Bone Mineral Density Math and Beer's Law
How do we find the bone mineral density of bones?Copyright 2007 Glitzy queen00, Wikimedia Commons http://commons.wikimedia.org/wiki/File:Ap_lateral_elbow.jpg
Students revisit the mathematics required to find bone mineral density, to which they were introduced in lesson 2 of this unit. They learn the equation to find intensity, Beer's law, and how to use it. Then they use the associated activity to investigate real world applications prior to completing a sheet of practice problems that use the Beer's law equation.
Students see the real-world connection with this material right away. Beer's law and the other equations that they learn in this lesson are mathematical tools that biomedical engineers use every day when they make bone mineral density readings. Environmental engineers also use light intensity measurements when designing solar panels and other forms of energy storage. Almost every type of engineer knows and uses Beer's law.
After this lesson, student should be able to:
- Apply their knowledge of logarithms to solve Beer's law problems.
- Explain the breath of the applications of logarithms.
CCSS.Math.Content.HSA-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
Grades 9-12
Do you agree with this alignment?CCSS.Math.Content.HSA-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Grades 9-12
Do you agree with this alignment?CCSS.Math.Content.HSA-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Grades 9-12
Do you agree with this alignment?CCSS.Math.Content.HSA-SSE.A.1 Interpret expressions that represent a quantity in terms of its context
Grades 9-12
Do you agree with this alignment?CCSS.Math.Content.HSA-SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients.
Grades 9-12
Do you agree with this alignment?CCSS.Math.Content.HSA-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
Grades 9-12
Do you agree with this alignment?CCSS.Math.Content.HSA-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions.
Grades 9-12
Do you agree with this alignment?CCSS.Math.Content.HSF-BF.B.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Grades 9-12
Do you agree with this alignment?CCSS.Math.Content.HSF-LE.A.4 For exponential models, express as a logarithm the solution to ab<sup>ct</sup> = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Grades 9-12
Do you agree with this alignment?
STEL-3H Analyze how technology transfer occurs when a user applies an existing innovation developed for one function to a different purpose.
Grades 9-12
Do you agree with this alignment?
CCSS.Math.Content.HSF-BF.B.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Grades 9-12
Do you agree with this alignment?
- BMD Math Presentation (ppt)
- BMD Math Presentation (pdf)
- BMD Math Practice Problems (doc)
- BMD Math Practice Problems (pdf)
- Practice Problem Answers (doc)
- Practice Problem Answers (pdf)
- Log Quiz 2 (doc)
- Log Quiz 2 (pdf)
- Log Quiz 2 Answers (doc)
- Log Quiz 2 Answers (pdf)
- Pamphlet Poster Paper Instructions (doc)
- Pamphlet Poster Paper Instructions (pdf)
Calculation of Bone Mineral Density:
The basic equations for dual-photon absorptiometry can be derived from a number of underlying assumptions. First, it is assumed that the material is composed of varying amounts of only two substances (in this case bone and soft tissue). Second, it is assumed that scatter can be ignored. Under these circumstances, for any given photon energy, the number of photons striking the detector (N) can be calculated from the number of incident photons (No) using Beer's law.
Beer's law:
where μs and μb represent the mass attenuation coefficients (cm2/g) of soft tissue and bone (respectively) and Ms and Mb represent the area densities (g/cm2) of the two tissue types.
I = I o e -μl
I = Intensity.
I0 = Intensity with no object.
μ = attenuation coefficient (depends upon material and x-ray energy).
l = length of the x-ray path.
Lecture Information
Present to students the BMD Math Presentation (PPT), a PowerPoint® file. Then have students complete the BMD Math Practice Problems (PDF). Administer the Log Quiz 2 (PDF). Then students are ready to conduct the associated activity, Light Intensity Lab.
To answer the Challenge Question introduced in lesson 1, students next must Go Public. Refer to the Pamphlet Poster Paper Instructions (PDF) for how students are to present all that they have learned about logarithms and bone mineral density during the course of the entire unit.
Recall from lesson 1 that you are working for a company that makes small specimen cabinet x-ray machines for medical research. Today you will learn how to do some calculations with the data that you might collect from such a machine.
Post-Introduction Assessment
Practice Problems: At lesson end or as homework, assign students to complete the BMD Math Practice Problems (PDF). Review their answers to assess the depth of their understanding.
Lesson Summary Assessment
Quiz: At lesson end, administer the 13-question, multiple-choice Log Quiz 2 (PDF), which covers all types of logarithmic problems covered in this unit. Review students' answers to gauge their comprehension.
Student Presentations: In order to answer the unit challenge question first posed in lesson 1, have students create posters, pamphlest or papers, as described in the Pamphlet Poster Paper Instructions (PDF).
Contributors
Kristyn Shaffer; Megan Johnston
Supporting Program
VU Bioengineering RET Program, School of Engineering, Vanderbilt University
Acknowledgements
The contents of this digital library curriculum were developed under National Science Foundation RET grant nos. 0338092 and 0742871. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.
Copyright
2013 by Regents of the University of Colorado; original © 2006 Vanderbilt University