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2018-08-03 10:25

Posting: # 19131
Views: 1,949

 Dose-proportionality versus dose-linearity [PK / PD]

Dear Colleagues,

I would like to re-open again the never ending discussion regarding the concepts of dose-proportionality versus dose-linearity. Based on some internal discussion I tried to summarize my current understanding of this topic in bullet points where I would be happy to get the opinion of the form members on these considerations.

The concept of dose-proportionality is based on the premise that a dose of zero leads to an exposure of zero (i.e. line going through the origin). In contrast, the concept of dose-linearity does not require having a line going through the origin.

Currently I am aware of only two situations where the concept of dose-linearity is favored over the concept of dose-proportionality namely

Situation 1: Endogenous substances (e.g. estradiol) where the individual PK metrics are not adjusted for the endogenous background prior evaluation of the dose-response relationship. This means a dose of zero leads to an AUC > 0 due to the endogenous background

Situation 2: A compounds that is invariably lost upon administration which would result also in a line not going through the origin.

Statistical considerations:
Typically, dose-proportionality is an equivalence problem and can be evaluated based on the work of Smith et al (2000). In contrast, dose-linearity is sometimes evaluated by fitting a quadratic equation and evaluate whether or not this model provided a better fit than just a straight line.

In addition, I became aware of cases where dose-proportionality was not shown but dose-linearity was claimed based on the approach above. I think this approach is flawed as the 1) quadratic model approach is also driven by precision and sample size and not necessarily just by degree of non-linearity and 2) the absence of evidence is not the evidence of absence.

My summary:
I am having serious troubles thinking of a situation which fails to show dose proportionality (for lack of evidence) and yet still shows linear PK and therefore:

I would recommend using the concept of dose-proportionality (e.g. Smith et al, 2000) unless there is biological reason not to do so (e.g. endogenous substances or substances that are invariably lost upon administration). If you have to stick with the concept of dose-linearity and the fitted line of the quadratic model does not go through the origin and there is no underlying rationale for that (e.g. endogenous substance) than I would consider this as an inadequate model for the data at hand (e.g. internal validation failed).

Best regards & looking for an interesting discussion


2018-08-03 11:21
(edited by ElMaestro on 2018-08-03 11:43)

@ martin
Posting: # 19132
Views: 1,511

 Dose-proportionality versus dose-linearity

[2006, at an EU agency, in a meeting room on the 4th floor.....]

Chairman: " it looks like we are done with the agenda. I'd like to thank all of you for attending. Minutes from this meeting will be circulated first to you and then to the committee within 30 days. Unless there are more questions I'd like to wrap up this meeting, and I expect to see you all again in three months."
(Silent room... 1 second... 2 seconds.... someone raises a finger)
Chairman: "Yes, Romain?"
Romain: "I have just a little outstanding topic that I feel we must address."
Chairman: "Oh for fucks sa..., I mean: Romain, please go ahead."
Romain: "It concerns the distinction between pharmacokinetic linearity and pharmacokinetic proportionality, a matter which I am sure you all agree has to be a top priority because we are entrusted to work on only the most serious matters."
The chairman closes his eyes and bangs his head into the table.
Jerome: "Dang, there goes my flight home."
Melissa: "Haven't we wasted...I mean spent... about 5000 hours discussing this during the last 4 meetings?!"
Romain: "If you don't agree with me then it is because you don't understand. Didn't you read the 87 papers I sent you before this meeting? Villeroy & Boch clearly showed in 1967 that..."
Aaron: "Dear God, please wake me up from this nightmare."
Chairman, eyes still closed: "Since it is past 3:30pm the agency's staff are no longer at work. You will need to sort out your own hotels and flights."
Romain: "We need to do something!"
Chairman, slowly opening his eyes, now a distinctly new look on his face: "You are absolutely right, Romain. You are absolutely right."
A leftover tomato from the lunch cart smashes into Romain's forehead. Kathi throws a handbag into his face. Someone pokes Romain in the left eye with chopsticks. 16 people closing in on Romain. Then sheer mayhem. A flurry of punches flying through the air, along with a few chairs. Eyeglasses, socks and a laptop charger go airborne. A bit of blood spatter here and there.
Melissa: "Ewww Romain, this shirt was all white, now it has matrix all over it, you filthy fucking bastard."
Someone's dentures hitting the wall. The sound of a knee cap busting due to its sudden encounter with a baseball bat. A high-heeled shoe hammers into the projector in the ceiling. It crashes down on top of the chairman's head. Electric sparks, then a lot of smoke and the smell of burned scalp.

3 floors below, at the Department of Access Control and Safety. A bright orange light suddenly flashing on a dashboard. An alarm goes off.
Goon 1: "Oh... looks like we have another code red on the fourth floor."
Goon 2: "What's happening there?"
Goon 1: "Someone apparently mentioned linearity."
Goon 2: "Oh shit. Not again. Last time it took 15 people an hour to separate them."
Goon 1, smiling: "Not to mention the cleanup afterwards.... but hey, it's past 3:30pm."
Goon 2: "Nothing we can do, then. The overtime budget got exhausted two months ago."
Goon 2 reaches under the dashboard and disconnects a wire. The bright orange light disappears. The alarm stops.

[And so forth...]

if (3) 4

Best regards,

"(...) targeted cancer therapies will benefit fewer than 2 percent of the cancer patients they’re aimed at. That reality is often lost on consumers, who are being fed a steady diet of winning anecdotes about miracle cures." New York Times (ed.), June 9, 2018.

2018-08-06 19:39

@ ElMaestro
Posting: # 19155
Views: 1,368

 Dose-proportionality versus dose-linearity

Dear martin,

I must admit I am a bit relieved to see that apparently there is some general confusion around this topic ;-).

Your example 1. makes sense to me. Do you have an example for 2.?

Maybe we can go one step back and clarify nomenclature. When you talk about "linear PK" you apparently do not mean the same thing as dose proportionality. Are you using "linear PK" interchangebly to "dose linearity"? I see all kinds of definitions, some define linear PK to be dose proportionality, some don't. Of course it always depends on the underlying model (how many compartments?)... Is there a general definition of the three terms?


2018-08-07 13:04
(edited by martin on 2018-08-07 13:44)

@ Ben
Posting: # 19158
Views: 428

 Dose-proportionality versus dose-linearity

Dear Ben,

Its a good idea looking at the definitions first and here is my understanding

Linear pharmacokinetics: All transport processes follow a first order kinetic (e.g. absorption, distribution and/or elimination) where first order kinetic means that the concentration is changed at a rate proportional to the concentration (in contrast to non-linear kinetic which means that one or more transport process(es) follow other than first order kinetic). Please note that this implies that an IV administration given as infusion modeled as zero-order absorption is per definition non-linear PK but typically regarded as a non-relevant deviation from linear PK in case that processes after end of infusion follow a first order kinetic and AUC during infusion time is small compared to the total AUC.

Dose-proportionality: the relation between dose and exposure (e.g. AUC) is given as straight line passing through zero on the ordinate

Dose-linearity: the relation between dose and exposure (e.g. AUC) is given as straight line starting on the ordinate at any value greater or smaller than zero

On the premise of linear pharmacokinetics:
1) PK metrics such as AUC, Cmax, Ctrough increase proportional with dose (to be more general: on the premise of linear PK doubling of dose leads to doubling of any individual concentration at a given time point)
2) PK metrics tmax, t1/2, CLs, Vss, MRT and F are independent of dose
3) Concentrations after repeated dosing can be predicted from concentrations following a single dose (e.g. AUC0–τ at steady state = AUC0–inf after a single dose)

For this reason,
1) Linear pharmacokinetic implies dose proportionality and not necessarily vice versa
2) Linear pharmacokinetic implies PK linearity after repeated administration (i.e. assessed if AUC0–τ at steady state equals AUC0–inf after a single dose) and not necessarily vice versa

Taking the above points into consideration I think that linear PK does not imply dose-linearity.

Best regards


PS.: regarding an example for situation 2: you may think of a drug which is given far in excess of a receptor

PPS.: My understanding is that nearly all drugs are expected to exhibit nonlinear PK behavior when administered at "extremely" high doses (e.g. leading to saturable absorption or saturable metabolism which is studied in the field of toxicokinetics) which motivates formally assessing dose-proportionality only for the clinical relevant dose range.
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