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Hello,
I would like to thank you all for the fruitful and didactic
discussion. Regarding the two-compartment PK. However I would like to
ask another question. Since I did not receive any answer before, I
would like to clarify it.
I was asked to model a two-compartment PK model of a drug, with non-
linear, saturable inter-compartment exchange.
Suppose I don't have enough experimental data to answer the followign
question, what is the most probable cause of the saturation
phenomenon? Is it the concentration on either side of the compartment
or is it the concentration gradient. In other words, does the exchange
rate depend on (C_1) or on (C_1 - C_2) ?
Thank you
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David D wrote:
> I was asked to model a two-compartment PK model of a drug, with
> non-linear, saturable inter-compartment exchange.
> Suppose I don't have enough experimental data to answer the followign
> question, what is the most probable cause of the saturation
> phenomenon? Is it the concentration on either side of the compartment
> or is it the concentration gradient. In other words, does the
exchange
> rate depend on (C_1) or on (C_1 - C_2) ?
Lets try to think about this for a real life example. Levodopa is a
pro-drug for dopamine with saturable transport from plasma to brain
across the blood brain barrier. As far as I know the reverse movement of
levodopa from brain to plasma is not saturable.
If we ignore the conversion to dopamine (for simplicity) and just
consider the kinetics of levodopa it could be written like this:
dCp/dt = (R + CLbp*Cb - (Vmax/(Km+Cp)+CLe)*Cp)/Vp
dCb/dt= (Vmax*Cp/(Km+Cp) - CLbp*Cb)/Vb
R=Ratein of levodopa to plasma. CLbp=distribution clearance from brain
to plasma; Vmax,km=Saturable distribution process from plasma to brain;
CLe=first-order elimination of levodopa from plasma; Vp=apparent volume
of distribution of levodopa in plasma compartment; Vb=apparent volume of
distribution of levdopa in brain compartment.
Cp=levodopa conc in plasma compartment; Cb=levodopa conc in brain
compartment
Note that the 'driving force' for each transfer process is the
concentration in one of the compartments -- it is not the difference
between the concentrations. The difference in concs could be used if you
had first-order distribution processes in both directions -- but this is
just algebra and not mechanism.
dCp/dt = (R + CLbp*Cb - (CLbp+CLe)*Cp)/Vp
dCb/dt= (CLbp*Cp - CLbp*Cb)/Vb ; mechanism
or
dCb/dt= (CLbp*(Cp - Cb))/Vb ; algebra
Please note that just because these equations can be written does not
imply that a saturable process across the blood brain barrier could be
identified from measuring plasma concs alone. My guess is that the
saturable transfer of levodopa is only a minor part of the total
distribution clearance. Efforts to characterise plasma levodopa kinetics
in humans typically show a two compartment behaviour after an IV
infusion but no evidence of any non-linearity e.g. Chan PL, Nutt JG,
Holford NH. Pharmacokinetic and pharmacodynamic changes during the first
four years of levodopa treatment in Parkinson's disease. J Pharmacokinet
Pharmacodyn.
2005;32(3-4):459-84.
Nick
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New
Zealand
n.holford.at.auckland.ac.nz
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
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David,
There seems to be a typo when you tried to copy my equations because
at least one of the equations at
http://mathbin.net/814
is wrong. I wrote:
dCp/dt = (R + CLbp*Cb - (Vmax/(Km+Cp)+CLe)*Cp)/Vp
dCb/dt= (Vmax*Cp/(Km+Cp) - CLbp*Cb)/Vb
Your translatation wrote something like:
dCp/dt = (R + CLbpb - (Vmax/(Km+Cp)+CLe)*Cp) 1/Vp
dCb/dt= (Vmax*Cp/(Km+Cp) - CLbpCb) 1/Vb
Note especially that the expression CLbp*Cb became corrupted as Cbpb.
It also used arithmetic operators inconsistently e.g. sometimes 'x'
was used for multiplication and sometimes the arithmetic operator was
missing. Note that with only a minor change to suit Berkeley Madonna
conventions for defining a symbol for a differential equation then my
code is executable:
R=1
CLbp=1
Vmax=1
Km=1
CLe=1
Vp=1
Vb=1
init Cp=0
init Cb=0
d/dt(Cp) = (R + CLbp*Cb - (Vmax/(Km+Cp)+CLe)*Cp)/Vp
d/dt(Cb)= (Vmax*Cp/(Km+Cp) - CLbp*Cb)/Vb
Is it is possible to execute LaTeX code?
Nick
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New
Zealand
n.holford.-at-.auckland.ac.nz
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
[LaTeX code can be executed, i.e. intrepreted using appropriate
software, somewhat like BASIC - db]
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Formatted Nick's response can be found here:
http://mathbin.net/814
Thank you Nick, both for the detailed explanation and for the ref.
[Sorry, I think I got this and Nick's reply out of order - db]
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The following message was posted to: PharmPK
Nick,
Aren't your equations:
dCp/dt = (R + CLbp*Cb - (Vmax/(Km+Cp)+CLe)*Cp)/Vp
dCb/dt= (Vmax*Cp/(Km+Cp) - CLbp*Cb)/Vb
based on the assumption that the clearance terms CLbp and CLe have been
fitted to some measured concentrations? If so, and if the dosing
(concentration) range is limited, then single values may apply without
significant error. I know that you're well aware that fitting values to
parameters with simplified models based on appropriate assumptions can
easily cover up mechanisms that would only show up under a wider range
of
concentrations.
On the other hand, it seems that virtually all transport processes
have to
be saturable in some way. If there are carrier-mediated transporters
involved, then the expression levels of those transporters will
dictate how
much it would take to saturate them. If only passive diffusion is
involved,
then the surface area of the membrane would limit how much drug could
diffuse, assuming that concentrations high enough to achieve saturation
could be reached.
As far as ignoring the concentration gradient, I submit that doing so is
algebra, not mechanism, at least when only passive diffusion (with no
difference in ionization or binding on both sides of the membrane) is
involved. What is happening on a molecular scale? Brownian motion
results in
molecules moving around in all directions. When a membrane is
presented with
drug in solution on both sides, Brownian motion on each side will cause
molecules to enter the membrane. The side with the higher
concentration will
contribute more molecules into the membrane than the other side, so
the net
movement of drug will be from high to low concentration, in accordance
with
Fick's First Law. When (if) the concentrations become equal, the
movement in
each direction is the same, so the net movement is zero. I don't see how
(again, assuming only passive diffusion and no difference in
ionization or
binding) it could be any other way.
Best regards,
Walt Woltosz
Chairman & CEO
Simulations Plus, Inc. (NASDAQ: SLP)
42505 10th Street West
Lancaster, CA 93534-7059
U.S.A.
http://www.simulations-plus.com
E-mail: walt.aaa.simulations-plus.com
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Dear Dr. Woltosz,
You wrote:
If only passive diffusion is involved, then the surface area of the
membrane would limit how much drug could diffuse, assuming that
concentrations high enough to achieve saturation could be reached.
Could you please elaborate on how passive diffusion across a membrane
becomes saturated at high concentration?
Best regards,
Frederik Pruijn
Frederik B. Pruijn PhD MSc (Senior Research Fellow)
Experimental Oncology Group
Auckland Cancer Society Research Centre
Faculty of Medical and Health Sciences
The University of Auckland
Private Bag 92019
Auckland 1142
New Zealand
E-mail: f.pruijn.-a-.auckland.ac.nz
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Walt,
I think I agree completely with everything you say about general
principles of drug movement. But the original question was about how
to write a model -- one of those things that are always wrong but
sometimes useful. So of course the equations are only an useful
approximation to reality. There is no assumption made about fitting
concentrations. The assumption is only that this model describes
adequately what you want to use the model for e.g. simulating profiles
or estimating parameters.
Nick
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New
Zealand
n.holford.-a-.auckland.ac.nz
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
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The following message was posted to: PharmPK
Dear Frederick,
As I picture it in my mind (no guarantee this picture is correct, but
it's
intuitive), it's like water flowing through a sieve - the capacity to
flow
is not infinite. Bigger holes would help, but given a particular size
hole,
and a particular surface area, there would be a limit. Higher pressure
might
increase it somewhat, but we're not dealing with high pressures in
biological membranes.
So I would expect that if you could get enough molecules in solution
on one
side of a membrane and have an infinite sink on the other side, you
would
reach a concentration where the membrane just couldn't accommodate a
higher
transport rate. Does that make sense?
Best regards,
Walt
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Dear Walt,
With water flowing through a sieve gravity would be the driving force.
With diffusion it is the so-called concentration gradient. So, to
follow your analogy with water & sieve by increasing the concentration
on one side of a membrane we increase the "pressure". I can't see any
principal reason why this process would become saturated. Obviously,
at very high concentrations the structure and properties of the
membrane might change depending on the diffusant. The only limit I can
see is that of temperature on Brownian motion (described by the
diffusion coefficient D) of the diffusant molecules but net flux (J)
will not be limited according to J=\0x00D dc/dx.
I am not sure that this has direct biological/physiological relevance
but I'd like to hear other views.
Best regards,
Frederik Pruijn
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The following message was posted to: PharmPK
Dear Frederik,
We usually assume a constant diffusion coefficient, when we model
diffusion
processes. In general this might not be true. In particular this will
not
be true, if we talk about a theoretical increase of concentrations,
just to
look if the mass transfer via diffusion is saturable. In situations like
this, the diffusion coefficient depends on the concentration, diffusion
processes (and saturation) are modelled using a concentration dependent
diffusion coefficient. A model like this will be able to describe the
picture provided by Walt. I like this picture, and when he says, higher
pressure might increase the transport rate somewhat, he talks about
adding
a term for directed flow to the transport model.
Best regards,
Peter Wolna
Global Biostatistics
Merck Serono Development/Global Clinical Operations
Merck KGaA, Germany
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The following message was posted to: PharmPK
Dear Frederick,
Right - in the sieve example, gravity is the driving force. A taller
column
of water would have higher pressure, resulting in an increase in flow.
But a
wider reservoir of the same height would not, even though there would be
more water.
Perhaps the better analogy at this time of year would be the crowds
entering
a football stadium. There are only so many portals for them to go
through.
They all seek to get in as quickly as possible. In some instances,
everyone
lines up in an orderly queue, while in others, everyone crowds toward
the
front, trying to get ahead of as many people as possible. The latter
might
be the best analogy for a biological membrane. If the number of people
is
small, they go through about as fast as they can walk. If it becomes
large,
there is a lot of bumping and pushing and they shuffle along. The
speed of
the people on the inside of the gate determines how fast those going
through
the gate can proceed. Sometimes, the flow stops because the people
inside
can't move any faster through the tunnels to the seats. More portals and
tunnels would increase the total flow, but given a fixed number, there
is a
point where the portals and tunnels simply cannot accommodate a higher
flow
rate.
Best regards,
Walt
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Dear Peter and Walt,
Can you please provide any references for a concentration-dependent
diffusion coefficient in relation to passive diffusion across a
(biological) membrane?
In our modelling (i.e. fitting of experimental data) of (passive)
diffusion across 3D multicellular layers we use a constant diffusion
coefficient, which we validated experimentally (Hicks et al., CANCER
RESEARCH 63, 5970-5977, September 15, 2003).
Best regards,
Frederik Pruijn
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The following message was posted to: PharmPK
Dear Walt and Frederick,
I am not an expert for this field but as I understand you Walt, at
some point entropy is taking its
share. I.e. diffusion coefficient is constant only in the DILUTED
solutions and laminar
(non-turbulent) flow. Higher concentrations of solute and/or solution
flow turbulence cause changes
of diffusion coefficient. Another way to take into account this
phenomenon is to include some kind
of entropy term into (kinetic) equation. Am I right?
Zeljko Debeljak, PhD
Medical Biochemistry Specialist,
Osijek Clinical Hospital,
CROATIA
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The following message was posted to: PharmPK
Dear Frederik,
Like many ideas applied to modeling in pharmacology the idea of a
concentration-dependent diffusion coefficient comes from chemical
engineering. If you search for articles in that area, you will certainly
find a lot of references. But I'm sorry, I'm not aware of references
discussing this idea using biological membranes as an example.
With kind regards,
Peter
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