The continuation-ratio (CR) model is frequently used in dose response studies to model a three-category outcome as the dose levels vary. Design issues for a CR model defined on an unrestricted dose interval have been discussed for estimating model parameters or a selected function of the model parameters. This paper uses metaheuristics to address design issues for a CR model defined on any compact dose interval when there are one or more objectives in the study and some are more important than others. Specifically, we use an exemplary nature-inspired metaheuristic algorithm called particle swarm optimization (PSO) to find locally optimal designs for estimating a few interesting functions of the model parameters, such as the most effective dose (

In drug discovery, efficacy and toxicity of the drug are the two of the most important endpoints in early phase trials. In phase I clinical trials, a main goal is to describe the dose-limiting toxicities (

To model trinomial responses above, [

The continuation-ratio (CR) model is a more flexible alternative because it does not have the restrictive assumption as the PO model does as described in Chapter 9 in [

[

Recently a class of algorithms called nature-inspired metaheuristic algorithms has proved very popular in the optimization literature. [

PSO is a metaheuristic optimization algorithm inspired from the way animals, such as birds and fishes, search for food. The birds fly continuously in the sky to look for food on the ground. Each has its own perception where the food is (local optimum) but it communicates with the rest and collectively decide as a flock where the food is (global optimum). Accordingly, each bird flies toward the global optimum in the direction of the local optimum (not giving up completely where it thinks the food is). Birds are referred as particles and each bird represents a candidate solution to the optimization problem. Velocities and locations of each bird are adjusted at each iteration and if and when the flock converges, the perceived global optimum is found. In order to efficiently identify the optimal points, we initiate a flock of birds in the pre-defined search space. Let

The current literature on optimal designs is replete with single objective optimal designs. In practice, there are frequently two or more objectives of interest in a study and they may be of unequal interest to the researcher. There is some work on two and three-objective optimal design problems and we provide a brief review of them in this section.

When there are two objectives, the more important objective is the primary objective, and the other is the secondary objective. The sought dual-objective optimal design is one that optimizes the primary objective first, before optimizing the second objective and ensuring that the obtained design has a higher efficiency for the first objective. When there are multiple objectives, a similar idea applies; the objectives are first prioritized in terms of their importance and the sought design is the one that delivers higher efficiencies for the more important objectives. For example, in a model based dose response study, a researcher may wish to find a three-objective optimal design and in order of importance, the goals are to estimate the maximum tolerated dose,

For the simple case, when we have two-objective optimal designs and we have two linear and quadratic polynomials as competing regression models, [

The simplicity of the above approach is valid for dual-objective optimal design problems. When there are multiple-objectives, the efficiency plot becomes high dimensional and it becomes difficult to visually identify the correspondence between which compound optimal design is the sought constrained optimal design. Unlike the two-objective optimal design problems, there is no explicit easy analytical description of the multi-objective designs, especially when the models are nonlinear and complicated. Our experience is that standard algorithms to find them also becomes problematic.

We found a variety of optimal designs for the flexible continuation-ratio (CR) model, which has great potential for dose finding studies because it simultaneously models both probabilities of observing efficacy and adverse effect without having to assume the correlation between them.

There is work in the literature that simultaneously study efficacy and toxicity by postulating bivariate parametric models. For example, to model dose-response curves, [

The optimal designs that we are interested to construct is a three-objective compound optimal design that provides efficient estimates for the most effective dose (

[

In dose-response studies, side effects and drug toxicity can be very serious and need to be well controlled. One of the main targets in clinical trials is to find out the “dose that is closest to an acceptable level of toxicity”. The target dose is defined as Maximum Tolerated Dose (

The worth of a design is measured of the quality of the Fisher’s information matrix. This matrix is proportional to the negative of the expectation of the second derivatives of the total log likelihood function with respect to the model parameters. This section derives the matrix for the CR model and makes clear the explicit nature of the objective functions.

The response from a subject assigned to dose

Since

Locally

By the implicit function theorem, if the function

Thus, the locally

In the CR model, since

In what is to follow, we treat a design

[

1. The design

2. The design

3.

This is the equivalence theorem for

In the same paper, [

In practice, there are usually not one, but a couple of objectives of interest and it is desirable to incorporate them at the design stage. There are two typical approaches to construct a multiple-objective design: the compound and constrained optimal design ([

Suppose there are two competing objectives

For any user-selected constant

The constrained optimal design is harder to find than the compound optimal design. The latter is easier to determine because a concave combination of concave functionals is still concave for a fixed

For linear models, [

Extensions to constructing optimal designs for nonlinear models under 3 or more objective criteria using the Lagrange’s multiplier method were considered in [

In [

Mean Responses from the CR model with corresponding nominal values; directional derivatives of the compound criterion evaluated at the PSO-generated multiple-objective design on the unrestricted design space (middle); and directional derivatives of the compound criterion evaluated at the PSO-generated multiple-objective design on the design space [-2,7] (right).

The checking condition for a three-objective compound optimal design is just a weighted sum of the checking conditions for the optimal design for each objective, i.e. we have

We apply standard PSO to find the locally

Three-objective compound optimal designs for estimating the

( |
||||

(−3.3, 0.5, 3.4, 1) | −4.875 (0.102) | −1.139 (0.464) | 5.016 (0.322) | 7.874 (0.112) |

(−1, 0.5, 2, 1) | −2.790 (0.202) | −0.637 (0.513) | 3.683 (0.284) | - |

(0.4, 0.2, 2, 1) | −12.610 (0.366) | −3.918 (0.158) | −0.942 (0.470) | 8.727 (0.006) |

Three-objective compound optimal designs for estimating the

( |
|||

(−3.3, 0.5, 3.4, 1) | −2.000 (0.152) | 0.1045 (0.502) | 6.328 (0.345) |

(−1, 0.5, 2, 1) | −2.000 (0.330) | −0.156 (0.403) | 3.820 (0.267) |

(0.4, 0.2, 2, 1) | −2.000 (0.356) | −0.438 (0.319) | 7.000 (0.325) |

Different efficiency plots of compound optimal designs for the CR model with nominal values

Different efficiency plots of compound optimal designs for the CR model with nominal values:

Invariance of the optimal designs to linear transformation of the design is a desirable property for an optimal design.

To further investigate the influence of dose space on the compound optimal design, we verified the algorithm was still able to find compound optimal designs on an unrestricted space that may not be symmetric about 0 for the 3 sets of nominal values for the model parameters. For example, Table

In our PSO implementation, for each fixed

Can we find a three-objective optimal design

To answer the question we need to investigate the impact of different combinations of

Different efficiency plots of compound optimal designs for the CR model with nominal values:

( |
||||||

(−3.3, 0.5, 3.4, 1) | 0.55 | 0.35 | 0.1 | 0.63 | 0.63 | 0.87 |

(−1, 0.5, 2, 1) | 0.7 | 0.3 | 0 | 0.82 | 0.85 | 0.88 |

(0.4, 0.2, 2, 1) | 0.6 | 0.4 | 0 | 0.75 | 0.75 | 0.81 |

These efficiency plots provide an answer to the question posed at the beginning of the section. The answer is no, at least for the three sets of nominal values we show here; there is no compound optimal design

For each set of the nominal values, Table

In addition, we observe that for all the three nominal values sets, the compound optimal designs maintain high

We close this section by noting that the above approach of finding locally optimal designs can be directly extended to finding Bayesian optimal designs with one or multiple objectives. Other methods for finding Bayesian optimal designs for Phase I trials and do not require that each objective be a convex function of the information matrix are available. Some seminal work and notable work in this direction are [

As an application, we revisit the clinical trial in [

CPU time required to find PSO and DE-generated locally

Model | Algorithm | CPU time |

PSO(20,200) | 0.4 | |

DE(20,200) | 2.9 | |

PSO(100,200) | 1.4 | |

DE(80,400) | 57.4 | |

PSO(20,200) | 0.3 | |

DE(20,200) | 1.4 | |

PSO(80,300) | 1.2 | |

DE(60,200) | 11.0 | |

PSO(20,200) | 0.1 | |

DE(20,200) | 1.4 | |

PSO(100,300) | 2.5 | |

DE(40,300) | 6.5 |

CPU time required by PSO and DE to find locally

Model | Algorithm | CPU time |

PSO(60,100) | 4.5 | |

DE(20,200) | 2.3 | |

PSO(40,1000) | 2.8 | |

DE(40,200) | 4.6 | |

PSO(80,200) | 1.2 | |

DE(40,200) | 7.4 |

When tackling design or any optimization problems, especially high dimensional ones, where the solution is unknown, it is always not clear which metaheuristic algorithm to use. Solely relying on results from a single metaheuristic algorithms is risky since they do not guarantee convergence to the global optimum. It is thus important to run several metaheuristic algorithms and observe whether they all produce similar solution. If it is the case, the confidence in our solution is increased.

We now briefly compare performance of PSO with two other algorithms. We choose to compare CPU time required by PSO to find optimal designs with two of its competitors: Differential evolution (DE) and Cocktail algorithm (CA). Some of the models we used in the comparisons are listed in Tables

DE is a metaheuristic algorithm like PSO, which is a general purpose optimization algorithm. We use it to find locally

CA is a deterministic algorithm, which means that the same input results in the same answer, which in our case, results in the same optimal design. CA requires the design space to be discretized into a finite number of dose levels to approximate the continuous space. This approach can limit the ability of CA to find the optimal design because the discretized points may not correspond to a design point of the optimal design. However, our experience is that the CA-generated designs can usually still be highly

Cocktail algorithm (CA) is a deterministic algorithm which always produces the same output given one input. For PSO, we still use the same tuning parameters in section 5.2.1 such that the algorithm produces the locally D-optimal at 90% chance. Our empirical experience with CA is that although its generated designs can be very efficient even when the grid size

PSO and CA generated locally

Algorithm | CPU time | |||||||||

PSO(80,300) | 1.2 | 1 | −1 | −1 | 0 | 0 | 1 | 1 | ||

0 | 1 | 0 | 1 | 0 | 1 | |||||

weight | 0.1875 | 0.1875 | 0.1275 | 0.1275 | 0.1875 | 0.1875 | ||||

CA (10) | 0.1 | 0.997 | −1 | −1 | −0.1111 | −0.1111 | 0.1111 | 0.1111 | 1 | 1 |

0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | |||

weight | 0.1870 | 0.1870 | 0.6300 | 0.6300 | 0.6300 | 0.6300 | 0.1870 | 0.1870 | ||

CA (1000) | 94.6 | 1 | −1 | −1 | −0.0101 | −0.0101 | 0.0101 | 0.0101 | 1 | 1 |

0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | |||

weight | 0.1875 | 0.1875 | 0.6250 | 0.6250 | 0.6250 | 0.6250 | 0.1875 | 0.1875 |

Comparisons of the locally D-optimal design of the CR model with constant or non-constant slopes on

Model | Algorithm | # of support points | CPU time |

CR model with |
PSO(20,200) | 3 | 0.1 |

CA(100) | 5 | 1.7 | |

CA(1000) | 5 | 20.8 | |

CA(10000) | 4 | 71.3 | |

CR model with |
PSO(100,500) | 4 | 1.5 |

CA(100) | 5 | 2.9 | |

CA(1000) | 6 | 14.3 | |

CA(10000) | 5 | 122.7 |

We observe from Tables

Our work generalizes the work of [

We used simulation studies and investigated efficiencies of the PSO-generated designs when we applied two repair mechanisms to bring out of area points back to the dose space: one employs a random repair mechanism and brings outlying points back to the dose space and the other exploits a common property of optimal designs and bring them back to the boundary of the dose space. The latter option greatly expedites PSO in searching for locally D- and c-optimal designs for a variety models, including the continuation-ratio model. Additionally, we compared performance of PSO with boundary repair to two other popular algorithms: the Cocktail algorithm (CA) in the statistics literature [

The techniques developed here are broadly applicable to find types of optimal designs for other nonlinear models. For example, standardized maximin optimal designs were found for estimating parameters in Michaelis-types of models that study how substrate concentration affects reaction with the enzyme in [

The PSO codes to find designs reported in the paper can be directly amended to find different types of multiple-objective optimal designs for other models. For example, the codes were recently amended to find dual-objective Bayesian optimal designs for estimating a percentile and model parameters simultaneously in a Beta regression for modeling proportion of malformed purple sea urchins in the embryo after they were treated with a toxic agent [

When a metaheuristic algorithm does not produce a solution near to the optimum, it is common to use one of its variants. A variant is just one of many modifications of the algorithm that improves the original version in various ways. The variant can have improved convergence property, for a more targeted application, have better self-finding tuning parameters or the like. Popular metaheuristic algorithms, like PSO, probably have a couple of dozens of variants now. Examples of PSO variants are [

Another option is to hybridize the algorithm with one or more carefully selected metaheuristic algorithms so that the hybridized version combines the strengths of each algorithm. A common guiding principle is that the hybridized algorithm performs better than each of the individual algorithm, see [

As an example, a hybridized metaheuristic algorithm was used to extend the capabilities of the celebrated Simon 2-stage design for a Phase II trial [

Expanding on this framework and to further reduce the difficulty or uncertainty of specifying the alternative hypothesis, [

Hybridized metaheuristics was also used to find an optimal discriminiation design to ascertain the most plausible nonlinear model among several nonlinear models [

Not surprisingly, nature-inspired algorithms have been promptly applied to better understand various aspects of COVID-19. Various such algorithms, such as PSO, DE and Imperialist Competitive Algorithm (ICA) have been used to tackle the control and spread of COVID-19. ICA is based on human behavior and proposed by [

Pareto Optimization (PO) is a common approach to solve optimization problems with multiple objectives. [

The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Wong is also grateful for the Yushan Scholarship Award from the Ministry of Education in Taiwan and the hospitality of the Department of Statistics at the National Cheng Kung University in Tainan, where this manuscript was completed.