# PharmPK Discussion - Two-compartment model and volumes of distribution

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• On 10 Sep 2008 at 16:16:14, "david d" (mailing.lists.david.at.gmail.com) sent the message
`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/V1where  k10 and k12 are elimination rate and inter-compartment rate,respectively. C1(0) is the initial concentration in the centralcompartment, 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)*k21Now, suppose that we know that in a certain condition, Vss of thepatient increased by some fraction (i.e. 50%). What PK parameters inthe above equation need to be changed in order to account for thischange?Thank youDavid[Increasing the k12/k21 ratio would give a higher Vss but I don'tthink your equations are correct, seehttp://www.boomer.org/c/p4/c19/c1902.htmlhttp://www.boomer.org/c/p4/c19/c1904.htmlandhttp://www.boomer.org/c/p4/c19/c1905.html- db]`
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• On 10 Sep 2008 at 10:12:07, "Serge Guzy" (GUZY.at.xoma.com) sent the message
`The following message was posted to: PharmPKThe mass balance is not correctUusally we perform a mass balance based on the amount and the equationsaredA1/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 toconfusion) and has the units of Volume per unit of time:K12=Q/V1K21=Q/V2Now you getdA1/dt= - k10*A1 -k12.A(1) .V1/V1 +k21.A(2) .V2/V2dA2/dt= k12.A(1) .V1/V1 -k21.A(2) .V2/V2dA1/dt= - k10*A1 -Q.C1 +Q.C2dA2/dt= Q.C1 -Q.C2T=0, A(1)=D or C1(0)=D/V1VSS=V1+V2Therefore, as an example if VSS increase by 50%, it could be the resultof increasing both V1 and V2 by 50% but apparently there are infinitenumber of other possibilities.Hope it helpsSerge GuzyPresident, CEO; POP-PHARM; Inc.`
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• On 11 Sep 2008 at 11:04:17, "Boris Gorelik" (boris.-a-.optimata.com) sent the message
`The following message was posted to: PharmPKIt seems to me that there is a problem in definitions.k12 and k21 should be called "rate contstants" and not "rate", while themost 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.C2For convinience we can write it down as follows:dA1/dt=-k10*A1 + Q(C2-C1)Dividing it by V1, we getdA1/(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 initiallywritten by David.Now, the question is: what are the primary parameters (i.e. those thatare most tightly connected to the underlying physiological mechanisms)and what are the secondary ones (i.e. calculated). The A,Q,V are usuallyconsidered to be primary, while C,k12,k21 are the secondary. However, inmany PK/PD calculations the secondary parameters are more convinient towork 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 thetitle 'Two compartment distribution' in the annual index - db]`
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• On 11 Sep 2008 at 18:36:06, Nick Holford (n.holford.at.auckland.ac.nz) sent the message
`The following message was posted to: PharmPKHi,Boris Gorelik wrote: > Now, the question is: what are the primary parameters (i.e. thosethat > are most tightly connected to the underlying physiologicalmechanisms) > and what are the secondary ones (i.e. calculated). The A,Q,V areusually > considered to be primary, while C,k12,k21 are the secondary.However, in > many PK/PD calculations the secondary parameters are moreconvinient to > work with.I agree in part with what Boris has written but not completely. From atheoretical perspective it is concentration (or chemical activity) thatis the driving force for mass transfer and for chemical reactions -- itis not amount -- so I cannot agree that 'C' is secondary and 'A' isprimary. Note also that concentration is not a parameter -- it is avariable.Parameterisation is important for intepretation and for estimation andshould be distinguished from the algebraic convenience of performingsome calculation.Interpretation of parameters is more useful when the parameters can berelated to physiological or pharmacological mechanisms. Parametersestimated in this way can have their variability explained better byother covariates e.g. renal function will change clearance of renallycleared drugs but there is no physiological entity resembling a rateconstant so parameterisation in terms of a rate constants will alwaysrequire an empirical application of renal function.Differences in estimation can sometimes be observed with differentparameterisations because of the dependence on numerical issues relatedto such matters as derivatives. But this is only a challenge for bettercomputer hardware and software and not of fundamental importance.So my bottom line preference is to parameterise in terms of quantitiesthat can be mapped to some mechanistic or physical reality. This meansusing volumes and clearances for distribution and mass transferkinetics.Nick--Nick Holford, Dept Pharmacology & Clinical PharmacologyUniversity of Auckland, 85 Park Rd, Private Bag 92019, Auckland, NewZealandn.holford.aaa.auckland.ac.nz tel:+64(9)923-6730 fax:+64(9)373-7090http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford`
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• On 12 Sep 2008 at 01:58:05, (Luis.Pereira.-a-.mcphs.edu) sent the message
`Dear Nick and BorisAs I may have said before, particularly in theoretical terms, I cannotagree with the statement "it is concentration ... that is the drivingforce for mass transfer ... - it is not amount". This idea is indeeddeeply rooted in our education about kinetics by means of a simplisticinterpretation 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 aresomething rather different from the "concentration gradient" thatwe're taught to drive reactions and transfer processes. We should keepin mind that the mass action law in full generality is formulated foractivities or fugacities. Molecules don't choose to go from crowdedspaces to others where they see few of their kind. It's aprobabilistic phenomenon, binary in nature, just asymptoticallyGaussian for a large number of molecules. Although strictly forelementary reactions, the law states that the rate of any chemicalreaction is proportional to the product of the masses of the reactingsubstances, with each mass raised to a power equal to the coefficientthat occurs in the chemical equation (by Guldberg in the late eighteenhundreds). In other words, the product of the concentrations of thereactants include the power of their stoichiometry, stemming from thefundamental molecular or massic relationship. Since the earlyapplications were in Chemistry, volumes were accurately measured andtherefore concentrations may be used, as stated in the majority ofcurrent printed references. But, for instance, in physics, the law isdeduced without invoking chemical potentials at all. Denoting by N_ithe quantities of substances A_i in moles, the free energy G of themixture corresponds to the sum of the free enthalpies g_i of theconstituents and the mixing entropy S_m: G=sum(N_ixg_i)-TxS_m. Masstransfer is really about the second law of thermodynamics.Dealing with biological systems, the danger of the concentrationrationale resides precisely on the fact that there's no way to measurethe volume. We define the aqueous volume that would be needed to havethe same mass and still get the sampled concentration, calling itapparent. Therefore, what drives the transfer process can only be theactual mass, since the concentration depends on our own definition ofvolume.On the issue of parameterization, I totally agree that it makes adifference, some times crucial, on computation and estimation, butcertainly not in interpretation. Unless the basic argument is putforward about people not being comfortable interpreting reciprocaltimes, while being much at ease dealing with time units, let's say interms of a half-life for straightforward clinical applications, thenI'll agree. But the real interpretation of the underlying phenomenonmust be exactly the same regardless of the parameterization. So, afirst order rate constant is just the fraction of all the molecules atthe source 'traveling' via a process or pathway per unit time. Forexample, if a rate constant is found to be 0.03h^-1, then 3% of thematerial in the source travels by this pathway per hour. Thisinformation is highly relevant since it's entirely independent ofmass, volume or concentration. It's intrinsic to the kinetics of theprocess. E.g., according to Agatha Christie, the time constant for thecooling of a cadaver is 3.5 hrs. The time constant (the reciprocal ofthe rate constant) is yet another simple index of how rapidly aprocess will respond to a step stimulus. And it is extensively used inthe physiological literature. I concede that renal clearance isubiquitous, but that doesn't mean it's the best way to characterizethe renal function. It always reports to a volume cleared per unittime, volume of blood, plasma, serum or better said volume ofdistribution, which is naturally an apparent volume and something toalways 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, PharmacometricsMassachusetts College of Pharmacy and Health SciencesChildrens Hospital Boston / Harvard Medical School179 Longwood Ave, Boston, MA  02115`
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• On 12 Sep 2008 at 10:00:19, "Walt Woltosz" (walt.aaa.simulations-plus.com) sent the message
`The following message was posted to: PharmPKLuis,If it's only mass that maters, and not concentration, how would youtreat acondition wherein only half of the mass is in solution, perhapsbecause ofprecipitation?It seems to me that mass in solution (i.e., concentration) is whatmattersin the vast majority of pharmaceutical applications (barringendocytosis).Best regards,Walt WoltoszChairman & CEOSimulations Plus, Inc. (NASDAQ: SLP)42505 10th Street WestLancaster, CA  93534-7059U.S.A.http://www.simulations-plus.comE-mail: walt.aaa.simulations-plus.com`
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• On 12 Sep 2008 at 23:38:19, "Boris Gorelik" (boris.aaa.optimata.com) sent the message
`The notion of Luis about chemical potentials is correct. However inthe case of simple pharmacokinetics, where the stoichiometry of thediffusion through a membrane is 1:1 and there is no pressure applied,the concentration gradient is a pretty good and close approximation ofthe gradient of chemical potential.Boris`
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• On 12 Sep 2008 at 18:30:20, (Luis.Pereira.-a-.mcphs.edu) sent the message
`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 yousay, if the kinetics of the process follows let's say zero order, thenthe rate is constant and the solid material acts as an infinite sourceto maintain its concentration equal to solubility. If, just as anotherexample, the kinetics is first order, then the rate depends on theamount still remaining in solution to be let's say transferred acrossa membrane. Of course we can multiply and divide the right-hand sideof the relationship by a volume term and then we convert mass intoconcentration and the first order rate constant into a clearance justreparameterizing the problem (dM/dt = k.M = kV(M/V) = CL.C) which isPERFECTLY OK. If the volume is known it doesn't matter saying that theconcentration (or the mass) is the driving force for the process. Theless material we have the slower it goes, and for a given physicalvolume, the less material the more dilute the solution and the slowerit goes as well. What I tried to say was that the concentration wemeasure in an aliquot of plasma is related to the amount in thathomogeneous environment, which we may even extend to the rest of thebody, by an apparent volume, which accommodates all kinds of binding,partitioning and distribution issues. But then we end up with twounknowns, mass and volume. I even recall you alluding to the unstirredand heterogeneous nature of the intestinal lumen at the site ofabsorption, and the variable nature of the fraction absorbed which isabsolutely true. So, the precipitated mass is still conceptually thedriving force for the transfer process but just several orders ofmagnitude less relevant that the amount in solution. Which is not tosay that this is a perfect Newtonian solution in a beaker of a knowvolume with a semi-permeable membrane interface. The amount in theliquid phase acts much more as a driving force than the amount in thesolid phase. But unless we sample the local concentration, weshouldn't say the concentration of the solution we prepared in orderto administer the drug, or the virtual concentration in the averagevolume of the intestine, or, on the other side of the border, theconcentration of the drug in the blood stream, are the driving forcesfor the respective mass transfer processes. Individual moleculeswonder around, randomly walking, and the more they are the moredriving power they'll have.I hope I was a bit clearer. But again, on this platform, who is?Best regardsLuis--Luis M. Pereira, Ph.D.Assistant Professor, PharmacometricsMassachusetts College of Pharmacy and Health SciencesChildrens Hospital Boston / Harvard Medical School179 Longwood Ave, Boston, MA  02115`
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• On 12 Sep 2008 at 12:37:10, Nick Holford (n.holford.-at-.auckland.ac.nz) sent the message
`Luis,Thanks for the update on thermodynamics -- but I think you miss thepoint about the difference between concentration and amount. The totalamount of mass in the universe is fixed (more or less). It is theconcentration of mass in space time that makes things interesting. Onsimiliar lines, my previous comment was to say that amount is not aprimary variable determining pharmacokinetics. Concentration is areasonable approximation for any realistic view of pharmacokineticsAND pharmacodynamics. I dont see any danger in this viewpoint.However, your interpretation of clearance as 'volume cleared per unittime' is true only in a mathematical sense. It ignores the physicalconcept that clearance is describing -- and this can lead to dangerousconclusions. Clearance is the proportionality factor betweenconcentration and rate of elimination. The units are secondary to thedefinitions of concentration and rate of elimination. The naive ideathat clearance is volume cleared per unit time led one quite renownedclinical pharmacologist to say that if the clearance is 10 L/h and theapparent volume of distribution is 100 L then all the dose is clearedin 10 hours. This appeared in an early edition of one well knowntextbook of clinical pharmacology but was quickly removed from latereditions :-)Nick--Nick Holford, Dept Pharmacology & Clinical PharmacologyUniversity of Auckland, 85 Park Rd, Private Bag 92019, Auckland, NewZealandn.holford.-at-.auckland.ac.nzhttp://www.fmhs.auckland.ac.nz/sms/pharmacology/holford`
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