From: "Jiannguo Li" <LIJ4@gunet.georgetown.edu>

Subject: Correct codes of modeling Michaelis-Menten elimination using concentration-time data

Date: Fri, 29 Sep 2000 11:23:07 -0400

Dear NM users,

In the NONMEM Users Guide -Part VIII (March, 1998, Page 321), the suggest code for modeling Michaelis-Menten elimination using concentration-time data for one compartment disposition should be:

DADT(1)=3D-VM*(A(1)/S1)/(KM+A(1)/S1) (Equation 1)

The S1 was defined in the $PK). If you use this code to model the alcohol data (Wilkinson PK et al (1976), "blood ethanol concentrations during and following constant-rate intravenous infusion of alcohol", Clinical Pharmacology and therapeutics, 19:213-223), the resulting parameter estimates (Vm and KM) were far from what reported in the paper for each of six subjects. However, a little bit change of the code:

DADT(1)=3D-VM*A(1)/(KM+A(1)/S1) (Equation 2)

(i.e. the amount A(1) in the numerator was not scaled by S1. The similar results of the parameter estimates were obtained. I think the amount A(1) in the numerator of M-M equation was implicitly scaled by S1 if S1 had been defined in $PK block, just like the case of defining a deferential equation for a first-order elimination ( DADT(1)=3D-Ke*A1(1)), it is incorrect to explicitly code A1(1) as A(1)/S1 if S1 had been defined in $PK. Consequently, the equation (2) should be the correct code. This is confirmed by simulation using the reported Vm and Km of the subject 1(Vm=3D0.253 mg/(ml*hr) , Km=3D0.108 (mg/ml, dose=3D79.6*79.6 (gm), infusion time=3D2 hours). The simulated predictions using equation 2 were coincident with those report in the paper for the subject 1. The data for the first subject and the control files of simulation using both equations 1 and 2 for the first subject are attached for inspections. Any comments are welcome.

Jianguo Li, PhD

Center for Drug Development Science

Georgetown University

filename=SIM103.CTL

filename=SIM101.CTL

filename=SUBJ1.prn

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From: Lewis B Sheiner <lewis@c255.ucsf.edu>

Subject: Re: Correct codes of modeling Michaelis-Menten elimination usingconcentration-time data

Date: Fri, 29 Sep 2000 09:29:09 -0700

There is no scaling in DES. The reason for the difference is that the standard definition of Vm is a maximum rate, i.e. amount/time. The Users Guide equation uses that convention. If you do not scale A(1) by S(1) then the Vm estimate you get is really Vm/Vd, with units of concentration per time. This is the parameter Dr Li uses (e.g. "Vm=0.253 mg/(ml*hr)"); it is no mistake to redefine a parameter in this way if one chooses, so long as the definition is clearly stated and the usage is consistent. However I believe it is inadvisable in this instance to do so. To minimize confusion, we should try, I think, to stick to the the conventions of the fields from which we borrow.

LBS.

_______________________

_/ _/ _/_/ _/_/_/ _/_/_/ Lewis B Sheiner, MD (lewis@c255.ucsf.edu)

_/ _/ _/ _/_ _/_/ Professor: Lab. Med., Biophmct. Sci., Med.

_/ _/ _/ _/ _/ Box 0626, UCSF, SF, CA, 94143-0626

_/_/ _/_/ _/_/_/ _/ 415-476-1965 (v), 415-476-2796 (fax)

*****

From: alison@c255.ucsf.edu

Subject: Re: Correct codes of modeling Michaelis-Menten elimination using concentration-time data

Date: Fri, 29 Sep 2000 11:15:41 -0700 (PDT)

Jianguo Li compared these two differential equations:

DADT(1)=-VM*(A(1)/S1)/(KM+A(1)/S1) (Equation 1)

DADT(1)=-VM*A(1)/(KM+A(1)/S1) (Equation 2)

He got better results with the second.

He said:

I think the amount A(1) in the numerator of M-M equation was implicitly scaled by S1 if S1 had been defined in $PK block, just like the case of defining a deferential equation for a first-order elimination ( DADT(1)=-Ke*A1(1)), it is incorrect to explicitly code A1(1) as A(1)/S1 if S1 had been defined in $PK.

Lewis Sheiner responded:

There is no scaling in DES. The reason for the difference is that the standard definition of Vm is a maximum rate, i.e. amount/time. The Users Guide equation uses that convention. If you do not scale A(1) by S(1) then the Vm estimate you get is really Vm/Vd, with units of concentration per time. This is the parameter Dr Li uses (e.g. "Vm=0.253 mg/(ml*hr)"); it is no mistake to redefine a parameter in this way if one chooses, so long as the definition is clearly stated and the usage is consistent. However I believe it is inadvisable in this instance to do so. To minimize confusion, we should try, I think, to stick to the the conventions of the fields from which we borrow.

I just want to amplify what Lewis said a bit. PREDPP uses scale S1 only in the ERROR block, after compartment amounts have been computed. The reserved value F in the ERROR block is a scaled compartment amount. (In this case, F = A(1)/S1.) The kinetics implemented in the differential equation can be written in terms of amounts A(1), or concentration A(1)/S1. It is simply a question of the units of KM and VM.

WIth this change to $PK in the first control stream, he can leave Equation 1 as-is and get the same results as with Equation 2:

VM=THETA(1)*THETA(3)

(instead of VM=THETA(1))

Or, he can leave the $PK block (and definition of VM) as-is, and change the differential equation thus:

DADT(1) =- VM*V*(A(1)/S1)/(KM+A(1)/S1)

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From: "Jiannguo Li" <LIJ4@gunet.georgetown.edu>

Subject: Re: Correct codes of modeling Michaelis-Menten elimination using concentration-time data

Date: Fri, 29 Sep 2000 15:44:03 -0400

Now, we understand that the implicit unit of Vm is always "amount/time" in order to use the code suggested by the help guide. Since the unit of initial estimates of Vm from the Hans-Woolf plot or the Woolf-Augustinsson-Hofstee plot using concentration time data is "amount/(time*volume)", the NM users may need to be reminded to adjust the initial estimates of Vm by the initially selected volume of distribution (V) for the use of the code.

Jianguo Li, PhD

Center for Drug Development Science

Georgetown University