# PharmPK Discussion - Comparison of first and second order rate constants

PharmPK Discussion List Archive Index page
• On 19 Jan 2004 at 16:35:19, Toufigh Gordi (tgordi.-at-.buffalo.edu) sent the message
`Dear all,I have constructed a PK/PD model similar to what follows:dA1/dt= -A1*C*K12dA2/dt= A1*C*K12 - A2*K20I need to compare the time course of the system with respect to thetransfer from compartment 1 to 2 and the elimination from compartment2. The main objective is to determine which process is faster: transferfrom A1 to A2, or elimination from A2. My problem is that K12 is asecond order rate constant, dependent on A1 and C, whereas K20 is afirst order rate constant, dependent on A2. Thus K12 has the units of1/(C*time), while K20 has the unit of 1/time. Can anybody suggest ageneral approach in comparing the two rates? References to publishedmaterial are highly welcome.Regards,Toufigh Gordi[What is the differential equation for 'C'? Does C vary with timewithin a particular experiment? - db]`
Back to the Top

• On 20 Jan 2004 at 08:48:05, kim.travis.aaa.syngenta.com sent the message
`I would have thought that a simple and functionally-relevant way ofdoingthis would be to see how your model behaves. I assume that the chemicalstarts in compartment 1 - if not then what follows may not apply.  Ifyourmodel predicts only low levels in compartment 2 at any timepoint thentranfer out of 2 is fast compared to transfer from 1 to 2.  If chemicalbuilds up in compartment 2 then the opposite is true.  You could dothis bysimulation, or as the model is so very simple you could analyticallyderivethe maximum concentration in compartment 2 over time, given knowninitialconditions.There is a danger in overinterpretting the parameters of such models andforgetting that it is how the whole model and data behaves that isimportant,Kim`
Back to the Top

• On 20 Jan 2004 at 11:15:32, "J.H.Proost" (J.H.Proost.aaa.farm.rug.nl) sent the message
`Dear Dr. Gordi,With respect to your question about the model:> I have constructed a PK/PD model similar to what follows:>> dA1/dt= -A1*C*K12> dA2/dt= A1*C*K12 - A2*K20I have three comments:1. I do not understand the rationale of this model. As was asked byDavid Bourne: what is C?It seems that your equations refer to an absorption model, in which therate of absorption is dependent also on the concentration (or whatever)of something else?2. The question how to compare the rate constants is simple. Theabsorption rate constant (or 'drug entry constant') is C*K12, which hasthe unit of reciprocal time.3. The question why one would compare the rate constants is not clearto me. Perhaps there may be a valid reason for this; if so, pleaseexplain. If C*K12 can be regarded as an absorption rate constant, andK20 as an elimination rate constant, a comparison of the numeric valuesis meaningless. Both constants refer to completely different processes.'C*K12' is the rate of removal of drug from comp. 1 divided by theamount in comp. 1, and 'K20 is the rate of removal of drug from comp. 2divided by the amount in comp. 2. Both rate constants are dependent onthe volumes of the corresponding compartments. What would we learn fromthe knowledge which parameter, C*K12 or K20, is largest?Best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics and Drug DeliveryUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlandstel. 31-50 363 3292fax  31-50 363 3247Email: j.h.proost.aaa.farm.rug.nl`
Back to the Top

• On 20 Jan 2004 at 14:13:13, Toufigh Gordi (tgordi.-at-.buffalo.edu) sent the message
`It seems that the problem is more complex than I thought. I hadoriginally given these equations:"dA1/dt= -A1*C*K12dA2/dt= A1*C*K12 - A2*K20"Here comes additional information about the model. The full modeldescribes the effect of a drug on the number of organisms. C istime-variable drug concentrations, which could be simplified asfollowing a mono-exponential elimination rate (although in reality itcould be much more complex, but still follow a mono-exponential decay).A1 is the number of viable organisms, which due to a direct drug effectare transferred to an "injured" form (A2). Thus, A2 is the number ofinjured organisms, which are then cleared from the body. The full modelis again more complex than the one presented here, but I don't find itof relevance to my aim, i.e. comparing the rate of transfer from A1 toA2 and A2 out. The observed number of organisms are the sum of A1 andA2 since there is no practical way of separating the viable and injuredorganism. To complicate the situation even more, there is anothertransfer to A2: from another viable, but not visible, organismcompartment, with another second-order rate constant.During the model-building, I tried a model without any "injured"compartment but the fits gets much better when it is added to thesystem, which would imply the A2 to 0 is slower than A1 to A2. However,I would like to have a more solid evidence of this by a possiblecomparison of the two rates. So the question is: is there any way ofcomparing a first-order (in this case K20) rate constant and asecond-order (K12) rate constant?I think it is a good idea, as proposed by Kim Travis, to simulate themodel and see how the numbers change.Thank you!T. Gordi`
Back to the Top

• On 20 Jan 2004 at 12:40:41, Walt Woltosz (walt.-at-.simulations-plus.com) sent the message
`The following message was posted to: PharmPK At 02:15 AM 1/20/2004,Hans Proost wrote:"1. I do not understand the rationale of this model. As was asked byDavid Bourne: what is C?It seems that your equations refer to an absorption model, in which therate of absorption is dependent also on the concentration (or whatever)of something else?"I also wonder what C refers to in the equation.For passive diffusion, which governs the absorption for most drugs, theabsorption rate is dependent on two concentrations - one on each sideof the membrane. Let's remember that the modern definition ofabsorption is crossing the apical membrane into the enterocytes, notreaching the portal vein or systemic circulation (this was emphasizedrepeatedly at the Oral Drug Delivery Course in Vail, Colorado lastweek).For passive diffusion, Ficks' Law says that molecules cross a membraneat a rate that is determined by the difference in concentration acrossthe membrane:J = D * (C1 - C2)2. The question how to compare the rate constants is simple. Theabsorption rate constant (or 'drug entry constant') is C*K12, which hasthe unit of reciprocal time.The practice of assuming an absorption rate "constant" is a simplifyingassumption that, in our experience, rarely holds true. In fact, Ka isnever a constant, although in some situations, it can be treated as onewith acceptable error.Consider the equation for Fick's Law above and what can cause the flux(J) to change.D: for absorption, this is the product of the local effectivepermeability (Peff) and the surface/volume ratio. The local effectivepermeability, Peff, changes with position in the gastrointestinaltract. It is different in duodenum, jejunum, ileum, and colon for mostdrugs. The differences are caused by different surface area, differenttight junction gap (for paracellular transported drugs), different pH(ionization effects), and different transporter expression (for influxor efflux). Sometimes the differences are small. Usually the colon Peffis significantly different from the small intestine values.C1 and C2: assume C1 is the lumen concentration, and C2 is theconcentration in the enterocytes. When C1 >> C2, you can treat C2 aszero. But how long does that last? As the drug is absorbed, C1 dropsand C2 increases. When they become equal, absorption stops. When theenterocyte concentration is higher than lumen, the flux is reversed,and drug is exsorbed/secreted into the lumen (a phenomenon that hasbeen observed for iv doses). The old-fashioned approach of assuming anabsorption rate "constant" ignores this, and it ignores the fact thatthe absorption rate steadily decreases as C1 decreases and C2increases. In our experience, using a concentration gradient (Fick'sLaw) approach provides far more accurate and consistent results -enabling a single model to predict plasma concentration-time resultsfor different doses without a change in model parameters.Walt WoltoszChairman & CEOSimulations Plus, Inc. (SIMU)1220 W. Avenue JLancaster, CA 93534-2902U.S.A.http://www.simulations-plus.comPhone: (661) 723-7723FAX: (661) 723-5524E-mail: walt.aaa.simulations-plus.com`
Back to the Top

Want to post a follow-up message on this topic? If this link does not work with your browser send a follow-up message to PharmPK@boomer.org with "Comparison of first and second order rate constants" as the subject

Copyright 1995-2010 David W. A. Bourne (david@boomer.org)