Disease Progress Models

Diane R Mould

Projections Research IncPhoenixville 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

SolidOrgan

TransplantRejection?

Administer Drug or PlaceboCadaveric Donor?Matched or Unmatched?First Transplant?

Up-regulationOf CD25+

T Cells

Measure CD25+ T Cells

Immune Inflammatory

Response

Measure IL6, TNFalpha

Cell Death

Administer Drug or Placebo

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)

tStS 0)(

80

85

90

95

100

105

0 13 26 39 52 65 78 91 104 117time

stat

us

Linear Disease Progression Model with Temporary (“Offset”)

Placebo or Active Drug Effect (adapted from Holford 1997 & 1999)

E(t) = ·Ce,A(t) or E(t) = Emax ·Ce,A(t) / (EC50 +

Ce,A(t))

80

85

90

95

100

105

110

115

0 13 26 39 52 65 78 91 104 117time

sta

tus

Natural History

Temporary Improvement

tCEStSAeOFF

)()(,0

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

80

85

90

95

100

105

0 13 26 39 52 65 78 91 104 117time

sta

tus

Natural History

Modified Disease Slopes

Linear Disease Progression Model with Disease Modifying (“Slope”) Active Drug

Effect adapted from Holford 1999

tCEStSAeSLOPE

])([)(,0

Alternative Drug Effect Mechanisms Superimposed on a Linear Natural History Disease Progression Model

adapted from Holford 1999

Natural History

80

85

90

95

100

105

110

115

0 13 26 39 52 65 78 91 104 117

time

stat

us

Symptomatic (Offset) Improvement

Modified Disease Progress Slopes

Asymptotic Progress Model

• Non-Zero Asymptote (S0, Sss, Tprog)

– Progression to “burned out” state (Sss)

• Zero Asymptote (S0, Tprog)– Spontaneous recovery

tTCE progAeTPeStS )(/2ln0

,)(

tT

SS

tTprogprog eSeStS

/2ln/2ln

01)(

Dealing with Zero and Non-Zero Asymptote

Models• Both Models can be altered to include– Offset Pattern

– Slope Pattern

Here the slope pattern is on the “burned out state”

– Both Offset and Slope Patterns

tT

SS

tT

AeOFF

progprog eSeSCEtS

/2ln/2ln

0,1)()(

tT

SSAeOFF

tT progprog eSCEeStS /2ln

,

/2ln0 1)()(

Zero Asymptote Model

0

20

40

60

80

100

120

0 13 26 39 52 65 78 91 104time

stat

us

SymptomaticProtectiveBothNatural History

tTPCEAeOFF

AeTPeSCEtS ))(/()2ln(0,

0,)()(

Non-Zero Asymptote Model

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8 9 10Time

Sta

tus

Natural HistoryProtective SssProtective TPSymptomatic

tTPCEssAeSS

tTPCEAeOFF

AeTPAeTP eSCEeSCEtS ))(/()2ln(0,,

))(/()2ln(0,

0,0, 1))(()()(

PSG DATATOP Cohort

-0 2 4 6

YEAR

-0

50

100

UPD

RS

-0 2 4 6

YEAR

-0

20

40

UPD

RS

-0 2 4 6

YEAR

-0

20

40

60

UPD

RS

-0 2 4 6

YEAR

-0

10

20

30

40

50

UPD

RS

-0 2 4 6

YEAR

-0

20

40

60

UPD

RS

-0 2 4 6

YEAR

-0

20

40

60

UPD

RS

-0 2 4 6

YEAR

-0

10

20

30

UPD

RS

-0 2 4 6

YEAR

-0

20

40

60

UPD

RS

TotalTotal 11 22 33

44 55 66 77

ID

PhysiologicalModels of Disease Progress

Either of these can change with timeto produce disease progression

BaselineStatus

Kloss0Ksyn0

KsynStatus

Kloss

Physiological Models of Disease Progress

tlosstloss eMaxprogKKloss 50

)2ln(

0 1)1(1

Ae

Ae

CC

CPDI

,

,

501

Disease is caused by build up or loss of a particular endogenoussubstance

Drug action can be described using delay function such as an effect compartment

SPDIkKdt

dSlosssyn

Disease Progression Due to Decreased Synthesis

0 200 400 600 800 1000

0

50

100

150

UntreatedInhibit LossStimulate Synthesis

Time

Sta

tus

Disease Progression Due to Increased Loss

0 200 400 600 800 1000

0

50

100

150

UntreatedInhibit LossStimulate SynthesisS

tatu

s

Time

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 2 4 6 8 10

Years

0.92

0.93

0.94

0.95

0.96

0.97

BM

D Status30 mg/d60 mg/d150 mg/d

Models Describing Growth

Aedeathgrowth CRkRkdt

dR,

Kgrowth

Kdeath

First Order Kinetics for Input!Drug Stimulates Loss of Response (R )

Ce,A

StimulatoryResponse

Gompertz Growth Function Models

RrKRsKdt

dRr

RsKCEC

CEKRsRsRrK

dt

dRs

RSSR

SOAe

AeSRRS

,

,max 50

max1

Describes the Formation of Two Responses: Sensitive (Rs) and Resistant (Rr) Defines a Maximal ResponseDrug Effect is Delayed via Link Model and Limited via Emax Model

0

100

200

300

400

500

600

0 2 4 6 8 10 12 14 16 18 20

Time (days)

Ce

ll C

ou

nt Responsive Cell Population - No drug

Responsive Cell Population - Low Dose

Responsive Cell Population - High Dose

Growth Curves for 3 Treatments - Untreated, Low and High Dose

Regrowth!

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

))(exp()( t-HtS

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

• Then the Cumulative Hazard is

• Survival is

kth )(

KttS exp)(

)(ln

)(0

tSKt

KtKdttHt

Hazard, Cumulative Hazard and Survival

• In This Example Hazard Remains Constant

• Cumulative Hazard (Risk) Increases With Time

• Surviving Fraction Drops

0

2

4

6

8

10

0 4 8 12 16 20 24

Time (h)

Haz

ard

0

0.2

0.4

0.6

0.8

1

Su

rviv

al

Hazard

Cumulative Hazard

Survival

0

2

4

6

8

10

12

0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552

Time (h)

WB

C

0

0.2

0.4

0.6

0.8

1

Pro

ba

bil

ity

WBC Survival Hazard Cumulative Hazard

0

2

4

6

8

10

12

0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Comparing Hematopoietic Factors Using Hazard Functions Better

Survival

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