Two Compartment Body Model and V d Terms by Jeff Stark

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1 Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic parameers (ke,, Cl) are no affeced by he amoun of he dose. Of course, a change in dose will be refleced by a proporional change in plasma concenraion. (2) Immediae disribuion and equilibrium of he drug hroughou he body. Considering hese assumpions, we can easily describe he vs profile of a drug afer an iv bolus injecion (he same principles apply o oher roues of adminisraion as well) by he exponenial equaion: k () e e 0 ose k e e ln m-ke Since immediae disribuion is assumed, he whole body is reaed as one uni. The slope of he graph represens he oal eliminaion consan of he drug ou of he body regardless of he eliminaion pahway: k e k ren + k me + k bil +.. This one comparmen model can ofen be used o predic vs profile and oher pharmacokineic parameers. The ruh is, however, ha very few drugs show immediae disribuion and equilibrium hrough he body. If disribuion is minimal, he one comparmen can be an adequae approximaion. (We wan o use he simples model we can). If no, we mus aler he model o beer fi he daa. The "nex sep up" is he Two-Comparmen body Model which includes a peripheral comparmen ino which he drug may disribue. Mulicomparmenal/Two Comparmen Body Model 1

2 THE TWO COMPARTMENT MOEL i.v. bolus dose Cenral k 10 Eliminaion k 12 k 21 Peripheral Alhough hese comparmens do no necessarily have a physiological significance, common designaions are: Comp 1 (cenral) - blood and well perfused organs, e.g. liver, kidney, ec.; "plasma" Comp 2 (peripheral) - poorly perfused issues, e.g. muscle, lean issue, fa; "issue" A ypical vs profile for his model is: Obviously an equaion wih only one exponenial erm canno describe his curve; Two are required. () α ae + β be a, b, α, and β are called "hybrid consans". This form is much simpler han he analogous equaion expressed in erms of he "hybrid consans" (k 10, k 12, k 21 ) and dose. α and β may be wrien in raher long expressions of he microconsans. A and b, in urn, can be expressed in erms of α, β, microconsans, and dose (and Vc - more laer on he volume erm. The hybrid consans a, b, α, and β may be found by "feahering" which allows us o separae disribuion and eliminaion (similar o he way we found k a and k e in oral adminisraion). Mulicomparmenal/Two Comparmen Body Model 2

3 Procedure: 1) Graph vs on ln (or log) scale 2) Exrapolae he erminal slope back o he y-axis. Call his exrapolaed line '; he y- inercep is b. 3) For several ime poins deermine (-') and plo hese poins. Connec hem. The slope of his line is -α and he y-inercep is a. The slope of he erminal segmen is -β. ( ) ln ( ) a b mα (-' ) mβ The iniial concenraion afer an i.v. bolus dose can be given by: 0 a + b Why is he slope so seep for he disribuion phase of he vs curve? Once a drug eners he body, eliminaion begins. Afer an i.v. bolus injecion o he cenral comparmen, here is disribuion ino he peripheral comparmen and eliminaion from he cenral comparmen. Thus, he concenraion decreases rapidly a firs. isribuion ino he peripheral comparmen coninues unil he free concenraion in he cenral comparmen (plasma) is equal o he free concenraion in he peripheral comparmen (issue), i.e. here is a ne flow of drug ou of he plasma (along he concenraion gradien) unil seady-sae is reached. In his siuaion, seady-sae is no mainained and is only momenary. Afer seady-sae, a concenraion gradien is again creaed - his ime in he opposie direcion - by he coninual eliminaion of drug from he cenral comparmen. In response o his, drug begins o flow back ino he cenral comparmen where i is eliminaed. Thus, in he β-phase, he concenraion of drug in he peripheral comparmen is greaer han ha in he cenral comparmen. The concenraions in boh comparmens decrease proporionally as eliminaion from he plasma coninues. This may be viewed as a "pseudo seady-sae": he body wans o reach an equilibrium beween he amoun of drug in each comparmen (X c and X p for cenral and peripheral) bu canno due o eliminaion (k 10 ). Mulicomparmenal/Two Comparmen Body Model 3

4 k10 Xc Xc Eliminaion k12 k21 no ne change a seady sae ne flow back o cenral comparmen in he β-phase Xp Xp This "pseudo seady-sae" in he β-phase may be apparen in comparing he concenraion in he issue o ha in he plasma. Plasma ln Con. Tissue } Same slopes in β-phase; conc. higher in issue max he free concenraion. In plasma is equal o he free concenraion in he issue. This is he seady-sae. Volume of disribuion erms Recall ha V d relaes he amoun of drug in he body o he concenraion in he plasma. For a one-comparmen model we saw an iniial concenraion of: 0 Mulicomparmenal/Two Comparmen Body Model 4

5 which can be solved for V d o give 0 In he one comparmen model here is a single V d erm (since we assumed immediae disribuion and equilibrium). Thus, he amoun of drug in he body a any ime, X, may be a X ime, C p by a similar equaion, V d is he same wheher we are ineresed in he iniial concenraion a 0 or a laer ime poins. For a drug which requires someime for disribuion hrough he body (and, hus, fis a 2- comparmen model), his is no he case. There will be differen V d erms depending on where we are in he C p vs profile. This is no as complicaed as i may appear. Consider he iniial plasma concenraion for drug described by C p () ae -α + be -β, 0 Vc Since he drug is only in he cenral comparmen a ime 0 (i.e. no disribuion has aken place ye), he volume erm relaing C p0 o dose is V d, he volume of disribuion of he cenral comparmen. If we know he dose and measure C p0, we can calculae V c, Vc a + b 0 (subsiuing C po a+ b) This erm is useful if we wan o predic iniial (peak) plasma concenraion following i.v. bolus dose. This will be valid only a 0;. 0 Vc Noe: For a one comparmen model, we expressed Cl TOT in erms of a rae consan and V d, ClTOT ke In he 2 comparmen model, we have several V d erms o choose from in wriing relaionship like his. Since Cl TOT and V d are sill independen (his fac has no changed), we mus use differen rae consans depending on he choice of V d. Using V c, we may wrie: ClTOT k10 Vc Mulicomparmenal/Two Comparmen Body Model 5

6 where k 10 is he eliminaion rae consan from he cenral comparmen and V c is he volume of disribuion of he cenral comparmen. V d afer he iniial ime poin (0_. A any ime poin afer 0, V d may vary. Thus, V d will be a funcion of, V d (), jus as we see for he amoun of drug in he body, X, and he plasma concenraion, C p. in he erminal/β-phase a seady sae Vc a 0 X or X C p Alhough we can wrie an equaion o deermine V d a any ime poin, i is no always pracical o carry ou hese calculaions (he equaion includes AUC and AUC a he given ime poin). We can, however, deermine a wo specific poins wihou using he V d () equaion. These are useful V d erms ha involve parameers already deermined previously (e.g. α, β, ec.) a seady sae A seady-sae, he free concenraion in plasma is equal o he free concenraion in he issue. The V d a his poin, V dss, is useful as i gives us he sum of he volume erms of he cenral and peripheral comparmens a equilibrium. V dss relaes he amoun of drug in he body a seadysae X ss, o he plasma concenraion, C pss,. ss 2 2 X ss aβ + bα Vc + Vp C 2 pss ( aβ + bα) The poin of equilibrium is unique in ha here is one overall eliminaion rae consan, ke, jus as we saw in he one-comparmen model. This k e may be wrien in erms of he microconsans (k 10, k 12, k 21 ) used in he wo comparmen model. This allows us o wrie Mulicomparmenal/Two Comparmen Body Model 6

7 ClTOT ke ss in he β-phase. Again, he erm in he β-phase (also referred o as he pos-disribuion or erminal phase) relaes he amoun of drug in he body o he plasma concenraion. X β C p where is some ime poin in he β-phase. This is a unique siuaion since he concenraion of drug in he issue is in a dynamic equilibrium wih he plasma concenraion ("pseudo seadysae"). ue o his dynamic equilibrium, X and C p decrease proporionally and V db is consan (observe he equaion above and he previous graphs. The Cl TOT expression in he β-phase will involve he rae consan β, ClTOT β β Solving his equaion for V dβ gives ClTOT β β Recall ha Cl TOT may be deermined using a raher simple equaion ClTOT, AUC where is he dose given and AUC is he oal area under he curve (in he vs profile). Subsiuing his ino he equaion for V dβ gives β. AUC β Since V dβ may be calculaed using he AUC, i is ofen called V darea, V dβ V darea. The choice of erminology may be based on he equaion used o deermine his erm. There is one more V d erm ha should be included in his discussion. Unlike he previous erms, however, his one has no physical meaning. I is merely an approximaion and does no accoun for any disribuion. This erm is V dexrap, so named because he calculaion involves exrapolaing he erminal slope back o he y-axis o find b, Mulicomparmenal/Two Comparmen Body Model 7

8 The real o a + b b ln Terminal Slope m-β V dexrap /b V dexrap relaes he concenraion erm b o he dose a ime 0. Since he iniial concenraion is higher han b (Recall 0 a + b), he calculaion resuls in overesimaing V d. In oher words, V dexrap is calculaed by preending he vs profile fis a one-comparmen model even hough we know oherwise. To summarize: exrap > area > ss > Vc β eermining he loading dose For an i.v. bolus injecion, he iniial plasma concenraion may be prediced using 0 V. d In a muliple dosing regimen, a loading dose can reduce he ime needed o reach seady sae levels. If we wan o achieve C pss wihou a delay, L C pss V, d we need o be able o deermine he loading dose. Mulicomparmenal/Two Comparmen Body Model 8

9 L C pss This equaion may be solved for L, L C pss. If we know he drug in quesion follows he vs profile of a wo comparmen model, which erm do we use o deermine an appropriae loading dose? V c? If V c is used, L C pss Vc Since V d is small and only involves he cenral comparmen. Thus, our L calculaed will no be large enough. V dexrap? Since V dexrap is a gross overesimaion of a meaningful V d value, using i would resul in a loading dose ha is oo big: L C pss exrap. This could lead o overshooing he range of he herapeuic window and resul in oxic side effecs. V dss? This is he bes choice. L C pss ss If V dss is no known, a value beween V dβ and V c may be adequae. Mulicomparmenal/Two Comparmen Body Model 9

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