Transcript DISEASE PROGRESSION MODELS

Disease Progress Models
Diane R Mould
Projections Research Inc
Phoenixville PA
Clinical Pharmacology
=
Disease Progress + Drug
Action*
*It also follows that
“Drug Action” = Drug Effect + Placebo
Effect
PKPD Models
• Pharmacokinetic (dose, concentration,
time)
– drug disposition in individuals & populations
– disease state effects (renal & hepatic
dysfunction)
– intervention effects (hemodialysis)
– concurrent medication effects
– pharmacogenetic influences
• Pharmacodynamic (dose or concentration,
effect, time)
– physiologic & biomarkers
– surrogate endpoints
– clinical effects and endpoints
Disease Progression Model
• Quantitative model that accounts
for the time course of disease
status, S(t):
– “Symptoms” - measures of how a patient feels
or functions (“clinical endpoints”)
– “Signs” - physiological or biological
measurements of disease activity
(“biomarkers”)
• “Surrogate Endpoints” (validated markers
predictive of, or associated with Clinical Outcome)
• “Outcomes” (measures of global disease status, such
as pre-defined progression or death)
Motivations for
Disease Progression Models
• Visualization of the time course of disease in
treated and untreated conditions
• Evaluation of various disease interventions
• Simulation of possible future courses of disease
• Simulation of clinical trials
Model Building Process
• Talk to a Disease Specialist
• Draw pictures of time course of disease
• Translate into disease progress model
• Explain the models/parameters to the
Specialist
• Ask Disease Specialist for advice on
factors influencing parameters
• Translate into models with appropriate
parameters and covariates
Example Construction of a
Disease Model
Solid
Organ
Transplant
Administer Drug or Placebo
Cadaveric Donor?
Matched or Unmatched?
First Transplant?
Rejection?
Administer
Drug or Placebo
Up-regulation
Of CD25+
T Cells
Measure CD25+ T Cells
Immune
Inflammatory
Response
Measure IL6, TNFalpha
Cell Death
Components of a Disease
Progression Model
• Baseline Disease State, So
• Natural History
• Placebo Response
• Active Treatment Response
S(t) = Natural History + Placebo +
Active
Linear (Natural History) Disease
Progression Model
(adapted from Holford 1999)
105
S (t )  S 0    t
status
100
95
90
85
80
0
13
26
39
52
65
time
78
91
104 117
Linear Disease Progression Model with
Temporary (“Offset”) Placebo or Active
Drug Effect
(adapted from Holford 1997 & 1999)
S (t )  S 0  EOFF (C e, A )    t
115
E(t) =  ·Ce,A(t)
or E(t) = Emax ·Ce,A(t) / (EC50 + Ce,A(t))
110
105
status
100
Temporary Improvement
95
90
Natural History
85
80
0
13
26
39
52
65
time
78
91
104
117
Handling Pharmacokinetic
Data for Disease Progress
Models
• Use actual measured concentrations
– This is easy to do
• Use a “Link” model to create a lag
between observed concentrations and
observed effect
– This is more “real” as the time course
for change in disease status is usually
not the same as the time course of the
drug
Prednisone Treatment
of Muscular Dystrophy
Griggs et al. Arch Neurol (1991); 48: 383-388
AZT Treatment of HIV
Sale et al, Clin Pharm Ther (1993) 54:556-566
Tacrine Treatment of
Alzheimer’s Disease
Holford & Peace, Proc Natl Acad Sci 89 (1992):11466-11470
Tacrine Treatment of
Alzheimer's Disease
• Baseline Disease State: So
• Natural History: So +  ·t
• Placebo Response:  p·Ce,p(t)
• Active Treatment Response: a ·C
e,A(t)
Holford & Peace, Proc Natl Acad Sci 89 (1992):11466-11470
Linear Disease Progression Model with Disease
Modifying (“Slope”) Active Drug Effect
adapted from Holford 1999
S (t )  S 0  [ E SLOPE (C e, A )   ]  t
105
100
status
Modified Disease Slopes
95
90
85
Natural History
80
0
13
26
39
52
65
time
78
91
104
117
Alternative Drug Effect Mechanisms Superimposed on a Linear Natural
History Disease Progression Model
adapted from Holford 1999
115
110
Symptomatic (Offset) Improvement
Modified Disease Progress Slopes
status
105
100
95
90
85
Natural History
80
0
13
26
39
52
65
time
78
91
104 117
Asymptotic Progress Model
• Zero Asymptote (S0, Tprog)
– Spontaneous recovery
S (t )  S 0  e


 ln 2 / ETP ( Ce , A )  T p ro g  t
• Non-Zero Asymptote (S0, Sss, Tprog)
– Progression to “burned out” state (Sss)
S (t )  S 0  e
 ln 2 / T prog  t

 S SS  1  e
ln 2 / T prog  t

Dealing with Zero and NonZero Asymptote Models
• Both Models can be altered to include
– Offset Pattern
S (t )  E
OFF
(C )  S  e
e, A
0
 ln 2 / T
prog
t

 S  1 e
SS
ln 2 / T
prog
 t
– Slope Pattern
S (t )  S 0  e
 ln 2 / T prog  t

  EOFF (Ce, A )  S SS   1  e
ln 2 / T prog  t
Here the slope pattern is on the “burned out
state”
– Both Offset and Slope Patterns


120
Zero Asymptote Model
100
Symptomatic
Protective
Both
Natural History
status
80
60
40
20
0
0
13
26
39
52
time
65
S (t )  EOFF (Ce, A )  S0  e
78
91
104
 ln(2 ) /( ETP ( Ce , A ) TP0 )t
Non-Zero Asymptote Model
40
35
30
Status
25
20
Natural History
Protective Sss
Protective TP
Symptomatic
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
Time
S (t )  EOFF (Ce, A )  S0  e
 ln(2 ) /( ETP ( Ce , A ) TP0 )t

 ( ESS (Ce, A )  S ss,0 )  1  e
 ln(2 ) /( ETP ( Ce , A ) TP0 )t

PSG DATATOP Cohort
ID
Total
1
2
3
50
20
4
6
-0
-0
2
4
6
2
4
6
-0
2
4
YEAR
YEAR
YEAR
YEAR
4
5
6
7
60
30
UPDRS
UPDRS
40
40
20
20
-0
4
YEAR
6
20
2
4
YEAR
6
40
20
-0
-0
6
60
10
-0
2
20
-0
-0
60
-0
30
10
UPDRS
2
40
20
-0
-0
UPDRS
40
UPDRS
UPDRS
UPDRS
50
-0
UPDRS
40
60
100
-0
-0
2
4
YEAR
6
-0
2
4
YEAR
6
Physiological
Models of Disease Progress
Ksyn0
Baseline
Status
Ksyn
Status
Kloss0
Kloss
Either of these can change with time
to produce disease progression
Physiological Models of
Disease Progress
dS
 K syn  k loss  PDI  S
dt
Disease is caused by build up or
loss of a particular endogenous
substance
ln(2 )
t 


Kloss  K loss0  1  ( Maxprog  1)  1  e t 50loss  



PDI  1 
Ce , A
C 50  Ce, A
Drug action can be described using delay
function such as an effect compartment
Disease Progression Due to
Decreased Synthesis
150
100
Untreated
Inhibit Loss
Stimulate Synthesis
50
0
0
200
400
Time
600
800
1000
Disease Progression Due to
Increased Loss
150
100
Untreated
Inhibit Loss
Stimulate Synthesis
50
0
0
200
400
600
Time
800
1000
Disease Progress Models
• Alzheimer’s Disease
– Linear: Drug effect symptomatic
• Diabetic Neuropathy
– Linear: Drug effect both?
• Parkinson’s Disease
– Asymptotic: Drug effect both?
• Osteoporosis
– Inhibition of Bone Loss (estrogen)
Bone Mineral Density Change with
Placebo and 3 doses of Raloxifene
0.97
0.96
BMD
0.95
Status
30 mg/d
60 mg/d
150 mg/d
0.94
0.93
0.92
0
2
4
6
Years
8
10
Models Describing Growth
dR
 k growth  R  k death  R  C e, A
dt
Kdeath
Response
Stimulatory
Kgrowth
Ce,A
First Order Kinetics for Input!
Drug Stimulates Loss of Response (R )
Gompertz Growth Function
Models



E max  C e, A 
dRs
  K SO   Rs
 K RS  Rr    Rs   max  Rs    K SR  1 

dt


 EC50  C e, A 
dRr
 K SR  Rs  K RS  Rr
dt
Describes the Formation of Two Responses: Sensitive (Rs) and Resistant (Rr)
Defines a Maximal Response
Drug Effect is Delayed via Link Model and Limited via Emax Model
Growth Curves for 3 Treatments Untreated, Low and High Dose
600
500
Cell Count
400
Responsive Cell Population - No drug
300
Responsive Cell Population - Low Dose
Responsive Cell Population - High Dose
200
Regrowth!
100
0
0
2
4
6
8
10
Tim e (days)
12
14
16
18
20
Using Survival Functions to
Describe Disease Progress
• Empirical means of evaluating the
relationship between the drug effect
and the time course of disease
progress
• Links the pharmacodynamics to
measurement of outcome
Survival Function
• S(t) = P(T > t)
• Monotone, Decreasing Function
• Survival is 1 at Time=0 and 0 as Time
Approaches Infinity.
• The Rate of Decline Varies According to
Risk of Experiencing an Event
• Survival is Defined as
S (t )  exp( -H(t ))
Hazard Functions
• Hazard Functions Define the Rate of
Occurrence of An Event
– Instantaneous Progression
– PKPD Model Acts on Hazard Function
• Cumulative Hazard is the Integral of the
Hazard Over a Pre-Defined Period of Time
– Describes the Risk
– Translates Pharmacodynamic Response into a
Useful Measure of Outcome
• Assessment of Likely Benefit or Adverse Event
• Comparison With Existing Therapy
Hazard Functions
• Define “T” as Time To Specified Event
(Fever, Infection, Sepsis following
chemotherapy)
– T is Continuous (i.e. time)
– T is Characterized by:
• Hazard: Rate of Occurrence of Event
• Cumulative Hazard or Risk
• Survival: Probability of Event NOT Occurring
Before Time = t
Hazard Functions
• Hazard is Assumed to be a Continuous
Function
– Can be Function of Biomarkers (e.g.
Neutrophil Count)
• Hazard Functions can be Adapted for Any
Clinical Endpoints Evaluated at Fixed Time
Points (e.g. During Chemotherapy Cycle)
• The Hazard Function is Integrated Over
Time to Yield Cumulative Probability of
Experiencing an Event by a Specified Time
(Risk).
Using Hazard Functions in
PK/PD Models
• If Hazard Function is Defined as a
Constant Rate “K” Such that
h(t )  k
• Then the Cumulative Hazard is
t
H (t )   Kdt  Kt
0
Kt   ln S (t )
• Survival is
S (t )  exp Kt 
Hazard, Cumulative Hazard
and Survival
1
Hazard
Cumulative Hazard
8
0.8
Survival
0.6
4
0.4
2
0.2
Survival
6
Hazard
• In This Example
Hazard Remains
Constant
• Cumulative
Hazard (Risk)
Increases With
Time
• Surviving
Fraction Drops
10
0
0
0
4
8
12
Time (h)
16
20
24
Comparing Hematopoietic Factors Using
Better
Hazard Functions
12
12
11
Survival
0.9
10
10
0.8
0.8
0.7
88
66
0.5
0.4
0.4
44
0.3
0.2
0.2
22
0.1
00
00
00
24
24
48
48
72
72
96 120
120 144
144 168
168 192
192 216
216 240
240 264
264 288
288 312
312 336
336 360
360 384
384 408
408 432
432 456
456 480
480 504
504 528
528 552
552
96
Time (h)
WBC
Survival
Hazard
Cumulative Hazard
Probability
WBC
0.6
0.6
Summary
• Accounting for Disease Progress is Important
For the Analysis of Drug Effects
–
–
–
–
–
Better Able to Discern True Effect
Improves Reliability of Simulation Work
Developing New Drug Candidates
Visualize the Drug Use Better
Convert Data into Understanding!
• Issues Associated With Building Disease
Progress Models
– Lack of Available Data for Untreated Patients
– Time Required to Collect Data
– Variability Inherent in Data May Require Large
Numbers of Subjects to Determine Parameters
Accurately