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Error model

From: Leonid Gibiansky <LGibiansky>
Date: Wed, 31 Mar 2010 07:50:12 -0400

Dear Martin
As you may notice, I changed the subject line. The original thread had
nothing to do with the error model; it was about MM model. The control
stream that I used was copy-paste from the real project that worked just
fine both for simulations and estimation. Modeling requires common sense
and diagnostics: the same model that is good for one dataset can be
terrible for the other one. Moreover, for any error model you propose I
can present you with the hypothetical situation that would violate the
model assumption. That is why modeling is interactive process: you try
one model (whether it is the error model variation or number of
compartments, or type of nonlinearity) look on the diagnostics, correct
the model, etc, until you are happy with the outcome. The problem that
you pointed out is obvious, and indeed, manifest itself sometimes: I've
seen it on several real data sets. If you face it, you just need to
correct the error model to be in agreement with your data.

For the log-transform, I would like to re-iterate that this is simply a
trick to implement exponential error model in nonmem. What you and Nick
say is that the proportional (or additive+proportional) model is good
enough in most cases, and I would agree with it. But in some rear cases
(I've seen in in the problem with noisy data for the PD biomarkers), the
true exponential is much better, and then you have no choice except to

As to the bioanalytical data with negative concentrations, I do not
believe that you will get them (on any FDA-submitted analysis) any time
soon. Moreover, this could be irrelevant to the use of the additive part
of the error model: more often that not, this additive part is much
larger than the assay error, so it comes from some other sources, and I
guess, those "other sources" cannot result in negative values. In those
cases, error models with positive predictions would be more mechanistic.


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

Martin Bergstrand wrote:
> Dear Leonid,
> As I have pointed out once before on NMusers
> ( the error
> model that you are using can be very problematic. The RUV model only have
> the desired properties as long as THETA(7) is larger than TY (TY=IPRED in
> your example). If TY << THETA(7) this error model will give rise to an
> almost infinite RUV and hence completely unrealistic predictions (eg. DV:
> 10^-50 to 10^50). If you don't understand what I mean you can take the
> dataset associated with this model and add an observations at a very late
> time point (where TY typically is << TY). If you simulate out this sample a
> number of times you will see that DV takes on values in an almost infinite
> range. Even though this is perhaps primarily a problem during simulation but
> it is of course also potentially harmful to estimations.
> In contrast to Nick I do sometimes see the benefit of modeling a
> log-transformed DV since it in many cases improve the runtimes and "model
> stability" of NONMEM. However even though I have been playing around with it
> quite a lot I haven't found a really good RUV model for the so typical
> "combined error" structure of bioanaytical data. I feel that the downside of
> simulating some negative DVs with the additive + proportional RUV model for
> non transformed data generally is less of a problem. The sad part of this is
> as Nick and others has pointed out numerous times that it is a constructed
> problem. Isn't it about time that we as the customers of bioanalytical
> analysis enforce a better data reporting standard that doesn't imply non
> random censoring of data?
> Best regards,
> Martin Bergstrand, MSc, PhD student
> -----------------------------------------------
> Pharmacometrics Research Group,
> Department of Pharmaceutical Biosciences,
> Uppsala University
> -----------------------------------------------
> martin.bergstrand
> -----------------------------------------------
> Work: +46 18 471 4639
> Mobile: +46 709 994 396
> -----Original Message-----
> From: owner-nmusers
> Behalf Of Leonid Gibiansky
> Sent: den 31 mars 2010 00:07
> To: Nick Holford
> Cc: nmusers
> Subject: Re: [NMusers] Parallel first order and Michaelis-Menten elimination
> Nick,
> The form that I use directly follows from the TMDD model suggested in
> 2001 in:
> Mager DE, Jusko WJ. General pharmacokinetic model for drugs exhibiting
> target-mediated drug disposition. J. Pharmacokinetic and Pharmacodynamic
> Vol. 28, pp. 507-532 (2001).
> Derivation of the MM model from the TMDD equations can be found in:
> Gibiansky L, Gibiansky E, Kakkar T, Ma P: Approximations of the
> target-mediated drug disposition model and identifiability of model
> parameters. J Pharmacokinet Pharmacodyn. (2008) 35(5):573-91.
> (this is not the only paper that discusses the TMDD/MM correspondence,
> but I like it, for the obvious reasons :) ). TMDD model assumes that all
> elimination occurs in the same volume as the central volume.
> You can check that in this case the nonlinear part of elimination is
> volume-proportional (so, mass-proportional) with VM being WT-independent
> [in VM*A(1)/(KM+A(1)/V1) ].
> As to the error model with negative predictions, I will start to use it
> as soon as I get my first dataset with negative concentrations :) Note
> that in many if not all cases, most of the residual error comes not from
> the assay error but from other unexplained factors. While assay error
> could lead to negative concentrations, "other unexplained factors"
> cannot result in negative values, so I prefer to use the error model
> that provides only positive values. Of course, this is the matter of
> preferences / style. Everyone has their own favorite models, including
> the error models.
> Best !
> Leonid
> --------------------------------------
> Leonid Gibiansky, Ph.D.
> President, QuantPharm LLC
> web:
> e-mail: LGibiansky at
> tel: (301) 767 5566
> Nick Holford wrote:
>> Leonid,
>> I accept that the unit volume parameterization of VM is quite reasonable
>> if you think that all elimination occurs in the same volume as the
>> distribution volume. This is the usual test tube model that gives rise
>> to the unit volume belief system. It is not a realistic view of
>> elimination from the human body
>> I would also accept that if all elimination occurs in some blood cell
>> distribution volume (e.g.white cells) then this will be highly
>> correlated with the drug distribution volume and the unit volume
>> parameterization will appear to work fine but will fail if there is some
>> covariate that determines blood cell volume distribution differently
>> from drug distribution volume.
>> However, I don't accept that the elimination of a biological can be
>> independent of weight if we refer to the actual mass eliminated per unit
>> time. The first order part of the model for elimination which uses
>> clearance is certainly not independent of weight so why would you
>> imagine that the mixed order elimination process would be independent of
>> weight?
>> Note that the oldest example of a mixed order elimination process is for
>> ethanol (Widmark essentially invented the science of pharmacokinetics
>> with a mixed order elimination model). Ethanol elimination largely
>> occurs in the liver while it distributes in total body water. Liver
>> metabolism scales allometrically in a different way from total body
>> water so I do not believe that the mixed order elimination of ethanol
>> should be described with the unit volume belief system. I much prefer
>> the unit body belief system.
>> Sorry - I was confused by your residual error model which at first sight
>> seemed to be a transform both sides model. However, the model you use
>> restricts all residual errors to be non-negative which is not a
>> realistic description of any non-censored residual error process. All
>> real world uncensored measurement errors should have an error
>> distribution on both sides of zero when the true value is zero.
>> Nick
>> Leonid Gibiansky wrote:
>>> 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
>>> --------------------------------------
>>> Leonid Gibiansky, Ph.D.
>>> President, QuantPharm LLC
>>> web:
>>> 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
>>>>> -------------------
>>>>> $MODEL
>>>>> 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)
>>>>> $ERROR
>>>>> TY = A(1)/V1
>>>>> IPRED=TY
>>>>> W = SQRT(THETA(7)**2/TY**2+THETA(8)**2)
>>>>> Y = IPRED*EXP(W*ERR(1))
>>>>> $THETA
>>>>> .....
>>>>> $OMEGA
>>>>> .....
>>>>> $SIGMA
>>>>> 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
>> --
>> 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 Wed Mar 31 2010 - 07:50:12 EDT

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