BCRM
Príklad 1: Neuenschwander et al.
Dávky lieku
dose<-c(1,2.5,5,10,15,20,25,30,40,50,75,100,150,200,250)Kostra CRM, ktorá reprezentuje preddefinované pravdepodobnosti vedľajších účinkov:
p.tox0<-c(0.010,0.015,0.020,0.025,0.030,0.040,0.050,0.100,0.170,0.300,0.400,0.500,0.650,0.800,0.900)Cieľová pravdepodobnosť vedľajších účinkov
target.tox<-0.30Dáta z prvých 5 kohort, spolu 18 pacientov
data<-data.frame(patient=1:18,dose=rep(c(1:4,7),c(3,4,5,4,2)),tox=rep(0:1,c(16,2)))
data## patient dose tox
## 1 1 1 0
## 2 2 1 0
## 3 3 1 0
## 4 4 2 0
## 5 5 2 0
## 6 6 2 0
## 7 7 2 0
## 8 8 3 0
## 9 9 3 0
## 10 10 3 0
## 11 11 3 0
## 12 12 3 0
## 13 13 4 0
## 14 14 4 0
## 15 15 4 0
## 16 16 4 0
## 17 17 7 1
## 18 18 7 1Najskôr skúsime použiť mocninový model s 1 neznámym parametrom.
Apriórne rozdelenie pre \(\alpha\) je lognormálne.
Stredná hodnota je 0 a smerodajná odchýlka 1.34 (na logaritmickej škále).
Ďalšia odporúčaná dávka je vypočítaná pomocou aposteriórnej strednej hodnoty.
Power.LN.bcrm<-bcrm(stop=list(nmax=18),data=data,
p.tox0=p.tox0, dose=dose, ff="power", prior.alpha=list(3,0,1.34^2), target.tox=target.tox,
constrain=FALSE, #povolujeme zvysit/znizit dalvku o viac ako 1 hladinu
sdose.calculate="median",pointest="mean")##
## Stopping: Reached maximum sample sizeprint(Power.LN.bcrm)##
## Estimation method: exact
## Target toxicity level: 0.3
## Model: 1-parameter power
## Prior: Lognormal( Mean:0, Variance:1.7956)
##
## Standardised doses (skeleton):
## 1 2.5 5 10 15 20 25 30 40 50 75 100
## 0.010 0.015 0.020 0.025 0.030 0.040 0.050 0.100 0.170 0.300 0.400 0.500
## 150 200 250
## 0.650 0.800 0.900
##
## Unmodified (unconstrained) CRM used
##
## Posterior mean estimate of probability of toxicity used to select next dose
##
## Toxicities observed:
## Doses
## 1 2.5 5 10 15 20 25 30 40 50 75 100 150 200 250
## n 3 4 5 4 0 0 2 0 0 0 0 0 0 0 0
## Toxicities 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0
##
## Posterior estimates of toxicity:
## Doses
## 1 2.5 5 10 15 20 25 30 40
## Mean 0.0702 0.0866 0.1010 0.1130 0.1250 0.1460 0.1650 0.244 0.333
## SD 0.0558 0.0630 0.0686 0.0731 0.0769 0.0831 0.0879 0.102 0.109
## Median 0.0561 0.0723 0.0865 0.0995 0.1110 0.1330 0.1530 0.237 0.330
## Doses
## 50 75 100 150 200 250
## Mean 0.467 0.5580 0.641 0.757 0.8650 0.9330
## SD 0.108 0.0996 0.088 0.066 0.0398 0.0205
## Median 0.471 0.5640 0.648 0.764 0.8700 0.9360
## Doses
## Quantiles 1 2.5 5 10 15 20 25 30 40
## 2.5% 0.00493 0.00787 0.0110 0.0142 0.0175 0.0244 0.0316 0.0702 0.130
## 25% 0.02860 0.03910 0.0488 0.0579 0.0667 0.0833 0.0990 0.1690 0.255
## 50% 0.05610 0.07230 0.0865 0.0995 0.1110 0.1330 0.1530 0.2370 0.330
## 75% 0.09710 0.11900 0.1380 0.1540 0.1690 0.1960 0.2190 0.3120 0.408
## 97.5% 0.21300 0.24400 0.2690 0.2900 0.3080 0.3400 0.3660 0.4620 0.552
## Doses
## Quantiles 50 75 100 150 200 250
## 2.5% 0.249 0.347 0.450 0.608 0.773 0.886
## 25% 0.395 0.493 0.586 0.717 0.842 0.922
## 50% 0.471 0.564 0.648 0.764 0.870 0.936
## 75% 0.544 0.629 0.704 0.804 0.893 0.948
## 97.5% 0.668 0.735 0.792 0.865 0.928 0.965
##
## Next recommended dose: 40plot(Power.LN.bcrm) 
Interpretácia:
toxicities observed: z dát vidíme, že vedľajšie účinky mali dvaja pacienti, ktorým bola podaná dávka 25
metóda odporúča ako ďalšiu dávku 40mg
To isté s ohraničením, že nesmieme zvýšiť dávku o viac ako 1 hladinu
Power.LN.bcrm2<-bcrm(stop=list(nmax=18),data=data,
p.tox0=p.tox0,dose=dose,ff="power", prior.alpha=list(3,0,1.34^2),target.tox=target.tox,
constrain=TRUE,sdose.calculate="median",pointest="mean")##
## Stopping: Reached maximum sample sizeprint(Power.LN.bcrm2)##
## Estimation method: exact
## Target toxicity level: 0.3
## Model: 1-parameter power
## Prior: Lognormal( Mean:0, Variance:1.7956)
##
## Standardised doses (skeleton):
## 1 2.5 5 10 15 20 25 30 40 50 75 100
## 0.010 0.015 0.020 0.025 0.030 0.040 0.050 0.100 0.170 0.300 0.400 0.500
## 150 200 250
## 0.650 0.800 0.900
##
## Modified (constrained) CRM used, starting dose: 1
##
## Posterior mean estimate of probability of toxicity used to select next dose
##
## Toxicities observed:
## Doses
## 1 2.5 5 10 15 20 25 30 40 50 75 100 150 200 250
## n 3 4 5 4 0 0 2 0 0 0 0 0 0 0 0
## Toxicities 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0
##
## Posterior estimates of toxicity:
## Doses
## 1 2.5 5 10 15 20 25 30 40
## Mean 0.0702 0.0866 0.1010 0.1130 0.1250 0.1460 0.1650 0.244 0.333
## SD 0.0558 0.0630 0.0686 0.0731 0.0769 0.0831 0.0879 0.102 0.109
## Median 0.0561 0.0723 0.0865 0.0995 0.1110 0.1330 0.1530 0.237 0.330
## Doses
## 50 75 100 150 200 250
## Mean 0.467 0.5580 0.641 0.757 0.8650 0.9330
## SD 0.108 0.0996 0.088 0.066 0.0398 0.0205
## Median 0.471 0.5640 0.648 0.764 0.8700 0.9360
## Doses
## Quantiles 1 2.5 5 10 15 20 25 30 40
## 2.5% 0.00493 0.00787 0.0110 0.0142 0.0175 0.0244 0.0316 0.0702 0.130
## 25% 0.02860 0.03910 0.0488 0.0579 0.0667 0.0833 0.0990 0.1690 0.255
## 50% 0.05610 0.07230 0.0865 0.0995 0.1110 0.1330 0.1530 0.2370 0.330
## 75% 0.09710 0.11900 0.1380 0.1540 0.1690 0.1960 0.2190 0.3120 0.408
## 97.5% 0.21300 0.24400 0.2690 0.2900 0.3080 0.3400 0.3660 0.4620 0.552
## Doses
## Quantiles 50 75 100 150 200 250
## 2.5% 0.249 0.347 0.450 0.608 0.773 0.886
## 25% 0.395 0.493 0.586 0.717 0.842 0.922
## 50% 0.471 0.564 0.648 0.764 0.870 0.936
## 75% 0.544 0.629 0.704 0.804 0.893 0.948
## 97.5% 0.668 0.735 0.792 0.865 0.928 0.965
##
## Next recommended dose: 30plot(Power.LN.bcrm2)
Mocninový model nepopisuje dáta dostatočne: skúsime dvojparametrický logistický model.
Zároveň zmeníme aj kostru crm, aby sme lepšie zohľadnili vývoj vedľajších účinkov
p.tox1<-c(0.063, 0.125, 0.188, 0.250, 0.313, 0.375, 0.438, 0.500, 0.563, 0.625, 0.688, 0.751)
mu<-c(2.27,0.26)
Sigma<-rbind(c(1.98^2,-0.13),c(-0.13,0.40^2))
Logit.LN.bcrm2<-bcrm(stop=list(nmax=18),data=data,
p.tox0=p.tox1,dose=dose[1:12],ff="logit2",prior.alpha=list(4,mu,Sigma),target.tox=target.tox,
constrain=TRUE,sdose.calculate="median",pointest="mean")
print(Logit.LN.bcrm2)
plot(Logit.LN.bcrm2)Nasimulujeme 10 pokusov veľkosti 36 (veľkost kohorty 3).
Máme ohraničenie na zvyšovanie dávky, začíname na najnižšej dávke.
Skutočné pravdepodobnosti vedľajších účinkov su p.tox0
Power.LN.bcrm.sim<-bcrm(stop=list(nmax=36),p.tox0=p.tox0,dose=dose,ff="power"
,prior.alpha=list(3,0,1.34^2),target.tox=target.tox,constrain=TRUE
,sdose.calculate="median",pointest="mean",start=1,simulate=TRUE,nsims=10,truep=p.tox0)## Simulated trial: 10print(Power.LN.bcrm.sim)## Operating characteristics based on 10 simulations:
##
##
## Sample size 36
##
## Doses
## No dose 1 2.5 5 10 15
## Experimentation proportion NA 0.0833 0.0833 0.0833 0.0833 0.0833
## Recommendation proportion 0 0.0000 0.0000 0.0000 0.0000 0.0000
## Doses
## 20 25 30 40 50 75
## Experimentation proportion 0.0833 0.0833 0.0833 0.125 0.133 0.0667
## Recommendation proportion 0.0000 0.0000 0.0000 0.200 0.500 0.2000
## Doses
## 100 150 200 250
## Experimentation proportion 0.00833 0 0 0
## Recommendation proportion 0.10000 0 0 0
##
## Probability of DLT
## [0,0.2] (0.2,0.4] (0.4,0.6] (0.6,0.8] (0.8,1]
## Experimentation proportion 0.792 0.2 0.00833 0 0
## Recommendation proportion 0.200 0.7 0.10000 0 0plot(Power.LN.bcrm.sim) 
Porovnáme CRM s 3+3 návrhom (obmedzíme sa len na prvých 12 dávok)
Toto dlho trva, keďže sa generuju všetky možné priebehy 3+3 návrhu
Power.LN.bcrm.compare.sim<-bcrm(stop=list(nmax=36),p.tox0=p.tox0[1:12],dose=dose[1:12]
,ff="power",prior.alpha=list(3,0,1.34^2),target.tox=target.tox,constrain=TRUE
,sdose.calculate="median",pointest="mean",start=1,simulate=TRUE,nsims=50
,truep=p.tox0[1:12],threep3=TRUE)## Simulated trial: 10## Simulated trial: 20## Simulated trial: 30## Simulated trial: 40## Simulated trial: 50## Calculating operating characteristics of a standard 3+3 trial for comparison...## 12% complete## 16% complete## 20% complete## 24% complete## 28% complete## 32% complete## 36% complete## 40% complete## 44% complete## 48% complete## 52% complete## 56% complete## 60% complete## 64% complete## 68% complete## 72% complete## 76% complete## 80% complete## 84% complete## 88% complete## 92% complete## 96% complete## 100% completeprint(Power.LN.bcrm.compare.sim,threep3=TRUE)## Operating characteristics based on 50 simulations:
##
##
## Sample size 36
##
## Doses
## No dose 1 2.5 5 10 15
## Experimentation proportion NA 0.0833 0.0833 0.0833 0.0833 0.0833
## Recommendation proportion 0 0.0000 0.0000 0.0000 0.0000 0.0000
## Doses
## 20 25 30 40 50 75
## Experimentation proportion 0.0833 0.0833 0.0867 0.123 0.167 0.0317
## Recommendation proportion 0.0000 0.0000 0.0000 0.180 0.560 0.2000
## Doses
## 100
## Experimentation proportion 0.00833
## Recommendation proportion 0.06000
##
## Probability of DLT
## [0,0.2] (0.2,0.4] (0.4,0.6] (0.6,0.8] (0.8,1]
## Experimentation proportion 0.793 0.198 0.00833 0 0
## Recommendation proportion 0.180 0.760 0.06000 0 0
##
##
## ******************** 3+3 operating characteristics *****************
## Mean Minimum Maximum
## Sample size 36.23669 3 72
##
## Doses
## < 1 1 2.5 5 10 15
## Experimentation proportion NA 0.0898 0.09040 0.09110 0.09140 0.0918
## Recommendation proportion 0.00117 0.0026 0.00456 0.00701 0.00991 0.0170
## Doses
## 20 25 30 40 50 75
## Experimentation proportion 0.0929 0.0978 0.110 0.113 0.0848 0.0372
## Recommendation proportion 0.0259 0.0917 0.208 0.346 0.2070 0.0623
## Doses
## 100
## Experimentation proportion 0.0093
## Recommendation proportion 0.0172
##
## Probability of DLT
## [0,0.2] (0.2,0.4] (0.4,0.6] (0.6,0.8] (0.8,1]
## Experimentation proportion 0.869 0.122 0.0093 0 0
## Recommendation proportion* 0.713 0.269 0.0172 0 0
##
## * Amongst those trials that recommend an MTD
##
## Doses
## 1 2.5 5 10 15 20 25
## Average number of patients 3.100 3.1400 3.1800 3.2100 3.2500 3.320 3.510
## Average number of DLTs 0.031 0.0471 0.0636 0.0804 0.0975 0.133 0.175
## Doses
## 30 40 50 75 100
## Average number of patients 3.990 4.240 3.330 1.560 0.412
## Average number of DLTs 0.399 0.721 0.999 0.624 0.206plot(Power.LN.bcrm.compare.sim,threep3=TRUE)## Warning: position_dodge requires non-overlapping x intervals
Dvojparametrický logistický model, apriórne rozdelenie dvojrozmerné lognormálne
sdose<-log(dose/250)
mu<-c(2.15,0.52)
Sigma<-rbind(c(0.84^2,0.134),c(0.134,0.80^2))
TwoPLogistic.mean.bcrm<-bcrm(stop=list(nmax=18),data=data,sdose=sdose
,dose=dose,ff="logit2",prior.alpha=list(4,mu,Sigma),target.tox=target.tox
,constrain=FALSE,pointest="mean")## Warning in bcrm(stop = list(nmax = 18), data = data, sdose = sdose, dose =
## dose, : Exact method slow for 2-parameter model, suggest using rjags (MCMC)##
## Stopping: Reached maximum sample sizeprint(TwoPLogistic.mean.bcrm)##
## Estimation method: exact.sim
## Target toxicity level: 0.3
## Model: Two-parameter logistic
## Prior: Log Multivariate Normal## Mean Vector:## [1] 2.15 0.52
##
## Variance-Covariance Matrix:
## [,1] [,2]
## [1,] 0.7056 0.134
## [2,] 0.1340 0.640
##
## Standardised doses (skeleton):
## 1 2.5 5 10 15 20
## -5.5214609 -4.6051702 -3.9120230 -3.2188758 -2.8134107 -2.5257286
## 25 30 40 50 75 100
## -2.3025851 -2.1202635 -1.8325815 -1.6094379 -1.2039728 -0.9162907
## 150 200 250
## -0.5108256 -0.2231436 0.0000000
##
## Unmodified (unconstrained) CRM used
##
## Posterior mean estimate of probability of toxicity used to select next dose
##
## Toxicities observed:
## Doses
## 1 2.5 5 10 15 20 25 30 40 50 75 100 150 200 250
## n 3 4 5 4 0 0 2 0 0 0 0 0 0 0 0
## Toxicities 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0
##
## Posterior estimates of toxicity:
## Doses
## 1 2.5 5 10 15 20 25 30 40
## Point estimate 0.0116 0.0295 0.0613 0.127 0.192 0.253 0.309 0.36 0.449
## Doses
## 50 75 100 150 200 250
## Point estimate 0.521 0.648 0.728 0.818 0.866 0.894
##
## Next recommended dose: 25plot(TwoPLogistic.mean.bcrm)
Príklad 2
dose<-c(1,2.5,5,10,15,20,25,30)
p1<-c(.02, .06, .08,.12, .20, .30, .40, .50)
#p1<-c(.01, .05, .09, .14, .18, .22, .26, .30)
#p1<-c(.10, .20, .30, .40, .50, .60, .70, .80)
#p1<-c(.20, .30, .40, .50, .60, .65, .70, .75)
target.tox<-0.30Ručné zadávanie dát
Doposiaľ sme podali liek 12 pacientom, dvaja mali vedľajšie účinky (dávka 4)
data<-data.frame(patient=1:12,dose=rep(c(1:4),c(3,3,3,3)),tox=rep(0:1,c(10,2)))
pbcrm<-bcrm(stop=list(nmax=24),data=data,p.tox0=p1,dose=dose,ff="power",prior.alpha=list(3,0,1.34^2),target.tox=target.tox,constrain=FALSE
,sdose.calculate="median",pointest="mean",plot=TRUE)
print(pbcrm)
plot(pbcrm)Nasimulujeme 10 pokusov veľkosti 36
pbcrm.sim<-bcrm(stop=list(nmax=36),p.tox0=p1,dose=dose,ff="power"
,prior.alpha=list(3,0,1.34^2),target.tox=target.tox,constrain=TRUE
,sdose.calculate="median",pointest="mean",start=1,simulate=TRUE,nsims=10,truep=p1)## Simulated trial: 10print(pbcrm.sim)## Operating characteristics based on 10 simulations:
##
##
## Sample size 36
##
## Doses
## No dose 1 2.5 5 10 15
## Experimentation proportion NA 0.0833 0.0833 0.0833 0.0833 0.158
## Recommendation proportion 0 0.0000 0.0000 0.0000 0.0000 0.000
## Doses
## 20 25 30
## Experimentation proportion 0.217 0.233 0.0583
## Recommendation proportion 0.600 0.400 0.0000
##
## Probability of DLT
## [0,0.2] (0.2,0.4] (0.4,0.6] (0.6,0.8] (0.8,1]
## Experimentation proportion 0.492 0.45 0.0583 0 0
## Recommendation proportion 0.000 1.00 0.0000 0 0plot(pbcrm.sim)
Porovnanie crm a 3+3 návrhu na základe 50 simulácií
pbcrm.compare.sim<-bcrm(stop=list(nmax=36),p.tox0=p1,dose=dose
,ff="power",prior.alpha=list(3,0,1.34^2),target.tox=target.tox,constrain=TRUE
,sdose.calculate="median",pointest="mean",start=1,simulate=TRUE,nsims=50
,truep=p1,threep3=TRUE)## Simulated trial: 10## Simulated trial: 20## Simulated trial: 30## Simulated trial: 40## Simulated trial: 50## Calculating operating characteristics of a standard 3+3 trial for comparison...## 18% complete## 24% complete## 29% complete## 35% complete## 41% complete## 47% complete## 53% complete## 59% complete## 65% complete## 71% complete## 76% complete## 82% complete## 88% complete## 94% complete## 100% completeprint(pbcrm.compare.sim,threep3=TRUE)## Operating characteristics based on 50 simulations:
##
##
## Sample size 36
##
## Doses
## No dose 1 2.5 5 10 15 20
## Experimentation proportion NA 0.09 0.0933 0.0883 0.117 0.235 0.227
## Recommendation proportion 0 0.00 0.0000 0.0000 0.020 0.220 0.480
## Doses
## 25 30
## Experimentation proportion 0.108 0.0417
## Recommendation proportion 0.220 0.0600
##
## Probability of DLT
## [0,0.2] (0.2,0.4] (0.4,0.6] (0.6,0.8] (0.8,1]
## Experimentation proportion 0.623 0.335 0.0417 0 0
## Recommendation proportion 0.240 0.700 0.0600 0 0
##
##
## ******************** 3+3 operating characteristics *****************
## Mean Minimum Maximum
## Sample size 23.12627 3 48
##
## Doses
## < 1 1 2.5 5 10 15 20
## Experimentation proportion NA 0.1590 0.1720 0.172 0.176 0.160 0.105
## Recommendation proportion 0.00461 0.0377 0.0618 0.121 0.244 0.288 0.176
## Doses
## 25 30
## Experimentation proportion 0.0447 0.0110
## Recommendation proportion 0.0529 0.0146
##
## Probability of DLT
## [0,0.2] (0.2,0.4] (0.4,0.6] (0.6,0.8] (0.8,1]
## Experimentation proportion 0.839 0.150 0.0110 0 0
## Recommendation proportion* 0.752 0.228 0.0146 0 0
##
## * Amongst those trials that recommend an MTD
##
## Doses
## 1 2.5 5 10 15 20 25
## Average number of patients 3.2800 3.620 3.770 4.030 3.920 2.830 1.32
## Average number of DLTs 0.0656 0.217 0.301 0.484 0.785 0.848 0.53
## Doses
## 30
## Average number of patients 0.350
## Average number of DLTs 0.175plot(pbcrm.compare.sim,threep3=TRUE)## Warning: position_dodge requires non-overlapping x intervals