[R] Michaelis-menten equation

joerg van den hoff j.van_den_hoff at fz-rossendorf.de
Wed Jul 20 12:56:25 CEST 2005

I believe the following is correct:
first of all, as peter daalgaard already pointed out, your data Cp(t)
are following a straight line
very closely, i.e. 0.-order kinetics
for your diff. eq. this means that you are permanently in the range cp
>> Km so that
dCp/dt = - Vm/Vd = const. =: -b and, therefore, Cp = a - b*t
you can't get any reliable information concerning Km from the fit. the
solution of the diff. eq (according to Maxima),
namely t + const. = -(Km*log(Cp) + Cp)/(Vm/Vd) tells you the same:
Km*log(cp) << Cp in your data.
in any case (even in the range Km ~= Cp) you can't determine Vm _and_ Vd
separately according to your diff. eq., you only get the ratio b =
Vm/Vd. this does make sense:
what you are measuring  is the decreasing plasma concentration, you
don't have any information
concerning the "relevant" volume fraction, i.e. the Vd, in you data.
therefore any variation in the effective
max. velocity can either be ascribed to a variation of Vm or to a
modified Vd. in other words:
you should view Vm* = Vm/Vd as your  "effective" Vm.
res <- lm ( y~x ) yields a=8.561, b=Vm*= 0.279
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
0.00863 0.09340 0.13700 0.26500 0.22900 1.37000

i.e. 0.265 percent deviation on average between data and fit. I believe
that is the max. information you can get from your data
((the result for "b" is  accidently is not so far away
from the ratio of your results Vm = 10.04, Vd = 34.99 which actually
must be completely unstable
(double both parameters and nothing happens to the fit).
you state you simulated the data with km=4.69? using the above Vm and
Vd, the resulting data are (not unexpectedly)
quite different from those you used as input to the fit. maybe you made
an error somewhere "upstream"?
in conclusion: don't try to fit Vm _and_ Vd separately, check whether
your simulated data are correct, keep in mind that if km<<Cp, you can't
fit Km (at least not reliably).

Chun-Ying Lee wrote:
> Hi,
>    We are doing a pharmaockinetic modeling.  This model is
> described as intravenous injection of a certain drug into
> the body.  Then the drug molecule will be eliminated (or decayed)
> from the body.  We here used a MM eq. to describe the elimination
> process and the changes of the drug conc..  So the diff. eq. would
> be: dCp/dt = -Vm/Vd * Cp/(Km+Cp).  Vd is a volume of distribution.
> We used lsoda to solve the diff. eq. first and fit the diff. eq.
> with optim first (Nelder-Mead simplex) and followed by using nls 
> to take over the fitting process of optim.  However, we can not
> obtain the correct value for Km if we used the above model.  The
> correct Km can be obtained only when we modeled the diff eq. with
> dCp/dt= -Vm/Vd * Cp/(Km/vd + Cp).  Now we lost.  The data were
> from simulation with known Vm and Km.  Any idea?  Thanks.
> regards,
> --- Chun-ying Lee
>>it is not clear to me what you are trying to do:
>>you seem to have a time-concentration-curve in PKindex and you seem 
>>to set up a derivative of this time dependency according to some 
>>model in dCpdt. AFAIKS this scenario is  not directly related to the 
>>situation described by the Michaelis-Menten-Equation which relates 
>>some "input" concentration with some "product" concentration. If Vm and
>>Km are meant to be the canonical symbols,
>>what is Vd, a volume of distribution? it is impossible to see (at least
>>for me) what exactly you want to achieve.
>>(and in any case, I would prefer "nls" for a least squares fit 
>>instead of 'optim').
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