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Hello.
Consider two-compartment model. The concentration in the first
(central) compartment is defined by :
dC1/dt= - k10*C1 - (C1-C2)*k12, such that C1(0)=D/V1
where k10 and k12 are elimination rate and inter-compartment rate,
respectively. C1(0) is the initial concentration in the central
compartment, D is the dose and V1 is the volume of compartment 1.
the concentration in the second compartment is defined by:
dC2/dt=-(C2-C1)*k21
Now, suppose that we know that in a certain condition, Vss of the
patient increased by some fraction (i.e. 50%). What PK parameters in
the above equation need to be changed in order to account for this
change?
Thank you
David
[Increasing the k12/k21 ratio would give a higher Vss but I don't
think your equations are correct, see
http://www.boomer.org/c/p4/c19/c1902.html
http://www.boomer.org/c/p4/c19/c1904.html
and
http://www.boomer.org/c/p4/c19/c1905.html
- db]
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The following message was posted to: PharmPK
The mass balance is not correct
Uusally we perform a mass balance based on the amount and the equations
are
dA1/dt= - k10*A1 -k12.A(1) +k21.A(2)
dA2/dt= k12.A(1) -k21.A(2)
The inter-compartment rate is defined as Q (not k12 which would lead to
confusion) and has the units of Volume per unit of time:
K12=Q/V1
K21=Q/V2
Now you get
dA1/dt= - k10*A1 -k12.A(1) .V1/V1 +k21.A(2) .V2/V2
dA2/dt= k12.A(1) .V1/V1 -k21.A(2) .V2/V2
dA1/dt= - k10*A1 -Q.C1 +Q.C2
dA2/dt= Q.C1 -Q.C2
T=0, A(1)=D or C1(0)=D/V1
VSS=V1+V2
Therefore, as an example if VSS increase by 50%, it could be the result
of increasing both V1 and V2 by 50% but apparently there are infinite
number of other possibilities.
Hope it helps
Serge Guzy
President, CEO; POP-PHARM; Inc.
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The following message was posted to: PharmPK
It seems to me that there is a problem in definitions.
k12 and k21 should be called "rate contstants" and not "rate", while the
most apropriate term for Q, in my opinion, is clearance or "flow".
Now, let's take the following equation proposed by Serge:
dA1/dt= - k10*A1 -Q.C1 +Q.C2
For convinience we can write it down as follows:
dA1/dt=-k10*A1 + Q(C2-C1)
Dividing it by V1, we get
dA1/(V1*dt) = dC1/dt = -k10*A1/V1 + Q/V1(C2-C1)
Following the definition that k12=Q/V1:
dC1/dt = -k10*C1 + k12(C2-C1), which is exactly what was initially
written by David.
Now, the question is: what are the primary parameters (i.e. those that
are most tightly connected to the underlying physiological mechanisms)
and what are the secondary ones (i.e. calculated). The A,Q,V are usually
considered to be primary, while C,k12,k21 are the secondary. However, in
many PK/PD calculations the secondary parameters are more convinient to
work with.
Also see the following discussion on PharmPK:
http://www.boomer.org/pkin/PK08/PK2008310.html
[This link may not be permanent so you might want to search for the
title 'Two compartment distribution' in the annual index - db]
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The following message was posted to: PharmPK
Hi,
Boris Gorelik wrote:
> Now, the question is: what are the primary parameters (i.e. those
that
> are most tightly connected to the underlying physiological
mechanisms)
> and what are the secondary ones (i.e. calculated). The A,Q,V are
usually
> considered to be primary, while C,k12,k21 are the secondary.
However, in
> many PK/PD calculations the secondary parameters are more
convinient to
> work with.
I agree in part with what Boris has written but not completely. From a
theoretical perspective it is concentration (or chemical activity) that
is the driving force for mass transfer and for chemical reactions -- it
is not amount -- so I cannot agree that 'C' is secondary and 'A' is
primary. Note also that concentration is not a parameter -- it is a
variable.
Parameterisation is important for intepretation and for estimation and
should be distinguished from the algebraic convenience of performing
some calculation.
Interpretation of parameters is more useful when the parameters can be
related to physiological or pharmacological mechanisms. Parameters
estimated in this way can have their variability explained better by
other covariates e.g. renal function will change clearance of renally
cleared drugs but there is no physiological entity resembling a rate
constant so parameterisation in terms of a rate constants will always
require an empirical application of renal function.
Differences in estimation can sometimes be observed with different
parameterisations because of the dependence on numerical issues related
to such matters as derivatives. But this is only a challenge for better
computer hardware and software and not of fundamental importance.
So my bottom line preference is to parameterise in terms of quantities
that can be mapped to some mechanistic or physical reality. This means
using volumes and clearances for distribution and mass transfer
kinetics.
Nick
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New
Zealand
n.holford.aaa.auckland.ac.nz tel:+64(9)923-6730 fax:+64(9)373-7090
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
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Dear Nick and Boris
As I may have said before, particularly in theoretical terms, I cannot
agree with the statement "it is concentration ... that is the driving
force for mass transfer ... - it is not amount". This idea is indeed
deeply rooted in our education about kinetics by means of a simplistic
interpretation of the law of mass action and Fick's laws of diffusion.
But the in vivo drug concentrations we deal with on a daily basis are
something rather different from the "concentration gradient" that
we're taught to drive reactions and transfer processes. We should keep
in mind that the mass action law in full generality is formulated for
activities or fugacities. Molecules don't choose to go from crowded
spaces to others where they see few of their kind. It's a
probabilistic phenomenon, binary in nature, just asymptotically
Gaussian for a large number of molecules. Although strictly for
elementary reactions, the law states that the rate of any chemical
reaction is proportional to the product of the masses of the reacting
substances, with each mass raised to a power equal to the coefficient
that occurs in the chemical equation (by Guldberg in the late eighteen
hundreds). In other words, the product of the concentrations of the
reactants include the power of their stoichiometry, stemming from the
fundamental molecular or massic relationship. Since the early
applications were in Chemistry, volumes were accurately measured and
therefore concentrations may be used, as stated in the majority of
current printed references. But, for instance, in physics, the law is
deduced without invoking chemical potentials at all. Denoting by N_i
the quantities of substances A_i in moles, the free energy G of the
mixture corresponds to the sum of the free enthalpies g_i of the
constituents and the mixing entropy S_m: G=sum(N_ixg_i)-TxS_m. Mass
transfer is really about the second law of thermodynamics.
Dealing with biological systems, the danger of the concentration
rationale resides precisely on the fact that there's no way to measure
the volume. We define the aqueous volume that would be needed to have
the same mass and still get the sampled concentration, calling it
apparent. Therefore, what drives the transfer process can only be the
actual mass, since the concentration depends on our own definition of
volume.
On the issue of parameterization, I totally agree that it makes a
difference, some times crucial, on computation and estimation, but
certainly not in interpretation. Unless the basic argument is put
forward about people not being comfortable interpreting reciprocal
times, while being much at ease dealing with time units, let's say in
terms of a half-life for straightforward clinical applications, then
I'll agree. But the real interpretation of the underlying phenomenon
must be exactly the same regardless of the parameterization. So, a
first order rate constant is just the fraction of all the molecules at
the source 'traveling' via a process or pathway per unit time. For
example, if a rate constant is found to be 0.03h^-1, then 3% of the
material in the source travels by this pathway per hour. This
information is highly relevant since it's entirely independent of
mass, volume or concentration. It's intrinsic to the kinetics of the
process. E.g., according to Agatha Christie, the time constant for the
cooling of a cadaver is 3.5 hrs. The time constant (the reciprocal of
the rate constant) is yet another simple index of how rapidly a
process will respond to a step stimulus. And it is extensively used in
the physiological literature. I concede that renal clearance is
ubiquitous, but that doesn't mean it's the best way to characterize
the renal function. It always reports to a volume cleared per unit
time, volume of blood, plasma, serum or better said volume of
distribution, which is naturally an apparent volume and something to
always keep in mind.
Hoping this stimulates our thinking and nothing else more primordial,
accept my best regards.
Luis
--
Luis M. Pereira, Ph.D.
Assistant Professor, Pharmacometrics
Massachusetts College of Pharmacy and Health Sciences
Childrens Hospital Boston / Harvard Medical School
179 Longwood Ave, Boston, MA 02115
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The following message was posted to: PharmPK
Luis,
If it's only mass that maters, and not concentration, how would you
treat a
condition wherein only half of the mass is in solution, perhaps
because of
precipitation?
It seems to me that mass in solution (i.e., concentration) is what
matters
in the vast majority of pharmaceutical applications (barring
endocytosis).
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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The notion of Luis about chemical potentials is correct. However in
the case of simple pharmacokinetics, where the stoichiometry of the
diffusion through a membrane is 1:1 and there is no pressure applied,
the concentration gradient is a pretty good and close approximation of
the gradient of chemical potential.
Boris
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Hi Walt,
I understand what you say and I even think you provide the answer.
Excluding pinocytosis/endocytosis, carrier-mediated transport,
convection, advection, and all other possible complications, as you
say, if the kinetics of the process follows let's say zero order, then
the rate is constant and the solid material acts as an infinite source
to maintain its concentration equal to solubility. If, just as another
example, the kinetics is first order, then the rate depends on the
amount still remaining in solution to be let's say transferred across
a membrane. Of course we can multiply and divide the right-hand side
of the relationship by a volume term and then we convert mass into
concentration and the first order rate constant into a clearance just
reparameterizing the problem (dM/dt = k.M = kV(M/V) = CL.C) which is
PERFECTLY OK. If the volume is known it doesn't matter saying that the
concentration (or the mass) is the driving force for the process. The
less material we have the slower it goes, and for a given physical
volume, the less material the more dilute the solution and the slower
it goes as well. What I tried to say was that the concentration we
measure in an aliquot of plasma is related to the amount in that
homogeneous environment, which we may even extend to the rest of the
body, by an apparent volume, which accommodates all kinds of binding,
partitioning and distribution issues. But then we end up with two
unknowns, mass and volume. I even recall you alluding to the unstirred
and heterogeneous nature of the intestinal lumen at the site of
absorption, and the variable nature of the fraction absorbed which is
absolutely true. So, the precipitated mass is still conceptually the
driving force for the transfer process but just several orders of
magnitude less relevant that the amount in solution. Which is not to
say that this is a perfect Newtonian solution in a beaker of a know
volume with a semi-permeable membrane interface. The amount in the
liquid phase acts much more as a driving force than the amount in the
solid phase. But unless we sample the local concentration, we
shouldn't say the concentration of the solution we prepared in order
to administer the drug, or the virtual concentration in the average
volume of the intestine, or, on the other side of the border, the
concentration of the drug in the blood stream, are the driving forces
for the respective mass transfer processes. Individual molecules
wonder around, randomly walking, and the more they are the more
driving power they'll have.
I hope I was a bit clearer. But again, on this platform, who is?
Best regards
Luis
--
Luis M. Pereira, Ph.D.
Assistant Professor, Pharmacometrics
Massachusetts College of Pharmacy and Health Sciences
Childrens Hospital Boston / Harvard Medical School
179 Longwood Ave, Boston, MA 02115
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Luis,
Thanks for the update on thermodynamics -- but I think you miss the
point about the difference between concentration and amount. The total
amount of mass in the universe is fixed (more or less). It is the
concentration of mass in space time that makes things interesting. On
similiar lines, my previous comment was to say that amount is not a
primary variable determining pharmacokinetics. Concentration is a
reasonable approximation for any realistic view of pharmacokinetics
AND pharmacodynamics. I dont see any danger in this viewpoint.
However, your interpretation of clearance as 'volume cleared per unit
time' is true only in a mathematical sense. It ignores the physical
concept that clearance is describing -- and this can lead to dangerous
conclusions. Clearance is the proportionality factor between
concentration and rate of elimination. The units are secondary to the
definitions of concentration and rate of elimination. The naive idea
that clearance is volume cleared per unit time led one quite renowned
clinical pharmacologist to say that if the clearance is 10 L/h and the
apparent volume of distribution is 100 L then all the dose is cleared
in 10 hours. This appeared in an early edition of one well known
textbook of clinical pharmacology but was quickly removed from later
editions :-)
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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