# Bioequivalence and Bioavailability Forum

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martin
Senior

Austria,
2010-01-13 13:38

Posting: # 4595
Views: 13,561

## extent of accumulation [NCA / SHAM]

Dear HS!

I have a question regarding estimating the extent of accumulation following repeated study drug administration. I have data from a single dose PK study and I would like to estimate the extent of accumulation in AUCs in the case of repeated study drug administrations.

According to Cawello (2003, page 61-62) "The extent of accumulation can be calculated by the ratio of the dosing interval to the half-life". Accumulation index R = 1/(1-2) where ε = dosing interval divided by half-life.

I suppose the accumulation is based on two assumptions
• Accumulation refers to accumulation in steady state conditions. Cawello (2003, page 58) "After about 5 half-lives, the equilibrium is practically achieved"
• The formula for estimating the extent of accumulation is based on the assumption of a one-compartmental model following an IV bolus.
I have to questions in this context:
1. In the case that the decline over time can be described by a biexponential decline (two-compartmental model) following an IV bolus with an initial and terminal phase which half-life should be used to estimate the extent of accumulation? I think the terminal half-life is not suitable as the theory of accumulation was developed on assumption of a one-compartmental model. right? I think the effective or non-compartmental half-life calculated as log(2) × MRT would be more suitable. Do you agree?
2. In the case of only 2 to 4 repeated administration, steady state may not be reached according to Cawello (2003, page 58). In this case I would estimate the accumulation in AUCs based on the principle of superposition. Do you agree?

Best regards

Martin
ElMaestro
Hero

Denmark,
2010-01-13 16:15

@ martin
Posting: # 4598
Views: 10,626

## extent of accumulation

Hi Martin,

» I suppose the accumulation is based on two assumptions
»
» - Accumulation refers to accumulation in steady state conditions. (...)

What is the intended interpretation of accumulation in steady state?

Best regards
EM.

I could be wrong, but…

Best regards,
ElMaestro

No, I still don't believe much in the usefulness of IVIVCs for OIPs when it comes to picking candidate formulations for the next trial. This is not the same as saying I don't believe in IVIVCs.
martin
Senior

Austria,
2010-01-13 17:45

@ ElMaestro
Posting: # 4600
Views: 10,668

## extent of accumulation

Hi ElMaestro !

The extent of the expected accumulation is needed for interpretation of a non-clinical repeated dose-toxicity study. One objective of these kind of studies is whether systemic exposure increases or decreases after repeated study drug administration compared with the systemic exposure after first administration.

Accumulation (e.g. higher AUC after repeated than after first administration) occurs when the dosing interval is "too short" with respect to the half-life of the compound investigated. On the other hand, an increase in drug exposure after repeated dosing may be due to damage to the eliminating organs and subsequent reduction in clearance which may indicate a safety concern.

The dosing interval in these kind of studies is frequently chosen based on practical/medical considerations. The expected extent of accumulation would therefore be extremely useful in interpretation of study results.

best regards

martin

PS.: One drawback of estimation the expected accumulation using data from a previous single dose PK study is, of course, that a potential study effect is not taken into account. Dose-proportionality plays also an important role as usually a very high dose is investigated in these kind of studies which makes life really complicated.
Helmut
Hero

Vienna, Austria,
2010-01-13 19:14

@ martin
Posting: # 4603
Views: 10,621

## Too short tau?

Dear Martin!

» Accumulation (e.g. higher AUC after repeated than after first administration) occurs when the dosing interval is "too short" with respect to the half-life of the compound investigated.

Veto!
If you get AUCτ significantly (!) larger than AUC that’s not simply accumulation, but nonlinear PK (superposition principle does not hold). If your dosing interval is “too short”, you will get higher concentrations than you like (especially in toxicokinetics), but that's another story.

Regards,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
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martin
Senior

Austria,
2010-01-13 19:45

@ Helmut
Posting: # 4605
Views: 10,597

## Too short tau?

Dear HS!

The question of linear PK (dose-proportionality) is also addressed in these kind of studies using data after first study drug administration (usually several doses are studied).

Thanks, wonderful idea to check AUCinf from previous single dose PK study with AUCtau obtained from the repeated dose toxicity study (i.e. AUC following last study drug administration); I totally agree!

AUCtau=AUCinf in the case of accumulation.
AUCtau>AUCinf indicate nonlinear PK.

Interpretation of study results may get complicated in the case that dose-proportionality holds after first study drug administration but AUCtau after repeated study drug administration is larger than AUCinf. Any idea what this result could indicate?

best regards

martin
Helmut
Hero

Vienna, Austria,
2010-01-13 20:02

@ martin
Posting: # 4606
Views: 10,977

## Too short tau?

Dear Martin!

» The question of linear PK (dose-proportionality) is also addressed in these kind of studies using data after first study drug administration (usually several doses are studied).
»
» Thanks, wonderful idea to check AUCinf from previous single dose PK study with AUCtau obtained from the repeated dose toxicity study (i.e. AUC following last study drug administration)…

Well, that’s not my invention.
The problem of between-study comparisons is unresolved (variability!). I don’t know how reproducible results from your highly standardized nice rodents are, but in humans a serious comparison is always done in the same study (single dose profile > saturation > steady state profile; paired test – assuming no period effects, of course).

» AUCtau=AUCinf in the case of accumulation.
» AUCtau>AUCinf indicate nonlinear PK.

Exactly.

» Interpretation of study results may get complicated in the case that dose-proportionality holds after first study drug administration but AUCtau after repeated study drug administration is larger than AUCinf.

OK, I’m not an expert in pre-clinical PK. But for a new drug in humans IMHO the assessment would depend on the clinical indication. If the drug is intended for long-term use I would definitely concentrate on MD data (single dose only supportive). For an analgetic given only occasionally, I would not bother too much on nonlinear PK in steady state.

» Any idea what this result [linear PK in SD, nonlinear PK in steady state] could indicate?

Well, any one of the effects I mentioned in my previous post (time dependent PK, enzyme inhibition, capacity limited excretion, etc.).
Especially with biologics I have seen really weird stuff: Concentrations increased, stayed limited time at some ‘pseudo steady state’, and subsequently decreased to ‘true state state’. In this case the drug behaved as expected only for limited time (the first ‘steady state’) until an enzyme system was induced, which finally led to another (lower) steady state. That’s the tricky thing with predictions of plasma levels from single dose data.

A simple example (modified from Danielsson & Weiner, PK model 22):

Single dose study; 1 h infusion of 40 units (bottom left). It was decided based on a simulation (top ) to start the multiple dose study with a 1 h infusion of 120 units as a loading dose, followed by 0.5 h infusions of 40 units (τ = 8 h).
What we see, is auto-induction (increased clearance) which leads to a steady state of only ~50% () of the expected one. Unless we have performed the experiment, we never can be sure!

‘Hard core pharmacokineticists’ even support such a statement:

Unless we have performed steady state studies – at different dose levels –
we do know nothing about the PK of a drug.

Regards,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
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ElMaestro
Hero

Denmark,
2010-01-13 20:47

@ martin
Posting: # 4607
Views: 10,630

## extent of accumulation

Dear Martin,

» Accumulation (e.g. higher AUC after repeated than after first administration) occurs when the dosing interval is "too short" with respect to the half-life of the compound investigated. On the other hand, an increase in drug exposure after repeated dosing may be due to damage to the eliminating organs and subsequent reduction in clearance which may indicate a safety concern.

Sorry, I once again failed to explain my question. You mentioned "accumulation in steady state conditions"; this challenged my walnut-sized brain when I read it as I generally understand steady state as the state where accumulation is absent.

Re. Q1: I think the equation is to be used mainly in case you have reason to expect a monophasic elimination of first order. Hence, applying it to a system with two elimination constants could be disputable. However, using the terminal constant for the purpose would be the conservative approach in a regulators view, I expect. Simulations might help you a good piece of the way.

EM.
Helmut
Hero

Vienna, Austria,
2010-01-13 18:16

@ martin
Posting: # 4601
Views: 11,054

## extent of accumulation

Dear Martin!

» I have data from a single dose PK study and I would like to estimate the extent of accumulation in AUCs in the case of repeated study drug administrations.

Fine – assuming linear PK (no time dependent PK, enzyme induction/inhibition, capacity limited excretion, etc.).

» According to Cawello (2003, page 61-62)…
» I suppose the accumulation is based on two assumptions…

Right.

» Cawello (2003, page 58) "After about 5 half-lives, the equilibrium is practically achieved"

“Practically” in this context means 1-½5=96.875% of steady state. Some people are only satisfied with seven half-lives (99.2%), whilst others opt for a ‘magic’ level of 95% (4.322 half-lives).

» 1. In the case that the decline over time can be described by a biexponential decline (two-compartmental model) following an IV bolus with an initial and terminal phase which half-life should be used to estimate the extent of accumulation? I think the terminal half-life is not suitable as the theory of accumulation was developed on assumption of a one-compartmental model. right?

Yes, but think about the ‘predominant half-life’ in such a case. I would suggest to fire up some PK software and start a little simulation, e.g., two-compartment open, parametrized in macro constants (A, B, α, β), D=1, V=5, k21=0.5, α=1, β=0.1, duration of infusion 2. We get:
A=0.1111, B=0.8889, t½,α=0.6931, t½,β=6.931, AUC=A/α+B/β=1, AUMC=10, MRT=[(AUMC-length of infusion/2)/AUC)=9. We now can estimate the ‘important fraction’ based on A/α and B/β and see that ≈89% of accumulation will be ‘caused’ by the slower process (β).
I would not worry too much, because:
• If α approaches β we come closer to a one-compartment model
• The slower elimination gets compared to distribution, the higher the fraction of B/β of AUC will be. In other words, you can be quite confident basing your design on β. If we change β from 0.1 to 0.01 in the example above the fraction increases from ≈89% to ≈99%. Steady state is reached after 347 based on β and after 343 based on MRT – no big deal.
• Quite often the slower phase is not ‘visible’ at lower doses due to analytical limitations. Even if a very deep compartment exists, the AUC-ratio of the fast/slow phases shows that accumulation is essentially driven by the faster one. Remark: there are pharmacokineticists claiming that a one-compartment model does not exist at all.

» 2. I think the effective or non-compartmental half-life calculated as log(2) × MRT would be more suitable. Do you agree?

Maybe. In the example this would be 6.238. I have to think it over…
In my example it doesn’t make any difference (based on 5×t½,β 34.6, based on MRT 31.2; with a τ of 24 we are sufficiently close to steady state after the second dose).
On the other hand everything depends on the “quality of the estimate”, whether it may be λz or MRT. At David’s PKPD-list a quite heated ‘chicken-and-the-egg’ discussion on the ‘correct’ [sic!] parametrization in PK (rate constant vs. clearance) started on Nov 19th, 20091 – and is still ongoing.2 Where do we get the MRT from? In NCA λz is involved anyhow. To get a ‘reliable’ estimate of AUC most people are happy with AUCt/AUC≥80%. For the same precision of AUMC one must sample to 95% of AUC (don’t ask me for the reference, please – you have seen my office; it must be somewhere ). Leslie Benet even advocates a coverage of 95–99% if AUMC is aimed at!

» 2. In the case of only 2 to 4 repeated administration, steady state may not be reached according to Cawello (2003, page 58).

Well, 2–4 half-lives in the one-compartment model correspond to 75%, 87.5%, 93.75% of steady state.

» In this case I would estimate the accumulation in AUCs based on the principle of superposition. Do you agree?

What All of the above (linear PK!) is based on superposition, or AUC (SD) = AUCτ (steady state). I don’t get your point. Why are we performing steady state studies at all? Essentially to study whether the drug (with the given posology) follows linear PK or not. The latter may have clinical implications (in the worst case mandatory therapeutic drug monitoring). We only can make the assessment of linear PK if we are sufficiently close to steady state conditions.

1. PharmPK Discussion - A question of clearance: Part 1 (18 Nov 2009 – 30 Dec 2009)
2. PharmPK Discussion - A question of clearance: Part 2 (2 Jan 2010 – 3 Feb 2010)

Regards,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
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martin
Senior

Austria,
2010-01-13 19:14

@ Helmut
Posting: # 4602
Views: 10,638

## principle of superposition

Dear HS!

Thank you for your detailed answer. I think I got your point; the formula for the accumulation index R is simply a formalism of the principle of superposition or overlay technique (Gibaldi and Perrier, 1982, appendix E).

In the case of 4 subsequent administrations I can calculate the accumulation index but it is biased as only 93.75 of steady state is reached (on assumption of a one-compartmental model). For this reason, there is no way to get an unbiased estimate for the accumulation index in the case of 4 administration?

best regards

martin

PS.: There is a draft guideline on repeated dose-toxicity studies EMEA/CHMP/SWP/488313/2007 quoting "Generally, once a day administration is adequate. In some cases more frequent administration in animals than anticipated in clinical use may be appropriate". When studying for example a compound with a rather long terminal half-life this may cause some problems as for example Cmax of a real safe dose may reach the toxic level simply due to accumulation.

Helmut
Hero

Vienna, Austria,
2010-01-13 19:34

@ martin
Posting: # 4604
Views: 10,692

## principle of superposition

Dear Martin!

» […] the formula for the accumulation index R is simply a formalism of the principle of superposition or overlay technique (Gibaldi and Perrier, 1982, appendix E).

This was already introduced as the “Law of Corresponding Areas” (Gesetz der korrespondierenden Flächen) back in 1953!* So far for our American friends.

» In the case of 4 subsequent administrations I can calculate the accumulation index but it is biased as only 93.75 of steady state is reached (on assumption of a one-compartmental model). For this reason, there is no way to get an unbiased estimate for the accumulation index in the case of 4 administration?

You never get an unbiased estimate, unless your ‘are’ in true steady state (continuous infusion). Which percentage of steady state we accept as reasonable is merrily a convention. Don’t mix up dosing for at least four half-lives with four administrations!

» PS.: There is a draft guideline on repeated dose-toxicity studies EMEA/CHMP/SWP/488313/2007 quoting "Generally, once a day administration is adequate. In some cases more frequent administration in animals than anticipated in clinical use may be appropriate". When studying for example a compound with a rather long terminal half-life this may cause some problems as for example Cmax of a real safe dose may reach the toxic level simply due to accumulation.

Interesting. I have to read it over – but from your quote I don’t get the idea. If we are interested in steady state, the design IMHO must be based on the half-life – not on a formalistic ‘once-a-day’ schedule. For drugs with half-lives of a couple of weeks daily dosing is nonsense. Just my two ¢.

• Dost FH. Der Blutspiegel: Kinetik der Konzentrationsabläufe in der Kreislaufflüssigkeit.
Leipzig: Thieme-Verlag; 1953. p. 244.

Regards,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
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