# PharmPK Discussion - Writing differential equations (mass or concentration)

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• On 6 Mar 2009 at 18:32:01, "Pang, Yinuo" (yinuo.pang.aaa.spcorp.com) sent the message
`The following message was posted to: PharmPKDear all,I was wondering if someone out there could help me to understand what isthe best way to write differential equations: should I use mass terms orconcentration terms?For a one-compartment model, one can start with either mass orconcentration and will get the same equation using V1 as the convertingfactor, since there's only one volume in the model.A1=total amount, C1=conc, V1=apparent volume of the compartmentStart with mass:(1)  dA1/dt = - k10 * A1(2)  dA1/dt / V1 = -k10 * A1 / V1(3)  dC1/dt = -k10 * C1Start with concentration:(4)  dC1/dt = -k10 * C1However, when we talk about a two-compartment model, since there are twodifferent volume terms involved, V1 and V2, the differential equationsare no longer inter-changeable (Note the difference between eqn (5) and(10), or eqn (7) and (8)):A1=total amount in compartment 1, C1=conc in compartment 1, V1=apparentvolume of compartment 1A2=total amount in compartment 2, C2=conc in compartment 2, V2=apparentvolume of compartment 2k12=rate constant from compartment 1 to 2, k21=rate constant fromcompartment 2 to 1Start with mass:(5)  dA1/dt = -k12 * A1 + k21 * A2(6)  dA1/dt / V1 = -k12 * A1 / V1 + k21 * A2 / V1(7)  dC1/dt = -k12 * C1 + k21 * A2 / V1Start with concentration:(8)  dC1/dt = -k12 * C1 + k21 * C2(9)  dC1/dt * V1 = -k12 * C1 * V1 + k21 * C2 * V1(10) dA1/dt = -k12 * A1 + k21 * C2 * V1I've seen usage of eqn (5)-(7) in many publications, but I've also seeneqn (8) in some textbooks as well. I'm a little confused regarding whichone is valid, and why.Then, if we move on to a 2nd-order process, let's say that 1+2=3, andeverything happens within one compartment:*********     *********     kon     ********** A1,C1 *  +  * A2,C2 *  ---------> * A3,C3 **********     *********  <--------- *********                            koffV=apparent volume of the compartmentkon=association rate constant, koff=dissociation rate constantStart with mass:(11)  dA1/dt = -kon * A1 * A2 + koff * A3(12)  dA1/dt / V = -kon * A1 * A2 / V + koff * A3 / V(13)  dC1/dt = -kon * A1 * C2 + koff * C3(14)  Or: dC1/dt = -kon * C1 * A2 + koff * C3Start with concentration:(15)  dC1/dt = -kon * C1 * C2 + koff * C3(16)  dC1/dt * V = -kon * C1 * C2 * V + koff * C3 * V(17)  dA1/dt = -kon * A1 * C2 + koff * A3(18)  Or: dA1/dt = -kon * C1 * A2 + koff * A3In this case, depending on whether we use mass or concentration to startwith, we get different equations again. (Note the difference betweeneqn(11) and (17)/(18), or eqn(13)/(14) and (15). It seems thateqn(15)-(18) are the ones used in publications, which makes sense inchemical kinetics where concentrations are used in writing rateequations.I guess when we talk about mass transfer from one compartment toanother, we should use mass terms, but when there's formation of newentities (reactive system), concentration terms should be used. Butagain, I did see eqn(8) in some textbook (Shargel and Yu, AppliedBiopharmaceutics and Pharmacokinetics), and still wondering if this is avalid expression, and how do we explain the difference.I may have just trapped myself in a loop and couldn't get out. Ifsomeone is willing to spend some time explaining all this to me, thatwill be very much appreciated!!Thanks,yinuo`
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• On 7 Mar 2009 at 18:14:25, "Shawn Spencer" (shawn.spencer.-at-.famu.edu) sent the message
`The following message was posted to: PharmPKPang Yinuo,It's rare to have closed 2-compartment non-eliminating PK equations intextbooks and in the literature.Chemical kinetics receives a slightly different treatment thanpharmacokinetics, because with the former, you may be dealing with aclosedreaction chamber, where the volume of a "second compartment" or reactiveelement may be shared.  With PK however, your kinetic processes maydescribemembrane transfer or better yet, tissue binding, where the volume is notshared by the mass.  Right away, you will recognize that to describe the"concentration of drug bound to tissue" is not meaningful, andconsequentlyare often termed "apparent" volumes for this reason (although I don'tlikethat term personally).So, volumes are not input variables in PK models, but rather areparameterized wherever possible.The correct approach is to directly derive your models in terms of yoursystem under observation.dA1/dt = -k12 * A1 + k21 * A2dC1/dt = -k12 * A1/V1 + k21 * A2 / V2are both correct, ...for a closed system.Hope it helps.-Shawn D. Spencer, Ph.D., R.Ph.Assistant Professor of BiopharmaceuticsFlorida A&M College of PharmacyTallahassee, FL 32307shawn.spencer.at.famu.edu`
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• On 8 Mar 2009 at 10:28:49, Leonid Gibiansky (gibiansky.aaa.gmail.com) sent the message
`YinuoThe book (Shargel and Yu, Applied Biopharmaceutics and Pharmacokinetics)contains number of misprints and errors. The 2-compartment modelequations (4.4)-(4.5) that is presented there (chapter 4) do not satisfymass balance equations, and should not be used. So, your system (1)-(3)is correct, (5)-(7) is correct, (8)-(10) is not correct, (11)-(14) isnot correct (rate of drug-target complex production is proportional toconcentrations of components, not to their amounts), (15)-(18) iscorrect if you assume equal volumes for A1, A2 and A3This is not the only case when books and published journal articlescontain errors or misprints, so it is always helpful to checkassumptions behind equations and the underlying biology.ThanksLeonid--Leonid Gibiansky, Ph.D.President, QuantPharm LLCweb:    www.quantpharm.come-mail: LGibiansky at quantpharm.com`
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