apr 1999 19:15:39 -0400
From: Harold Boxenbaum
Organization: Z
X-Accept-Language: en
MIME-Version: 1.0
To: PharmPK.at.pharm.cpb.uokhsc.edu
Subject: Re: PharmPK flip-flop kinetics
Peter et al,
There are a few references on flip-flop models in my file on this subject,
presented in chronological order:
(1) JT Dolusio, JC LaPiano & LW Dittert. Pharmacokinetics of ampicillin
trihydrate, sodium ampicillin, and sodium dicloxacillin following intramuscular
injection. J Pharm Sci 60: 715-719 (1971);
(2) PR Byron & RE Notari. Critical analysis of "flip-flop" phenomenon in
two-compartment pharmacokinetic model. J Pharm Sci 65:1140-1144 (1976);
(3) KKH Chan & M Gibaldi. Assessment of drug absorption after oral
administration. J Pharm Sci 74:388-393 (1985); and
(4) H Boxenbaum. Pharmacokinetic tricks and traps: Flip-flop models. J Pharm
Pharmaceutical Sci 1(3): 90-91 (1998 or 1999).
The later article is available gratis on-line by signing up for a subscription
to the totally electronic journal, Journal of Pharmacy and Pharmaceutical
Sciences (possibly available by starting at www.ualberta.ca etc.).
A flip-flop model is simply one in which rate of drug absorption approximates
rate of drug elimination (e.g., growth hormone). This occurs when rate of
absorption is "rate-limiting," viz., rate of absorption is the slowest in a
catenary-like process. A simplified way of visualizing this is that rate of
absorption is very much slower than rate of elimination (i.e., absorption
rate-limited). Although often not appreciated, the plasma concentration-time
profile of a drug exhibiting flip-flop properties parallels that of the rate of
absorption plot (see reference 5 above). Consequently, virtually all but IV
plasma concentration-time profiles of growth hormone parallel rate of
absorption, e.g., sub-Q.
If you do not have an IV, it is always dangerous to model. Metzer et al (can't
find the reference) showed a long time ago that a two-compartment disposition
model with an absorption process -- total of 3 exponentials -- may often
collapse to a biexponential function. So the parameters you obtain from a
biexponential fit are incorrect. Even before this, Sharney, Wasserman &
Gevirtz demonstrated the principle of collapsing (Representation of certain
mammillary N-pool systems by two pool models, Am J Med Elect 3:249-260
(1964)). A NCA may therefore be your best bet. If you do have a flip-flop,
the terminal exponential rate constant will be the terminal exponential
absorption rate constant.
I have analyzed over 100 rate of absorption plots from animals and humans
utilizing a variety of dosage forms. In most cases, absorption is
tri-exponential-ish. That is, rate of absorption starts at
zero (by definition), rises to a maximum, declines relatively rapidly, and then
declines more slowly. Modeling absorption by assuming either a zero-order or
first-order process is often deceiving. As Metzler et al demonstrated, you may
get a good fit of data to your equations, but the parameters are hybridized and
therefore incorrect. In those cases where I've investigated data from models
assuming zero or first-order kinetics (where I have IV data to calculate rate
of absorption), this type of absorption rarely exists. The trap is you can
still get good fits. For example, a 4 exponential equation can collapse into
an apparent two exponential equation. Therefore good fits do not validate your
model, but only indicate consistency. When one models with sums of
exponentials, one can simply assume consistency (best to assume this for the
equation and not the model). In most cases, there is no reason to assume
validation. If you want to model with sums of exponentials, simple do so and
report the results in terms of macroscopic parameters (coefficients and
exponents) as well as model-independent parameters, e.g., mean systemic or oral
clearance, which is always dose over AUC.
Good luck. Harold.
---
Mime-Version: 1.0
Date: Mon, 5 Apr 1999 10:23:19 -0400
To: PharmPK.-at-.pharm.cpb.uokhsc.edu
From: shoaf.at.clinpharm.niaaa.nih.gov
Subject: Re: PharmPK flip-flop kinetics
Without iv data (or subcu/im if you know absorption from the sight is
rapid) you cannot tell if the parameters you determine are equivalent to
alpha or beta. You can get excellent fits but what they mean....?
See "The effect of age and diet on sulfadiazine/trimethoprim disposition
following oral and subcutaneous administration to calves."
Shoaf SE, et al. J Vet Pharmacol Ther. 1987 Dec;10(4):331-45
as an example of true flip-flop kinetics (1 week old calves) versus getting
apparent(sp?) absorption/elimination rates.
Susan Shoaf, Ph.D.
Acting Chief
Unit of Pharmacokinetic Studies
NIAAA/LCS
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- On 9 Apr 1999 at 22:33:52, Daro Gross (maildrop.-at-.iname.com) sent the message
Flip-flop kinetics can be modelled more precisely with better convergence
when using a biological model I developed that is derived from a different
area of mathematical modeling: "black box modeling."
The model use developed from extensive data gathered from both clinical and
basic research results and has been highly accurate in predicting
appropriate medication, effective dosage levels, side-effects, and
metabolic pathways.
I started out in medicine and picked up advanced mathematical modeling
techniques so am prejudiced towards medical solutions, not mathematical
tools. If I can answer any questions, please contact me.
Daro Gross
P.S. I am willing to donate whatever research results I have available to
me so please feel free to email with any problem or question. I have a
large amount of research that remains unpublished, and I do not expect to
catch up soon.
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- On 11 Apr 1999 at 22:14:41, Maria Durisova (exfamadu.-a-.savba.sk) sent the message
Dear Peter,
This reply was also sent to PharmPK:
In the equations given above, the functions X1(t), X2(t), and
C(t) have no argument, there are no initial conditions, and there
is no drug input.
Set (1) of the differential equations (DE) can be used to
describe the drug fate after the single extravascular dose D
dX1(t)/dt=-Ka*X1(t) + D*delta(t) X1(0)=0
dX2(t)/dt= Ka*X1(t) - Kel*X2(t) X2(0)=0 (1)
where delta(t) is the Dirac delta pulse and the product
D*delta(t) is the drug input. This set of the DE has for D>0,
C(t)=X2(t)/V, and A=D*Ka/V, the three analytical solutions given
by Eqs. 2, 3, and 4:
C(t)=A*(exp(-Kel*t)-exp(-Ka*t))/(Ka-Kel) (2)
for Ka>Kel
C(t)=A*(exp(-Ka*t)-exp(-Kel*t))/(Kel-Ka) (3)
for Ka
C(t)=A*t*exp(-Ka*t) (4)
for Ka=Kel
It is obvious that the absorption rate constant Ka and the
elimination rate constant Kel of set (1) of the DE can be the
parameters of the negative or positive exponential terms in
Eqs.2, and 3. This is the consequence of the fact that for set
(1) of the DE, the relation between Ka and Kel (i.e. whether
Ka>Kel, or Kaonly important for the analytical solutions of this set of the
DE, given by Eqs. 2, 3, 4. Or in other words, for set (1) of the
DE, the transfer from the DE to all the three analytical
solutions is unambiguous. However, the transfer from the
analytical solutions given by Eqs. 2 and 3 to the DE is not
unambiguous. Such a transfer is unambiguous only for the solution
given by Eq. 4, i.e. for Ka=Kel. If the transfer from the
analytical solutions given by Eqs. 2 and 3 to the DE was
unambiguous, it would be possible to define the general
unambiguous meaning of the parameters of these analytical
solutions.
Considered from the system approach point of view, set (1) of the
DE is the model of the system describing the drug fate after the
extravascular single dose D. It is a SECOND-ORDER model. On the
other hand, Eqs. 2, 3, and 4 are the models of the measured
response of this system (e.g. the measured drug
concentration-time profile in blood), i.e. they are the models of
data. Furthermore, set (1) of the DE is the model of such
a system which is considered to consist of the two FIRST-ORDER
subsystems connected in serial. The first subsystem has the mean
residence time
MRTa=1/Ka (5)
and describes the absorption process. The second subsystem has the
mean residence time
MRTel=1/Kel (6)
and describes the elimination process. The whole system describes
the fate of the drug (absorption and elimination) and has the
mean residence time
MRT=MRTa + MRTel. (7)
In many experiments with a single extravascular dose of
a drug under study there is no information available whether
Ka>Kel, Kaconcentration-time profile is measured in blood or plasma. This
profile is the response of the whole system describing the drug fate,
i.e. this profile is the SUM of the responses of the absorption
and elimination subsystem. Thus in these experiments it is not
possible to decide which of the model parameters represents Ka
and which Kel. Analogously as the number 6 can be composed as
6 = 4 + 2, or 6 =2 + 4, or 6 = 3 + 3,
and so on (see Eqs.5,6,7).
Best regards,
Ladislav Dedik
and
Maria Durisova
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