Comparing Cancer Risks between Radiation and Dioxin Exposure Based on Two-Stage Model Tsuyoshi Nakamura Faculty of Envir

1. Comparing Cancer Risks between Radiation and Dioxin Exposure Based 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 Model 2. Historical Aspects 3. Estimation Method 4. Radiation by JANUS 5. Dioxin by Kociba Summary Conclusion

2. Two-Stage Model Three States of Cells Normal, Intermediate and Malignant N I M Four Parameters for Rates m1 : First Mutation Rate for NI b : Clonal Expansion Rate for I d : Death Rate for I m2 : Second Mutation Rate for IM

3. History Mathematical tool based on Molecular biology to study Mechanistic processes in Cancer development (Moolgavkar, Venzon, Knudson, 70’s) Special Feature Explicit 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

4. Problems Unidentifiability All parameters are not identifiable. Reparameterization or Assumption is necessary. Non-Convergence MLE of the identifiable parameters are still often hardly obtainable, because of the peculiar shape of the likelihood surface (Portier et at. 1997). Non-Standard Algorithm Lack of Confidence in Results Lack of Comparison among Studies

5. 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)= Si,jProb{M(t)=(i,j,0)}

6. Probability Generating Function P(i,j,k|t)=Prob{M(t)=(i,j,k) | M(0)=(1,0,0) } G(u,v,w|t)= Si,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)= Si,j,k Q(i,j,k|t)uivjwk S(t)=Si,jP(i,j,0|t)=G(1,1,0|t) Differential Equations It follows that (Portier et al 1996) dG(t)/dt= m1G(t)H(t)-m1G (t) dH(t)/dt= bH(t)2+d-(b+d+m2)H(t) G(0)=1, H(0)=1

7. Survivor Function S(t) X0 = Number of N-cells, Large and Constant u = NI Rate per Cell per unit Time ==> m1=uX0 S(t)=exp{-L(t)}, L(t)=h{t(R+y)/2+log[{R-y+(R+y)e-Rt}/2R]} L is Cumulative Hazard with new parameters h=m1/b, y=b-d-m2 and R2=(b-d-m2)2+4bm2 Original likelihood y: Net Proliferation Rate r=m1m2=h(R2-y2)/4:Overall Mutation Rate l(h,y,r) based on L(t) is termed Original likelihood. Non-convergence is frequent !

8. L(t|d=0)= [ ] Conditional likelihood Put d=0 then m*1, b* and m*2 are employedto emphasize these parameters are valid only when d=0 l(m*1,b*,m*2) based on L(t|d=0) is termed Conditional likelihood. Looks Better Shape!

9. Transformation Conditional likelihood converges better! Biological interpretation of parameters is ? It ignores the death of the I-cells. Biological parameters estimated by m*1, b* and m*2 are h=m*1/b*, y=b*-m*2and r=m*1m*2 (Nakamura and Hoel2002) Thus, MLE of h, y and r are obtained from Conditional Likelihood ! Practically y=b*, since m*2 is small

10. Comparison on Experimental Data Conditional vs Original JANUS data for Radiation Risk study On g 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

11. Illustration of Two-Stage Model b m2 m1=uX0 d Cited from Moolgavkar(1999) Statitics for the Environment4, Wiley

12. Control mice 3707 with 1894 Cancer Conditional Likelihood: l =-13692.7, ||U||< 0.001 Parameter Estimate SE log m*1 -4.7618 0.10524 log b*-4.8182 0.03767 log m*2 -12.898 0.13539 logr -17.660 0.1760 logh0.05632 0.1244 Original likelihood: l = -13692.7, ||U||<0.001 Parameter Estimate SE logh 0.05632 0.16436 logy -4.8185 0.04563 logr-17.660 0.2086 Initial Trial Values are assigned as h0=m*1/b*, y0=b* andr0=m*1m*2

13. Regression Model logq=a+bDose (Contol + g) 7402 mice with 4133 Cancer ConditionalLikelihood: l=-29446.65, ||U||=0.002 Const.a (SE) Slope b (SE) logm*1 -4.931 (0.0817) 0.00717 (0.00115) logb*-4.851 (0.0278)-0.000345 (0.000071) logm*2 -12.43 (0.1014) -0.002934 (0.001126) logr -17.37 (0.1181) 0.00424 (0.000241) -------------------------------------------------------------------------------------------------------------------------------------------------- Original likelihood: l=-29446.69, ||U||=0.2542 Const.a (SE) Slope b (SE) logh-0.0797 (0.1318) 0.00749 (0.00131) logy-4.852 (0.0373)0.000345 (0.000077) logr -17.37 (0.1536) 0.00424 (0.000266) All Estimates are of p<0.01

14. Effect of Exposure on Mutation and Promotion 1) r=m1m2=uX0m2 2) X0 is Constant not affected by Exposure 3) Effect of exposure on u and that on m2 are the same ( Moolgavkar et al ,1999), 4) logr=a+bDose ==> Dose effect on u and that on m2 is b/2 Dose Effect on Mutation Rate and Net proliferation Rate may be obtained from Conditional likelihood without Additional Assumption!

15. Log Cumulative Hazards Two-Stage (H) vs K-M(V) Dose 0 :Subjects 3707 Cancer 1894 V H

16. Log Cumulative Hazards Two-Stage (H) vs K-M(V) Dose 86 : Subjects 1376Cancer 960 V H

17. Log Cumulative Hazards Two-Stage (H) vs K-M(V) Dose 756 : Subjects 396Cancer 190 V H

18. Regression Coefficients for Dioxin 205 rats,31 cancer, logq=a+blog(1+Dose) Conditional Likelihood:l= -206.77,||U||=0.0004 Const. SE Slope SE logm*1 -3.780 0.7075non logb*-3.961 0.1062 0.0680 0.01497 logm*2-20.82 1.371 non logr -24.601.259 non Original Likelihood:l= -207.012, ||U||= 0.0012 Const.a SE Slope SE logh0.0865 0.8192non logy-3.979 0.10830.06580.01466 logr -24.32 1.216 non Original Likelihood:l=-207.585, ||U||=5.7489 Incomplete-convergent case Const. SE Slope SE logh-0.47240.7395 non logy-3.773 0.045890.06310.0200 logr -27.140.3368 non

19. 100 10 1 0 Log Cumulative Hazards for Dioxin Doses week

20. 756 400 197.6 86.31 43.15 0 Log Cumulative Hazards for Radiation Doses week