Relating Dose Exposure and Patient
Variability to Content Uniformity
Tim Kramer
May, 2013
Company Confidential
© 2013 Eli Lilly and Company
Overview
• Plasma Concentration Profile Examples
• Estimating Total, Between and Within Patient
Variability
• First approximations
• Accounting for multiple doses per patient
• Accounting for dose inaccuracies
• Estimating Patient Impact of Dose Variability
• Setting Dose Variability Specifications
• Conclusions
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
2
Representative Plasma
Concentration Time Profiles
•
•
•
•
•
MBSW, May 2013 – Tim Kramer
One patient
Three separate doses
Single dose-administration profiles
Semi-log scale
Open symbols represent terminal point
selection
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3
Dose (mg)
Dose (mg)
MBSW, May 2013 – Tim Kramer
Dose (mg)
AUC(0-?)
(ng•h/mL)
AUC(0-? )
(ng•h/mL)
Dose (mg)
AUC(0-?)
(ng•h/mL)
AUC(0-? )
(ng•h/mL)
AUC(0-? )
(ng•h/mL)
Areas Under Curve for Multiple
Studies
Company Confidential © 2013 Eli Lilly and Company
Dose (mg)
4
Cmax (ng/mL)
Cmax(ng/mL)
Cmax (ng/mL)
Cmax for Multiple Studies
Dose (mg)
Dose (mg)
MBSW, May 2013 – Tim Kramer
Dose (mg)
Cmax (ng/mL)
Cmax (ng/mL)
Dose (mg)
Dose (mg)
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Some Questions…
• We observe variability between patient
exposures (AUC and Cmax) that is not
explained by the nominal doses
• How much smaller would the patient variability be
if there wasn’t dose variability?
• How much dose variability is there?
• How to separate patient variability from dose
variability?
• What would be the worst case impact of dose
variability?
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
6
Simple Regression Slopes and Standard
Errors for log(Exposure) vs. log(Dose)
AUC
Cmax
Study
Slope
Std. Error.
Slope
Std. Error.
1
0.91
0.05
0.76
0.04
2
1.08
0.06
0.88
0.04
3
1.03
0.05
1.11
0.04
4
1.02
0.06
1.42
0.05
5
0.97
0.18
1.9
0.29
Proportionality constants for AUC are all very close to 1.
Proportionality constants for Cmax vary but appear to be
between 0.5 and 2.
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
7
Simple Regression Slopes of
log(Exposure) vs. log(Dose)
Slope (Log Exposure vs. Log Dose)
2.5
2
1.5
Estimate + 1*S.E.
1
Estimate - 1*S.E.
Estimate
0.5
0
AUC
AUC
AUC
AUC
AUC Cmax Cmax Cmax Cmax Cmax
Study Study Study Study Study Study Study Study Study Study
1
2
3
4
5
1
2
3
4
5
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
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Progress Report…
• Proportionality constant for these studies varied
between
• 0.9 and 1.1 for AUC
• 0.7 and 1.9 for Cmax
• Still need to determine reasonable values for
• Between subject variability
• Within subject variability
• Dose variability
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
9
One Patient, Three Doses
Dose (mg)
AUC
(0 to ∞),
ng/mL
Cmax
(ng/mL)
X
6.63
0.286
4X
27.3
1.259
15X
124
8.323
Representative pattern for an
individual patient
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
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AUC(0-Inf)
(ng•h/mL)
Multiple Patients, Multiple Doses
Dose (mg)
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
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(ng•h /m L)
AU C(0-inf)
Individual Patient for One Study,
Multiple Doses
Same
subject
Dose (mg)
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
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( n g • h / m L)
A U C (0 -inf)
Individual Patient for One Study,
Multiple Doses
Same
subject
Dose (mg)
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
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More Complicated Model
Log(Exposure(Subject i,Dose d)) =
α + β*Log(Dose d) + i + id
where
i represents a random subject effect
and
id represents unexplained error
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
14
Counts of Doses Per Subject by
Study
Doses per
Subject
Study 1
Study 2
Study 3
Study 4
Study 5
1
16
34
1
23
24
2
10
0
0
28
0
3
0
0
15
0
0
Only studies 1, 3 and 4 can be used to estimate between
and within subject variability separately.
Studies 2 and 5 can only estimate the combined effect of
between and within subject variability.
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
15
More Complicated Regression
Results
Between Within
Between Within
Total SD
Prop.
Subjects Subjects
Study Exposure
Subjects Subjects
as % of
Constant
as % of as % of
SD
SD
Base
Base
Base
1
0.890
0.454
0.248
57%
28%
68%
3
1.065
0.360
0.094
43%
10%
45%
1.046
0.265
0.190
30%
21%
39%
4
AUC
2
1.084
63%
5
0.966
48%
1
0.751
0.292
0.344
34%
41%
57%
3
1.116
0.254
0.158
29%
17%
35%
1.433
0.258
0.249
29%
28%
43%
4
Cmax
2
0.883
37%
5
1.895
94%
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
16
Quantifying the Variability
% of Base calculated as 𝑒 𝜎 − 1 since the
regression was log(exposure) versus log(dose):
log 𝑦 = 𝛼 + 𝛽 ⋅ log 𝐷𝑜𝑠𝑒 + 𝜀
𝑦 = 𝑒 𝛼+𝛽⋅log
𝐷𝑜𝑠𝑒 +𝜀
= 𝛼 ′ ⋅ 𝐷𝑜𝑠𝑒𝛽 ⋅ 𝑒 𝜀
When  ~ N(0,σ2), e has median of 1
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
17
Quantifying The Impact Of Dose
Variability
• For an absolutely consistent dose, we see
variability in the exposure (Cmax, AUC) due to
within subject and between subject variability.
Denote this total variability by σ2.
• When the dose varies, we need to determine the
degree to which the total variability changes
assuming a linear relationship between the
actual dose and the resulting exposure.
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
18
Simulated Example
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Company Confidential © 2013 Eli Lilly and Company
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Comparing The Effect Of Dose
Variability
If response is
proportional to dose
with proportionality
constant k, and the
exposure variance
for a fixed dose is
𝜎𝐬 𝟐 , and the
variance of dose is
𝜎𝐝 𝟐 , then the total
variability is
𝟐
𝒌𝟐 ⋅ 𝜎𝐝 + 𝜎𝐬 𝟐 .
For this example, 𝒌 = 𝟏, 𝝈𝒔 = 𝟏, and 𝝈𝒅 = 𝟑 so the
𝟐
total variance, 𝝈𝑻 , i𝐬 𝟏𝟎. Hence, 𝝈𝑻 = 𝟑.16.
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
20
(Most) Complicated Model
Previous analysis of blood plasma concentration
studies treated doses as if they were exactly
nominal.
If true doses varied
(like blue dots) then
the apparent variability
between and within
patients would be
overstated.
How to correct?
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
21
(Most) Complicated Model
Log(Exposure(Subject i,Dose d)) =
α + β*Log(Dose d * 𝜹id,1) + i + id,2 where
id,1 represents dose inaccuracy,
i represents a random subject effect,
and id,2 represents unexplained error
𝜹id,1 ~ N(1,σd2)
How to assign dose inaccuracies to the
individual results?
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
22
Single Dose Per Subject Approach
Exposure
For each nominal
dose, generate dose
errors according to
expected order
statistics from normal
distribution. Adjust
doses according to
observed exposure.
Dose (mg)
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
23
Single Dose Per Subject Approach
Nominal Dose Adjustment
Exposure
Order
Percentile
σd = 1%
σd = 2%
σd = 3%
19.85
1
0.083
98.6%
97.2%
95.9%
23.75
2
0.167
99.0%
98.1%
97.1%
23.84
3
0.250
99.3%
98.7%
98.0%
28.02
4
0.333
99.6%
99.1%
98.7%
29.49
5
0.417
99.8%
99.6%
99.4%
29.61
6
0.500
100.0%
100.0%
100.0%
34.91
7
0.583
100.2%
100.4%
100.6%
47.45
8
0.667
100.4%
100.9%
101.3%
48.77
9
0.750
100.7%
101.3%
102.0%
52.31
10
0.833
101.0%
101.9%
102.9%
57.09
11
0.917
101.4%
102.8%
104.1%
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
24
(Most) Complicated Regression
Results: Single Dose Per Subject
AUC
Study
2
5
Cmax
σd
Slope
0%
1.084
Total SD
as % of
σ
Base
0.4916
63%
0.883
Total SD
as % of
σ
Base
0.3164
37%
1%
1.085
0.4847
62%
0.883
0.3153
37%
2%
1.086
0.4778
61%
0.883
0.3143
37%
4%
1.089
0.4643
59%
0.883
0.3127
37%
8%
1.093
0.4377
55%
0.883
0.3107
36%
0%
0.966
0.3935
48%
1.895
0.6615
94%
1%
0.980
0.3856
47%
1.913
0.6496
91%
2%
0.993
0.3777
46%
1.930
0.6377
89%
4%
1.016
0.3615
44%
1.959
0.6141
85%
8%
1.056
0.3286
39%
2.003
0.5684
77%
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
Slope
25
(Most) Complicated Regression:
Multiple Doses Per Subject
Perform two regressions:
• Fit regression model (no dose uncertainty):
Log(Exposure(Subject i,Dose d)) =
α + β*Log(Dose d) + i + id
• Treat as one subject all individuals who only had
single dose
• Use residuals for each dose after fitting random
subject effect to assign dose uncertainty as
before
• Fit second regression model (with dose
uncertainty)
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
26
(Most) Complicated Regression
Results: AUC
Study
σd
1
1
1
3
3
3
4
4
4
0%
4%
8%
0%
4%
8%
0%
4%
8%
MBSW, May 2013 – Tim Kramer
Between Within
Between Within
Total SD
Prop.
Subjects Subjects
Subjects Subjects
as % of
Constant
as % of as % of
SD
SD
Base
Base
Base
0.890
0.892
0.894
1.065
1.064
1.061
1.046
1.063
1.076
0.45
0.45
0.44
0.36
0.35
0.34
0.26
0.27
0.27
0.25
0.22
0.19
0.09
0.08
0.08
0.19
0.16
0.14
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57%
57%
56%
43%
42%
40%
30%
31%
31%
28%
24%
21%
10%
8%
9%
21%
18%
15%
68%
65%
62%
45%
43%
41%
39%
37%
35%
27
(Most) Complicated Regression
Results: Cmax
Study
σd
1
1
1
3
3
3
4
4
4
0%
4%
8%
0%
4%
8%
0%
4%
8%
MBSW, May 2013 – Tim Kramer
Between Within
Between Within
Total SD
Prop.
Subjects Subjects
Subjects Subjects
as % of
Constant
as % of as % of
SD
SD
Base
Base
Base
0.751
0.753
0.754
1.116
1.118
1.118
1.433
1.442
1.445
0.29
0.29
0.29
0.25
0.25
0.24
0.26
0.27
0.28
0.34
0.32
0.30
0.16
0.14
0.12
0.25
0.20
0.15
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34%
34%
34%
29%
28%
27%
29%
31%
32%
41%
38%
35%
17%
15%
13%
28%
22%
16%
57%
55%
52%
35%
33%
31%
43%
40%
37%
28
Analysis Summary
Exposure
Measure
Slope
Within
Subject as
% of Base
Between
Subjects as
% of Base
Total
AUC
0.9 to 1.1
10 to 30%
30 to 50%
35 to 70%
Cmax
0.7 to 2.0
15 to 40%
25 to 35%
35 to 95%
Exposure
Measure
Slope
Within
Subjects
SD
Between
Subjects
SD
Total SD
AUC
0.9 to 1.1
0.10 to 0.25 0.25 to 0.45 0.30 to 0.50
Cmax
0.7 to 2.0
0.15 to 0.35 0.25 to 0.30 0.30 to 0.65
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
29
Impact of Dose Variability: Patient
Population, Single Dose
• Increase to standard deviation seen by patient
population:
(𝑘 2
2
2
2
2
2
⋅ 𝜎𝑑 + 𝜎𝑤 + σb )/(𝜎𝑤 + 𝜎b )
Case
σd
Slope
σw
σb
Ratio
“Typical”
2%
1
0.15
0.30
100.2%
#2
3%
1
0.15
0.30
100.4%
#3
4%
1
0.15
0.30
100.7%
#4
4%
2.0
0.15
0.30
102.8%
#5
4%
2.0
0.10
0.30
103.2%
“Worst”
4%
2.0
0.10
0.25
104.3%
• Impact of replicated dose would be less
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
30
Impact of Dose Variability: Individual
Patient, Single Dose
• Increase to standard deviation seen by
individual patient:
(𝑘 2
2
⋅ 𝜎𝑑 + 𝜎𝑤 2 )/𝜎𝑤 2
Case
σd
Slope
σw
Ratio
“Typical”
2%
1
0.15
101%
#2
3%
1
0.15
102%
#3
4%
1
0.15
103%
#4
4%
2.0
0.15
113%
“Worst”
4%
2.0
0.10
128%
• Impact of replicated dose would be less
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
31
So What Have We Learned?
• On a population basis, very little impact from
dose variability…worst case is on the order of
4% increase in total standard deviation* by
patient population.
• On an individual basis, worst case could be as
high as a 28% increase in standard deviation but
the likely increase is a few percentage points
* Increase in standard deviation relative to the one that
would be observed for absolutely correct doses
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
32
Other Motivations for Dose
Uniformity Specification?
• From variability perspective, σd=4% seems
adequate…little impact on overall patient
variability
• Readily meets compendial expectations of being
between 75% and 125% (6σ capable)
efficacy
MBSW, May 2013 – Tim Kramer
safety
Company Confidential © 2013 Eli Lilly and Company
33
Other Motivations for Dose
Uniformity Specification?
• Potential signal for process upset
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
34
Conclusions
• 4% dose standard deviation will likely have
minimal impact on variability in exposure
experienced by patient
• Efficacy and safety ensured by 4% dose
standard deviation…±6σ is contained within
(75%,125%)
• No strong patient driver for tighter specifications
for content uniformity
• Monitoring content uniformity may provide signal
of process upset
MBSW, May 2013 – Tim Kramer
Company Confidential © 2013 Eli Lilly and Company
35