A system for determining maximum tolerated dose in clinical trial

Figures & data

Figure 1. Graph illustration for target toxicity tolerance probability in Theorem 2.3. (a) The target toxicity tolerance probability θ in system $(\mathbb{Y},\mathit{F},\theta )$; (b) The target toxicity tolerance probability ${\theta }^{\ast }$ in system $({\mathbb{Y}}^{\ast },{\mathit{F}}^{\ast },{\theta }^{\ast })$.

Figure 2. Graph illustration for $\mathit{F}$ in Theorem 2.3, where $x_1<x_2$. (a) $\mathit{F}$ in system $(\mathbb{Y},\mathit{F},\theta )$; (b) ${\mathit{F}}^{\ast }$ in system $({\mathbb{Y}}^{\ast },{\mathit{F}}^{\ast },{\theta }^{\ast })$.

Figure 3. Graph illustration for (a) the target toxicity tolerance probability with form (10(10) $\begin{array}{rl}{\theta }_{\alpha }& ={\theta }_{\alpha }\left(y\right)\\ & =\left\{\begin{array}{ll}1,& \mathrm{i}\mathrm{f}y\le y_0,\\ {\theta }_0+(1-{\theta }_0){\left(\frac{y_1-y}{y_1-y_0}\right)}^{\alpha },& \mathrm{i}\mathrm{f}{y}_0<y<y_1,\\ {\theta }_0,& \mathrm{i}\mathrm{f}y\ge y_1,\end{array}\right.\end{array}$(10) ) and (b) the negative derivative curve of the target toxicity tolerance probability.

Table 1. An Illustration of Proposition 3.3. The parameters for the true model are ${\text{β}}_0=-1$, ${\text{β}}_1=1$ and $\sigma =1$. The parameters for the target toxicity tolerance probability are $y_0=-1$, $y_1=1$ and ${\theta }_0=0.05$.

α	$y^{\ast }$	MTD(${\theta }_{\alpha }$)	α	$y^{\ast }$	MTD(${\theta }_{\alpha }$)
1	1.00	0.35	10	−0.34	−0.84
2	0.72	0.23	50	−0.79	−1.40
3	0.34	−0.04	100	−0.88	−1.51
4	0.14	−0.25	1000	−0.98	−1.63
5	0.00	−0.41	10000	−1.00	−1.64

Table 2. Setting of simulation study for the normal toxicity response.

Number of cohorts in one trial	10
Number of patients in each cohort	3
Number of patients in one trial	30
Number of simulations	200
Number of dose levels	6
Target toxicity probability	$y_0=-2$
	$y_1=4$
	${\theta }_0=0.01,0.10,0.30$
	$\alpha =0.2,1,5$
Model parameter	${\text{β}}_0=1$
	${\text{β}}_1=2$
	${\sigma }^2=0.25,1,2$

Table 3. Simulation results by generating 200 data sets for each scenario: each cell consists of two values $s\left(t\right)$, where s is the percentage that the dosage $d_i$ is recommended as an MTD, and t a percentage that the dosage is used in the experiment to search for the MTD. For each row, the dosage with largest percentage in MTD selection is boldfaced.

			True	Dose level
${\theta }_0$	${\sigma }^2$	α	MTD	$d_1:-1.47$	$d_2:-1.10$	$d_3:-0.69$	$d_4:-0.42$	$d_5:0.00$	$d_6:0.42$
0.01	0.25	0.02	−0.78	0.5 (11.25)	25.5 (25.00)	67.5 (52.25)	6.5 (11.0)	0.0 (0.5)
		1	−0.94	1.0 (11.80)	64.0 (49.40)	35.0 (36.80)	0.0 (2.0)
		5	−1.16	12.5 (22.35)	87.5 (74.75)	0.0 (2.90)
	1	0.02	−0.19	0.0 (15.00)	1.5 (10.95)	7.5 (14.35)	42.5 (27.65)	43.5 (27.45)	5.0 (4.60)
		1	−0.55	0.0 (14.75)	5.0 (13.75)	44.5 (31.75)	47.0 (33.00)	3.5 (7.20)	0.0 (0.15)
		5	−1.11	25.0 (38.30)	68.0 (51.65)	7.0 (9.65)	0.0 (0.40)
	2	0.02	0.24	2.0 (21.65)	5.5 (13.45)	12.5 (16.40)	30.5 (22.70)	40.0 (20.10)	9.5 (5.70)
		1	−0.31	3.5 (22.25)	7.0 (14.50)	23.5 (20.65)	42.0 (26.60)	22.0 (14.25)	2.0 (1.75)
		5	−1.26	61.0 (69.15)	37.0 (27.30)	2.0 (3.45)	0.0 (0.10)
0.10	0.25	0.02	−0.77	0.5 (10.65)	21.5 (24.05)	71.5 (52.10)	6.5 (12.55)	0.0 (0.65)
		1	−0.93	1.0 (11.20)	62.5 (46.95)	36.5 (39.55)	0.0 (2.30)
		5	−1.14	1.0 (19.70)	88.5 (76.40)	0.5 (3.90)
	1	0.02	−0.18	0.0 (11.25)	1.5 (11.00)	5.5 (13.55)	38.0 (26.20)	47.5 (30.60)	7.5 (7.40)
		1	−0.53	0.0 (11.50)	5.5 (13.15)	37.0 (30.50)	52.5 (35.60)	5.0 (8.95)	0.0 (0.30)
		5	−1.05	12.5 (26.45)	71.0 (56.75)	16.0 (15.90)	0.5 (0.90)
	2	0.02	0.27	0.0 (14.10)	0.0 (10.00)	0.5 (10.60)	6.5 (13.10)	34.5 (21.85)	58.5 (30.35)
		1	−0.27	0.5 (14.45)	1.5 (11.00)	13.5 (16.85)	39.5 (25.95)	39.5 (26.40)	5.5 (5.35)
		5	−1.09	35.0 (47.35)	53.5 (41.35)	11.5 (10.20)	0.0 (1.10)
0.30	0.25	0.02	−0.75	0.0 (10.25)	19.0 (22.45)	70.5 (52.10)	10.5 (14.35)	0.0 (0.85)
		1	−0.90	1.0 (10.70)	51.0 (41.50)	47.0 (44.20)	1.0 (26.45)
		5	−1.10	6.0 (14.90)	92.0 (76.55)	2.0 (8.45)	0.0 (0.10)
	1	0.02	−0.13	0.0 (10.15)	1.5 (10.80)	4.5 (13.35)	32.0 (23.40)	53.0 (32.60)	9.0 (9.70)
		1	−0.47	0.0 (10.35)	2.0 (11.70)	31.5 (25.45)	55.0 (38.35)	12.0 (13.45)	0.0 (0.70)
		5	−0.92	4.0 (14.35)	50.5 (43.45)	40.0 (38.10)	1.5 (4.05)	0.0 (0.05)
	2	0.02	0.34	0.0 (10.45)	0.0 (10.00)	0.5 (10.40)	5.0 (12.15)	26.5 (18.90)	68.0 (38.10)
		1	−0.17	0.0 (10.40)	1.0 (11.00)	6.0 (13.50)	34.0 (23.90)	52.5 (31.85)	6.5 (9.35)
		5	−0.85	7.5 (19.95)	46.0 (40.05)	39.0 (30.90)	7.0 (8.30)	0.5 (0.80)

Figure 4. Graph illustrations for the $\theta \left(y\right)$ functions discussed in Table 4.

Table 4. Choices for the target toxicity tolerance probability functions by (10(10) $\begin{array}{rl}{\theta }_{\alpha }& ={\theta }_{\alpha }\left(y\right)\\ & =\left\{\begin{array}{ll}1,& \mathrm{i}\mathrm{f}y\le y_0,\\ {\theta }_0+(1-{\theta }_0){\left(\frac{y_1-y}{y_1-y_0}\right)}^{\alpha },& \mathrm{i}\mathrm{f}{y}_0<y<y_1,\\ {\theta }_0,& \mathrm{i}\mathrm{f}y\ge y_1,\end{array}\right.\end{array}$(10) ).

$(y_0,y_1)$	${\theta }_0$	${\theta }^{\ast }$	$y^{\ast }$	α
$(-2,4)$	0.10	0.3	$-1$	8.2496
$(-2,4)$	0.10	0.3	1	2.1699
$(-2,4)$	0.10	0.3	3	0.8394
$(-2,4)$	0.01	0.3	$-1$	6.7344
$(-2,4)$	0.01	0.3	1	1.7714
$(-2,4)$	0.01	0.3	3	0.6853

Table 5. Simulation results of the recommended MTD levels, as well as dosage allocation percentages for different choices of ${\theta }^{\ast }$ and $y^{\ast }$.

		Dose level
${\sigma }^2$	$y^{\ast }$	$d_1$	$d_2$	$d_3$	$d_4$	$d_5$	$d_6$
		${\theta }_0=0.01$
0.25	−1	24.5 (33.35)	75.5 (65.65)	0.0 (1.00)
	1	2.0 (12.80)	83.0 (64.90)	15.0 (21.60)	0.0 (7.00)
	3	0.5 (11.65)	51.0 (41.85)	47.0 (42.75)	1.5 (3.75)
1	−1	49.0 (59.75)	48.0 (37.40)	3.0 (2.75)	0.0 (0.10)
	1	1.0 (15.55)	10.5 (17.55)	73.0 (47.55)	15.5 (18.00)	0.0 (1.35)
	3	0.0 (14.70)	2.0 (11.75)	33.0 (25.00)	50.5 (35.00)	14.5 (12.75)	0.0 (0.80)
2	−1	83.0 (86.20)	15.5 (12.80)	1.5 (1.00)
	1	5.0 (22.90)	18.0 (19.15)	39.0 (29.95)	0.0 (22.70)	4.0 (5.05)	0.0 (0.25)
	3	2.0 (21.65)	6.0 (14.15)	18.5 (18.95)	40.0 (25.40)	30.5 (17.35)	0.25 (2.50)
		${\theta }_0=0.10$
0.25	−1	31.0 (38.05)	69.0 (61.10)	0.0 (0.85)
	1	2.5 (12.10)	86.0 (68.20)	11.5 (19.20)	0.0 (0.50)
	3	1.0 (10.90)	53.0 (43.45)	45.0 (42.65)	1.0 (3.00)
1	−1	47.5 (56.80)	51.0 (40.45)	1.5 (2.65)	0.0 (0.10)
	1	1.0 (12.50)	17.5 (20.95)	67.0 (49.20)	14.5 (16.25)	0.0 (1.10)
	3	0.5 (11.70)	3.5 (11.85)	32.5 (27.00)	55.0 (37.50)	8.5 (11.25)	0.0 (0.70)
2	−1	76.5 (79.35)	23.0 (19.00)	0.5 (1.65)
	1	2.0 (16.70)	17.5 (20.45)	45.5 (33.55)	32.0 (23.75)	3.0 (5.40)	0.0 (0.15)
	3	0.0 (14.15)	1.0 (10.65)	6.0 (13.90)	36.5 (25.15)	51.0 (28.75)	5.5 (7.40)

Table 6. Simulation results of the recommended MTD levels, as well as dosage allocation percentages for ‘dichotomous’ cases. Two decision criteria are employed: criterion I is to choose next dosage level at a nearest dose, and criterion II is to choose next dosage level at the lower bound.

		Dose level
	${\sigma }^2$	$d_1$	$d_2$	$d_3$	$d_4$	$d_5$	$d_6$
Criterion I	0.25	0.0 (10.30)	0.0 (10.00)	0.0 (10.00)	22.5 (21.85)	77.5 (47.70)	0.0 (0.15)
	1	0.0 (10.50)	0.0 (10.05)	3.5 (12.45)	56.0 (39.40)	40.0 (25.80)	0.5 (1.80)
	2	0.5 (14.50)	3.5 (12.85)	21.0 (21.15)	50.5 (31.90)	24.0 (17.85)	0.5 (1.75)
Criterion II	0.25	0.0 (10.15)	0.0 (10.10)	0.5 (10.45)	91.5 (60.30)	8.0 (8.95)	0.0 (0.05)
	1	0.0 (10.10)	0.0 (10.40)	15.0 (18.85)	48.2 (32.20)	8.0 (10.95)	0.0 (0.50)
	2	1.5 (15.15)	14.0 (18.75)	26.0 (25.15)	52.0 (31.90)	6.5 (8.05)	0.0 (0.70)