Table Of ContentClinical Laboratory
Mathematics
CM
Mark D. Ball, Ph.D., SC(ASCP)
Specialty Chemistry Development Coordinator
Pathology Laboratories
Northwestern Memorial Hospital
Chicago, Illinois
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Library of Congress Cataloging-in-Publication Data
Ball, Mark D.
Clinical laboratory mathematics / Mark D. Ball.—1st ed.
p. ; cm. — (Pearson clinical laboratory science series)
Includes index.
ISBN-13: 978-0-13-234437-1
ISBN-10: 0-13-234437-8
I. Title. II. Series: Pearson clinical laboratory science series.
[DNLM: 1. Clinical Laboratory Techniques--methods--Problems and
Exercises. 2. Mathematics--methods--Problems and Exercises. 3. Problem
Solving--Problems and Exercises. QY 18.2]
610.72'4--dc23
2012036467
10 9 8 7 6 5 4 3 2 1
ISBN 10: 0-13-234437-8
ISBN 13: 978-0-13-234437-1
Contents
Foreword vii
Preface viii
Reviewers ix
Chapter 1 arithmetiC and algebra 1
Addition 2
Subtraction 3
Multiplication 4
Division 5
Fractions 6
Percentages 12
Algebra 15
Ratios 20
Appendix Mind Over Calculator: A Few Tips for Calculating
Without Electronics 27
Chapter 2 exponential notation
and logarithms 35
Exponents and Logarithms 36
Negative Exponents 37
Exponential Notation (Scientific Notation) 37
Algebraic Rules for Exponents 39
Algebraic Rules for Logarithms 39
The Logarithmic Scale 41
Logarithmic Transformation of Ratios 43
The Natural Logarithm 44
The Usefulness of Logarithms 45
Chapter 3 rounding and the signifiCanCe
of figures 52
Simple Rounding 53
Figure Significance 54
Significant Figures in the Results of Calculations 57
Significant Figures in Exponential Expressions and Logarithms 60
Absolute and Relative Uncertainty 62
iii
iv Table of conTenTs
Rounding Error 62
Keeping Figure Significance in Perspective 63
Chapter 4 systems of measurement 68
United States Customary System of Units 69
The Metric System 69
International System of Units 73
Equivalencies Between Systems 73
The Mole 74
Dimensional Analysis 75
The Ratio Method 77
Temperature Scales 77
Chapter 5 solutions and ConCentrations 80
Expressing Concentration 81
Specific Gravity 84
The pH Scale 85
Converting Between Units 87
Chapter 6 dilutions 93
Simple Dilutions 93
Serial Dilutions 96
Chapter 7 proportionality, graphs, and rates
of Change 102
Proportionality 103
Straight Lines 103
Standard Curves 110
Nonlinear Graphs 112
Online Appendix
Forms of the Equation of a Line
Chapter 8 statistiCs 118
The Central Tendency 119
Dispersion 121
The Normal Distribution 123
Regression 125
Judging Goodness-of-Fit 129
Correlation 132
Coefficient of Determination 134
Significance Testing 135
Online Appendices
Degrees of Freedom
The Reasoning Behind Linear Regression
Classical and Inverse Calibrations
Table of conTenTs v
Data Weighting
How the Range of x Values Affects Uncertainty in a Regression Line
The Reasoning Behind the Coefficients of Correlation and Determination
Arithmetic Means, Geometric Means, and Log-Normal Distributions
Chapter 9 Chemistry 153
Analytical Spectroscopy 154
Enzyme Kinetics 157
pH Buffering 164
Acid-Base Disorders 168
Anion Gap 169
Osmolarity and Osmolality 170
Osmolality Gap/Osmolarity Gap 171
Lipid Calculations 172
Creatinine Clearance 173
Online Appendices
Elementary Chemical Kinetics
Models of Enzyme Catalysis
An Example of the Physiological Significance of K
M
Linear Transformations of the Michaelis-Menten Equation
Sample Calculation of an Acetic Acid / Acetate Buffer
The Bicarbonate Buffer System of Blood
The Formula for Calculating Plasma Osmolarity
Making Sense of the Equation for Creatinine Clearance
Chapter 10 hematology 180
Manual Cell Enumeration 181
Hematocrit 183
Erythrocyte Indices 183
Rule of Three 184
Red-Cell Distribution Width 184
Reticulocyte Production Index 185
Enumerating Reticulocytes 186
Osmotic Fragility 187
International Normalized Ratio 188
Correction of WBC Count for Nucleated RBCs 189
Online Appendix
International Sensitivity Index (ISI)
Chapter 11 Quality Control 194
Levey-Jennings Charts 195
Westgard Multirules 197
Multirules for Three Controls 200
Resolution of Out-of-Range Controls 201
Rationale Behind the Rules 201
Random and Systematic Error 202
vi Table of conTenTs
Chapter 12 method evaluation 208
Diagnostic Value 209
Quality Assurance for Methods and Instruments 214
Determining Reportable Range 217
Determining Reference Ranges 218
Online Appendices
ROC Curves and Likelihood Ratios
Polynomials and the Polynomial Method for Evaluating Nonlinearity
Capturing the Curves in Linearity Testing
Frequently Asked Questions and Common Misunderstandings 223
Online Advanced Topics
Equilibrium Constants
Effect Size
Ion Product of Water
Activity as Opposed to Concentration
Estimation of Glomerular Filtration Rate by Means of Exogenous Tracers
Answer Key 233
Glossary 292
Index 299
Foreword
Clinical Laboratory Mathematics is part of Pearson’s Clinical Laboratory Science series of textbooks,
which is designed to balance theory and practical applications in a way that is engaging and useful to stu-
dents. The author of Clinical Laboratory Mathematics presents highly detailed technical information
and effective tools that will help beginning learners envision themselves as members of a healthcare team,
while helping advanced learners and practitioners continue their education. The synergy of theoretical and
practical information in this text enables learners to analyze data and synthesize conclusions. Additional
applications and instructional resources are available at www.myhealthprofessionskit.com.
We hope that this book, as well as the entire series, proves to be a valuable educational resource.
Elizabeth A. Gockel-Blessing (formerly Zeibig), PhD, MLS(ASCP)CM
Clinical Laboratory Science Series Editor, Pearson Health Science
Interim Associate Dean for Student and Academic Affairs, Department of Clinical Laboratory
Science, Doisy College of Health Sciences, Saint Louis University
vii
Preface
Clinical Laboratory Mathematics is a comprehensive textbook on the mathematical techniques and
theories of clinical laboratory science. It is written for students at any point on the trajectory toward an
undergraduate or graduate degree in the discipline, from an associate’s degree to a doctorate. Students and
practitioners of related disciplines will also find the book useful: pathologists, medical students, nurses,
pharmacists, biochemists, biomedical engineers, and physician assistants.
Going well beyond the notion of “relevance,” this book tries to convey the conviction that learning
mathematics is not only helpful, but often critical, in the high-technology milieu of a clinical laboratory. It
repeatedly highlights the reasons for developing a battery of mathematical tools: (1) to handle unfamiliar
mathematical problems that arise in the course of laboratory work; (2) to follow the reasoning in seminars,
papers, and discussions; (3) to detect mathematical errors made by individuals; (4) to recognize instrument
malfunctions or method anomalies through mathematical irregularities; (5) to adapt new methods, ideas,
and technologies that require some mathematical competence; and (6) to shift smoothly into research-
oriented work, whether in the form of short-term projects in a routine laboratory, long-term projects in
a research laboratory, or method development at a diagnostics company.
Therefore, the book integrates real-world examples of mathematical tools at work in the clinical
laboratory. To achieve this goal, practice problems are strategically designed to have the student confront
scenarios involving mathematical questions that have both context and consequence. Such problems offer
the student a chance to think under the circumstances that a laboratory professional might encounter on
the job, requiring him or her to solve a mathematical problem while coming to appreciate the importance
of correct calculation and the repercussions of error.
The book supports both self-guided study and the more traditional lecture-discussion format. Meet-
ing the needs of either approach, or of any approach in-between, is a matter not only of organizing the top-
ics logically, but also of liberally cross-referencing so that students see connections and common motifs.
This technique promotes comprehension while lessening the burden of brute memorization.
The book includes online resources (www.myhealthprofessionskit.com) intended to meet the needs
of advanced users: (1) chapter appendices, which elaborate topics introduced in the main text, and (2)
advanced topics, which emerge from frequently asked questions and from the main text.
Because some instructors start their courses with a review of arithmetic, and because some students
seek such a review, the first chapter deals with addition, subtraction, multiplication, division, fractions,
decimals, percentages, algebra, and ratios. Furthermore, it includes strategies for speeding up calculations
without relying on electronics. Subsequent chapters cover increasingly complex and specialized topics,
with the online appendices carrying those topics to the greatest depth.
viii
Reviewers
James E. Daly, MEd, MT(ASCP) Kathleen Paff, MA, MT(ASCP)
Lorain County Community College Kellogg Community College
Elyria, Ohio Battle Creek, Michigan
Amy Gatautis, MBA, MT(ASCP)SC Travis M. Price, MS, MT(ASCP)
Cuyahoga Community College Weber State University
Cleveland, Ohio Ogden, Utah
Amy Kapanka, MS, MT(ASCP)SC Susan Schoffman, MPH, MT(ASCP), CLS(NCA)
Hawkeye Community College Tulsa Community College
Cedar Falls, Iowa Tulsa, Oklahoma
Pamela Lonergan MS, MT(ASCP)SC Dick Y. Teshima, MPH, MT(ASCP)
Norfolk State University University of Hawaii at Manoa
Norfolk, Virginia Honolulu, Hawaii
Leslie Lovett, MS, MT(ASCP) Darius Y. Wilson, EdD
Pierpont Community and Technical College Southwest Tennessee Community College
Fairmont, West Virginia Memphis, Tennessee
Stephen Olufemi Sodeke, PhD, MA Patricia Wright, MT(ASCP)
Tuskegee University Southeastern Community College
Tuskegee, Alabama Whiteville, North Carolina
ix
