Dose proportionality is a common phrase used pharmacokinetics. Early in the pre-clinical development process, we evaluate dose proportionality in animal species. Then if the drug advances to clinical trials, one of the first assessments in humans is to evaluate dose proportionality. Why is it so important? What do we learn from understanding dose proportionality? What does it mean?

Dose proportionality occurs when increases in the administered dose are accompanied by proportional increases in a measure of exposure like AUC or C_{max}. Thus an evaluation of dose porportionality usually includes exposure analysis of 3 or more doses to produce a graph similar to the one below:

As shown, a linear regression (or other statistical construct) is used to determine if there is a linear increase in exposure with increases in dose. In general, there are 3 possible outcomes:

- Exposure is dose proportional (as shown in the figure)
- Exposure is less than dose proportional (points would fall below the line)
- Exposure is greater than dose proportional (points would fall above the line)

The explanation of dose proportionality does little to explain what it means. To understand the utility of dose proportionality, we need to look at the equation for clearance (see previous post):

Equation 1

By rearranging this equation we get the following:

Equation 2

This equation suggests that AUC is linearly related to Dose. The proportionality factor is 1/CL. Thus if exposure is dose proportional, it means that clearance is constant over that same dose interval. This is the reason that it is important to test for dose proportionality.

So when testing dose proportionality for your next drug, remember that you are really testing to see if there is constant clearance.

Does the plot show proportionality or just linearity? Dose increases five-fold from 100 to 500 but AUC increases eight-fold. If proportional, would it not be five-fold also?

Hans, Thank you for the question. Reading approximate numbers from the graph using the blue circles, I read an AUC of ~35,000 ng*h/mL at a dose of 100 mg. Then I read an AUC of ~160,000 ng*h/mL at a dose of 500 mg. The increase in dose is 5-fold, as you mention. The increase in exposure is 160,000/35,000 = 4.6-fold increase in exposure. In this example, the increase in exposure (4.6) is of similar magnitude to the increase in dose (5), making this drug dose-proportional across the range 100-500 mg. The best-fit line has slightly different values, but the overall message is still the same.

– Nathan

Hi,

One question linked to dose-proportionnality. In case of supra-dose proportionnality, which model would you recommend to fit the data and simulate AUC or clearance at the non tested doses:

1) Power model?

AUC = A*Dose exp(B)

2) Michaelis-menten model?

Cl = Clo – Vm*Cmax/ Km + Cmax

Thank you for your help.

Xavier

I don’t think a single answer will work for all situations. I would use the model that best describes the data. If you have data across multiple doses, fit the various models and determine which one best represents the data. If you have limited data, try to use the most physiologically relevant model that is supported by the existing data.

Excellent blog, I wish I had found it earlier. One curiosity about the point you make here “when testing dose proportionality for your next drug, remember that you are really testing to see if there is constant clearance”, or more appropriately, about the AUC. From the equations, I deduce that as long as the AUC increases proportionally to the dose, the Clearance is left the same. But what if the PK-profile curve used to determine the AUC actually changes shape with escalating doses? I guess it’s unlikely for it to keep exact proportionality over a dose range… But have you ever seen it? What may be the cause? (Just curious, I do not have a specific example at hand)

Thank you for your kind comments. I have never seen a situation in which a plasma concentration-time curve changes shape with changing dose, except when saturation of absorption occurs (usually in toxicology studies only). But even if the shape changes, the theory holds true. Dose proportionality occurs when exposure increases linearly with increases in doses administered. Very good question!

How do I know two solid dosages are dose proportional?

To test the proportionality of 2 solid dosage forms, you would conduct a pharmacokinetic study with each dosage form and compare the AUCs for each dose level and each dosage form.

Hi,

Why do we generally go for dose normalisation when we talk about doe proportionality?

Thanks in advance

Excellent question, thank you for asking.

Dose normalization is another way of assessing dose proportionality. If the exposure increases linearly with dose, then dose normalized exposure should be the same across multiple dose levels. For example, let’s assume that we give 100 mg and calculate an AUC of 500 ng*h/mL, then we give 200 mg and calculate an AUC of 1000 ng*h/mL. If you divide the AUC by dose, you would get 5 ng*h/(mL*mg) for the 100 mg dose and 5 ng*h/(mL*mg) for the 200 mg dose. Since the dose-normalized AUCs are the same, we can conclude that the exposure is dose-proportional.

Thank you so much for the answer

I am having a hard time understanding this concept, and proper dosing w/o ANY BSA OR HEIGHT INFO in my studies/problems? Could you please explain?

Cindy,

In some studies (usually preclinical animal studies) doses are normalized to body size using body surface area, weight, or other calculations. This is done to normalize the dose between large and small animals. This is not commonly done in clinical studies, except in the case of pediatric patients (normally neonates), or in oncology settings (highly toxic compounds). You can still test for dose proportionality by using the normalized dose (e.g., mg/kg) and looking for proportional increases in exposure with increases in dose.

Nathan

Will you consider a formulation with dose proportionate , if there is a variation only film coating concentration where as the main core tablets remain same for sequential increase in LC

Ex : 25 mg , 50 mg, 75 mg & 100 mg tablet in 250 mg Avg weight of the tablet where the film coating concentration is only varying

Dose proportionality is a function of the pharmacokinetic response following administration of different dosage strengths. If you administer the various tablets you outline and drug exposure increases proportional to the dose increase, then you would have dose proportionality for those tablets.

Have you been meet this situation,that when dose doubles and the AUC increased but less than twice such as a constant as 1.5? For example, the dose were 10, 20,40,80 mg/kg, and the AUC were 100,150, 225 and 338 respectively. In this situation is it dose proportional?

Qian, I have not met that situation previously. In that case the increase in dose is predictable, but not proportional. I’m intrigued why you think that this occurs.

Nathan

Thanks, Nathan. During the preclinical preliminary TK study, I met the similar situation, I am not sure if it is aslo proportional. Or a situation approach the saturation.

Are there some criteria to choose right number of doses? Thank’s for your answer.

You need at least 3 doses to test for proportionality. I recommend 5 dose levels if possible. Also you may want doses that cross at least 1 log (eg, 0.1 to 1 mg).

Can you recommend some books on this subject?

Many PK books contain some information on dose proportionality; however, I strongly recommend the following 2 manuscripts:

1. K. Gough, M. Hutchison, O. Keene, B. Byrom, S. Ellis, L. Lacey, J McKellar. “Assessment of dose proportionality: Report from the statisticians in the pharmaceutical industry/pharmacokinetics UK joint working party” Drug Information Journal, Vol 29, pp 1039 – 1048, 1995.

2. B. Smith, F. Vandenhende, K. DeSante, N. Farid, P. Welch, J Callaghan, S. Forgue, “Confidence interval criteria for assessment of dose proportionality” Pharmaceutical Research, Vol 17. No 10, pp 1278 – 1283, 2000.

Why can not I use ANOVA to carry out a study on the two dose levels as it described in B. Smith, F. Vandenhende, K. DeSante, N. Farid, P. Welch, J Callaghan, S. Forgue, “Confidence interval criteria for assessment of dose proportionality” Pharmaceutical Research, Vol 17. No 10, pp 1278 – 1283, 2000.?

Katerina, you may use the ANOVA on 2 dose levels as described in the manuscript you mention. This method is less robust than the power model method, and less reliable than datasets with more dose levels. There are many methods of determining dose proportionality; however, in my opinion the most accurate and reliable method is the power model with data from at least 3 dose levels.

Nathan, it is known that there are dose proportionality for reference formulation. Is it true that there will be dose proportionality for test formulation with the same active substance?

Katerina,

You cannot assume dose proportionality for two different formulations based only on the active substance. However, you can assume dose proportionality if the two formulations have equivalent dissolution profiles at all dose strengths using accepted in vitro systems.

Ok, but I can’t do in vitro study. I have two different dosage forms with the same active substance, for one of witch dose proportionality is known. Are there any variants to assume dose proportionality without doing in vitro study? And is it possible to estimate dose proportionality by WinNonLin?

Dose proportionality must be verified by some sort of experimental test. The two accepted methods are to compare dissolution rates (in vitro) or compare systemic exposure after administration to humans (in vivo). There is no purely mathematical way to test for proportionality without any data from an experiment.

Thank you for your answer. Is it possible to analyze dose proportionality by WinNonLin, if I will do in vivo study?

Yes. You would compare the AUC and Cmax of the 2 formulations relative to the increases in dose. This can be done using a power model relating dose and each PK parameter. If the exponent is 1, then dose proportionality is concluded.

Hi Nathan, if a drug’s PK profile fit a two-compartment model and its exposure was found to be linear (not necessarily proportional), when dose reduced half, can I conclude that the Cmax will reduce half?

Thanks!

Good question. The Cmax will increase using the same linear relationship that was derived from previous doses regardless of the specific PK model used to describe concentration-time data.

Hello Nathan: if there are 5 levels offered of a MR dosage form of an existing drug e.g. 15, 30, 60,75 and 90 mg some folks think in terms of doing a BE study comparing 6x 15 mg with 1 x 90 mg. If BE on a dose-normalized basis then dose proportionality is concluded.

What do think of that ?

ANGUS

Angus,

The approach you propose of testing the proportionality between the high and low dose is a common one, but not one of my favorites. I prefer to use the power model with all dose levels. My preference is based on using all available data rather than relying on 2 of 5 data points to declare proportionality.

Nathan

Hello Nathan,

Thanks your wonderful elucidations! If I want to evaluate dose proportionality of three doses with identical formulation, How do dose proportionality using WinNonlin or SPSS? Could you please share your experiences regarding dose proportionality?

Best regards

Ping Du

The most common method for assessing dose proportionality is to use the Power model. The power model is the following: Cmax (or AUC) = B0*Dose^(B1). If there is dose proportionality, then B1 will be close to 1. To calculate this in software, it is easiest to convert the power model to a linear model using logarithmic transformation, to get the following: ln(Cmax) = B0 + B1*ln(Dose), which has the form y = b + mx. You can then perform a linear regression with ln(Dose) as the independent X-value and ln(Cmax or AUC) as the dependent Y-value. If the 90% confidence interval for the slope include 1.0, then you can consider the exposure to be dose proportional. Some people require that the 90% confidence intervals lie within a specific range (e.g. 80 – 125%); however, the range for the confidence intervals is dependent upon the range of doses used in the study. The linear regression can be performed by nearly any statistical software package that can perform linear regression.

For a complete evaluation, please review the following paper: “Confidence Interval Criteria for Assessment of Dose Proportionality” Brian P. Smith, et. al., Pharmaceutical Research, Vol 17, No. 10, 2000, pp 1278 – 1283 (link).

Does anyone know if the SAS code (Proc. Mixed) for Brian Smith’s dose proportionality procedure (see above) is published. Brian’s procedure also allows the calculation of within subject variance.

Hi Nathan,

How to justify, more than dose proportional increase in exposure (significantly) with increase in dose.

Thank you,

Shiva

Shiva,

When increases in exposure are more than dose proportional it usually indicates that drug clearance has been saturated.

Nathan

Nat: I seem to recall that diltiazem displays more exposure than is proportional to dose and the reason for this is it inhibits its own metabolism.

Nice example Angus. Thank you.

Hello Nathan,

Excellent blog..!

One thing I am still not getting is why we go for dose proportionality ??means how it is showing that drug is effective or not..Or maybe I am not knowing what it exactly means that exposure is dose proportional???Can you please explain??

Thanks!

Thank you Nivedita.

Dose proportionality does not directly relate to the effectiveness of a drug. It simply means that across a range of doses, the clearance is constant. Therefore, the amount of drug exposure is proportional to the size of the administered dose. This is useful when you need to increase or decrease a dose to achieve a safe and effective drug level for a patient. If a drug does not show dose proportionality, then small changes in dose could cause large swings in drug concentrations, which could lead to safety or efficacy issues.

Nathan

Hi Nathan,

I have two questions related to dose proportionality:

1. How to interpret situations where exposure vs dose is proportional as shown in the figure above however the y-intercept is not zero ?

2. When doing dose-proportionality, the dose levels should have the same dosing regimen? For example, I can only compare 10, 50, 100 mg BID and not 10 BID, 50QD, 100 once weekly because there may be accumulation.

Thanks for this wonderful blog.

Hi Nathan,

For a dose proportionality problem, when we use the equivalence criteria together with power model, do we need to seriously consider the sample size calculation?

And, when DP is not a main concern in a study, can we still use the method （the equivalence criteria together with power model）regardless of the power requirement??

Tony

Hi Tony,

Sample size should always be considered in any statistical analysis. Very small sample sizes can give results that are not applicable to a broader population. That being said, the power model can be used for dose proportionality assessments even when it isn’t the primary focus of the study. And evaluation of the confidence interval around parameter estimates is still valuable.

Nathan

Yes， that is the situation， in one of my study which overall is a general PK study, DP is not the primary outcome variable, but the clinician wants to explore the dose linearity. So , we also use power model together with equivalence criteria. I think it is acceptable even though there is no sample size consideration.

The dose proportionality study should have a sample size (for BE) based on the within subject variance calculated from a prior study with your drug formulation.

Hi Angus, Can I say that , if the study is designed specifically for DP, then we need to consider form sample size determination?

What is your opinion for those studies where DP is exploratory? same as Nathan?

Many thanks!