- On 3 Apr 1999 at 22:09:41, "Bonate, Peter, Quintiles" (pbonate.-at-.qkcm.quintiles.com) sent the message

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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 - On 3 Apr 1999 at 22:10:20, "Nick Holford" (n.holford.at.auckland.ac.nz) sent the message

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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 - On 5 Apr 1999 at 22:46:20, David_Bourne (david.-a-.pharm.cpb.uokhsc.edu) sent the message

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[Three replies - db]

X-Sender: st005899.at.brandywine.otago.ac.nz

Mime-Version: 1.0

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

---

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 - On 9 Apr 1999 at 22:33:52, Daro Gross (maildrop.-at-.iname.com) sent the message

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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. - On 11 Apr 1999 at 22:14:41, Maria Durisova (exfamadu.-a-.savba.sk) sent the message

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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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