In order to comprehensively evaluate the effects of toxicities to patients, Bekele and Thall (2004) [13] first proposed the toxicity burden as a weighted sum of individual toxicity types and grades, where the severity weights reflect the relative health impact of each grade and type of toxicity. The proposed Ti3+3 design utilizes type-specific and overall toxicity burdens similar to Bekele and Thall (2004), which is described next.

Let d ∈ { 1 , … , T } denote a set of ascending doses explored in a phase I trial. Assume multiple types of treatment-related toxicity j ∈ { 1 , … , J } are observed for patients, and each type of toxicity is classified into toxicity grades k ∈ { 0 , … , K } using a standard reference, such as the CTCAE [1]. For example, neurotoxicity and GI toxicity are two different types of adverse events, each consisting of five grades with grade 0 denoting no toxicity, 1-2 moderate toxicity, 3-4 severe toxicity, and 5 death. Toxicities with grade 5, corresponding to treatment-related death, require trial suspension and direct intervention from the safety committee and therefore are not considered in the proposed design.

Specifically, let Y i j ∈ { 0 , … , K } denote the observed toxicity grade of type j for patient i; and let { X i = d } denote the event that patient i is treated at dose d. Denote p d j k = P r ( Y i j = k | X i = d ) the toxicity probability of grade k for type j at dose d, and apparently ∑ k = 0 K p d j k = 1, 0 ≤ p d j k ≤ 1. The proposed Ti3+3 design relies on a weight matrix W that is elicited through consultation with clinicians. Let W = { w j k } be a standardized matrix of weights, W = w 10 … w 1 K ⋮ ⋱ ⋮ w J 0 … w J K , where ∑ j , k w j k = 1. Here w j k ≥ 0 quantifies the relative health impact assigned by physicians for the toxicity event of type j and grade k. Denote the row sum c j = ∑ k = 0 K w j k . The magnitudes of { c j }’s reflect the relative severity of different types of toxicity events; within each type j, the magnitudes of { w j k }’s reflect the relative average severity of different grades of toxicity events. These interpretations are exploited in the definitions of the toxicity burdens. In Appendix A, we describe an algorithm to guide the elicitation of the W matrix by statisticians and clinicians. To reflect the belief that higher grade of a type of toxicity is more impactful to a patient’s health, we assume the monotonicity 0 = w j 0 < ⋯ < w j K for any type j, which implies an increasing toxicity burden for high toxicity grades.

Next, we define a type-specific toxicity burden for toxicity type j at dose d as (2.1) T B d j = ∑ k = 0 K w j k w j K p d j k , where T B d j ∈ ( 0 , 1 ) is analogous to the toxicity probability in DLT-based dose-finding trials. To see this, T B d j in (2.1) is a weighted sum of the type-specific toxicity probabilities of different grades, with the weight equal to 1 for the highest grade K, and w j k w j K for grade k < K. Since w j k < w j K for k < K, w j k w j K < 1. Therefore, (2.1) implies that T B d j is a re-scaled probability of toxicity of the highest grade K for type j at dose d, converting all the lower grade toxicities to the highest grade by a weight factor of w j k w j K . Note that T B d j is a parameter. To estimate it, we consider the following statistics, T B ˆ i d j , the observed type-specific toxicity burden for patient i treated at dose d, given by (2.2) T B ˆ i d j = ∑ k = 0 K w j k w j K I ( Y i j = k ) I ( X i = d ) , where I ( · ) is an indicator function and I ( Y i j = k ) = 1 , if patient i experiences toxicity type j with grade k , 0 , otherwise.

Here T B ˆ i d j is based on observed data { Y i j , X i }. Since a patient may experience multiple adverse events associated with multiple types and grades of toxicity, we only use the most severe grade of each type in defining the toxicity burdens. In other words, grade “k” in (2.2) is the most severe toxicity grade among all the toxicity events of type j for patient i. In addition, for patient i, T B ˆ i d = ∑ j = 1 J c j ∑ k = 0 K w j k w j K I ( Y i j = k ) I ( X i = d ) is the observed toxicity burden for patient i. Assuming a total of n d patients have been treated at the dose d, then T B d j can be estimated by (2.3) T B ˆ d j = 1 n d ∑ i = 1 n d T B ˆ i d j . It is trivial to show that T B ˆ d j is unbiased, i.e., E ( T B ˆ d j ) = T B d j , assuming w j k is given and fixed.

Finally, given the type-specific toxicity burdens T B d j , the overall toxicity burden at dose d can be defined as (2.4) T B d = ∑ j = 1 J c j T B d j = ∑ j = 1 J c j ∑ k = 0 K w j k w j K p d j k , where c j = ∑ k = 0 K w j k are constants that reflect the relative severity between toxicity types. Apparently, ∑ j c j = ∑ j , k w j k = 1. Similarly, the observed overall toxicity burden at dose d is calculated as follows (2.5) T B ˆ d = ∑ j = 1 J c j T B ˆ d j .

An interval dose-finding design like i3+3 needs to specify the target toxicity probability (of DLTs), p T , and an equivalence interval ( E I) to facilitate the dose-finding decisions. Similarly, the proposed Ti3+3 design needs a target toxicity burden ( T T B) and an associated E I for T T B. To start, we define a type-specific T T B j for toxicity type j. The T T B j can be viewed as the toxicity target for the MTD if only toxicity events of type j are considered as outcome; similarly, the E I j is the equivalence interval for the MTD when only type j toxicity events are modeled. For example, assume that there are two types of toxicities J = 2, and the targets for type-specific toxicity burden could be set to T T B 1 = 0.3 and T T B 2 = 0.25. In addition, denote E I j = ( T T B j − ϵ 1 j , T T B j + ϵ 2 j ) the equivalence interval of T T B j for toxicity type j, which is an interval centered around the T T B j . The upper bound and lower bound of the E I j reflect the highest and lowest value of toxicity burden that the clinicians would consider to be acceptable for MTD if only type j events are considered. Similar to how the definition of the DLT is tailored case-by-case for conventional dose-finding trials, these values should be elicited with the clinical team based on the specific context of a trial. Given the specified T T B j and Equation 2.4, an overall target toxicity burden, denoted as T T B, is defined as a weighted sum of T T B j by (2.6) T T B = ∑ j = 1 J c j T T B j . And similarly, the equivalence interval for the overall toxicity burden, E I, is derived as a weighted average of E I j with weights c j . That is, E I = ∑ j = 1 J c j E I j .

In summary, the main effort in applying the proposed Ti3+3 is in the initial setup, which requires the specification of the weight matrix, W, the target toxicity burden, T T B j , and the equivalence interval, E I j , for each toxicity type j. Once they are determined, dose finding proceeds based on a simple algorithm next.

Assume patients are assigned sequentially in cohorts, starting with the lowest dose. The next cohort of patients will not be enrolled until toxicity outcomes have been observed for the present cohort. Suppose dose d is the current dose used to treat patients and n d patients have been treated at the dose, Ti3+3 extends the dose-finding algorithm in the i3+3 design to accommodate the new toxicity burden endpoints. The dose-finding algorithm of Ti3+3 are first applied to toxicity type j to generate a type-specific decision, denoted as A j ∈ { “ E ” , “ S ” , “ D ” }, where “E”, “S”, and “D” denote “Escalation”, “Stay”, and “De-escalation”, respectively, and to the overall toxicity burden to generate A 0 ∈ { “ E ” , “ S ” , “ D ” }. The next cohort of patients is assigned to the minimum dose level indicated by decisions A j ’s and A 0 .

Below we introduce the dose-finding algorithm.

Algorithm 1 requires computing a new quantity T B ˆ d , − 1 j , which is defined as a hypothetical observed toxicity burden assuming the patient in the cohort who experienced the lowest toxicity burden would experience no toxicity at all. In other words, if we remove all the toxicity events from the patient with the lowest burden from the data at dose d, T B ˆ d , − 1 j is the new observed toxicity burden for type j. This idea is similar to that of the i3+3 design, the difference being that i3+3 only considers DLT while Ti3+3 considers different grades and types of toxicity.

Table 1 summarizes the decisions of Ti3+3. The type-specific decision B j ( A j ) for each toxicity type j (or overall burden if j = 0) is listed, and the final decision is to assign the next cohort to dose ( d + min j { B j } ). Similar to i3+3, in Ti3+3, there is a special rule. When T B ˆ d j > E I j and T B ˆ d , − 1 j < E I j , it indicates that removing the toxicity events from a single patient in the observed data renders the observed toxicity burden from being above the equivalence interval to below the interval. In other words, changing one-patient worth of information of the data would result in a reversal of the decision from de-escalation (since T B ˆ d j > E I j ) to escalation since ( T B ˆ d , − 1 j < E I j ). This implies that the information in the observed data is sparse and small change of the data results in reversal of decisions. Therefore, Ti3+3, in this case, does not de-escalate due to lack of confidence in the data, and instead, continues to treat patients at the current dose, i.e., “S” stay. The other rules in Table 1 are straightforward, following the idea of de-escalation if observed toxicity burden is above the E I, stay if inside the E I, or escalation if below the E I.

In Table 2, we provide three examples to illustrate the proposed algorithm. Suppose there are two types of toxicities, and a pre-determined standardized weight matrix is given below W = 0 0.03 0.11 0.17 0.42 0 0.03 0.03 0.07 0.14 . Moreover, assume T T B 1 = T T B 2 = T T B = 0.30 and the E I 1 = E I 2 = E I = ( 0.25 , 0.33 ). Based on the observed toxicity types and grades from patients, T B ˆ i d and T B ˆ d can be calculated based on (2.5). In case 1, n d = 3 patients are treated at the current dose d, and T B ˆ d = 0.34, which is greater than the upper bound 0.33. However, since T B ˆ d , − 1 = 0.24 falls below the E I, according to the Ti3+3 algorithm, the decision is, A 0 = “ S ”, stay at the current dose. Therefore, even though T B ˆ d = 0.34 is above the EI, the decision is to enroll more patients at the same dose d since T B ˆ d , − 1 is below E I. We see that there are only three patients at dose d and the data is sparse. In the second case, n d = 5 and T B ˆ d = 0.34, which is greater than 0.33. And T B ˆ d , − 1 = 0.28 falls inside of the E I, therefore, A 0 = “ D ”, de-escalate to the next lower dose ( d − 1 ). In the third case, the two quantities T B ˆ d , T B ˆ d , − 1 are the same and below the EI, and the decision is then A 0 = “ E ”, escalate to the dose ( d + 1 ). The examples demonstrate the simplicity of the proposed decision rules using the overall toxicity burden. Algorithm 1 applies these rules to each toxicity type j as well.

In addition, the Ti3+3 design consists of a few safety rules for practical and ethical concerns. Again, these safety rules are applied iteratively to each type-specific and overall toxicity burdens.

The calculation of the posterior distributions P r ( T B d j > T T B j | data ) and P r ( T B d > T T B | data ) are discussed below.

Once all patients finish their followup at the end of a trial, the Ti3+3 design first selects the type-specific MTD, denoted as d j ∗ , and the MTD based on the overall toxicity burden, denoted as d 0 ∗ .

Next, we propose a working statistical model to calculate the posterior probabilities P r ( T B d j > T T B j | data ) and P r ( T B d > T T B | data ) and select the MTD based on the observed data. Recall p d j k denote the probability of toxicity grade k for type j at dose d. For a given dose d, assume different types of toxicities are independent. Let p d j = { p d j 0 . … , p d j K } represent the vector of the probabilities associated with different toxicity grades for type j at dose d, and y d j = { y d j 0 . … , y d j K } the vector of patient counts across different grades, i.e., y d j k = ∑ i I ( Y i j = k ) I ( X i = d ). Then y d j is assumed to follow a the multinomial sampling distribution given by (2.7) y d j | p d j ∼ Multinomial ( p d j , n d ) , where for any j, ∑ k = 0 K p d j k = 1, p d j k ∈ ( 0 , 1 ), and ∑ k = 0 K y d j k = n d . Assume a conjugate Dirichlet prior distribution of p d j , i.e., (2.8) p d j ∼ Dirichlet ( α ) , where α = ( α 0 , … , α K ) are positive values. We set α 0 = ⋯ = α K = 0.1 for different toxicity types across doses. Following Morita et al. (2011) [23], since ∑ k = 0 4 α k = 0.5, a small value, the Dirichlet prior in (2.8) is deemed vague and has little impact on the posterior distribution given by (2.9) p d j | y d j ∼ Dirichlet ( y d j 0 + α 0 , … , y d j K + α K ) . Therefore, the posterior of T B d j and T B d can be computed numerically by sampling f j ( p d j | y d j ) in 2.9. To calculate the posterior probabilities used in safety rules, we have (2.10) P r ( T B d j > T T B j | data ) ≈ 1 S ∑ s = 1 S I ( ∑ k = 0 K w j k w j K p d j k ( s ) > T T B j ) , and (2.11) P r ( T B d > T T B | data ) ≈ 1 S ∑ s = 1 S I ( ∑ j = 1 J c j ∑ k = 0 K w j k w j K p d j k ( s ) > T T B ) , where p d j k ( s ) is the s-th random draw from f j ( p d j | y d j ).

To impose monotonicity assumption of dose-toxicity relationship, we apply isotonic regression to the posterior means of T B d j and T B d via the pool adjacent violators algorithm [24]. Let T B ˜ d j and T B ˜ d be the isotonic transformed posterior means for all dose levels. Among all the tried doses ( n d > 0) for which satisfy the safety rules, the estimated MTDs based on the type-specific and overall toxicity burden are defined as (2.12) d j ∗ = arg min d | T B ˜ d j − T T B | , j = 0 , 1 , … , J . If more than one dose of d j ∗ exists, only one dose is selected based on following rules:

Lastly, the final MTD is the smallest dose among the estimated type-specific and overall MTDs, i.e., (2.13) d ∗ = m i n { d j ∗ , j = 0 , . . , J } .

We simulate clinical trials to assess the operating characteristics of the Ti3+3 design. We first compare to the CRM-MC method by Lee et al. (2011) [16]. The Ti3+3 design requires close collaboration between statisticians and clinicians to define the numerical weights W and the E I’s at the study design stage. For simulation purpose, we adopt the general setting of the bortezomib trial described in Lee et al. (2011) [16]. Under the bortezomib trial, two main types ( J = 2 ) of toxicities are identified as related to the treatment. The first type ( j = 1 ) is neuropathy and the second ( j = 2 ) is low platelet count. The standardized matrix of weights W, provided in Lee et al. (2011), is given by: W = 0 0.03 0.11 0.17 0.42 0 0.03 0.03 0.07 0.14 . The first and second rows correspond to the weights for grade 0 to grade 4 of each type, neuropathy and low platelet count, respectively. The DLT is defined as the occurrence of a grade 3 or 4 neuropathy or a grade 4 low platelet count. Given this matrix and the definition of T B, a patient has a T B ˆ i d value 0.30 for a grade 3 neuropathy or 0.27 for a grade 4 low platelet count. And a patient experiencing a grade 3 neuropathy has similar T B ˆ i d = 0.30 as a patient experiencing a grade 2 neuropathy plus a grade 3 low platelet count T B ˆ i d = 0.31. These severity weights reflect the relative clinical importance between different toxicity types and grades, and less severe toxicities are given lower weights. In addition, c 1 = ∑ k = 0 4 w 1 k = 0.73 and c 2 = ∑ k = 0 4 w 2 k = 0.27, which implies that overall neuropathy is a much more severe type of toxicity than low platelet count.

To implement the Ti3+3 design, the T T B 1 , T T B 2 , and T T B are set at 0.30, which suggests (by comparing to W) that the occurrence of a grade 3 or higher neuropathy, or a grade 4 low platelet count are considered clinically unacceptable. Moreover, the equivalence intervals used are E I 1 = E I 2 = E I = ( 0.25 , 0.35 ). For the CRM-MC method, the toxicity burden of patient i is calculated as T i = 0.03 I ( Y i 1 = 1 ) + 0.11 I ( Y i 1 = 2 ) + 0.17 I ( Y i 1 = 3 ) + 0.42 I ( Y i 1 = 4 ) + 0.03 I ( Y i 2 = 1 , 2 ) + 0.07 I ( Y i 2 = 3 ) + 0.14 I ( Y i 2 = 4 ), where Y i 1 and Y i 2 are the grades of toxicity type 1 and 2, respectively, for the patient. CRM-MC applies primary and secondary constraints P r ( T i ≥ 0.25 | d ) ≤ 0.10 and P r ( T i ≥ 0.17 | d ) ≤ 0.25, respectively, to decide the dose for next patients. The vector of scaled doses is obtained via backward substitution with dose 3 as the prior guess of MTD, 0.08 as the indifference interval parameter, 0.69 as the prior median of the slope parameter, and a probit model with an intercept equal to 3. The prior distributions of the probit model slope parameter β and thresholds ( γ l − γ l − 1 ) follow independent exponential distributions with mean 1, as suggested by the authors. Refer to Lee et al. (2011) for more details.

Following the original bortezomib trial, Lee et al. (2011) implemented a simulation with a sample size 18, cohort size 1, and a starting dose at level 3. We consider a more common simulation setup and apply the following parameters to both designs. The sample size is fixed at 30 with a total of 10 cohorts. A total of five dose levels are investigated, and both designs, Ti3+3 and CRM-MC will start at the lowest dose level. We construct a total of eight scenarios, including the first four scenarios from Lee et al. (2011) and additional four scenarios. According to Ti3+3, the true MTD is defined as the highest dose with the T B d j falls below or inside the equivalence interval of T T B j for all type-specific and overall toxicity burdens, that is, the dose with T B d j ≤ 0.35 for j = 1 , 2 and T B d ≤ 0.35. Different from Ti3+3, the CRM-MC design selects MTD as the highest dose that satisfies the primary and secondary toxicity constraints. Therefore, CRM-MC and Ti3+3 sometimes would consider different doses in the same scenario as the true MTD, for example, in scenario 1, Ti3+3 considers dose level 3 as the true MTD while based on CRM-MC, dose level 2 is the true MTD. A full description of all dosing scenarios is provided in Appendix B. For each scenario, we simulate 1,000 trials.

The simulation results are presented in Table 3, which shows the percentage of recommending a particular dose (Selection %) and the percentage of patients assigned to each dose level (Allocation %). A desirable design should demonstrate a good balance between the ability to correctly identify the MTD and patient safety. Compared with the CRM-MC design, Ti3+3 has higher percentage of correct selection (PCS) among scenarios where both design consider the same dose as the true MTD (scenarios 2 , 3 , 4 , 6, and 8). In terms of patient allocation, Ti3+3 seems to perform better at assigning patients to the target dose (PCA) in scenarios 2-6, and is less likely to assign patients to a dose above the MTD (POA) in majority of the eight scenarios. Overall, the operating characteristics of our proposed Ti3+3 design are comparable to the CRM-MC design, which relies on a model-based inference for dose assignment decisions.

Additionally, we evaluate the performance of Ti3+3 by comparing it with the gBOIN design by Mu et al. (2019) [19]. The gBOIN design generalizes the BOIN design and provides a unified framework to incorporate non-binary toxicity outcomes. We follow the simulation settings described in Section 3.2 in Mu et al. (2019). Only one type of toxicity is considered, and the standardized weight matrix is defined as W = 0 0 0.16 0.33 0.50 . In words, grades 0 and 1 are of no concern, grade 2 toxicity is considered equivalent to half grade 3 toxicity, and grade 4 toxicity is equivalent to one and half grade 3 toxicity. Based on Mu et al. (2019), the T T B is 0.31, and the recommended default escalation and de-escalation boundaries are λ e ∗ = 0.249 and λ d ∗ = 0.377. The same set of target burden and E I is adopted in Ti3+3. A total sample of 30 patients and a cohort size of 3 is used in the simulation, with a starting dose at dose level 1. The details of the ten dose-toxicity scenarios is provided in Appendix B.

Table 4 shows the results based on 4,000 simulated trials. In general, the two designs demonstrate comparable operating characteristics, and Ti3+3 shows superior performance in terms of trial safety. Even though gBOIN yields higher PCS in scenarios 1 through 7, the differences in PCS are no greater than 0.05 except in scenario 7. And Ti3+3 generates higher PCS in scenarios 8, 9 and 10. It is worthnoting that Ti3+3 is less likely to recommend a overly toxic dose as the MTD across all scenarios. The patient allocation of the two designs are very close as shown in Table 4. The Ti3+3 yields higher or similar PCA in 7 out of 10 scenarios, and shows consistently lower POA across all scenarios. The POS and POA are two important safety metrics considered in early phase trials, and Ti3+3 shows relatively strong performance in the safety metrics.