Elsevier

Mathematical Biosciences

Volume 304, October 2018, Pages 62-78
Mathematical Biosciences

Designing group dose-response studies in the presence of transmission

https://doi.org/10.1016/j.mbs.2018.07.007Get rights and content

Abstract

Dose-response studies are used throughout pharmacology, toxicology and in clinical research to determine safe, effective, or hazardous doses of a substance. When involving animals, the subjects are often housed in groups; this is in fact mandatory in many countries for social animals, on ethical grounds. An issue that may consequently arise is that of unregulated between-subject dosing (transmission), where a subject may transmit the substance to another subject. Transmission will obviously impact the assessment of the dose-response relationship, and will lead to biases if not properly modelled. Here we present a method for determining the optimal design – pertaining to the size of groups, the doses, and the killing times – for such group dose-response experiments, in a Bayesian framework. Our results are of importance to minimising the number of animals required in order to accurately determine dose-response relationships. Furthermore, we additionally consider scenarios in which the estimation of the amount of transmission is also of interest. A particular motivating example is that of Campylobacter jejuni in chickens. Code is provided so that practitioners may determine the optimal design for their own studies.

Introduction

A group dose-response experiment involves exposing subjects to a range of doses of a substance (for example, an infectious agent, or bacteria or a drug) and measuring their responses (for example, if they became colonised) [4]. These experiments are routinely used to characterise the relationship between the dose of a substance and the response in a subject, known as the dose-response relationship.

Studies of this type have been widely used throughout pharmacology [27], toxicology [5] and in clinical trials [3], and methods for characterising the dose-response relationship developed [28]. However, a recent study by Conlan et al. noted a potential issue with such analyses when considering infectious agents [9]: in some cases, subjects may transmit their dose to other subjects, hence complicating the analysis. The motivating example is of Campylobacter jejuni in chickens.

The Campylobacter genus of bacteria is the most common cause of food-borne diarrhoeal disease in developed and developing countries – surpassing Salmonella and Shigella spp. [14]. Group dose-response experiments with C. jejuni in chickens are a useful tool in understanding the dose-response and transmission characteristics of the bacteria, allowing sensible measures to be put in place to contain, or eradicate, the infection in livestock used for human consumption. Chickens are social animals, and thus ethically are required to be co-housed [2]. Conlan et al. [9] noted that previous statistical analyses of the dose-response characteristics of C. jejuni in chickens had neglected the potential for transmission between co-housed subjects – resulting in incorrect estimation of the dose-response relationship.

The presence of transmission in these experiments leads to an “all-or-nothing” response if the subjects are observed too late – that is, once at least one subject is infected within a group, transmission to the initially uninfected chickens leads to more chickens being colonised than is representative of the administered dose. This yields a lower estimated ID50 (i.e., the dose required to infect 50% of the population, on average), and steeper slope-at-half-height – common statistics used to characterise dose-response curves [9]. To limit between-subject dosing, one might attempt to sample the chickens after a very short period of time following initial dosing. However, there exists a latent period between a chicken being challenged and it becoming colonised (i.e., it presenting its response), thus this also provides inaccurate assessment of the number of colonised subjects. Finally, a chicken is “observed” via post-mortem caecal sampling, meaning that only one observation of each subject is possible.

Studies of this form – grouped dose-response experiments with the potential for between-subject dosing – are common, and given the ethical, financial and physical constraints associated with such studies, determining their optimal experimental design in order to obtain the most information about the dose-response relationship is important. One must consider the allocation of the number of subjects to groups, possibly different doses, and the associated time(s) to observe the process, in order to gain the most information about the dose-response relationship. In particular, using these optimal design tools, we can quantify the trade-off in information between allocating many individuals to few groups (doses), or few individuals to many groups (doses). We furthermore give consideration to scenarios in which the estimation of the transmission rate is also of interest – highlighting the potential for these tools to inform design of experiments where the purpose is understanding the transmission dynamics of a pathogen (e.g., avian influenza as in [26]).

We work within a Bayesian framework, allowing for use of prior information concerning the various components of the dose-response study, and transmission dynamics. Our method involves a novel continuous-time Markov chain model for the dynamics within such a study, combined with recently-developed methods for Bayesian optimal experimental design [20], [21]. MATLAB code is provided so that practitioners may determine the optimal design for their own studies.

Section snippets

Modelling of group dose-response experiments

The first step in determining the optimal experimental design for these experiments is determining suitable models to represent the dynamics amongst a group of subjects. In determining a suitable model, we must ensure we account for the experimental aspects we wish to determine as part of our optimal designs. First and foremost, we are interested in the optimal doses to allocate to subjects in order to gain the most information about the dose-response relationship. Hence, we must represent the

Results

Recall, we consider two scenarios: 1) where we have an informative prior distribution on the model parameters, and 2) where we have an uninformative prior distribution. For both scenarios, we consider the optimal designs with respect to both 1) the EKLD, and 2) the MAPE. Furthermore, we also establish the optimal designs when we are interested in either 1) the dose-response parameters only, 2) the dose-response and the transmission rate parameter, and 3) the transmission rate parameter only.

Discussion

First of all, we wish to reiterate that the results here are prior-specific, and therefore different trends may be apparent when considering other prior distributions than those we have considered here.

The designs returned under the two different utility functions in Figs. 3 and 4 show distinct differences. In particular, the EKLD designs consistently prefer more groups (with less replicates in each), whereas the MAPE designs prefer more replicates within less groups. The designs under the EKLD

Conclusion

Group dose-response challenge experiments are routinely used to assess safe, effective, or hazardous doses of a substance. However, the possibility for transmission can lead to incorrect estimation of the dose-response relationship. Here, we have utilised optimal experimental design theory to demonstrate how to pre-determine a suitable experimental design in order to target different aspects of the dose-response relationship, or transmission dynamics.

Within the experimental design framework,

Authors’ contributions

DJP produced code and results, and drafted the manuscript. NGB, JVR and JT conceived the study, and helped draft the manuscript. All authors gave final approval for publication.

Acknowledgements

JVR acknowledges the support of the ARC (Future Fellowship FT130100254; CoE ACEMS) and the NHMRC (CRE PRISM2). The authors would like to thank Andrew Conlan and Andrew Grant (Department of Veterinary Medicine, University of Cambridge), for their advice regarding the modelling, and the technical aspects of dosing with infectious bacteria, respectively.

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