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Re: Parallel first order and Michaelis-Menten elimination

From: Leonid Gibiansky <LGibiansky>
Date: Sun, 28 Mar 2010 15:45:07 -0400

Hi Nick

This form of equations can be derived from the target-mediated drug
disposition equations. In that, VM=Kint*Rmax, where Kint is the
internalization rate and Rmax is the concentration of the target
(receptor). Target-mediated clearance is believed to be carried out in
the central volume (in that particular form of the equations), and thus,
VM is coming from the enzyme theory equations (maximum reaction rate).
In my experience, if you follow parametrization that I use, VM is
independent on weight (and thus, random effect on volume does not
correlate with the random effect on VM - if the one is needed- unlike
the parameterization that you propose). For biologics, if we believe
that the mechanism of non-linear clearance is TMDD, it is more
mechanistic to use parameterization suggested in my e-mail.

Alternatively, the same MM equation can be derived from TMDD using
slightly different assumptions. In that form, VM=ksyn, where ksyn is the
target production rate per unit volume. Both forms (Vm=ksyn or
VM=Kint*Rmax) interpret MM constants in terms of mechanistic parameters
of the TMDD system.

As to the error model, I used non-transformed variables but borrowed the
error model from the log-transformed case. I like it because it would
not deliver negative results on simulations.

While the log-transformation may or may not provide much benefits, it is
the only way to implement true exponential (rather than proportional)
error model in nonmem. This is purely technical
(mathematical-statistical-numerical method-related) problem, no biology
behind this transformation, so I cannot see much sense to argue for or
against using this trick.

Any way, the question was about MM part, not the error model. We can
supplement MM model with any more conventional error model of personal
choice (hopefully supported by the data and confirmed by the VPC).

Best !

Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
e-mail: LGibiansky at
tel: (301) 767 5566

Nick Holford wrote:
> Leonid,
> Thanks for the code example which illustrates one side of a religious
> debate which took place a few weeks ago on PharmPK. The essence of this
> debate was should one normalize PK parameters to a unit volume or to a
> unit body.
> The unit volume believers feel that the rate constant is the 'natural'
> way to describe pharmacokinetics while the unit body believers feel that
> clearance is more 'natural'. Both groups agree that the two systems are
> just reparameterizations and make identical numerical predictions.
> Your coding of Vmax for the mixed order elimination process has the
> implicit units of mass/time per unit volume e.g. mg/h/L. This is the
> unit volume belief system.
> I am a unit body believer so I would code this system differently with a
> very simple change- substituting A(1) with C1 to multiply the mixed
> order expression. I have also changed VM to VMUB to indicate that the
> dimensions of the Vmax parameter are per unit body i.e. mg/h per body.
> DADT(1) = -K10*A(1)-C1*VMUB/(KM+C1)-K12*A(1)+K21*A(2)
> It could also be written like this to emphasize that the mixed order
> process has the same units as CL (for unit body believers) when C1 tends
> to 0:
> DADT(1) = -C1*(CL+VMUB/(KM+C1) - K12*A(1)+K21*A(2)
> I note also that your residual error model implies that the DV has been
> log transformed. This reflects yet another belief system which I think
> you have shown has little, if any, practical merit. I prefer to keep the
> DV in the original units.
> Best wishes,
> Nick
> Leonid Gibiansky wrote:
>> ADVAN6 ADVAN8 or (nm7) ADVAN13
>> The code is below
>> Leonid
>> -------------------
>> NCOMP = 2
>> COMP = (CENTRAL) ;1
>> COMP = (PERIPH) ;2
>> $PK
>> CL= THETA(1)*EXP(ETA(1))
>> V1= THETA(2)*EXP(ETA(2))
>> Q = THETA(3)*EXP(ETA(3))
>> V2= THETA(4)*EXP(ETA(4))
>> VM= THETA(5)*EXP(ETA(5))
>> KM= THETA(6)
>> K10 = CL/V1
>> K12 = Q/V1
>> K21 = Q/V2
>> S1 = V1
>> S2 = V2
>> $DES
>> C1 = A(1)/S1
>> DADT(1) = -K10*A(1)-A(1)*VM/(KM+C1)-K12*A(1)+K21*A(2)
>> DADT(2) = K12*A(1)-K21*A(2)
>> TY = A(1)/V1
>> W = SQRT(THETA(7)**2/TY**2+THETA(8)**2)
>> Y = IPRED*EXP(W*ERR(1))
>> .....
>> .....
>> 1 FIX ; ~ERR
>> --------------------------------------
>> Leonid Gibiansky, Ph.D.
>> President, QuantPharm LLC
>> web:
>> e-mail: LGibiansky at
>> tel: (301) 767 5566
>> chenyuhong
>>> Dear All,
>>> I am working with a Biologic and would like to have a PK model with
>>> parallel first order and Michaelis-Menten elimination. Any suggestion
>>> about which subroutine I am supposed to use? If you can share an
>>> example for the control stream, that will be a great help.
>>> Thanks,
>>> Yuhong
> --
> Nick Holford, Professor Clinical Pharmacology
> Dept Pharmacology & Clinical Pharmacology
> University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand
> tel:+64(9)923-6730 fax:+64(9)373-7090 mobile:+64(21)46 23 53
> email: n.holford
Received on Sun Mar 28 2010 - 15:45:07 EDT

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