# Mathematical Modelling for Optimal Vaccine Dose Finding: Maximising Efficacy and Minimising Toxicity

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## Abstract

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## 1. Introduction

- i.
- Assumed statistical efficacy model.
- ii.
- Trial size.
- iii.
- Method of trial dose selection.

- When the method of trial dose selection is fixed, how dose-optimisation approaches are affected by the assumed statistical efficacy model and trial size.
- When trial size is fixed, how dose-optimisation approaches are affected by the assumed statistical efficacy model and method of trial dose selection.

## 2. Materials and Methods

#### 2.1. Overview of Simulation Study Methodology

#### 2.2. Efficacy, Toxicity, and Utility

#### 2.2.1. Dose Efficacy

#### 2.2.2. Dose Toxicity

#### 2.2.3. Dose Utility

- Weight
_{Efficacy} - DisabilityWeight
_{Toxicity0} - DisabilityWeight
_{Toxicity1} - DisabilityWeight
_{Toxicity2} - DisabilityWeight
_{Toxicity3}

_{Efficacy}> DisabilityWeight

_{Toxicity2}, then the protection that may be gained from an efficacious vaccine response would outweigh the discomfort of the grade 2 event. Conversely, if Weight

_{Efficacy}< DisabilityWeight

_{Toxicity3}, then the protection that may be gained from an efficacious vaccine response would be outweighed by the discomfort of the grade 3 event. The disability weight for each grade was increasing (i.e., a grade 2 adverse event was worse than a grade 1 adverse event) (Table 2).

_{Efficacy}would vary depending on the disease’s severity, prevalence, and level of confidence in the surrogate of protection. Hence, in this work, we chose Weight

_{Efficacy}to be similar relative to DisabilityWeight

_{Toxicity3}(Table 2). This ensures that both maximising efficacy and minimising toxicity are important and prevents the optimal dose from being one that is optimal with regards to only one of these goals. Practically, Weight

_{Efficacy}could be chosen based on epidemiological models [34].

Weight | Value | Source |
---|---|---|

Weight_{Efficacy} | 0.133 or 0.266 | Chosen to be equal to either DisabilityWeight_{Toxicity3} or twice DisabilityWeight_{Toxicity3} |

DisabilityWeight_{Toxicity0} | 0.000 | Chosen to be 0, as no discomfort/toxicity is caused |

DisabilityWeight_{Toxicity1} | 0.006 | [35] |

DisabilityWeight_{Toxicity2} | 0.051 | [35] |

DisabilityWeight_{Toxicity3} | 0.133 | [35] |

#### 2.3. Scenarios

#### 2.4. Dose-Optimisation Approaches

- i.
- An assumed efficacy model (saturating, peaking, or weighted);
- ii.
- A trial size (10/30/60/100);
- iii.
- A method of trial dose selection (with either retrospective or continual modelling).

#### 2.5. Additional Details

_{10}scale, although we did not otherwise assume units. For viral vector vaccines, these units would likely be viral particles or infectious units. Additionally, we consistently used a dose range of 0–10 on the log

_{10}scale. This was purely for convenience and could be rescaled to the minimum and maximum possible dose for any given vaccine. This is referred to as the ‘dosing space’.

#### 2.6. Objective 1: When the Method of Trial Dose Selection Is Fixed, How Dose-Optimisation Approaches Are Affected by the Assumed Statistical Efficacy Model and Trial Size

- i.
- Efficacy model: saturating, peaking, or weighted;
- ii.
- Trial dose-selection method: full uniform exploration;
- iii.
- Trial size: 10, 30, 60, or 100.

_{10}-scale dosing space, we would have assigned test doses at 0, 2, 4, 6, 8, and 10. This method of dose selection is reasonable as a naive method, as it would ensure that all areas of the dosing space were evenly explored. As these data would then be a representative sample of all possible doses, this should have allowed for good model calibration and hence a good suggestion of optimal dose.

#### 2.6.1. Metrics for Comparison between Approaches

#### Simple Regret

#### Inaccuracy

#### Average Regret

#### 2.7. Objective 2: When Trial Size Is Fixed, How Dose-Optimisation Approaches Are Affected by the Assumed Statistical Efficacy Model and Method of Trial Dose Selection

- i.
- Efficacy model: saturating, peaking, or weighted;
- ii.
- Trial size: 30;
- iii.
- Trial dose-selection method: full uniform exploration, standard fully continual modelling, balanced exploration (softmax) fully continual modelling, or three-stage (softmax).

- Conducting a small trial on a select set of doses;
- Gathering efficacy and toxicity data from this experiment;
- Updating the efficacy and toxicity models based on these data;
- Using the models to select either the next set of doses to test or to select the final dose to predict as ‘optimal’.

#### 2.7.1. Fully Continual Standard

#### 2.7.2. Fully Continual, Balanced Exploration (Softmax)

#### 2.7.3. Three-Stage (Softmax)

- Stage 1.
- a.
- ⅓ of the trial population is dosed following the full uniform exploration approach outlined in objective 1.
- b.
- Efficacy and toxicity models are calibrated using these data and pseudo-data [3.7.5].

- Stage 2.
- a.
- The second ⅓ of the population is dosed according to the utility predictions of the combined efficacy and toxicity models, using the softmax selection method with relatively high exploration.
- b.
- Efficacy and toxicity models are calibrated using these data, data from step one, and downweighted pseudo-data.

- Stage 3.
- a.
- The final ⅓ of the population is dosed according to the utility predictions of the combined efficacy and toxicity models, using the softmax selection method with relatively low exploration.
- b.
- Efficacy and toxicity models are calibrated using all collected data, with pseudo-data being ignored. The predicted optimal dose is selected according to the utility predictions of the combined efficacy and toxicity models.

#### 2.7.4. Dose-Escalation/De-Escalation Rules

_{10}scale (that is to say the middle dose). A dose could not be in excess of ½ a log above of the maximum previously tested dose or more than ½ a log below the minimum previously tested dose. For example, dose 10 (10

^{10}) could not be tested unless a dose of at least 9.5 (10

^{9.5}) had been previously tested. This was suggested to reduce the risk of unexpected higher-grade toxicities.

#### 2.7.5. Pseudo-Data

#### 2.7.6. Comparison between Approaches/Trial Designs

## 3. Results

#### 3.1. Objective 1: When the Method of Trial Dose Selection Is Fixed, How Dose-Optimisation Approaches Are Affected by the Assumed Statistical Efficacy Model and Trial Size

#### 3.2. Objective 2: When Trial Size Is Fixed, How Dose-Optimisation Approaches Are Affected by the Assumed Statistical Efficacy Model and Method of Trial Dose Selection

#### 3.2.1. Qualitative Analysis

#### 3.2.2. Quantitative Ranking

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Visual depiction of the process of conducting simulation studies used in this work to assess mathematical-modelling-based dose-optimisation approaches. The aim was to evaluate dose-optimisation approaches (red), in particular the effect of changing the assumed dose–efficacy model, trial size, and trial dose-selection method. These were tested by simulating clinical trials (purple) based on ‘scenarios’ (blue). Repeated simulation of clinical trials was conducted for different dose-optimisation approach/scenario pairs, and metrics related to how effectively optimal dose was located were calculated. These were tabulated and compared to assess whether the assumed dose–efficacy model, trial size, and trial dose-selection method influence the consistency of dose optimsation.

**Figure 3.**Visual example of model averaging. When the saturating Akaike weight is 0, the predicted efficacy curve is defined entirely by the peaking model (blue). When the saturating Akaike weight is 1, the predicted efficacy curve is defined entirely by the saturating model (green). If both models are equally as likely, given the available data, then the saturating Akaike weight and the peaking Akaike weight are both 0.5, and the predicted efficacy curve is the midpoint of the saturating and peaking curves (orange).

**Figure 4.**Visual example of ordinal dose toxicity. The plot shows the proportion of individuals that would experience different adverse event grades for each dose. In this example, at low doses, grade 0 (blue) adverse events are most likely. By dose 6, grade 1 (yellow) and grade 2 (green) adverse events are likely but grades 0 and 3 are also possible. By the maximum dose, approximately 50% of individuals would experience a grade 3 adverse event, and almost all others would experience grade 2 events.

**Figure 5.**Three examples of the 14 tested scenarios. For each scenario, we show dose efficacy, dose toxicity, and the resultant dose–utility plots. Optimal dose is also given. For the toxicity plots, grade 0, 1, 2, and 3 adverse event probabilities are represented by blue, orange, green, and red, respectively.

**Figure 7.**Percentage simple regret (PSR) for all scenarios by assumed efficacy model and trial size. Trial dose selection method was full uniform exploration. A lower PSR denotes a more optimal final dose. Individual points represent PSR for a single simulated clinical trial using one dose-optimisation approach for one of the 14 scenarios. The middle line of each boxplot is the median value; the box marks the 25th and 75th percentiles, and the whiskers mark the 5th and 95th percentiles of the data. Black lines represent the 95% confidence interval for the median of each distribution [47]. The majority of these distributions of PSR were different to a statistically significant extent at the p = 0.05 threshold according to the Kolmogorov–Smirnov test due to the large number of simulations conducted (100 per approach/scenario pairing). For further details on statistical significance see Supplementary S12.

**Figure 8.**Inaccuracy (

**a**) and absolute inaccuracy (

**b**) for all scenarios by assumed efficacy model and trial size. Trial dose-selection method was full uniform exploration. The closer the inaccuracy/absolute accuracy was to 0, the more accurate the prediction of utility was at the predicted optimal dose. Individual points represent inaccuracy/absolute inaccuracy for a single simulated clinical trial using that dose-optimisation approach for one of the 14 scenarios. The middle line of each boxplot is the median value; the box marks the 25th and 75th percentiles, and the whiskers mark the 5th and 95th percentiles of the data. Black lines represent the 95% confidence interval for the median of each distribution [47]. The majority of these distributions of absolute inaccuracy were different to a statistically significant extent at the p = 0.05 threshold according to the Kolmogorov–Smirnov test due to the large number of simulations conducted (100 per approach/scenario pairing). For further details on statistical significance, see Supplementary S12.

**Figure 9.**Percentage average regret for all scenarios by assumed efficacy model and trial size. Trial dose-selection method was full uniform exploration. Individual points represent percentage average regret for a single simulated clinical trial using that dose-optimisation approach for one of the 14 scenarios. The middle line of each boxplot is the median value; the box marks the 25th and 75th percentiles, and the whiskers mark the 5th and 95th percentiles of the data. Black lines represent the 95% confidence interval for the median of each distribution [47]. The majority of these distributions of PAR were not different to a statistically significant extent at the p = 0.05 threshold according to the Kolmogorov–Smirnov test. For further details on statistical significance, see Supplementary S12.

**Figure 10.**Percentage simple regret (PSR) for all scenarios by assumed efficacy model and trial dose-selection method. Trial size was 30. Individual points represent PSR for a single simulated clinical trial using that dose-optimisation approach for one of the 14 scenarios. The middle line of each boxplot is the median value; the box marks the 25th and 75th percentiles, and the whiskers mark the 5th and 95th percentiles of the data. A lower PSR denotes a more optimal final dose. Black lines represent the 95% confidence interval for the median of each distribution [47]. The distributions of PSR for the approaches that assumed a saturating model were different to the distributions of the approaches that assumed a peaking or weighted efficacy mode to a statistically significant extent at the p = 0.05 threshold according to the Kolmogorov–Smirnov test. For further details on statistical significance, see Supplementary S12.

**Figure 11.**Inaccuracy (

**a**) and absolute inaccuracy (

**b**) for all scenarios by assumed efficacy model and trial dose-selection method. Trial size was 30. Individual points represent inaccuracy/absolute inaccuracy for a single simulated clinical trial using that dose-optimisation approach for one of the 14 scenarios. The middle line of each boxplot is the median value; the box marks the 25th and 75th percentiles, and the whiskers mark the 5th and 95th percentiles of the data. The closer inaccuracy/absolute accuracy is to 0, the more accurate the prediction of utility is at the predicted optimal dose. Black lines represent the 95% confidence interval for the median of each distribution [47]. The majority of these distributions of absolute inaccuracy were not different to a statistically significant extent at the p = 0.05 threshold according to the Kolmogorov–Smirnov test. For further details on statistical significance, see Supplementary S12.

**Figure 12.**Percentage average regret for all scenarios by assumed efficacy model and trial dose-selection method. Trial size was 30. Individual points represent percentage average regret for a single simulated clinical trial using that dose-optimisation approach for one of the 14 scenarios. The middle line of each boxplot is the median value; the box marks the 25th and 75th percentiles, and the whiskers mark the 5th and 95th percentiles of the data. A lower percentage average regret denotes better outcomes for trial participants. Black lines represent the 95% confidence interval for the median of each distribution [47]. The majority of these distributions of PAR were not different to a statistically significant extent at the p = 0.05 threshold according to the Kolmogorov–Smirnov test. For further details on statistical significance, see Supplementary S12.

Adverse Reaction Grade | General Description |
---|---|

0 | None. |

1 | Mild. Does not interfere with normal activity. |

2 | Moderate. Interference with normal activity. Little or no treatment required. |

3 | Severe. Prevents normal activity. Requires treatment. |

**Table 3.**Copeland scores and rankings for all approaches with a trial size of 30 across all scenarios. Ordering is by aggregate rank. Aggregate rank was calculated as the sum of ranks for simple regret, absolute inaccuracy, and average regret. Aggregate score was the mean of scores for simple regret, inaccuracy, and average regret.

Aggregate of Simple Regret, Absolute Inaccuracy, and Average Regret | Simple Regret | Absolute Inaccuracy | Average Regret | |||||
---|---|---|---|---|---|---|---|---|

Approach | Rank | Score | Rank | Score | Rank | Score | Rank | Score |

Weighted, Fully Continual, Balanced | 8 | 0.570 | 1 | 0.564 | 3 | 0.522 | 4 | 0.625 |

Peaking, Fully Continual, Standard | 12 | 0.572 | 7 | 0.498 | 4 | 0.517 | 1 | 0.701 |

Peaking, Softmax Three Stage | 12 | 0.536 | 4 | 0.552 ^{1} | 1 | 0.556 | 7 | 0.500 |

Peaking, Fully Continual, Balanced | 14 | 0.557 | 3 | 0.552 ^{1} | 6 | 0.510 | 5 | 0.610 |

Weighted, Fully Continual, Standard | 15 | 0.565 | 8 | 0.485 | 5 | 0.514 | 2 | 0.698 |

Weighted, Softmax Three Stage | 15 | 0.528 | 5 | 0.541 | 2 | 0.549 | 8 | 0.493 |

Saturating, Fully Continual, Standard | 20 | 0.543 | 10 | 0.447 | 7 | 0.492 | 3 | 0.691 |

Peaking, Full uniform exploration | 24 | 0.414 | 2 | 0.563 | 10 | 0.480 | 12 | 0.201 |

Saturating, Fully Continual, Balanced | 24 | 0.519 | 9 | 0.463 | 9 | 0.486 | 6 | 0.609 |

Saturating, Softmax Three Stage | 28 | 0.465 | 11 | 0.442 | 8 | 0.489 | 9 | 0.465 |

Weighted, Full uniform exploration | 28 | 0.400 | 6 | 0.516 | 11 | 0.480 | 11 | 0.203 |

Saturating, Full uniform exploration | 34 | 0.330 | 12 | 0.378 | 12 | 0.406 | 10 | 0.205 |

^{1}Scores are rounded to three decimal places, but ranks were calculated before rounding.

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## Share and Cite

**MDPI and ACS Style**

Benest, J.; Rhodes, S.; Evans, T.G.; White, R.G. Mathematical Modelling for Optimal Vaccine Dose Finding: Maximising Efficacy and Minimising Toxicity. *Vaccines* **2022**, *10*, 756.
https://doi.org/10.3390/vaccines10050756

**AMA Style**

Benest J, Rhodes S, Evans TG, White RG. Mathematical Modelling for Optimal Vaccine Dose Finding: Maximising Efficacy and Minimising Toxicity. *Vaccines*. 2022; 10(5):756.
https://doi.org/10.3390/vaccines10050756

**Chicago/Turabian Style**

Benest, John, Sophie Rhodes, Thomas G. Evans, and Richard G. White. 2022. "Mathematical Modelling for Optimal Vaccine Dose Finding: Maximising Efficacy and Minimising Toxicity" *Vaccines* 10, no. 5: 756.
https://doi.org/10.3390/vaccines10050756