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Hi,
I am trying to build a renal model of my compound. The compound is
completely excreted by kidney. In the kidney, it is reabsorbed by
different segment. There is passive and active reabsorption at different
segment in the kidney. There is no report on the passive permeability or
active transport Vmax and Km, but we know that at three different sites,
the reabsorption is about 20%, 65%, 10%. We have clinical data
containing IV Cplasma-time profile and accumulate urine drug amount over
a period. My question is would it be possible we find out the Vmax and
Km with our current data?
Here is my equation:
dAe/dt=fu*GFR*Cserum(t) - [Vmax/(km+Crenal(t))*Crenal(t)*SA1 +
Ppassive*SA2*Crenal(t)]
Ae - accumulated urine drug amount
SA1-surface area of active reabsorption site
SA2-surface area of passive reabsorption site
Ppassive-paracellular permeability (passive)
Vmax- unit mass per unit time per unit area
fu-unbound fraction of Mg
Crenal-Mg concentration in active reabsorption site
We do not have each subject's GFR, but we know healthy subject's GFR,
120 mL/min, fu is available. We do not know Crenal(t), because there is
no way to obtain it. We could assess Ppassive, using 65% of the total
fu*GFR and a rough assessment Crenal average. SA1 and SA2 are available.
How would be able to find out Vmax and Km using our Cplasma-time and
urine accumulation data?
Thank you!
Chunsheng
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Tricia,
It is not possible to say if your model is identifiable from the
information you give. But if you have a big concentration range in serum
and lots of urine samples and the unbound concs in blood are both higher
and lower than the km then you have a reasonable chance of estimating
Vmax and Km.
Renal concentrations will typically be delayed in relation to serum
concs. I have used an effect compartment model in the past to model the
delay in renal concs of frusemide relative to the time course of unbound
concs in blood.
I don't know why you say SA1 and SA2 are available. I assume you can
only make a guess at these numbers based on some other study. I would
ignore them unless you really have a way of measuring them in each
subject individually.
Good luck!
Nick
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Dear Nick,
Thank you so much for your comments/advice.
The compound I am working on is an endogenous compound. Our body can
maintain a relatively stable plasma concentration and it is pretty high,
in mM range.
The published clinical research used isotope to trace the compound that
were administered IV and orally. So we collected the literature data of
the isotope-labled-compound's Cplasma-time and urine accumulation
amount.
In this case, in the equation, would the Vmax and Km still reflect the
normal compound's Vmax and Km? (I think the transporter is supposed to
handle the normal and the isotope-labeled compound the same, but the
normal compound is pretty high in concentration, so the transporter may
be still saturated even the isotope-labeled compound concentration is
very low. So I am quite confused on what Vmax and Km I can get with the
isotop-labled-compound data?)
The SA1 and SA2 are my estimates according to literature. I agree with
you that we probably should ignore them and simply see them as some
scaling factors, since my estimates on them are not accurate and needs
to be adjusted anyway.
Thanks for your comments on the renal concentration. Could you please
kindly send me your publication on the compartmental model modeling
frusemide renal concs? Could you also please kindly suggest me some more
reading materials on this kind of study? I think I may need to establish
the relationship between plasma concentration and renal concentration
and the following is my thought:
In the equation I want to solve: dAe/dt=fu*GFR*Cserum(t) -
(Vmax/(km+Crenal(t))*Crenal(t)*SA1 + Ppassive*SA2*Crenal(t),
there are three things I do not know: Vmax, Km, Crenal(t). My clinical
data can tell me Ae, Cserum(t), I can guess Ppassive. I think in order
to successfully model the urine secretion, what I need to find out is
the relationship between Cserum(t) and the Crenal(t), right? And then I
can use the Cserum(t) and Ae data to fit the equation and find out the
parameters.
Please kindly let me know your comments/advice and I greatly appreciate
it!
Thank you so much again.
Sincerely,
Tricia (Chunsheng)
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Tricia,
It would be interesting to know what compound you are working with. It
is rather boring to have to deal with abstract ideas only without
knowing what real biological problem you are trying to investigate :-) I
would like to learn something too!
It is a long standing principle of tracer kinetic methodology that the
kinetics of tracer concentrations will always reflect first-order
processes even when combined with non-tracer concentrations at the same
time. Thus if you are using tracers you will not be able to estimate
Vmax and Km. You will therefore only be able to determine renal and
non-renal first order clearances with measurement of urine excretion
rates and serum concentration.
dC/dt= (Ratein - (CLr +CLnr)*C)/V ; serum tracer conc
dCe/dt= ln(2)/Teq*(C - Ce) ; effect compartment conc
dAu/dt=CLr*Ce ; urine excretion rate
In principle I agree it is a good idea to standardize your parameter
estimates to covariates such as SA1, SA2 and GFR. But you should
understand that these guesses at typical covariates do not change your
ability to describe the data. They can be useful for comparison of your
parameters with others in the literature that have been standardized in
a similar way based on actual measurements of the covariates.
The work I did with frusemide was only ever published as an abstract
(Holford & Brater 1985) a long time ago. But the idea of using an effect
compartment model is simple (see above). Teq is the effect compartment
equilibration half-life. Look in any review or textbook that deals with
PKPD. If you are stuck then contact me again.
Good luck!
Nick
Holford NHG, Brater DC, A Physiological Pharmacokinetic-Dynamic Model
for Furosemide applied to the Interaction with Ibuprofen, Naproxen, and
Sulindac. Clin.Pharmacol.Ther. 37:202 (1985). Abstract
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Dear Nick,
In your reply to Tricia you wrote:
"It is a long standing principle of tracer kinetic methodology that the kinetics
of tracer concentrations will always reflect first-order processes even when
combined with non-tracer concentrations at the same time. Thus if you are using
tracers you will not be able to estimate Vmax and Km."
In my understanding, tracer kinetics reflects the kinetics of the total of
tracer and non-tracer compounds, as concluded from the logical assumption that
all molecules behave the same (ignoring any isotope effect, i.e. that the tracer
behaves somewhat different due to the difference in molecular weight). This was
also mentioned by Tricia in her reply to your questions.
Your statement quoted above suggests that my understanding is incorrect. Could
you explain, or give some references? I would like to learn something too! :-)
best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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Hans,
Tracer kinetic methods for studying physiological systems were developed a long
time ago and I don't have any old reference books handy in which this theory is
fully developed. However this set of teaching slides refers to tracer kinetics
always being first-order:
http://www2.bio.ku.dk/isotopkursus/english/Course/Course_materials/Powerpoint/PDF/Isotopteknik_F20_ENG.pdf
e.g. slide 27
"Many processes may be described and analyzed as first-order processes. This
normally applies to tracer investigations of biological systems in 'steady
state', irrespective of the concentration dependence of the underlying process!!"
If you have access to a hard copy library you may be able to find a better justification.
I am certainly ready to be corrected on this point because it struck me as being
un-intuitive when I first heard it many years ago (late 1970s).
Nick
Nick Holford, Professor Clinical Pharmacology
Dept Pharmacology & Clinical Pharmacology, Bldg 503 Room 302A
University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New
Zealand
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The following message was posted to: PharmPK
Hi Nick,
I think the underlying idea is that tracer has small concentration
relative to the endogenous concentration. If so, it will follow
linearized equations. If tracer concentration is comparable or larger
than the endogenous concentration, it will follow nonlinear kinetics.
Leonid
Leonid Gibiansky,
President, QuantPharm LLC
web: www.quantpharm.com
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Dear Nick,
Thank you for your reply. As you state, the quote from that ppt:
"Many processes may be described and analyzed as first-order processes. This
normally applies to tracer investigations of biological systems in 'steady
state', irrespective of the concentration dependence of the underlying process!!"
is un-intuitive. I did not yet find any justification for this statement, and,
personally, I don't believe this to be true. It may be so that this statement is
a way to say that the kinetics of a tracer can be analysed irrespective of the
presence of endogenous compounds or other ways of administration (e.g.
combination of a tablet and tracer iv dose to determine bioavailability).
As long as concentrations are well below the Km values of all relevant kinetic
processes, this is indeed true. Note that this ppt refers to zero-order and
Michaelis-Menten kinetics on this slide 27 only; the remainder of the
presentation refers to linear, first-order kinetics, and I presume that the
quoted statement refers to first-order kinetics.
At higher concentrations this statement cannot be true from a theoretical point
of view, and also not from experimental evidence, e.g. in figure 3 from the
following paper
Proost JH, Beljaars L, Olinga P, Swart PJ, Kuipers ME, Reker-Smit C, Groothuis
GMM, Meijer DKF. Prediction of the pharmacokinetics of succinylated human serum
albumin in man from in vivo disposition data in animals and in vitro liver slice
incubations. Eur J Pharm Sci 2006; 27(2-3): 123-132.
In this paper we explicitly assumed that tracer kinetics follow the kinetics of
the unlabeled compound, and I still believe this is the only correct way. But of
course I'm open to revise my view with convincing arguments.
best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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Leonid,
In this context the term 'tracer' refers to use of very low amounts and thus
concentrations.
The basis of the principle is that in the presence of steady state (and more or
less constant) concentrations of an endogenous substance then even if the
endogenous conc is greater than the km the clearance in the presence of the
endogenous substance will be essentially constant.
Kinetics of a tracer will therefore appear to be first-order because indeed
clearance is not changed by the tracer concentration.
If you did the tracer study at a lower endogenous substance steady state then
the clearance of the tracer would appear to be faster.
So my response to Tricia still stands -- the disposition of tracer concs will
appear to be first-order and it will not be possible to estimate the mixed-order
elimination parameters from tracer data alone.
Best wishes,
Nick
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Dear Nick and others,
In addition to my previous message discussing the cited statement:
"Many processes may be described and analyzed as first-order processes.
This normally applies to tracer investigations of biological systems in
'steady state', irrespective of the concentration dependence of the
underlying process !!"
which is un-intuitive, and seeming conflicting with my earlier view that
tracer kinetics will show nonlinear kinetics in the case of
concentration of the unlabeled compound that are around or above the Km
values of the relevant kinetic processes.
In the following analysis I come to the conclusion that these two views
are not conflicting, if expressed adequately.
Assume a steady state, with constant concentration of the unlabeled
compound at a level that nonlinear conditions apply, i.e. around or
above Km (Michaelis-Menten constant) of a relevant process, e.g. elimination.
Now we can derive the following equations:
rate of elimination of unlabeled compound = Vmax * C / (Km + C), where C
is the concentration of unlabeled compound.
rate of elimination of tracer = Vmax * Ct / (Km + C), where Ct is the
concentration of the tracer (assumed to be negligeble compared to C).
(for an explanation of this approach see Proost JH, Beljaars L, Olinga
P, Swart PJ, Kuipers ME, Reker-Smit C, Groothuis GMM, Meijer DKF.
Prediction of the pharmacokinetics of succinylated human serum albumin
in man from in vivo disposition data in animals and in vitro liver slice
incubations. Eur J Pharm Sci 2006; 27(2-3): 123-132.)
If C is constant, the expression Vmax / (Km+C) is constant, and
represents the clearance of the tracer. So, the rate of elimination of
the tracer is proportional to Ct, and thus it follows first-order
kinetics. Actually, I am quite surprised by this result!
At the same time, the concentration of unlabeled compound will alter the
kinetics of the tracer compound: its elimination will be delayed
compared to the situation where the concentration of unlabeled compound
is well below Km, since the clearance of the tracer is Vmax / (Km + C).
So, it would be incorrect to say that the kinetics of the tracer are
independent of the unlabeled compound.
In non-steady-state conditions, the kinetics of a tracer do not follow
first-order kinetics, since C is not a constant.
In summary:
1) In steady-state conditions, the kinetics of a tracer follow
first-order kinetics, even if the system is nonlinear (saturation).
2) The kinetics of a tracer are dependent on the concentration of the
unlabeled compound.
So, we may conclude that both views are correct. The above statement is
also correctly formulated, but it does not tell the whole story, since
it does not make clear that the kinetics of the tracer and so the
results of the analysis (e.g. the half-life of the tracer) are dependent
on the concentration of the unlabeled compound.
I would appreciate your view on this analysis.
best regards,
Hans
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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Hans,
I saw your question which came around the time I had just responded to Leonid (see below).
The key thing assumption is that tracer concs are negligible compared
with non-tracer concs (typically an endogenous compound). If the
endogenous concs are constant then the 'clearance' will appear to be
constant when observing tracer conc disposition. This is simple to predict from theory:
CLnt = Vmax*Cnt/(Km+Cnt) ; 'clearance' at constant conc of
not-tracer (Cnt)
CLboth = Vmax*[Cnt+Ct]/(Km+[Cnt+Ct]) ;where Ct is tracer conc
If Cnt>>Ct then CLnt~=CLboth i.e. the 'clearance' is determined by Cnt
and Cnt (by assumption) does not change so 'clearance' does not change.
Best wishes,
Nick
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The following message was posted to: PharmPK
Dear Hans
In general, I agree with the analysis although I would questions the part
"rate of elimination of tracer = Vmax * Ct / (Km + C)"
I would rather use Vmax*KM*ct/(KM+C)^2 that can be obtained by more
careful expansion of the nonlinear term.
Leonid
--
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
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Hans,
I agree. I think we are saying the same thing. It depends on the
assumption that tracer really means concs that are negligible compared
with non-tracer concs.
NIck
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Analytically the tracer should not interfere with the analyte and the
analyte should not interfere with the tracer this should be demonstrated
during validation
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Dear Leonid,
Thank you for your comment. You commented on the equation:
> "rate of elimination of tracer = Vmax * Ct / (Km + C)"
>
> I would rather use Vmax*KM*ct/(KM+C)^2 that can be obtained by more careful
> expansion of the nonlinear term.
I'm not sure whether this is correct. Assuming that the tracer and not-tracer
have exactly the same values for Vmax and Km, we can consider any mixture of
tracer and not-tracer as a single compound, with the normal Michaelis-Menten
equation.
rate of elimination of nt + t = Vmax*[Cnt+Ct]/(Km+[Cnt+Ct])
=46rom this equation it follows that
rate of elimination of tracer = Vmax*Ct/(Km+[Cnt+Ct])
rather than your equation. If I'm wrong, I would like to learn the background of
your equation.
Best regards,
Hans
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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Dear Nick,
Thank you for your comments. We seem to agree on the question with respect to
the linear kinetic behavior of a tracer concentration in the presence of a
constant not-tracer concentration.
But I have a few comments on your derivation:
> CLnt = Vmax*Cnt/(Km+Cnt) ; 'clearance' at constant conc of
> not-tracer (Cnt)
> CLboth = Vmax*[Cnt+Ct]/(Km+[Cnt+Ct]) ;where Ct is tracer conc
The denominator of both equations should be [Cnt+Ct] (of course, if Ct<term Ct can be left out).
If the left-hand reflects clearance, the concentration term in the right-hand
part should be omitted:
CL = Vmax/(Km+[Cnt+Ct])
and clearance of tracer and not-tracer are equal, irrespective of the values of
Cnt and Ct.
In my view, 'CLboth' is not a valid term. I would say, 'CL', irrespective of
being not-tracer or tracer.
The equations can also be written:
rate of elimination of nt + t = Vmax*[Cnt+Ct]/(Km+[Cnt+Ct])
rate of elimination of not-tracer = Vmax*Cnt/(Km+[Cnt+Ct])
rate of elimination of tracer = Vmax*Ct/(Km+[Cnt+Ct])
Best regards,
Hans
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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Hans,
I can confirm we agree on the principle of first-order (I much prefer that to
linear!) elimination of tracer in the presence of constant non-tracer
concentration.
Thanks for spotting my error in putting Cnt and Cnt+Ct in the numerator for the
expressions for clearance.
I agree with the 3 expressions you have given for the rate of elimination:
rate of elimination of nt + t = Vmax*[Cnt+Ct]/(Km+[Cnt+Ct])
rate of elimination of not-tracer = Vmax*Cnt/(Km+[Cnt+Ct])
rate of elimination of tracer = Vmax*Ct/(Km+[Cnt+Ct])
or in terms of clearance which shows that the clearance is the same for all 3
cases:
clearance of nt + t = Vmax/(Km+[Cnt+Ct])
clearance of not-tracer = Vmax/(Km+[Cnt+Ct])
clearance of tracer = Vmax/(Km+[Cnt+Ct])
Once again the important point is that if Cnt is constant and Ct << Cnt then the
clearance of tracer then becomes:
clearance of tracer = Vmax/(Km+Cnt)
so it Cnt is constant the clearance of tracer will be constant even when Ct
varies time so that the elimination of Ct will appear to be first-order.
Best wishes,
Nick
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The following message was posted to: PharmPK
Dear Hans,
After Nick's e-mail, I think I agree with your expression. I derived the
equation for the quantity "increase of total endogenous + tracer
concentrations above the background level" while actually tracer is
different compound (measured by radioactivity or something else but
different from the endogenous compound). In this case, rate of
elimination of tracer is indeed Vmax * Ct / (Km + C).
Sorry for confusion
Leonid
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
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