18 replies to this topic

[Michael Kong]

Hi all,

I am working on a dose escalation study in a small pilot (18 subj) and was wondering whether dose proportionality could be assessed in Phoenix (for example, using the power model and CI approach) and if so, specifically how this can be done.

Thanks,

Mike

[Helmut Schütz]

Hi Mike!

I’m using following ASCII code:

remark      DOSE PROPORTIONALITY
remark      Chow/Liu 2000 pp 368 Model 3
remark      data in original (untransformed) scale (X/Y)
remark      weight = 1/X
remark      weights must be provided in column 3
remark      (c) Helmut Schuetz, BEBAC, 1070 Vienna
remark      2003-05-22: new
remark      2003-07-08: tested
remark      2007-09-26: nobounds added
remark      ============================================
model
remark      Dose Proportionality
remark      Model 3: E(Y)=a*X^b (power function)
remark               where a>0 and b#0
remark               alpha = coefficient
remark               beta = exponent ('curvature')
remark               weight = 1/X
commands
dnames 'dose' 'response'
npar 2
pname 'alpha' 'beta'
initial 1 1
nobounds
weight
end
func 1
f = alpha * x ** beta
end
eom

Dose in the first column, PK response in column 2, weight in column 3. You have to calculate the weight by column transformation and sort on subjects in column 4. Assess proportionality by the confidence interval of beta (1 included?).

[Michael Kong]

Thanks very much Helmut for the quick response!

I will give it a try though can you clarify what is required in column 4? What is the weight by column transformation?

Thanks again,

Mike

[Helmut Schütz]

Hi Mike!

can you clarify what is required in column 4? What is the weight by column transformation?

You don’t need values in columns >3, especially if your have a FIM (dose escalating) study, which is performed in a parallel design. If your design is a confirmatory dose proportionality study, most likely it’s a cross-over. You can keep subject ID, periods, and sequences in columns >3 in order to perform BE on dose-adjusted PK response(s). But it’s not needed to execute the code.

One of the requirements in any type of regressions analysis is homoscedasticity (equal variances of the regressor). If you set all values in the third column to 1 (=unweighted) and plot the model’s residuals versus dose most likely you will see a funnel-type distribution: small residuals at low doses and large ones at high doses. Chow and Liu (3rd ed 2009. Chapter 18.3, pp 563–573) suggest to weight data by 1/dose. In order to use my code, you have to have the dose in column 1 and PK response in column 2. Use PHX’s tools to extract this values form you dataset. Send to > Data > Column Transformation > Transformation Type: Arithmetic > Transformation: x^n (map the dose in column 1 to x) > Arithmetic Operations > n = -1, New Column Name: Weight > Execute

[Michael Kong]

Thanks Helmut - I was finally able to revisit this issue again. I am dealing with a FIM parallel study so I have included only dose, response, and weight. I am having trouble using the code though as I haven't used user define ascii models in wnl phoenix before. I loaded the code into the setup tab under wnl5 ascii format but am stuck - not quite sure how to link the dataset to the code. Your help again would be appreciated!

Thanks,

Mike

[Helmut Schütz]

Hi Mike

I am dealing with a FIM parallel study so I have included only dose, response, and weight.

Correct

I am having trouble using the code…

Yes, the setup is a little bit counterintuitive. If you attach the project file to a post, I can see what i can do (delete confidential information from the file!).

[Michael Kong]

No problem - here you go Helmut. Thanks again.. [file name=Power_Model.phxproj size=107936]http://www.pharsight.com/extranet/media/kunena/attachments/legacy/files/Power_Model.phxproj[/file]

[Helmut Schütz]

Hi Mike!

 

See attached. You don't have to copy the results of the column transformations (Including the new weight-column) to the Data section – you can map them directly. It's a liitle bit weired, but in Main of the ASCII-model, you have to map Dose to "Time" and Cmax to "Concentrations" (maybe in future releases "X" / "Y" or "Independent" / "Dependent" would be less confusing terms). You notice that there's no column to map the weights. Go to the "Model Selection" tab at the bottom. Change Weighting from "Uniform" to "User Defined". Below select "Source". Note that the Weight-column appears in the table on top and is already mapped. Now you could execute the workflow, but the result is terrible. This is caused by my initial estimates of 1 and 1 for alpha and beta. In the Setup go to "Initial Estimates" and set alpha to 100 and keep beta at 1. Execute. I get alpha 120.81 (CI 107.11 – 134.51) and beta 0.8532 (0.6407 – 1.0657). Not so bad. But have a look at the residuals. Maybe that's not the right model.

 

I would suggest to try fitting AUC data instead. Cmax in dose proportionality is of limited value, IMHO. [file name=Power_Model_Mapped.phxproj size=260205]http://www.pharsight.com/extranet/media/kunena/attachments/legacy/files/Power_Model_Mapped.phxproj[/file]

[Michael Kong]

Hi Helmut,

I noticed the code you used is for untransformed data though I thought log transformed data is typically used - any pitfalls to using one over the other? Since we usually assume a log normal distribution for AUC, Cmax, shouldn't we be assessing linearity using log transformed data? Great job on Bebac btw - it's been a great resource for me..

Mike

[Helmut Schütz]

Hi Mike!

I noticed the code you used is for untransformed data […]

Yes, have a look at the referenced Chow&Liu and the ones quoted within.

Since we usually assume a log normal distribution for AUC, Cmax, shouldn't we be assessing linearity using log transformed data?

Not quite. Contrary to AUC (Cmax is already disputable) the distribution here is not based on PK relationships. The power model is a pure empiric one with no pharmacokinetic meaning.

Great job on Bebac btw - it's been a great resource for me..
Mike

THX! You can search there for this topic also. ;-)

[Zancong Shen]

I know this is doable simply in WNL 5.2 using the linear mixed effect wizard. YOu can get both "a" and "b" with CI95% around "b", but haven't tried the phoenix on this.

PKbeginner

[Michael Kong]

Hi PKbeginner,

This option is also available in Phoenix but I am not sure how to set it up for this kind of analysis - can you provide a little detail?. My study is a FIM dose escalating parallel study, not a confirmatory crossover design.

Thanks,

Mike

[Zancong Shen]

I have some guidance for myself used in WNL 5.2. It is not a SOP but it works.

equation: ln(Y) = ln(a) + b*ln(dose)

 

1) Add Subject, Y (Cmax, AUC, etc.), Dose to Winnonlin input file; naturally log transform Dose to LnDose.

2) Click Tools--Linear Mixed Effect Wizard, add LnDose in regressors/covariates, subject in classification variables, and Y to dependent variables. In the model specification, add LnDose. Choose “Ln(x)” in “Dependent variables transformation”. Choose “95” in Fixed Effect Confidence Interval to specify 95% confidence interval.

3) Click next step to LinMix Variance Structure, move “Subject” from classification variables to Random Effect Model.

4) Click Calculate to execute.

5) In the workbook output, go to Final Fixed Parameters, the estimate for “int” will be the value for constant a, whereas the estimate for LnDose:LnDose will be value for slope b. the values in column “lower CI” and “upper CI” for LnDose:LnDose represent the CI95% for the slope b.

 

Hope this help and also works well in Phoenix.

PKbeginner

[Michael Kong]

Thanks! The results seem to match up with PROC Mixed analysis in SAS.

Cheers,

Mike

[Helmut Schütz]

Dear pkbeginner & Mike!

 

Remarks:

[ul][li]If we linearize the model we have to antilog "int" in order to get the coefficient of the power model back.[/li][li]How do you bring in the weights? I was not successful to map them... Your procedure is unweighted – which is not covered by methods in the literature.[/li][li]If opting for a transformation, why not simply use WNLs linear model 502 later on? Much easier to set up than LME in my opinion[/li][/ul]

[Zancong Shen]

I don't know how useful is to consider the weights, please anyone else can give a help? Maybe I didn't explain well in the previous thread.

My procedure is:

Dose proportionality assessments will be performed by statistical analysis using the power model described below:

Y = a*(dose)^b

where Y is the pharmacokinetic parameter (Cmax, AUC etc.), and a and b are the coefficient and exponent, respectively, of the power equation.

By taking logarithms, the power model can be analyzed using linear regression thus has the form:

ln(Y) = ln(a) + b*ln(dose)

For dose proportionality the slope of the regression line ( = 1 and for dose independence b = 0. The final resultant coefficient a value include error component. If b is close to unity (1.0) with CI95% within (0.8, 1.25), the relationship between dose and the pharmacokinetic parameter is concluded to be dose proportional for the dose range studied.

 

PKbeginner

[Michael Kong]

Hi Helmut, pkbeginner,

If we log-transform, do we still need to consider weight (residuals should be normal distributed)?

Mike

[Helmut Schütz]

Dear PKbeginner!

I don't know how useful is to consider the weights,

Please read the reference I gave in my second answer to Mike (on the first page of this thread).

By taking logarithms, the power model can be analyzed using linear regression thus has the form:
ln(Y) = ln(a) + b*ln(dose)

So far, so good. You can use the slope ( from the linearized model directly in the power model as the exponent. But: The intercept is ln(a) - not 'a' (look at your own formula). Therefore you have to antilog it.
Apart from the algebra consider this simple numerical example:
a = 10, b =1; x = 10,20,40 | y = a*x^b = 100,200,400
If you run LME, you get the slope right, but the intercept is 2.3025851, which is - as expected - exactly ln(10). To get the correct 'a' you have to antilog it (e^int=a=10). If you use the intercept 'as reported' your estimated power model would read 2.3025851*x^1 - which is simply wrong (estimated y-values 23,46,92 don't match 100,200,400). ;-)
If you want to use the linear model 502, you have to ln-transform the y-values also. You get the same results as in LME.
The drawback of LME is that you have no means to visualize your fit. Otherwise you would have noticed your error before. Revise your SOP.

[Zancong Shen]

Hi Helmut,

Certainly the intercept from the report needs to be antilog to the original "a" before it can be used to calculate the Y. As you have shown this will be corrected rightaway when one plots the power model curve against the individual data (can immediately tell the curve is unreasonably biased if not done so).

Thank you for pointing this out and the thorough and useful discussion.

 

PKbeginner