Pharmacodynamic Modeling of an Ordinal Categorical Response

1
Pharmacodynamic Modeling of an Ordinal
Categorical Response

• Robert D Johnson

2
Introduction

• Serum based pharmacodynamic assay generates
  continuous response.
• Drug inhibits biochemical conversion in serum.
• Initial modeling focused on continuous data.
  Less than optimal performance.
• Explored categorical methods (logistic regression)

3
Background

• Plasma concentration-time data and
  pharmacodynamic activity-time data collected from
  2 Phase II studies and a single Phase III study
  (Different Indications)
• Phase 3 Study Indication 1
• Phase 2 Studies Indication 2

4
Background (Studies/Procedures)

• Drug Administration
• Route IV infusion
• Indication 1 (Phase III Study) 2.0 mg/kg for 10
  minutes following by 0.05 mg/kg/hr for 24 hours
  starting at termination of 10-minute infusion.
• Indication 2 (Phase II Studies) 2.0 mg/kg for
  10 minutes followed by 0.05 mg/kg/hr for 20 hours
  starting four hours after termination of the
  10-minute infusion.

5
Background (Studies/Procedures)

• Blood samples collected for analysis of free drug
  concentrations in serum and for pharmacodynamic
  activity measurements.
• Blood sampling paradigm (scheduled times)
• Phase II studies predose (0), 4, 12, 24, 36, 48,
  and 72 hours
• Phase III study predose (0), 4, 24, 72, and 96
  hours
• Blood samples analyzed for free drug
  concentrations using a validated (250 16000
  ng/mL) ligand binding method with
  electrochemiluminescence detection (ECL).

6
Methods

• POP PK analysis of free drug concentrations in
  serum conducted prior to the PD (2-staged
  approach). Derived from 755 observations from
  476 patients.
• Individual predicted free drug concentrations
  (Cij) independent variable in PD analysis.
• Initial modeling efforts focused on continuous
  data.

7
Methods

• Base model Emax model with baseline component.
• Dataset 2535 observations from 476 patients.
• Patients had at least 1 free drug concentration.
  

8
Methods

• A Categorical modeling approach implemented due
  to problems with continuous data approach.
• Pharmacodynamic activity data converted to
  Categorical response (ordinal scale).
• Placebo patients excluded from analysis. Over
  70 of observations would fall into one of the
  pharmacodynamic activity categories.

9
Results (Continuous Data)
10
Results (Continuous Data)
11
Results (Continuous Data)
12
Results (Continuous Data)
13
Results (Continuous Data)
14
Test For Normality (Goodness of Fit Tests)
15
Test For Normality
16
Conclusions Continuous Data

• Visual inspection of diagnostic plots suggested
  potential violations of model assumptions.
• Population Method robust against violations of
  model assumptions - validity of inferences ?
• Categorical methods used in place of continuous
  data. Continuous data transformed into ordinal
  categorical scale (k 3 categories).

17
Background (binomial distributions)

• Most basic categorization Binomial random
  variable i.e. dichotomous response (success vs
  failure or occurrence vs lack of occurrence)
• Statistical framework for analysis binomial
  distribution
• For n independent Bernoulli trials with x number
  successes and (n-x) failures, the response is
  distributed as EiBin(n,p)

18
Background (multinomial distribution)

• Multinomial response (k3 or greater response
  categories)
• Statistical frame work (multinomial distribution)
• Based on n independent trials with xi (x1, ? ?
  ? , xk) mutually exclusive occurrences of event
  Ei EiMULT(n,p1,p2,1-p1-p2).
• Multinomial distribution extension of the
  binomial distribution.
• Multinomial distribution member of Regular
  Exponential Class family of distributions.

19
Background (multinomial distribution)
20
Methods Categorical Data

• Pharmacodynamic response transformed to ordinal
  scale
• Activity ? 80 ? Full Activity (Y 0)
• 20 lt Activity lt 80 ? Partial Activity (Y1)
• Activity ? 20 ? Full Inhibition (Y2)
• Pharmacodynamic data analyzed by logistic
  regression (multinomial).
• Link function logit function

21
Methods Categorical Data

• Joint probability density function (Likelihood
  Function)

22
Methods Categorical Data

• The following functions of free concentration
  were evaluated (base model).

23
Methods Categorical Data

• Explicit relationships for g1(Cij) and g2(Cij)

24
Methods Categorical Data

• Continuous covariates BWGT, CLCR, AGE, PTINR.
• Categorical covariates Study (Stdy 1, Stdy 2,
  Stdy 3, Race, and Gender).

25
Methods Categorical Data

• Modeling Methodology Probability that the
  pharmacodynamic activity falls into each
  category.
• Mixed effect modeling using PROC NLMIXED (SAS)
• Adaptive Gaussian Quadrature method numerically
  integrates the likelihood function with respect
  to the random effects (generate marginal density
  function with respect to fixed effects).

26
Methods Categorical Data

• Log-Likelihood expression
• r0 1 when activity falls in category 0 and zero
  otherwise.
• r1 1 when activity falls in category 0 and zero
  otherwise.
• r2 1 when activity falls in category 0 and zero
  otherwise.

27
Methods (Base Model SAS Code)
28
Results Categorical Data (Base Model)
29
Methods Covariate Screening

• Initial screening Walds approximation to
  likelihood ratio test.
• Full model comprised of all covariates included
  with respect to g1(Cij) and g2(Cij).
• Wald method run using original SAS version.
• Best 50 models based upon rank ordered SBC run in
  NLMIXED.
• Final model corresponded to NLMIXED run with
  lowest SBC.

30
Results Categorical Data (Final Model)
31
Results Categorical Data (Final Model)
32
Effect of Dosing Regimen(PK Related)
33
Effect of Renal Function(PK Related)
34
Effect of Renal Function(PK Related)
35
Effect of Body Weight(PD Related)
36
Effect of Body Weight(PD Related)
37
Statistical Analysis (Simulation Results)

• Physicians Interested in Time to Recover Back to
  Baseline Activity.
• Initial Analysis ANOVA following transformation
  of response variable.
• Residuals still violated model assumptions
  following transformation.
• Performed non parametric analysis of variance
  (Wilcoxon Rank Sum).

38
Statistical Analysis (Simulation Results)

• Performed pair wise comparisons with baseline
  (control) based upon 95 confidence interval
  approach.
• Overall error rate controlled using Bonferroni
  adjustment.
• Paired comparisons calculated using the following

39
Effect of Renal Function(Recovery Time)
40
Effect of Renal Function(Recovery Time)
41
Effect of Renal Function(Recovery Time)
42
Effect of Renal Function(Recovery Time)
43
Effect of Body Weight(Recovery Time)
44
Effect of Body Weight(Recovery Time)
45
Effect of Body Weight(Recovery Time)
46
Effect of Body Weight(Recovery Time)
47
Conclusions

• Violations of model assumptions observed with
  continuous data.
• Relationship between pharmacodynamic activity and
  free drug concentrations was analyzed using
  multinomial logistic regression (generalized
  logits approach) following conversion of the
  continuous data to an ordinal categorical scale.
• A nonlinear equation was the most parsimonious of
  the concentration functions used within the
  exponentials of the logit expressions and
  represented the base pharmacodynamic model.

48
Conclusions

• Final model included body weight and study
  (1999052) with respect to concentration function
  g1(Cij) and study (1999052 and 1999053) with
  respect to concentration function g2(Cij).
• Patients that received a bolus infusion recovered
  back to baseline conditions faster (58 hours)
  than those that received a bolus immediate
  infusion (68 hours) or bolus delayed infusion
  (70 hours).
• Patients with severe renal impairment (CLCR 30
  mL/min) took longer to recover to baseline for
  full activity (78 hours) than patients with
  normal renal function (68 hours).

49
Conclusions

• Patients with higher body weight took longer to
  recover to baseline for full activity. Patients
  weighing 160 kg took 74 hours, while patients
  weighing 40 kg took 62 hours to recover to
  baseline.
• The probability that a patient had full activity
  at baseline (conc 0 ng/mL) increased with a
  decrease in body weight for CABG patients with
  normal renal function for body weights ranging
  from 160 kg (probability 0.58) to 40 kg
  (probability 0.73).

50
Conclusions

• The probability that a patient was fully
  inhibited at high free drug concentration
  (conc  1000 ng/mL) increased with a decrease in
  body weight for CABG patients with normal renal
  function for body weights ranging from 160 kg
  (probability 0.56) to 40 kg (probability
  0.68).
• The rank order for baseline recovery times for
  patients with normal renal function in each study
  was 1999052 (66 hours AMI), 2000099 (74 hours
  CABG), and 1999053 (84 hours AMI).

51
Backup Slides
52
Results (Continuous Data)
53
Test For Normality (Goodness of Fit Tests)

• Null hypothesis (Ho) residuals follow a normal
  distribution.
• Empirical distribution function tests (EDF)
  Kolmogorov-Smirnov, Anderson-Darling, Cramér-von
  Mises.
• Compare percentiles of ordered residuals
  (increasing rank order) versus theoretical
  distribution (CDF normal).

54
Test For Normality (Goodness of Fit Tests)

• Generate test statistic and determine p-value of
  test statistic. Low p-value supports departure
  from theoretical distribution.
• Caution - Large sample sizes good power to
  detect minor departure from theoretical
  distribution. Use in conjunction with normal
  probability plot (visual inspection/analyst
  judgment).
• Anderson-Darling test statistic quadratic class
  of EDF statistics

55
Test For Normality (Goodness of Fit Tests)
where is the CDF for a normal
distribution.
56
Background (multinomial distribution)

• Mean, variance, covariance easily obtained from
  moment generating function.

57
Background (trinomial distribution)

• Expected value (moment estimator) is a vector of
  means.

58
Background (trinomial distribution)

• Variance (moment based estimator) of the
  distribution is a matrix. Off diagonal elements
  are the covariance between p1 and p2.

59
Background (trinomial distribution)

• Maximum likelihood estimates of mean
  uniformly unbiased minimum variance
  estimator.
• Maximum likelihood estimate of variance
• is asymptotically unbiased.

60
Effect of Dosing Regimen(PK Related)
61
Effect of Renal Function(PK Related)
62
Effect of Renal Function(PK Related)
63
Effect of Body Weight(PD Related)
64
Effect of Body Weight(PD Related)
65
Effect of Study(PK and PD)
66
Effect of Study(PK and PD)
67
Effect of Dosing Regimen(Recovery Time)
68
Effect of Dosing Regimen(Recovery Time)
69
Effect of Dosing Regimen(Recovery Time)
70
Effect of Study(Recovery Time)
71
Effect of Study(Recovery Time)
72
Effect of Study(Recovery Time)