Organs and Artificial Organs

CONTENTS OF CURRICULUM UNIT 11.07.10

  1. Unit Guide
  1. Rationale and Introduction
  2. Content Objectives
  3. Overview of the Circulatory System
  4. Dimensional Analysis Calculation Demonstrating Heart Reliability
  5. Introduction to Units of Measure
  6. Dimensional Analysis of Graphical Data: Ventricular Pressure vs. Volume Graph
  7. Dimensional Analysis for Evaluating Potential Power Sources for an Artificial Heart
  8. Dimensional Analysis does Problem Analysis
  9. Student Background and Challenges
  10. Strategies
  11. Classroom Activities and Lesson Plans
  12. Standards Alignment
  13. Teacher and Student Resources
  14. Endnotes
  15. Bibliography

Dimensional Analysis: A Mathematics Tool to Dissect the Circulatory System

Richard Cordia Taylor

Published September 2011

Tools for this Unit:

Dimensional Analysis for Evaluating Potential Power Sources for an Artificial Heart

Since an implanted artificial heart is ideally a portable device, the energy and power will likely be supplied by batteries. A standard AA NiMH rechargeable battery in my camera is rated as 1.2 V at 2500 mA•hr. These battery specifications involve voltage, current, and time. I can do a quick calculation, keeping track of units only, to get V•A•s = (J/C)•(C/s)•s=J, an energy, so this looks like a calculation worth making in detail. The energy in the battery is

image 11.07.10.10

The number seems substantial, but I need to put it in context to see if the camera battery could really be a good potential power source. To build this context, I do another dimensional calculation to determine how long my battery could power an artificial heart using energy at the rate of a natural left ventricle running at 60 bpm.

image 11.07.10.11

This calculation tells me that my camera battery is not really a viable power source. I'd need eight of them a day, and that's assuming that there was 100% efficiency in converting the battery energy into pumping work. One-hundred percent efficiency is not realistic. One artificial implantable heart already developed had only 13.5% efficiency and required a 12 W electrical power input to produce a 1.6 W pumping output. 1 1 (The power requirements for this heart seem prohibitive to permanent implantation with battery power.)

There are several important technical aspects of this calculation. First, I stated the battery energy as a ratio with an invented unit to make the meaning of the end calculation more clear. Second, I arranged the conversion factors in a way to make the units I did not want in the end result reduce out of the calculation—sometimes I'll use the conversion factor as given, but sometimes I'll need to use the reciprocal of the conversion factor. Third, I violated my own rule about writing time in seconds because in the context of this evaluation, hours were a more appropriate and clearer unit to express the results of this particular calculation.

Since camera batteries do not provide sufficiently dense energy storage for an artificial heart, are there other sources? Alternative energy sources such as implantable generators driven by skeletal muscles have been explored but theoretically might only produce 100 J/day. 1 2 In the 1970's, nuclear batteries of up to 50 W powered by plutonium-238 were implanted in dogs and primates for periods exceeding a year with few radiation related side effects. 1 3 Nuclear batteries promised decades of continuous operation without recharging or replacing, but plutonium has more dangerous uses, making it difficult to envision these other applications.

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