Target toxicity design for phase I dose-finding

Figures & data

Table 1. A decision table for a ‘3+3+3’ design (target toxicity is 0.3).

DLTs	# of subjects
	3	6	9
0	E	E	E
1	S	E	E
2	D	S	E
3	DU	D	S
4		DU	D
5		DU	DU
6		DU	DU
7			DU
8			DU
9			DU

Notes: Row represents number of DLTs. Column represents number of subjects. ‘E’: escalate dose level; ‘S’: stay at current dose level; ‘D’: de-escalate dose level; ‘DU’: de-escalate and never come back to this dose level again.

Table 2. Rejection regions and decision rules.

Conclusion	Rejection region	Decision rule
Higher than target	$(s,u]$	De-escalate(D)
Much higher than target	$(u,n]$	De-escalate and Unacceptable (DU)
Lower than target	$[0,r]$	Escalate(E)
Not sure	$(r,s]$	Stay(S)

Figure 1. Schema of dose-finding using decision table. $n_i$ is the cumulative number of subjects treated at dose level i; $n_{max}$ is the maximum number of subjects allowed at any dose level; d is the total number of dose levels. ‘treat more patients’ means additional cohort of patients should be treated to evaluate DLT. After that, it should go back to Step 1.

Figure 2. True toxicity curves and their distributions.

Table 3. Decision tables for ‘3+3’ designs (traditional 3+3, TT, mTPI and BOIN).

# of subjects
	3				6
DLTs	3+3	TT	mTPI	BOIN	3+3	TT	mTPI	BOIN
0	E	E	E	E	E	E	E	E
1	S	S	S	D	E	E	E	E
2	D	D	D	D	D	S	S	D
3	D	DU	DU	DU	D	D	S	D
4					D	DU	DU	DU
5					D	DU	DU	DU
6					D	DU	DU	DU

Notes: The target toxicity rate is 0.3. TT ‘3+3’: the actual type I errors at each stage are (1) ${\alpha }_1=0.343$, 0.494; (2) ${\alpha }_2=0.216$, 0.311; (3) ${\alpha }_u=0.027$, 0.079. mTPI: ${ϵ}_1={ϵ}_2=0.05$. BOIN: ${\varphi }_1=0.25$, ${\varphi }_2=0.35$.

Table 4. Decision tables for ‘3+3+6’ designs (TT, mTPI, mTPI-2 and BOIN).

# of subjects
DLTs	3				6				12
	TT	mTPI	mTPI-2	BOIN	TT	mTPI	mTPI-2	BOIN	TT	mTPI	mTPI-2	BOIN
0	E	E	E	E	E	E	E	E	E	E	E	E
1	S	S	S	D	E	E	E	E	E	E	E	E
2	D	D	D	D	S	S	S	D	E	E	E	E
3	DU	DU	DU	DU	D	S	D	D	E	S	S	E
4					DU	DU	DU	DU	S	S	S	D
5					DU	DU	DU	DU	D	D	D	D
6					DU	DU	DU	DU	D	DU	D	D
7									DU	DU	DU	DU
8									DU	DU	DU	DU
9									DU	DU	DU	DU
10									DU	DU	DU	DU
11									DU	DU	DU	DU
12									DU	DU	DU	DU

Notes: The target toxicity rate is 0.3. TT ‘3+3’: the actual type I errors at each stage are (1) ${\alpha }_1=0.343$, 0.494, 0.576; (2) ${\alpha }_2=0.216$, 0.311, 0.36; (3) ${\alpha }_u=0.027$, 0.079, 0.095. mTPI: ${ϵ}_1={ϵ}_2=0.05$. BOIN: ${\varphi }_1=0.25$, ${\varphi }_2=0.35$.

Table 5. Comparison of four ‘3+3’ designs and CRM.

		$p_{\mathrm{true}}$	$p_{\mathrm{MTD}}$	# of patients	# of DLTs
	$\theta =0.25$
3+3	Mean	0.450	0.812	15.084	2.707
	Sd	0.090	0.143	3.432	0.631
TT 3+3	Mean	0.499	0.804	14.373	2.612
	Sd	0.079	0.147	3.280	0.602
mTPI(3+3)	Mean	0.472	0.802	14.105	2.561
	Sd	0.077	0.149	3.309	0.585
BOIN	Mean	0.251	0.883	11.137	1.540
	Sd	0.079	0.109	3.253	0.316
CRM	Mean	0.484	0.826	15.000	2.732
	Sd	0.161	0.159	0.000	1.391
	$\theta =0.30$
3+3	Mean	0.279	0.845	13.804	2.897
	Sd	0.089	0.137	3.367	0.294
TT 3+3	Mean	0.343	0.840	13.061	2.761
	Sd	0.101	0.140	3.151	0.273
mTPI(3+3)	Mean	0.340	0.837	12.826	2.709
	Sd	0.095	0.141	3.161	0.257
BOIN	Mean	0.133	0.906	9.004	1.602
	Sd	0.085	0.104	2.430	0.185
CRM	Mean	0.350	0.876	15.000	3.117
	Sd	0.207	0.131	0.000	1.165
	$\theta =0.35$
3+3	Mean	0.264	0.643	11.143	2.935
	Sd	0.178	0.320	2.441	0.079
TT 3+3	Mean	0.268	0.639	10.554	2.798
	Sd	0.106	0.319	2.223	0.067
mTPI(3+3)	Mean	0.261	0.637	10.356	2.740
	Sd	0.096	0.318	2.221	0.061
BOIN	Mean	0.096	0.318	6.831	0.061
	Sd	0.252	0.343	1.331	0.104
CRM	Mean	0.376	0.692	15.000	3.887
	Sd	0.197	0.339	0.000	0.904

Note: The table reports the average and standard deviation of $p_{\mathrm{true}}$, $p_{\mathrm{MTD}}$, number of patients treated and number of DLTs.

Table 6. Comparison of four ‘3+3+6’ designs and CRM.

		$p_{\mathrm{true}}$	$p_{\mathrm{MTD}}$	# of patients	# of DLTs
	$\theta =0.25$
TT	Mean	0.575	0.794	20.922	4.282
	Sd	0.106	0.147	3.264	1.290
mTPI	Mean	0.568	0.791	20.695	4.257
	Sd	0.089	0.147	2.970	1.278
mTPI-2	Mean	0.558	0.808	20.238	4.021
	Sd	0.088	0.139	3.138	1.152
BOIN	Mean	0.244	0.908	15.551	2.108
	Sd	0.082	0.090	4.030	0.470
CRM	Mean	0.553	0.778	21.000	4.477
	Sd	0.162	0.163	0.000	1.648
	$\theta =0.30$
TT	Mean	0.407	0.823	20.551	4.725
	Sd	0.098	0.144	3.589	0.759
mTPI	Mean	0.389	0.830	19.774	4.515
	Sd	0.105	0.140	3.142	0.725
mTPI-2	Mean	0.375	0.841	19.348	4.323
	Sd	0.103	0.135	3.409	0.630
BOIN	Mean	0.122	0.922	12.771	2.222
	Sd	0.079	0.091	3.440	0.253
CRM	Mean	0.372	0.856	21.000	5.053
	Sd	0.280	0.136	0.000	1.416
	$\theta =0.35$
TT	Mean	0.292	0.602	17.888	4.902
	Sd	0.109	0.309	3.483	0.242
mTPI	Mean	0.289	0.621	16.984	4.631
	Sd	0.112	0.313	2.773	0.249
mTPI-2	Mean	0.290	0.629	16.441	4.422
	Sd	0.120	0.316	3.031	0.183
BOIN	Mean	0.222	0.727	9.505	2.296
	Sd	0.270	0.345	2.156	0.101
CRM	Mean	0.310	0.626	21.000	5.801
	Sd	0.250	0.310	0.000	1.128

Note: The table reports the average and standard deviation of $p_{\mathrm{true}}$, $p_{\mathrm{MTD}}$, number of patients treated and number of DLTs.

Table 7. Operating characteristics of TT 3+3 under three dose-toxicity scenarios.

	Dose level	1	2	3	4	5
Scenario 1	True toxicity	0.050	0.300	0.500	0.600	0.700
3+3	Selection(%)-1	0.536	0.387	0.050	0.001	0.000
	Selection(%)-2	0.272	0.578	0.142	0.007	0.001
	# of DLTs	0.233	1.510	1.062	0.180	0.014
	# of patients (N = 12.3)	4.803	5.130	2.097	0.285	0.021
TT 3+3	Selection(%)-1	0.319	0.531	0.131	0.013	0.000
	Selection(%)-2	0.259	0.573	0.154	0.013	0.000
	# of DLTs	0.213	1.484	1.129	0.185	0.022
	# of patients (N = 11.7)	4.218	4.989	2.232	0.309	0.030
mTPI 3+3	Selection(%)-1	0.218	0.537	0.224	0.017	0.000
	Selection(%)-2	0.232	0.563	0.188	0.016	0.000
	# of DLTs	0.191	1.391	1.169	0.180	0.005
	# of patients (N = 11.4)	3.975	4.842	2.307	0.288	0.006
BOIN 3+3	Selection(%)-1	0.612	0.226	0.025	0.003	0.000
	Selection(%)-2	0.322	0.550	0.119	0.007	0.002
	# of DLTs	0.223	1.023	0.515	0.104	0.009
	# of patients (N = 9.5)	4.848	3.456	1.083	0.177	0.015
Scenario 2	True toxicity	0.100	0.150	0.300	0.450	0.500
3+3	Selection(%)-1	0.182	0.406	0.241	0.045	0.000
	Selection(%)-2	0.167	0.324	0.367	0.116	0.025
	# of DLTs	0.445	0.686	1.177	0.759	0.175
	# of patients (N = 14.5)	4.206	4.563	3.774	1.686	0.357
TT 3+3	Selection(%)-1	0.134	0.329	0.351	0.123	0.011
	Selection(%)-2	0.154	0.308	0.374	0.132	0.029
	# of DLTs	0.407	0.633	1.051	0.744	0.182
	# of patients (N = 13.7)	3.945	4.176	3.579	1.680	0.363
mTPI 3+3	Selection(%)-1	0.121	0.303	0.360	0.145	0.027
	Selection(%)-2	0.153	0.321	0.357	0.132	0.036
	# of DLTs	0.379	0.610	1.067	0.733	0.167
	# of patients (N = 13.2)	3.828	4.017	3.486	1.602	0.348
BOIN 3+3	Selection(%)-1	0.294	0.308	0.100	0.008	0.000
	Selection(%)-2	0.326	0.359	0.252	0.053	0.010
	# of DLTs	0.396	0.482	0.548	0.216	0.021
	# of patients (N = 9.3)	3.909	3.162	1.713	0.468	0.060
Scenario 3	True toxicity	0.010	0.100	0.250	0.300	0.400
3+3	Selection(%)-1	0.115	0.387	0.263	0.151	0.000
	Selection(%)-2	0.034	0.232	0.400	0.191	0.143
	# of DLTs	0.025	0.505	1.128	0.808	0.448
	# of patients (N = 16.4)	3.405	4.698	4.524	2.607	1.182
TT 3+3	Selection(%)-1	0.034	0.269	0.320	0.247	0.050
	Selection(%)-2	0.032	0.239	0.338	0.253	0.138
	# of DLTs	0.031	0.438	1.049	0.791	0.493
	# of patients (N = 15.4)	3.177	4.257	4.188	2.664	1.203
mTPI 3+3	Selection(%)-1	0.035	0.225	0.340	0.239	0.087
	Selection(%)-2	0.035	0.239	0.338	0.240	0.148
	# of DLTs	0.035	0.442	1.056	0.705	0.455
	# of patients (N = 14.9)	3.201	4.128	4.056	2.418	1.125
BOIN 3+3	Selection(%)-1	0.263	0.437	0.200	0.050	0.000
	Selection(%)-2	0.064	0.341	0.396	0.154	0.045
	# of DLTs	0.043	0.417	0.706	0.375	0.101
	# of patients (N = 12.2)	3.789	4.242	2.844	1.113	0.261

Notes: Selection (%): probability of selection current dose level as the MTD (1- MTD estimated via algorithm in Figure 1; 2- MTD estimated via isotonic regression); # of DLTs: number of DLTs at current dose level; # of patients: number of patients treated at current dose level; N: total number of treated patients. The target toxicity rate is 0.3.

Table 8. Operating characteristics of TT 3+3+6 design under three dose-toxicity scenarios.

	Dose level	1	2	3	4	5
Scenario 1	Toxicity rate	0.050	0.300	0.500	0.600	0.700
TT 3+3+6	Selection(%)-1	0.373	0.532	0.085	0.003	0.000
	Selection(%)-2	0.264	0.613	0.116	0.007	0.000
	# of DLTs	0.322	2.534	1.765	0.299	0.045
	# of patients (N = 19.0)	6.336	8.565	3.552	0.525	0.060
mTPI 3+3+6	Selection(%)-1	0.358	0.551	0.081	0.002	0.000
	Selection(%)-2	0.243	0.613	0.135	0.009	0.000
	# of DLTs	0.272	2.616	1.907	0.302	0.011
	# of patients (N = 19.3)	6.132	8.928	3.840	0.471	0.015
mTPI-2 3+3+6	Selection(%)-1	0.419	0.500	0.070	0.003	0.000
	Selection(%)-2	0.252	0.629	0.113	0.005	0.001
	# of DLTs	0.338	2.546	1.539	0.264	0.013
	# of patients (N = 18.5)	6.642	8.385	3.030	0.450	0.021
BOIN 3+3+6	Selection(%)-1	0.648	0.199	0.003	0.000	0.000
	Selection(%)-2	0.314	0.589	0.095	0.002	0.000
	# of DLTs	0.439	1.429	0.595	0.089	0.000
	# of patients (N = 14.7)	8.613	4.818	1.158	0.144	0.000
Scenario 2	Toxicity rate	0.100	0.150	0.300	0.450	0.500
TT 3+3+6	Selection(%)-1	0.107	0.353	0.392	0.079	0.007
	Selection(%)-2	0.120	0.299	0.425	0.127	0.026
	# of DLTs	0.489	1.070	2.075	1.410	0.302
	# of patients (N = 22.1)	4.827	6.783	6.849	3.102	0.594
mTPI 3+3+6	Selection(%)-1	0.119	0.361	0.375	0.079	0.011
	Selection(%)-2	0.117	0.290	0.444	0.127	0.022
	# of DLTs	0.484	0.995	2.047	1.359	0.308
	# of patients (N = 21.5)	4.797	6.369	6.708	3.006	0.696
mTPI-2 3+3+6	Selection(%)-1	0.108	0.377	0.401	0.063	0.001
	Selection(%)-2	0.131	0.305	0.425	0.108	0.029
	# of DLTs	0.473	0.968	2.102	1.170	0.207
	# of patients (N = 21.0)	4.734	6.600	6.852	2.496	0.411
BOIN 3+3+6	Selection(%)-1	0.297	0.319	0.090	0.005	0.000
	Selection(%)-2	0.328	0.339	0.272	0.050	0.009
	# of DLTs	0.571	0.805	0.737	0.270	0.038
	# of patients ($N=$13.8)	5.700	5.052	2.379	0.588	0.081
Scenario 3	Toxicity rate	0.010	0.100	0.250	0.300	0.400
TT 3+3+6	Selection(%)-1	0.037	0.263	0.297	0.242	0.025
	Selection(%)-2	0.028	0.192	0.343	0.268	0.169
	# of DLTs	0 .035	0.604	1.705	1.569	0.807
	# of patients(N = 23.3)	3.420	6.030	6.702	5.127	2.028
mTPI 3+3+6	Selection(%)-1	0.025	0.231	0.310	0.281	0.051
	Selection(%)-2	0.043	0.191	0.357	0.257	0.152
	# of DLTs	0.030	0.490	1.648	1.429	0.860
	# of patients (N = 22.6)	3.306	5.436	6.729	5.049	2.163
mTPI-2 3+3+6	Selection(%)-1	0.038	0.278	0.328	0.239	0.028
	Selection(%)-2	0.033	0.223	0.343	0.269	0.132
	# of DLTs	0.042	0.617	1.653	1.361	0.677
	# of patients (N = 22.1)	3.462	5.964	6.627	4.491	1.635
BOIN 3+3+6	Selection(%)-1	0.285	0.471	0.138	0.054	0.000
	Selection(%)-2	0.067	0.369	0.364	0.154	0.046
	# of DLTs	0.046	0.715	0.880	0.400	0.124
	# of patients (N = 17.8)	5.553	7.128	3.474	1.404	0.312

Notes: Selection(%): probability of selection current dose level as the MTD (1- MTD estimated via algorithm in Figure 1; 2- MTD estimated via isotonic regression); # of DLTs: number of DLTs at current dose level; # of patients: number of patients treated at current dose level; N: total number of treated patients. The target toxicity rate is 0.3.

Table 9. Comparison of TT ‘3+3’ and TT ‘3+3+6’.

		Prob. MTD		Over MTD		# of Patients
Dose level	Truth	3+3	3+3+6	3+3	3+3+6	3+3	3+3+6
1	0.20	0.373	0.324	0.141	0.156	4.608	6.912
2	0.30	0.307	0.369	0.514	0.480	3.477	6.552
3	0.40	0.148	0.125	0.821	0.849	1.698	3.477
4	0.50	0.026	0.024	0.969	0.974	0.501	1.032
5	0.60	0.002	0.000	1.000	1.000	0.075	0.132

Notes: Truth: true toxicity rate; Prob. MTD: the probability of selecting current dose level as the MTD; Over MTD: the probability of selecting current dose level as over the MTD; # of Patients: the average number of subjects treated at the current dose level.