The main objective of cancer phase I clinical trials is to determine a maximum tolerated dose (MTD) of a new experimental treatment. In practice, most of these trials are designed so that three patients per cohort are treated at the same dose level. In this paper, we compare the safety and efficiency of trials using the escalation with overdose control (EWOC) scheme designed with three or only one patient per cohort. We show through simulations that the number of patients per cohort does not impact the proportion of patients given therapeutic doses, safety of the trial, and efficiency of the estimate of the MTD. Additionally, we present guidelines and tabulated values on the number of patients needed to design a phase I cancer clinical trial using EWOC to achieve a given accuracy of the estimate of the MTD.
Cancer phase I clinical trials are small studies whose main objective is to determine a maximum tolerated dose (MTD) of a new experimental drug or combination of known drugs for use in a phase II trial. Patients are typically accrued to the trial sequentially in cohorts of size
The work we present here has been motivated by the frequent requests by clinicians and review committees at Cancer Center Institutions the authors worked at on (1) the number of patients that should be included in each cohort, and (2) the number of patients required to conduct a phase I cancer clinical trial. Denote by
The number of patients that are enrolled in a cancer phase I clinical trial is typically between 12 and 40 and trial duration depends on the study design and length of the study cycle to resolve toxicity outcome. An increasing number of clinicians we work with inquire about the number of patients they need to accrue in order to estimate the MTD with an acceptable degree of accuracy. We are not aware of any published methodologies for sample size determination (SSD) in cancer phase I clinical trials based on power calculation or precision of some Bayes estimates using either frequentist or Bayesian adaptive designs. As a point of fact, most sample size recommendations are based on prespecified stopping rules, see, for example, [
In this paper, we address the SSD problem using the traditional approach; we estimate the sample size based on a desired accuracy of the Bayes estimate on the average. Specifically, we seek the smallest number of patients so that the posterior variance of the MTD on the average over all possible trials is no more than a specified margin. This procedure is not based on a specific stopping rule and consequently preserves the coherent nature of EWOC, see [
This paper is organized as follows. Section
EWOC is a Bayesian adaptive design permitting precise determination of the MTD while directly controlling the likelihood of an overdose. It is the first statistical method to directly incorporate formal safety constraints into the design of cancer phase I clinical trials. Zacks et al. [
EWOC has been used to design over a dozen of phase I studies approved by the Research Review Committee and the Institute Review Board of the Fox Chase Cancer Center, Philadelphia, Winship Cancer Institute, Atlanta, and Cedars Sinai Medical Center, Los Angeles (see [
We adopt the-logistic-based model to represent the dose-toxicity relationship the following:
An increasing number of clinicians inquire about the number of patients they need to accrue in the design of cancer phase I trials to achieve a specific goal. Sample size recommendation based on the expected number of patients treated at each dose level in “
Denote by
This is also known as the average length criteria (ALC) because for each realization of a trial
The simulation results presented below all assume that the feasibility bound
We simulate trials under different scenarios corresponding to different values of
Table
Estimated proportions of patients given doses in an
Sample size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
|
12 |
18 |
24 |
30 |
||||||||
1
|
3
|
Diff. |
1
|
3
|
Diff. |
1
|
3
|
Diff. |
1
|
3
|
Diff. | |
|
||||||||||||
|
0.9768 | 0.9829 | −0.0061 | 0.9826 | 0.9854 | −0.0028 | 0.9864 | 0.9890 | −0.0026 | 0.9897 | 0.9910 | −0.0013 |
|
0.9065 | 0.9352 | −0.0287 | 0.9290 | 0.9465 | −0.0175 | 0.9433 | 0.9598 | −0.0165 | 0.9542 | 0.9661 | −0.0119 |
|
0.7525 | 0.7004 | 0.0521 | 0.7958 | 0.7440 | 0.0518 | 0.8261 | 0.7897 | 0.0364 | 0.8569 | 0.8210 | 0.0359 |
|
0.3634 | 0.2521 | 0.1114 | 0.4306 | 0.3309 | 0.0997 | 0.4747 | 0.3997 | 0.0750 | 0.5265 | 0.4444 | 0.0821 |
|
||||||||||||
|
0.9340 | 0.9603 | −0.0262 | 0.9430 | 0.9492 | −0.0062 | 0.9517 | 0.9571 | −0.0054 | 0.9592 | 0.9629 | −0.0037 |
|
0.8153 | 0.8839 | −0.0687 | 0.8378 | 0.8760 | −0.0382 | 0.8610 | 0.8896 | −0.0287 | 0.8784 | 0.8958 | −0.0174 |
|
0.6297 | 0.6729 | −0.0431 | 0.6507 | 0.6643 | −0.0136 | 0.6831 | 0.6924 | −0.0092 | 0.7032 | 0.7105 | −0.0074 |
|
0.2752 | 0.2298 | 0.0454 | 0.3142 | 0.2659 | 0.0483 | 0.3478 | 0.3152 | 0.0327 | 0.3713 | 0.3477 | 0.0236 |
|
||||||||||||
|
0.8658 | 0.9351 | −0.0693 | 0.8659 | 0.9032 | −0.0373 | 0.8772 | 0.8926 | −0.0155 | 0.8838 | 0.8984 | −0.0146 |
|
0.7122 | 0.8463 | −0.1341 | 0.7190 | 0.8036 | −0.0846 | 0.7359 | 0.7900 | −0.0541 | 0.7438 | 0.7931 | −0.0493 |
|
0.5256 | 0.6533 | −0.1277 | 0.5219 | 0.6063 | −0.0845 | 0.5390 | 0.5865 | −0.0475 | 0.5448 | 0.5915 | −0.0467 |
|
0.2154 | 0.2102 | 0.0052 | 0.2279 | 0.2276 | 0.0003 | 0.2549 | 0.2371 | 0.0178 | 0.2626 | 0.2604 | 0.0022 |
|
||||||||||||
|
0.7855 | 0.7121 | 0.0734 | 0.8272 | 0.7762 | 0.0510 | 0.8542 | 0.8180 | 0.0362 | 0.8784 | 0.8504 | 0.0280 |
|
0.6939 | 0.5833 | 0.1106 | 0.7269 | 0.6541 | 0.0728 | 0.7477 | 0.6968 | 0.0509 | 0.7825 | 0.7376 | 0.0449 |
|
0.4885 | 0.5258 | −0.0373 | 0.5267 | 0.5478 | −0.0211 | 0.5474 | 0.5647 | −0.0173 | 0.5874 | 0.5906 | −0.0032 |
|
0.2773 | 0.1144 | 0.1629 | 0.2956 | 0.1759 | 0.1197 | 0.3050 | 0.2165 | 0.0885 | 0.3282 | 0.2506 | 0.0776 |
|
||||||||||||
|
0.7038 | 0.6680 | 0.0358 | 0.7433 | 0.7201 | 0.0231 | 0.7587 | 0.7541 | 0.0046 | 0.7748 | 0.7814 | −0.0066 |
|
0.6031 | 0.5135 | 0.0897 | 0.6247 | 0.5757 | 0.0490 | 0.6348 | 0.6103 | 0.0246 | 0.6489 | 0.6417 | 0.0072 |
|
0.4129 | 0.4371 | −0.0242 | 0.4348 | 0.4531 | −0.0184 | 0.4458 | 0.4722 | −0.0264 | 0.4580 | 0.4903 | −0.0323 |
|
0.2332 | 0.1125 | 0.1207 | 0.2388 | 0.1648 | 0.0740 | 0.2416 | 0.1835 | 0.0581 | 0.2489 | 0.2100 | 0.0389 |
|
||||||||||||
|
0.6383 | 0.6107 | 0.0276 | 0.6484 | 0.6550 | −0.0066 | 0.6573 | 0.6826 | −0.0252 | 0.6608 | 0.6859 | −0.0251 |
|
0.5377 | 0.4446 | 0.0932 | 0.5364 | 0.5042 | 0.0322 | 0.5389 | 0.5313 | 0.0076 | 0.5322 | 0.5349 | −0.0027 |
|
0.3605 | 0.3556 | 0.0049 | 0.3651 | 0.3819 | −0.0168 | 0.3680 | 0.3921 | −0.0241 | 0.3627 | 0.3912 | −0.0286 |
|
0.2008 | 0.1036 | 0.0972 | 0.1988 | 0.1467 | 0.0521 | 0.1982 | 0.1583 | 0.0400 | 0.1926 | 0.1665 | 0.0261 |
Estimated proportions of patients given doses in an
Sample size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
|
12 |
18 |
24 |
30 |
||||||||
1
|
3
|
Diff. |
1
|
3
|
Diff. |
1
|
3
|
Diff. |
1
|
3
|
Diff. | |
|
||||||||||||
|
0.4140 | 0.2552 | 0.1588 | 0.4940 | 0.4012 | 0.0928 | 0.5503 | 0.4825 | 0.0679 | 0.6103 | 0.5418 | 0.0685 |
|
0.2790 | 0.2224 | 0.0566 | 0.3542 | 0.3137 | 0.0406 | 0.4092 | 0.3814 | 0.0278 | 0.4692 | 0.4339 | 0.0353 |
|
0.1702 | 0.0455 | 0.1247 | 0.2273 | 0.1544 | 0.0729 | 0.2696 | 0.2153 | 0.0543 | 0.3172 | 0.2620 | 0.0552 |
|
0.0746 | 0.0448 | 0.0298 | 0.1090 | 0.0830 | 0.0260 | 0.1321 | 0.1132 | 0.0189 | 0.1567 | 0.1390 | 0.0177 |
|
||||||||||||
|
0.3104 | 0.1702 | 0.1402 | 0.3744 | 0.2755 | 0.0989 | 0.4140 | 0.3500 | 0.0641 | 0.4315 | 0.3768 | 0.0547 |
|
0.1988 | 0.1463 | 0.0525 | 0.2590 | 0.2008 | 0.0583 | 0.2965 | 0.2553 | 0.0412 | 0.3101 | 0.2770 | 0.0331 |
|
0.1173 | 0.0267 | 0.0906 | 0.1588 | 0.0920 | 0.0668 | 0.1897 | 0.1362 | 0.0535 | 0.2005 | 0.1548 | 0.0457 |
|
0.0499 | 0.0263 | 0.0236 | 0.0728 | 0.0474 | 0.0254 | 0.0909 | 0.0682 | 0.0226 | 0.0976 | 0.0770 | 0.0206 |
|
||||||||||||
|
0.2428 | 0.1095 | 0.1333 | 0.2883 | 0.1983 | 0.0900 | 0.3079 | 0.2444 | 0.0635 | 0.3331 | 0.2694 | 0.0636 |
|
0.1500 | 0.0924 | 0.0576 | 0.1932 | 0.1416 | 0.0516 | 0.2146 | 0.1716 | 0.0430 | 0.2336 | 0.1895 | 0.0441 |
|
0.0864 | 0.0157 | 0.0707 | 0.1171 | 0.0626 | 0.0546 | 0.1350 | 0.0881 | 0.0469 | 0.1484 | 0.1016 | 0.0468 |
|
0.0356 | 0.0153 | 0.0203 | 0.0531 | 0.0321 | 0.0209 | 0.0646 | 0.0452 | 0.0194 | 0.0721 | 0.0489 | 0.0231 |
|
||||||||||||
|
0.0582 | 0.0052 | 0.0530 | 0.1270 | 0.0736 | 0.0534 | 0.1886 | 0.1342 | 0.0544 | 0.2354 | 0.1852 | 0.0502 |
|
0.0220 | 0.0000 | 0.0220 | 0.0619 | 0.0431 | 0.0188 | 0.1039 | 0.0803 | 0.0235 | 0.1414 | 0.1145 | 0.0269 |
|
0.0017 | 0.0000 | 0.0017 | 0.0235 | 0.0092 | 0.0143 | 0.0478 | 0.0310 | 0.0168 | 0.0724 | 0.0529 | 0.0195 |
|
0.0000 | 0.0000 | 0.0000 | 0.0055 | 0.0014 | 0.0040 | 0.0165 | 0.0094 | 0.0072 | 0.0275 | 0.0180 | 0.0095 |
|
||||||||||||
|
0.0344 | 0.0034 | 0.0310 | 0.0729 | 0.0349 | 0.0380 | 0.1041 | 0.0630 | 0.0411 | 0.1288 | 0.0859 | 0.0429 |
|
0.0126 | 0.0000 | 0.0126 | 0.0336 | 0.0179 | 0.0158 | 0.0554 | 0.0360 | 0.0194 | 0.0718 | 0.0460 | 0.0258 |
|
0.0008 | 0.0000 | 0.0008 | 0.0120 | 0.0035 | 0.0085 | 0.0236 | 0.0131 | 0.0105 | 0.0336 | 0.0189 | 0.0147 |
|
0.0000 | 0.0000 | 0.0000 | 0.0027 | 0.0005 | 0.0022 | 0.0075 | 0.0037 | 0.0038 | 0.0119 | 0.0062 | 0.0057 |
|
||||||||||||
|
0.0219 | 0.0014 | 0.0205 | 0.0410 | 0.0166 | 0.0244 | 0.0559 | 0.0323 | 0.0236 | 0.0752 | 0.0474 | 0.0278 |
|
0.0079 | 0.0000 | 0.0079 | 0.0187 | 0.0095 | 0.0091 | 0.0272 | 0.0166 | 0.0106 | 0.0404 | 0.0262 | 0.0142 |
|
0.0005 | 0.0000 | 0.0005 | 0.0067 | 0.0020 | 0.0047 | 0.0112 | 0.0056 | 0.0056 | 0.0196 | 0.0111 | 0.0085 |
|
0.0000 | 0.0000 | 0.0000 | 0.0017 | 0.0003 | 0.0014 | 0.0037 | 0.0019 | 0.0018 | 0.0076 | 0.0037 | 0.0038 |
Estimated proportions of patients given doses in an
Sample size |
|||||||||
---|---|---|---|---|---|---|---|---|---|
12 | 18 | 24 | 30 | ||||||
0.05 |
1
|
0.1436 |
1
|
0.1625 |
1
|
0.1781 |
1
|
0.1920 | |
3
|
0.0924 |
3
|
0.1230 |
3
|
0.1459 |
3
|
0.1644 | ||
|
|
|
|
|
|
|
|
||
|
0.10 |
1
|
0.2956 |
1
|
0.3200 |
1
|
0.3405 |
1
|
0.3587 |
3
|
0.2860 |
3
|
0.3101 |
3
|
0.3322 |
3
|
0.3497 | ||
|
|
|
|
|
|
|
|
||
0.15 |
1
|
0.4115 |
1
|
0.4412 |
1
|
0.4640 |
1
|
0.4839 | |
3
|
0.3888 |
3
|
0.4239 |
3
|
0.4516 |
3
|
0.4714 | ||
|
|
|
|
|
|
|
|
||
0.20 |
1
|
0.4988 |
1
|
0.5340 |
1
|
0.5589 |
1
|
0.5801 | |
3
|
0.4517 |
3
|
0.4991 |
3
|
0.5333 |
3
|
0.5564 | ||
|
|
|
|
|
|
|
|
Estimated proportions of patients exhibiting DLTs, treated above the MTD, MSE, and bias of the MTD under designs 1
Sample size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
|
12 | 18 | 24 | 30 | ||||||||
1
|
3
|
Diff. |
1
|
3
|
Diff. |
1
|
3
|
Diff. |
1
|
3
|
Diff. | |
|
||||||||||||
Proportion of DLTs | 0.3389 | 0.3456 | −0.0067 | 0.3271 | 0.3346 | −0.0075 | 0.3192 | 0.3254 | −0.0062 | 0.3136 | 0.3173 | −0.0037 |
Proportion above the MTD | 0.4694 | 0.5095 | −0.0401 | 0.4493 | 0.4701 | −0.0208 | 0.4274 | 0.4419 | −0.0145 | 0.4052 | 0.4140 | −0.0088 |
MSE | 0.0457 | 0.0467 | −0.0010 | 0.0389 | 0.0365 | 0.0024 | 0.0338 | 0.0311 | 0.0027 | 0.0311 | 0.0274 | 0.0036 |
Bias | 0.2013 | 0.1907 | 0.0106 | 0.1857 | 0.1676 | 0.0181 | 0.1738 | 0.1556 | 0.0182 | 0.1663 | 0.1467 | 0.0196 |
|
||||||||||||
Proportion of DLTs | 0.3346 | 0.3289 | 0.0058 | 0.3305 | 0.3311 | −0.0006 | 0.3262 | 0.3306 | −0.0044 | 0.3211 | 0.3279 | −0.0068 |
Proportion above the MTD | 0.5219 | 0.5436 | −0.0217 | 0.5185 | 0.5512 | −0.0327 | 0.5073 | 0.5376 | −0.0303 | 0.4928 | 0.5293 | −0.0366 |
MSE | 0.0558 | 0.0661 | −0.0102 | 0.0468 | 0.0542 | −0.0074 | 0.0402 | 0.0447 | −0.0045 | 0.0367 | 0.0393 | −0.0026 |
Bias | 0.2164 | 0.2315 | −0.0151 | 0.1972 | 0.2076 | −0.0104 | 0.1825 | 0.1878 | −0.0054 | 0.1741 | 0.1760 | −0.0019 |
|
||||||||||||
Proportion of DLTs | 0.3200 | 0.3121 | 0.0080 | 0.3210 | 0.3164 | 0.0047 | 0.3200 | 0.3183 | 0.0017 | 0.3181 | 0.3190 | −0.0009 |
Proportion above the MTD | 0.5573 | 0.5505 | 0.0069 | 0.5618 | 0.5766 | −0.0148 | 0.5470 | 0.5839 | −0.0369 | 0.5527 | 0.5836 | −0.0310 |
MSE | 0.0734 | 0.0849 | −0.0115 | 0.0636 | 0.0742 | −0.0106 | 0.0559 | 0.0672 | −0.0113 | 0.0513 | 0.0593 | −0.0080 |
Bias | 0.2403 | 0.2625 | −0.0222 | 0.2206 | 0.2416 | −0.0210 | 0.2049 | 0.2271 | −0.0222 | 0.1952 | 0.2126 | −0.0174 |
|
||||||||||||
Proportion of DLTs | 0.2461 | 0.2222 | 0.0239 | 0.2545 | 0.2414 | 0.0131 | 0.2594 | 0.2511 | 0.0082 | 0.2630 | 0.2581 | 0.0049 |
Proportion above the MTD | 0.2444 | 0.1786 | 0.0658 | 0.2803 | 0.2474 | 0.0329 | 0.2807 | 0.2701 | 0.0105 | 0.2954 | 0.2843 | 0.0111 |
MSE | 0.0276 | 0.0316 | −0.0040 | 0.0262 | 0.0291 | −0.0029 | 0.0243 | 0.0266 | −0.0023 | 0.0232 | 0.0242 | −0.0011 |
Bias | 0.1371 | 0.1493 | −0.0122 | 0.1353 | 0.1403 | −0.0050 | 0.1311 | 0.1342 | −0.0031 | 0.1302 | 0.1292 | 0.0010 |
|
||||||||||||
Proportion of DLTs | 0.2583 | 0.2408 | 0.0175 | 0.2626 | 0.2548 | 0.0078 | 0.2669 | 0.2607 | 0.0063 | 0.2689 | 0.2646 | 0.0043 |
Proportion above the MTD | 0.2202 | 0.1352 | 0.0850 | 0.2493 | 0.1917 | 0.0576 | 0.2663 | 0.2294 | 0.0369 | 0.2673 | 0.2489 | 0.0184 |
MSE | 0.0278 | 0.0308 | −0.0030 | 0.0269 | 0.0291 | −0.0022 | 0.0254 | 0.0279 | −0.0025 | 0.0237 | 0.0266 | −0.0029 |
Bias | 0.1167 | 0.1336 | −0.0169 | 0.1153 | 0.1266 | −0.0113 | 0.1107 | 0.1218 | −0.0111 | 0.1070 | 0.1194 | −0.0124 |
|
||||||||||||
Proportion of DLTs | 0.2749 | 0.2692 | 0.0057 | 0.2798 | 0.2737 | 0.0060 | 0.2803 | 0.2761 | 0.0042 | 0.2809 | 0.2782 | 0.0027 |
Proportion above the MTD | 0.2047 | 0.1037 | 0.1010 | 0.2216 | 0.1646 | 0.0570 | 0.2447 | 0.1987 | 0.0461 | 0.2461 | 0.2098 | 0.0364 |
MSE | 0.0291 | 0.0281 | 0.0009 | 0.0275 | 0.0290 | −0.0015 | 0.0284 | 0.0298 | −0.0014 | 0.0273 | 0.0284 | −0.0012 |
Bias | 0.0941 | 0.1089 | −0.0148 | 0.0833 | 0.1051 | −0.0219 | 0.0814 | 0.0997 | −0.0184 | 0.0758 | 0.0942 | −0.0184 |
Estimated proportions of patients exhibiting DLTs, treated above the MTD, MSE, and bias of the MTD under designs 1
Sample size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
|
12 | 18 | 24 | 30 | ||||||||
1
|
3
|
Diff. |
1
|
3
|
Diff. |
1
|
3
|
Diff. |
1
|
3
|
Diff. | |
|
||||||||||||
Proportion of DLTs | 0.1866 | 0.1657 | 0.0209 | 0.2010 | 0.1863 | 0.0147 | 0.2108 | 0.1996 | 0.0112 | 0.2168 | 0.2080 | 0.0088 |
Proportion above the MTD | 0.0258 | 0.0001 | 0.0257 | 0.0529 | 0.0341 | 0.0188 | 0.0787 | 0.0603 | 0.0185 | 0.0974 | 0.0817 | 0.0157 |
MSE | 0.0099 | 0.0092 | 0.0008 | 0.0102 | 0.0104 | −0.0002 | 0.0103 | 0.0102 | 0.0001 | 0.0098 | 0.0107 | −0.0008 |
Bias | 0.0065 | 0.0097 | −0.0033 | 0.0195 | 0.0238 | −0.0043 | 0.0287 | 0.0331 | −0.0044 | 0.0397 | 0.0401 | −0.0003 |
|
||||||||||||
Proportion of DLTs | 0.2250 | 0.2121 | 0.0129 | 0.2286 | 0.2226 | 0.0059 | 0.2342 | 0.2270 | 0.0072 | 0.2369 | 0.2341 | 0.0027 |
Proportion above the MTD | 0.0184 | 0.0000 | 0.0184 | 0.0384 | 0.0188 | 0.0197 | 0.0535 | 0.0339 | 0.0196 | 0.0666 | 0.0450 | 0.0216 |
MSE | 0.0178 | 0.0140 | 0.0038 | 0.0169 | 0.0152 | 0.0017 | 0.0167 | 0.0152 | 0.0015 | 0.0168 | 0.0153 | 0.0015 |
Bias | −0.0449 | −0.0356 | −0.0092 | −0.0317 | −0.0278 | −0.0039 | −0.0271 | −0.0177 | −0.0093 | −0.0223 | −0.0173 | −0.0050 |
|
||||||||||||
Proportion of DLTs | 0.2609 | 0.2570 | 0.0039 | 0.2604 | 0.2593 | 0.0011 | 0.2643 | 0.2601 | 0.0042 | 0.2632 | 0.2636 | −0.0004 |
Proportion above the MTD | 0.0132 | 0.0001 | 0.0132 | 0.0291 | 0.0120 | 0.0171 | 0.0392 | 0.0236 | 0.0156 | 0.0481 | 0.0323 | 0.0158 |
MSE | 0.0289 | 0.0228 | 0.0061 | 0.0306 | 0.0251 | 0.0055 | 0.0321 | 0.0250 | 0.0071 | 0.0327 | 0.0267 | 0.0060 |
Bias | −0.0898 | −0.0794 | −0.0104 | −0.0870 | −0.0762 | −0.0109 | −0.0905 | −0.0718 | −0.0186 | −0.0863 | −0.0746 | −0.0117 |
|
||||||||||||
Proportion of DLTs | 0.1549 | 0.1387 | 0.0162 | 0.1678 | 0.1541 | 0.0136 | 0.1774 | 0.1663 | 0.0111 | 0.1846 | 0.1762 | 0.0084 |
Proportion above the MTD | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0001 | 0.0022 | 0.0009 | 0.0013 | 0.0052 | 0.0032 | 0.0020 |
MSE | 0.0347 | 0.0349 | −0.0003 | 0.0276 | 0.0273 | 0.0003 | 0.0226 | 0.0221 | 0.0004 | 0.0184 | 0.0192 | −0.0008 |
Bias | −0.1575 | −0.1622 | 0.0047 | −0.1323 | −0.1326 | 0.0003 | −0.1128 | −0.1125 | −0.0003 | −0.0977 | −0.0996 | 0.0019 |
|
||||||||||||
Proportion of DLTs | 0.2042 | 0.1933 | 0.0109 | 0.2099 | 0.2026 | 0.0073 | 0.2143 | 0.2083 | 0.0060 | 0.2179 | 0.2135 | 0.0044 |
Proportion above the MTD | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0012 | 0.0003 | 0.0008 | 0.0020 | 0.0013 | 0.0007 |
MSE | 0.0641 | 0.0592 | 0.0049 | 0.0578 | 0.0542 | 0.0037 | 0.0543 | 0.0503 | 0.0040 | 0.0514 | 0.0464 | 0.0050 |
Bias | −0.2192 | −0.2159 | −0.0034 | −0.2025 | −0.1983 | −0.0042 | −0.1911 | −0.1869 | −0.0041 | −0.1818 | −0.1776 | −0.0042 |
|
||||||||||||
Proportion of DLTs | 0.2502 | 0.2469 | 0.0032 | 0.2533 | 0.2505 | 0.0028 | 0.2555 | 0.2522 | 0.0033 | 0.2548 | 0.2532 | 0.0016 |
Proportion above the MTD | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0004 | 0.0002 | 0.0001 | 0.0013 | 0.0008 | 0.0005 |
MSE | 0.0979 | 0.0877 | 0.0102 | 0.0995 | 0.0900 | 0.0095 | 0.1011 | 0.0883 | 0.0128 | 0.0989 | 0.0887 | 0.0102 |
Bias | −0.2760 | −0.2673 | −0.0087 | −0.2750 | −0.2640 | −0.0110 | −0.2747 | −0.2595 | −0.0152 | −0.2685 | −0.2564 | −0.0121 |
Estimated proportions of patients exhibiting DLTs, treated above the MTD, MSE, and bias of the MTD under designs 1
|
Sample size |
|||||||
---|---|---|---|---|---|---|---|---|
12 | 18 | 24 | 30 | |||||
Proportion of DLTs |
1
|
0.2546 |
1
|
0.2580 |
1
|
0.2607 |
1
|
0.2616 |
3
|
0.2444 |
3
|
0.2523 |
3
|
0.2563 |
3
|
0.2595 | |
|
|
|
|
|
|
|
|
|
Proportion above the MTD |
1
|
0.1895 |
1
|
0.2001 |
1
|
0.2040 |
1
|
0.2067 |
3
|
0.1685 |
3
|
0.1888 |
3
|
0.1984 |
3
|
0.2029 | |
|
|
|
|
|
|
|
|
|
MSE |
1
|
0.0427 |
1
|
0.0394 |
1
|
0.0371 |
1
|
0.0351 |
3
|
0.0429 |
3
|
0.0395 |
3
|
0.0365 |
3
|
0.0344 | |
|
|
|
|
|
|
|
|
|
Bias |
1
|
0.0186 |
1
|
0.0190 |
1
|
0.0181 |
1
|
0.0193 |
3
|
0.0271 |
3
|
0.0261 |
3
|
0.0259 |
3
|
0.0244 | |
|
|
|
|
|
|
|
|
In this section, we present tabulated values for average posterior standard deviation of the MTD and average length HPD interval that are achieved for even sample sizes
Average posterior standard deviation and average length of HPD of the posterior distribution of the MTD that are achieved for a given sample size for
|
Mean SD | Length of 90% HPD | Length of 95% HPD |
---|---|---|---|
6 | 0.2453 | 0.7386 | 0.8161 |
8 | 0.2399 | 0.7238 | 0.8040 |
10 | 0.2351 | 0.7111 | 0.7925 |
12 | 0.2309 | 0.6985 | 0.7818 |
14 | 0.2281 | 0.6913 | 0.7755 |
16 | 0.2248 | 0.6821 | 0.7678 |
18 | 0.2221 | 0.6748 | 0.7608 |
20 | 0.2197 | 0.6673 | 0.7546 |
22 | 0.2176 | 0.6624 | 0.7500 |
24 | 0.2153 | 0.6557 | 0.7439 |
26 | 0.2136 | 0.6505 | 0.7395 |
28 | 0.2119 | 0.6455 | 0.7352 |
30 | 0.2102 | 0.6410 | 0.7313 |
32 | 0.2085 | 0.6350 | 0.7257 |
34 | 0.2072 | 0.6313 | 0.7221 |
36 | 0.2057 | 0.6262 | 0.7176 |
38 | 0.2050 | 0.6240 | 0.7162 |
40 | 0.2036 | 0.6200 | 0.7123 |
Set Generate ( Simulate a trial of Calculate the posterior variance Repeat steps (i)–(iii) for
The left hand sides of (
In the numerical results presented here, we took
Estimated mean posterior standard deviation as a function of the number of patients accrued to the trial for different target probabilities of DLT
Estimated mean length of HPD of the posterior distribution of the MTD as a function of the number of patients accrued to the trial for different target probabilities of DLT
A randomized phase I clinical trial of the combination bortezomib and melphalan as conditioning for autologous stem cell transplant in patients with multiple myeloma was designed using EWOC and the results published in [
Figure
All the possible dose sequences that could be realized for the first four patients and a selected situation for patient 5. It assumes no simultaneous treatment of patients.
The objectives of this paper are to provide a rational for the choice of cohort sizes and number of patients to accrue in a phase I cancer clinical trial when the Bayesian adaptive design EWOC is used. In these trials, patients are typically enrolled in cohorts of size three for no apparent reason other than being in agreement with the traditional “3 + 3” design and shortening the duration of the trial. We have shown through simulations that the two designs are equally safe and that no practical gain is achieved in terms of the efficiency of the estimate of the MTD. Depending on how important the length of the trial is to the clinician and the institution, we recommend using one patient per dose level to avoid seeing simultaneous toxic events when a group of patients is treated at the same dose level as was the case in a recent phase I trial of the drug TGN1412, see [
The simulation results were obtained by generating the toxicity responses using the logistic model (
In the second part of the paper, we addressed the SSD problem by giving tabulated values of the number of patients to accrue in a cancer phase I clinical trial as a function of the posterior standard deviation and length of the HPD interval of the MTD on the average over all possible trials. Although this aspect of the trial never received much emphasis in the literature due to the relatively small number of patients and logistical issues associated with such trials, we felt that providing a measure of the accuracy of the estimate of the MTD that can be achieved for a given sample size would help the clinicians understand what can and cannot be achieved during this phase of the trial. Our results show that in general, there is 17% decrease in the average posterior standard deviation of the MTD when the sample size increases from 6 to 40 patients and that for a sample size of 20 patients, the average posterior standard deviation of the MTD is about one-fifth the range of the dose levels. Although this decrease in the average posterior standard deviation seems modest, we note that this is dependent upon the use of prior distribution for the MTD. A more informative prior based on past data will result in smaller average posterior standard deviations and narrower HPD intervals.
This paper is supported in part by the National Center for Research Resources, Grant UL1RR033176, and is now at the National Center for Advancing Translational Sciences, Grant UL1TR000124 (M. Tighiouart and A. Rogatko), Grant 5P01CA098912-05 (A. Rogatko), and Grant P01 DK046763 (A. Rogatko). The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.