Carlsmith (2024): AI Forecasting for Existential Risk
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This arxiv preprint by Carlsmith (2024) addresses AI forecasting methodologies for assessing existential risks, providing analytical frameworks relevant to AI safety risk quantification and long-term outcome prediction.
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Abstract
In diffusion-controlled drug delivery, it is possible for drug molecules to bind to the carrier material and never be released. A common way to incorporate this phenomenon into the governing mechanistic model is to include an irreversible first-order reaction term, where drug molecules become permanently immobilised once bound. For diffusion-only models, all the drug initially loaded into the device is released, while for reaction-diffusion models only a fraction of the drug is ultimately released. In this short paper, we show how to calculate this fraction for several common diffusion-controlled delivery systems. Easy-to-evaluate analytical expressions for the fraction of drug released are developed for monolithic and core-shell systems of slab, cylinder or sphere geometry. The developed formulas provide analytical insight into the effect that system parameters (e.g. diffusivity, binding rate, core radius) have on the total fraction of drug released, which may be helpful for practitioners designing drug delivery systems.
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| Page | Type | Quality |
|---|---|---|
| AI-Augmented Forecasting | Approach | 54.0 |
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# Total fraction of drug released from diffusion-controlled delivery systems with binding reactions
Elliot J. Carr
[elliot.carr@qut.edu.au](mailto:elliot.carr@qut.edu.au "") School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.
## Abstract
In diffusion-controlled drug delivery, it is possible for drug molecules to bind to the carrier material and never be released. A common way to incorporate this phenomenon into the governing mechanistic model is to include an irreversible first-order reaction term, where drug molecules become permanently immobilised once bound. For diffusion-only models, all the drug initially loaded into the device is released, while for reaction-diffusion models only a fraction of the drug is ultimately released. In this short paper, we show how to calculate this fraction for several common diffusion-controlled delivery systems. Easy-to-evaluate analytical expressions for the fraction of drug released are developed for monolithic and core-shell systems of slab, cylinder or sphere geometry. The developed formulas provide analytical insight into the effect that system parameters (e.g. diffusivity, binding rate, core radius) have on the total fraction of drug released, which may be helpful for practitioners designing drug delivery systems.
Keywords: drug delivery, binding, reaction-diffusion, core-shell, release profile.
## 1 Introduction
Increasingly, mechanistic mathematical models of drug delivery (based on physical conservation laws) are being developed to improve understanding of the transport mechanisms that control the release rate, explore the effect of varying design parameters on the release profile and avoid costly and time consuming experiments \[ [1](https://ar5iv.labs.arxiv.org/html/2401.09644#bib.bib1 "")\]. In this field of research, mechanistic mathematical models of _diffusion-controlled_ drug delivery are typically based on Fick’s second law, where the drug concentration within the system evolves in space and time according to the diffusion equation and specified initial and boundary conditions \[ [8](https://ar5iv.labs.arxiv.org/html/2401.09644#bib.bib8 ""), [11](https://ar5iv.labs.arxiv.org/html/2401.09644#bib.bib11 ""), [6](https://ar5iv.labs.arxiv.org/html/2401.09644#bib.bib6 ""), [2](https://ar5iv.labs.arxiv.org/html/2401.09644#bib.bib2 ""), [3](https://ar5iv.labs.arxiv.org/html/2401.09644#bib.bib3 ""), [12](https://ar5iv.labs.arxiv.org/html/2401.09644#bib.bib12 ""), [7](https://ar5iv.labs.arxiv.org/html/2401.09644#bib.bib7 ""), [5](https://ar5iv.labs.arxiv.org/html/2401.09644#bib.bib5 ""), [4](https://ar5iv.labs.arxiv.org/html/2401.09644#bib.bib4 ""), [9](https://ar5iv.labs.arxiv.org/html/2401.09644#bib.bib9 ""), [10](https://ar5iv.labs.arxiv.org/html/2401.09644#bib.bib10 "")\]. Such purely-diffusive models yield a release profile F(t)𝐹𝑡F(t) (cumulative amount of drug released over the time interval \[0,t\]0𝑡\[0,t\] divided by the initial amount of drug lo
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