1. 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. 2 Three States of Cells Normal, Intermediate and Malignant Four Parameters for Rates  1 : First Mutation Rate for N  I  : Clonal Expansion Rate for I  : Death Rate for I  2 : Second Mutation Rate for I  M NIM Two-Stage Model

3. 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. 4 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). Problems Non-Standard Algorithm Lack of Confidence in Results Lack of Comparison among Studies

5. 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)=  i,j Prob{M(t)=(i,j,0)}

6. 6 Differential Equations It follows that (Portier et al 1996) dG(t)/dt=  1 G(t)H(t)-  1 G (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)u i v j w k 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)u i v j w k S(t)=  i,j P(i,j,0|t)=G(1,1,0|t)

7. 7 Survivor Function S(t) X 0 = Number of N-cells, Large and Constant  = N  I Rate per Cell per unit Time  ==>  1 =  X 0 S(t)=exp{-  (t)},  (t)  { t(R+  +log[{R-  +(R+  )e -Rt }/2R] }  is Cumulative Hazard with new parameters  =  1 / ,  =  -  2 and R 2 =(  -  -  2 ) 2 +4  2 Original likelihood  : Net Proliferation Rate  =  1  2 =  (R 2 -  2 )/4:Overall Mutation Rate l ( , ,  ) based on  (t) is termed Original likelihood. Non-convergence is frequent !

8. 8   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  (t|  =0)= [] Put  =0 then

9. 9 Transformation Conditional 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

10. 10 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

11. 11 Illustration of Two-Stage Model Cited from Moolgavkar(1999) Statitics for the Environment 4, Wiley   1  X 0 22 

12. 12 Control mice Control mice 3707 with 1894 Cancer Original likelihood : l = -13692.7, ||U||<0.001 Parameter Estimate SE log  0.056320.16436 log  -4.81850.04563 log  -17.6600.2086 Initial Trial Values are assigned as   =    /  ,   =   and   =   1    Conditional Likelihood : l =-13692.7, ||U||< 0.001 Parameter Estimate SE log   1 -4.76180.10524 log    -4.81820.03767  log   2 -12.8980.13539 log  -17.6600.1760 log  0.056320.1244

13. 13 Regression Model log  =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 (0.001126) 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

14. 14  =  1  2 =  X 0  2 2) X 0 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. 15 Log Cumulative Hazards Dose 0 : Subjects 3707 Cancer 1894 Two-Stage (H) vs K-M(V) V H

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

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

18. 18 Original Likelihood : l =-207.585, ||U||=5.7489 Incomplete-convergent case Const. SE Slope SE log   -0.47240.7395 non log  -3.7730.04589 0.0631 0.0200 log  -27.140.3368 non Regression Coefficients for Dioxin 205 rats,31 cancer, log  =a+blog(1+Dose) Conditional Likelihood : l = -206.77,||U||=0.0004 Const. SE Slope SE log     -3.7800.7075 non log   -3.9610.1062 0.0680 0.01497 log   2 -20.821.371 non log  -24.601.259 non Original Likelihood : l = -207.012, ||U||= 0.0012 Const.a SE Slope SE log   0.08650.8192 non log  -3.9790.1083 0.0658 0.01466 log  -24.321.216 non

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

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