Journal of Biosciences and Medicines, 2024, 12, 251-265 https://www.scirp.org/journal/jbm ISSN Online: 2327-509X ISSN Print: 2327-5081 DOI: 10.4236/jbm.2024.129023 Sep. 26, 2024 251 Journal of Biosciences and Medicines Quasi-Static, Poiseuille Flow of Analgesics from an Elastomeric Pump: Theoretical Determination of Infusion Times and Toxicity Conditions Clare B. Lipscombe 1 , Trevor C. Lipscombe 2 , Don S. Lemons 3 1 Division of General Surgery, Vanderbilt University Medical Center, Nashville, TN, USA 2 Catholic University of America Press, Washington DC, USA 3 Department of Physics, Bethel College, North Newton, KS, USA Abstract Background: Elastomeric pumps (elastic balls into which analgesics or antibi- otics can be inserted) push medicines through a catheter to a nerve or blood vessel. Since elastomeric pumps are small and need no power source, they fit easily into a pocket during infusion, allowing patient mobility. Elastomeric pumps are widely used and widely studied experimentally, but they have well- known problems, such as maintaining reliable flow rates and avoiding toxicity or other peak-and-trough effects. Objectives: Our research objective is to de- velop a realistic theoretical model of an elastomeric pump, analyze its flow rates, determine its toxicity conditions, and otherwise improve its operation. We believe this is the first such theoretical model of an elastomeric pump con- sisting of an elastic, medicine-filled ball attached to a horizontal catheter. Method: Our method is to model the system as a quasi-Poiseuille flow driven by the pressure drop generated by the elastic sphere. We construct an engi- neering model of the pressure exerted by an elastic sphere and match it to a solution of the one-dimensional radial Navier-Stokes equation that describes flow through a horizontal, cylindrical tube. Results: Our results are that the model accurately reproduces flow rates obtained in clinical studies. We also discover that the flow rate has an unavoidable maximum, which we call the “toxicity bump”, when the radius of the sphere approaches its terminal, un- stretched value—an effect that has been observed experimentally. Conclu- sions: We conclude that by choosing the properties of an elastomeric pump, the toxicity bump can be restricted to less than 10% of the earlier, relatively constant flow rate. Our model also produces a relation between the length of How to cite this paper: Lipscombe, C.B., Lipscombe, T.C. and Lemons, D.S. (2024) Quasi-Static, Poiseuille Flow of Analgesics from an Elastomeric Pump: Theoretical De- termination of Infusion Times and Tox- icity Conditions. Journal of Biosciences and Medicines, 12, 251-265. https://doi.org/10.4236/jbm.2024.129023 Received: August 25, 2024 Accepted: September 23, 2024 Published: September 26, 2024 Copyright © 2024 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access