1

Comparing Cancer Risks betweenComparing Cancer Risks between

Radiation and Dioxin ExposureRadiation and Dioxin Exposure

Based on Two-Stage ModelBased on Two-Stage Model

Tsuyoshi Nakamura

Faculty of Environmental Studies,

Nagasaki University

David G. Hoel

Dept. of Biometry and Epidemiology,Medical University of South Carolina

1. Two-Stage Model2. Historical Aspects3. Estimation Method4. Radiation by JANUS5. Dioxin by KocibaSummary Conclusion

2

NormalCellsIntermediate

CellsMalignantCells

É 1

É 2

ɬɿ

Three States of CellsNormal, Intermediate and Malignant

Four Parameters for Rates

1 : First Mutation Rate for NI

: Clonal Expansion Rate for I

: Death Rate for I

2 : Second Mutation Rate for I

M

N I M

Two-Stage Model

3

HistoryMathematical tool based on Molecula

r biology to study Mechanistic processes in Cancer development

(Moolgavkar, Venzon, Knudson, 70’s)

Special FeatureExplicit modeling of

Clonal expansion, Differentiation and Mutation of I-cells

as a Continuous Stochastic Process

Cancer Incidence Data(time, type, covariate)

(t, 1, x): endpoint

(t, 0, x): censored

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UnidentifiabilityAll parameters are not identifiable. Re

parameterization or Assumptionis necessary.

Non-ConvergenceMLE of the identifiable parameters

are still often hardly obtainable,because of the peculiar shape of the li

kelihood surface (Portier et at. 1997).

ProblemsProblems

Non-Standard Algorithm

Lack of Confidence in Results

Lack of Comparison among Studies

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Survivor function S(t) Probability of No Malignant Cell at t,

is obtained by solving a series of differential equations,

derived from Stochastic processes on Probability Generating Function

(Moolgavkar et al 1990; Kopp et al 1994; Portier et al 1996;)

Stochastic processes

M(t)=(x(t),y(t),z(t)) denote the number of

N-, I- and M-cells at t, respectively.

M(t): Continuous Markov Birth-Death process

S(t)= i,jProb{M(t)=(i,j,0)}

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Differential Equations

It follows that (Portier et al 1996)

dG(t)/dt= 1G(t)H(t)-1G (t)

dH(t)/dt= H(t)2+-(++2)H(t)

G(0)=1, H(0)=1

Probability Generating Function

P(i,j,k|t)=Prob{M(t)=(i,j,k) | M(0)=(1,0,0) }

G(u,v,w|t)= i,j,k P(i,j,k|t)uivjwk

and

Q(i,j,k|t)=Prob{M(t)=(i,j,k) | M(0)=(0,1,0) }

H(u,v,w|t)= i,j,k Q(i,j,k|t)uivjwk

S(t)=i,jP(i,j,0|t)=G(1,1,0|t)

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Survivor Function S(t) X0 = Number of N-cells, Large and Constant = NI Rate per Cell per unit Time==> 1=X0

S(t)=exp{-(t)},

(t){t(R++log[{R-+(R+)e-Rt}/2R]} is Cumulative Hazard with new paramete

rs =1/, =-2 and R2=(--2)2+42 Original likelihood

: Net Proliferation Rate 

=12=(R2-2)/4:Overall Mutation Rate

l(,,) based on (t) is termed

Original likelihood.

Non-convergence is frequent !

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1, and

2 are employed to emphasize

these parameters are valid only when =0

 

l(1,,

2 ) based on (t|=0) is termed

Conditional likelihood.

Looks Better Shape!

Conditional likelihood

*1

β* *t+ log μ*2 +β*exp{−(β* +μ*

2)t}β*+μ*

2

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

(t|=0)= [ ]

Put =0 then

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TransformationTransformationConditional likelihood converges better!

Biological interpretation of parameters is ? It ignores the death of the I-cells.

Biological parameters estimated by

1, and 2 are

=1, =-

2 and =1

2

(Nakamura and Hoel 2002)

Thus, MLE of , and are obtained

from Conditional Likelihood !

Practically=, since 2 is small

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Conditional vs Original

Comparison on Experimental Data

JANUS data for Radiation Risk study

On and Neutron in Mice

Argonne National Laboratory (1953-1970 )

Reliable Pathological information

Kociba data for Chronic Toxicity study

on TCDD in Rats

Dow Chemical (1978)

Reliable Pathological information

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Illustration of Two-Stage Model

Cited from Moolgavkar(1999)

Statitics for the Environment 4, Wiley

1X0

2

12

ControlControl mice mice 3707 with 1894 Cancer

Original likelihood: l = -13692.7, ||U||<0.001

　　 Parameter 　 Estimate 　 SE　log 0.05632 0.16436log-4.8185 0.04563

log -17.660 0.2086Initial Trial Values are assigned as

=/, = and =

1

Conditional Likelihood: l =-13692.7, ||U||< 0.001

　　 Parameter Estimate 　 SE 　 log

1 -4.7618 0.10524 log-4.8182 0.03767log

2 -12.8980.13539 log -17.660 0.1760 log 0.05632 0.1244

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Regression Modellog=a+bDose

(Contol + ) 7402 mice with 4133 Cancer

ConditionalLikelihood: l=-29446.65, ||U||=0.002

Const.a (SE) Slope b (SE)log

-4.931 (0.0817) 0.00717 0.00115)log -4.851 (0.0278) -0.000345 (0.000071) log

2 -12.43 (0.1014) -0.002934 (0.00112

6)log-17.37 (0.1181) 0.00424 (0.000241)--------------------------------------------------------------------------------------------------------------------------------------------------

Original likelihood: l=-29446.69, ||U||=0.2542

Const.a (SE) Slope b (SE)log-0.0797 (0.1318) 0.00749 (0.00131)

log -4.852 (0.0373) 0.000345 (0.000077)log-17.37 (0.1536) 0.00424 (0.000266)

All Estimates are of p<0.01

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=12=X02

2) X0 is Constant not affected by Exposure

3) Effect of exposure on and that on 2

are the same ( Moolgavkar et al ,1999),

4) log=a+bDose

==> Dose effect on and that on 2 is b/2

Effect of Exposure on

Mutation and Promotion

Dose Effect on Mutation Rate

and Net proliferation Rate

may be obtained from

Conditional likelihood

without Additional Assumption!

15-8

-6

-4

-2

0

2

-8 -6 -4 -2 0 2

Log Cumulative HazardsLog Cumulative Hazards

Dose 0 : Subjects 3707 Cancer 1894

Two-Stage (H) vs K-M(V)

V

H

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Dose 86 : Subjects 1376 　 Cancer 960

Log Cumulative HazardsLog Cumulative HazardsTwo-Stage (H) vs K-M(V)

-8

-6

-4

-2

0

2

-8 -6 -4 -2 0 2

V

H

17

Dose 756 : 　 Subjects 396 　 Cancer 190

Log Cumulative HazardsLog Cumulative HazardsTwo-Stage (H) vs K-M(V)

-8

-6

-4

-2

0

2

-8 -6 -4 -2 0 2

V

H

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Original Likelihood: l =-207.585, ||U||=5.7489

Incomplete-convergent case Const. SE Slope SElog-0.4724 0.7395 non log -3.773 0.04589 0.0631 0.0200log-27.14 0.3368 non

Regression Coefficients for Dioxin Regression Coefficients for Dioxin 205 rats,31 cancer, log=a+blog(1+Dose)

Conditional Likelihood: l = -206.77,||U||=0.0004 Const. SE Slope SElog

-3.780 0.7075 non log -3.961 0.1062 0.0680 0.01497log

2 -20.82 1.371 non log-24.60 1.259 non

Original Likelihood: l = -207.012, ||U||= 0.0012 Const.a SE Slope SElog0.0865 0.8192 non

log -3.979 0.1083 0.0658 0.01466log-24.32 1.216 non

19-14

-12

-10

-8

-6

-4

-2

0

2

150 250 350 450 550 650 750

100

10

1

0

Log Cumulative Hazards

for Dioxin Doses

week

20-15

-10

-5

0

5

0 50 100 150 200

Log Cumulative Hazards

for Radiation Doses

756

400

197.6

86.31

43.15

0

week