Heat Transfer in Medicine and Biology

Analysis and Applications Volume 2

H eat Transfer in Medicine and Biology Analysis and Applications Volume 2

Edited by

Avraham Shitzer Technion - Israel Institute of Technology Haifa, Israel

and

Rohert C. Eherhart University of Texas Health Science Center Dallas, Texas

PLENUM PRESS· NEW YORK AND LONDON

Library of Congress Cataloging in Publication Data

Main entry under title:

Heat transfer in medicine and biology.

Bibliography: p. Inc1udes index. 1. Body temperature. 2. Animal heat.3. Heat - Transmission. 4. Medical ther-

mography. I. Shitzer, Avraham, 1940- . 11. Eberhart, Robert C., 1937-[DNLM: 1. Biomedical Engineering. 2. Body Temperature Regulation. 3. Energy Transfer. QT 34 H437] QP135.H37 1984 ISBN 978-1-4684-8287-4 DOI 10.1007/978-1-4684-8285-0

© 1985 Plenum Press, New York

599'.01912 ISBN 978-1-4684-8285-0 (eBook)

Softcover reprint of the hardcover 1st edition 1985

A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013

All rights reserved

84-17698

No part of this book may be reproduced, stored in aretrieval system, or transmitted in any form or by any means, electronic, mechanicaI, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

CONTRIB UTORS

R. F. Boehm Department of Mechanical and Industrial Engineering, University of Utah, Salt Lake City, Utah

Thomas C. Cetas Division of Radiation Oncology, University of Arizona, Tueson, Arizona

lohn C. Chato Department of Mechanical and Industrial Engineering, University of Illinois, U rbana, Illinois

Michael M. Chen Department of Mechanical and Industrial Engineering, U niversity of Illinois, Urbana, Illinois

Robert M. Curtis Shiley, Inc., Irvine, California Kenneth R. Diller Department of Mechanical Engineering, Biomedical Engineering

Center, University of Texas, Austin, Texas Robert C. Eberhart Department of Surgery, University of Texas Health Science

Center, Dallas, Texas L. M. Hanna Department of Bioengineering, University of Pennsylvania, Philadel­

phia, Pennsylvania Linda]. Hayes Department of Aerospace Engineering and Engineering Mechanics,

University of Texas, Austin, Texas Rakesh K. ]ain Department of Chemical Engineering, Carnegie-Mellon University,

Pittsburgh, Pennsylvania T. ]. Love School of Aerospace, Mechanical, and Nuclear Engineering, University

of Oklahoma, Norman, Oklahoma lohn ]. McGrath Bioengineering Transport Processes Laboratory, Michigan State

University, East Lansing, Michigan Robert W. Olsen Department of Surgery, University of Texas Health Science Center,

Dallas, Texas P. W. Scherer Department of Bioengineering, University of Pennsylvania, Philadel­

phia, Pennsylvania Avraham Shitzer Department of Mechanical Engineering, Technion, Israel Institute

of Technology, Haifa, Israel George]. Trezek Department of Mechanical Engineering, University of California,

Berkeley, California A. ]. Welch Department of Electrical and Computer Engineering and Biomedical

Engineering Program, University of Texas, Austin, Texas

v

PREFACE TO VOLUME 2

This volume presents applications of heat transfer in medicine. In recent years this subject has received increased attention as many more medical applica­tions, both in the hyper- and hypothermic ranges, have been developed. Among the subjects covered in this volume are the heating of body tissues and organs, electrosurgery, skin bums, preservation of tissues by freezing and the application of cryosurgery, heat and mass transfer in the respiratory system, heat transfer in teeth, thermography, and temperature measurement. Also included are two appendices, one presenting thermophysical properties of biological tissues and the other introducing the principles of numerical tech­niques in bioheat transfer.

As in Volume 1, each of the chapters in this volume is written by a leading authority in the Held. The chapters all begin with a review of the state of the art, which is followed by a rigorous analytical exposition of the problem treated. Examples are given, wherever applicable, for the use of the results in actual situations.

For a quickly expanding Held of science, we see here only the beginning of the application of heat transfer analysis in medicine. In the coming years we may expect more problems to be deHned and analyzed and more fruitful collaboration between life scientists and physical scientists. It is our sincere hope that this book shall serve the purpose of providing the required founda­tion for this needed collaboration.

vii

AVRAHAM SHITZER

ROBERT C. EBERHART

CONTENTS OF VOLUME 2

Contents of Volume 1 ................................................... xiii N omenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Part IV: APPLICATIONS OF HEAT TRANSFER IN MEDICINE

Chapter 16 ANALYSIS OF HEAT TRANSFER AND TEMPERATURE DISTRIBUTIONS IN TISSUES DURING LOCAL AND WHOLE-BODY HYPERTHERMIA Rakesh K. Jain

1. Introduction .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Distributed Parameter Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Lumped Parameter Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4. Thermal Energy Absorbed during Ultrasound, Microwave, and

Radiofrequency Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5. Temperature Distributions during Normothermia. . . . . . . . . . . . . . . . . . . . . . . 2f) 6. Temperature Distributions during Hyperthermia . . . . . . . . . . . . . . . . . . . . . . . 31 7. Summary and Recommendations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Chapter 17 TEMPERA TURE FIELDS AND LESION SIZES IN ELECTROSURGERY AND INDUCTION THERMOCOAGULATION Avraham Shitzer

1. Introduction '" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2. Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3. Summary........................................................... 81

Chapter 18 ANALYSIS OF SKIN BURNS Kenneth R. Diller

1. Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2. Physiology of Skin .................................................. 86 3. Physiological Aspects of the Burn Injury .............................. 88 4. Determination of Burn Injury from the Temperature-Time History ...... 92 5. Cooling Therapy for Burn Wounds ................................... 119 6. Quantification of the Microscopic Response to Burns ... . . . . . . . . . . . . . . .. 123 7. Conclusion ......................................................... 129

ix

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Chapter 19 LASER IRRADIATION OF TISSUE A. J. Welch

1. Introduction .,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 2. Laser Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3. Laser Safety. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4. Medical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5. A Thermal Model of Laser Irradiation of Tissue . . . . . . . . . . . . . . . . . . . . . . . . 146 6. Measurement and Prediction of Thermal Damage in the Retina. . . . . . . . . . .L 72-7. Summary and Recommendations for Future Work ...................... 179

Chapter 20 PRESERVATION OF BIOLOGICAL MATERIAL BY FREEZING AND THAWING John J. McGrath

1. Introduction ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 185 2. Basic Aspects of Low-Temperature Preservation ................. ....... 187 3. Osmosis ............................................................ 194 4. General Responses of Biomaterials to Freezing and Thawing . . . . . . . . . . .. 201 5. Mechanisms of Freeze-Thaw Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6. Applied Cryobiology ................................................ 212 7. Thermodynamic Models and Cryobiology ... . . . . . . . . . . . . . . . . . . . . . . . . .. 218 8. Cryomicroscopy..................................................... 229 9. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Chapter 21 THERMAL ANALYSIS FOR CRYOSURGERY George J. Trezek

1. Introduction ....................................................... , 239 2. Background......................................................... 239 3. Bioheat Transfer Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 240 4. Maximum Lesion Size ............................................... 241 5. Rate of Lesion Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 243 6. Steady-State Results and Applications ................................. 245 7. Evaluating the Rate of Lesion Growth ................................ 248 8. Comparison of Lesion Growth Computational Methods . . . . . . . . . . . . . . . .. 252 9. Cryosurgical Atlas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 255

Chapter 22 ANALYSIS OF HEAT EXCHANGE DURING COOLING AND REWARMING IN CARDIOPULMONARY BYPASS PROCEDURES Rohert M. Curtis and George J. Trezek

1. Introduction ....................................................... , 261 2. Heat Exchanger Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 263 3. Whole-Body Heat Exchanger Models.................................. 274 4. Summary and Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . 286

Contents oi Volume 2 / xi

Chapter 23 HEAT AND WATER TRANSPORT IN THE HUMAN RESPIRATORY SYSTEM P. W. Scherer and L. M. Hanna

1. Introduetion ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 2. Anatomy of the Respiratory System ................................... 288 3. Physiologieal Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 4. Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 5. Summary........................................................... 303

Chapter 24 HEAT TRANSFER IN TEETH R. F. Boehm

1. Introduetion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 2. The Tooth .. ... . . ..... ... ... ... ... .... . .......... ... ... . .... ........ 307 3. Thermal Response in "Normal" Teeth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 312 4. Thermal Faetors due to Tooth Repair ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 318 5. Implieations of Preventive Proeesses .................................. 321 6. Fundamentals of Thermal Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 323 7. Future Direetions ................................................... 327

Chapter 25 ANALYSIS AND APPLICATION OF THERMOGRAPHY IN MEDICAL DIA GNOSIS T. J. Love

1. Introduetion ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 333 2. Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 334 3. Optieal Properties of Skin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 337 4. Control of the Clinie Environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 339 5. Applieations of Thermography ....................................... 341 6. Relationship of Blood Flow to Temperature Pattern. . . . . . . . . . . . . . . . . . .. 344 7. Summary........................................................... 350

Chapter 26 COMPUTER-AIDED TOMOGRAPHIC THERMOGRAPHY Michael M. Chen

1. Introduetion .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 353 2. The Governing Equation and Relevant Parameters. . . . . . . . . . . . . . . . . . . .. 354 3. Separating the Perfusion and Metabolie-Heating Terms. . . . . . . . . . . . . . . .. 355 4. Effeets of Metabolie Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 356 5. Effeets of Blood Perfusion ........................................... 359 6. A Numerieal Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 362 7. Conelusions......................................................... 368

xii / Contents of Volume 2

Part V: SELECTED TOPICS

Chapter 27 TEMPERATURE MEASUREMENT Thomas C. Cetas

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 2. Temperature Scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 3. Thermometer Probes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 4. Thermography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 5. Calibration ......................................................... 386 6. Thermometry in Therapeutic Electromagnetic Field-Induced Heating ... 387 7. Thermometry in Therapeutic Ultrasound-Induced Heating . . . . . . . . . . . . . . 389 8. Summary........................................................... 391

Chapter 28 SENSITIVITY ANALYSIS OF ERRORS INDUCED IN THE DETERMINATION OF TISSUE PERFUSION A vraham Shitzer and Robert C. Eberhart

1. Introduction ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 2. Analysis ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 3. Results ............................................................. 395 4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

Appendix 2 SELECTED THERMOPHYSICAL PROPERTIES OF BIOLOGICAL MATERIALS John C. Chato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Appendix 3 FINITE-DIFFERENCE AND FINITE-ELEMENT METHODSOFSOLUTION Avraham Shitzer, Linda J. Hayes, Robert W. Olsen, and Robert C. Eberhart

1. Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 419 2. Finite Difference .................................................. " 419 3. Finite Element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 4. Conclusion ......................................................... 428

Index... ............................................................... 431

CONTENTS OF VOLUME 1

Contents of Volume 2 ................................................... xiii Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xvii

Chapter 1 INTRODUCTION Robert C. Eberhart and Avraham Shitzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Part I: THERMOREGULATION IN HOMEOTHERMS

Chapter 2 REGULATION OF BODY TEMPERATURE IN MAN AND OTHER MAMMALS John Bligh

1. Introduction ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2. Control and Regulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3. Cybernetics: From Biology to Engineering and Back Again . . . . . . . . . . . . . 17 4. Principles of Engineering Regulation ................................. 18 5. Problems in Understanding Thermoregulation ......................... 19 6. Balance between Heat Production and Heat Loss. . . . . . . . . . . . . . . . . . . . . . . 20 7. Heat Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8. Heat Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 9. Sensors of Temperatures ............................................. 28

10. Relations between Thermoregulatory Effectors and Ambient Temperature 31 11. Relation between Thermoregulatory Effectors and Core Temperature .... 32 12. Peripheral Vasomotor Tone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 13. Creation of the Null Point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 14. Variations in the Thermosensor-Thermoregulatory Effector Relations .... 37 15. Central Synaptic Interference: Another Approach to Understanding

Thermoregulation ................................................... 39 16. The Central Regulator and Its Associated Peripherals-A Synthesis ...... 42 17. Autonomie and Behavioral Thermoregulation . . . . . . . . . . . . . . . . . . . . . . . . . . 43 18. Thermal Acclimatization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 19. Summary........................................................... 47

Chapter 3 TEMPERATURE REGULATION IN EXERCISING AND HEAT­STRESSED MAN L. B. Rowell and C. R. Wyss

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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2. Thermoregulation in Resting Man .................................... 53 3. Competitive Interaction between Thermoregulatory and

Nonthermoregulatory Reflexes in Man. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4. Problems in Developing Models of Human Temperature Regulation. . . . . 73

Chapter 4 THERMOREGULATION IN PATHOLOGICAL STATES J. M. Lipton

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2. Normal Body Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3. Fever............................................................... 81 4. Dysthermia Produced by CNS Lesions ................................ 87 5. Heat Illness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6. Hypothermia........................................................ 96 7. Influence of Drugs, Alcohol, and Anesthesia on Thermoregulation . . . . . .. 100

Chapter 5 THERMOREGULATION AND SLEEP H. Craig Heller and Steven F. Glotzbach

1. Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2. Daily Cycles of Body Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3. Changes in Body Temperatures and Thermoregulatory Responses

Associated with Sleep States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4. Effects of Temperature on Sleep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5. Summary and Conclusions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Part II: THERMAL MODELING OF TISSUES

Chapter 6 HEAT GENERATION, STORAGE, AND TRANSPORT PROCESSES Avraham Shitzer and Robert C. Eberhart

1. Introduction ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 2. Tissue Heat Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 138 3. Storage of Thermal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 141 4. Conduction of Heat through the Tissue ............................... 142 5. Transport (Convection) of Heat by the Circulatory Sysyem . . . . . . . . . . . . .. 142 6. Heat Exchange with the Environment (Boundary Conditions) . . . . . . . . . .. 144 7. Summary........................................................... 151

Chapter 7 THE TISSUE ENERGY BALANCE EQUATION Michael M. Chen

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 153 2. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Contents of Volume 1 / xv

Part 111: ANALYTICAL AND BIOHEAT TRANSFER STUDIES

Chapter 8 MEASUREMENT OF THERMAL PROPERTIES OF BIOLOGICAL MATERIALS John C. Chato

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 167 2. Temperature Measurements .......................................... 167 3. Property Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 174 4. Properties Related to Ultrasonic Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5. Properties Related to Electromagnetic Radiation Effects ................ 187 6. Summary........................................................... 189

Chapter 9 ESTIMATION OF TISSUE BLOOD FLOW H. Frederick Bowman

1. Introduction ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 2. Thermal Model-Tissue Heat Balance... ......... ... ........ ... ... .... 194 3. Classification of Perfusion Estimation Techniques . . . . . . . . . . . . . . . . . . . . .. 195 4. Methods of Hensel-Betz-Benzing and Müller-Schauenburg . . . . . . . . . . . .. 197 5. Modeling for Thermal Property and Perfusion Measurements. . . . . . . . . . .. 202 6. Analytical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 203 7. Solution Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 204 8. Significance of Expanded f(t) ........................................ 210 9. Application of the Transient Thermal Model to Derive Perfusion . . . . . . .. 211

10. Perfusion Measurement Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 212 11. The Thermal Diffusion Probe ........................................ 213 12. Experimental Verification ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 217 13. Comparison with Cameron's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 225 14. Summary........................................................... 226

Chapter 10 GENERAL ANALYSIS OF THE BIOHEAT EQUATION A vraham Shitzer

1. Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 231 2. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 232

Chapter 11 GREEN'S FUNCTION FORMULATION OF THE BIOHEAT TRANSFER PROBLEM Hans G. Klinger

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 245 2. Mathematical Formulation of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 246 3. The Calculation of Green's Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 248 4. Physical Interpretation of the Solution Procedure ....................... 249 5. Macroscopic Temperature Distribution .............................. " 250 6. Effect of Local Perfusion Symmetries on the Temperature Distribution. .. 254

xvi / Contents of Volume 1

Chapter 12 THERMAL MODELS OF SINGLE ORGANS Rohert C. Eherhart

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 261 2. Anatomy of the Microcirculation ..................................... 261 3. Heat Transfer Models in Specific Organs .............................. 273 4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 322

Chapter 13 MATHEMATICAL SIMULATION OF HUMAN THERMAL BEHA VIOR USING WHOLE-BODY MODELS Eugene H. Wissler

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 325 2. Equations of Change for Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 328 3. Boundary and Initial Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 331 4. Metabolie Heat Generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 332 5. Heat and Mass Transport in the Lungs ................................ 338 6. Physiological Control Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 340 7. Validation of the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 355 8. Analysis of the "Lost Bell" Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 364 9. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 369

Chapter 14 THERMAL INTERACTION WITH GARMENTS Avraham Shitzer and John C. Chato

1. Introduction .............. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 375 2. Heat Exchange with Clothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 375 3. Heat Exchange with Fluid-Cooled Garments in Contact with the Skin. .. 382 4. Cylindrical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 384 5. Rectangular Model .................................................. 388

Chapter 15 ON THE RELATIONSHIP BETWEEN TEMPERATURE, BLOOD FLOW, AND TISSUE HEAT GENERATION Avraham Shitzer

1. Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 395 2. Analysis ............................................................ 395 3. Discussion ............................................... '.' . . . . . . . . . 397 4. Conclusion ......................................................... 408

Appendix 1 REVIEW OF ELEMENTARY HEAT TRANSFER A vraham Shitzer and Rohert C. Eherhart

1. Introduction ........................................................ 411 2. Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 411 3. Radiation........................................................... 414 4. Convection ......................................................... 416

Index.................................................................. 419

NOMENCLATURE

Numbers in parentheses after the description refer to chapters and equations in which symbols are first used or are thoroughly defined-e.g., "(17-1r' refers to equation (1) of Chapter 17. Equations are not listed for some symbols in such general usage as to be familiar to all readers. Dimensions are given in terms of mass (M), moles, length (L), time (t), temperature (T), volts (V), and ohms. SI and cgs dimensions are given in text and tables, where appropriate. Symbols that appear infrequently or in one section only are not listed.

A A A A Ai

A ij

a a a B B B B B BM Bi b b

c c c c

area, L 2

heat conductance parameter (A3-20), MI Lt2 T constant in burn injury equation (17-1), t- 1

coefficient in general solution of bioheat equation (15-6) parameters in endurance time equations (13-66, 67, 69); para-

meters in shivering thermogenesis eq uations (13-60, 62, 64); parameter in glycogen depletion equation (13-68); attenu­ation parameter for vasoconstrictor outflow (13-45)

sensitivity coefficient of variable Xj with respect to Xi (28-2) tube or capillary spacing (14-37), L radius, thickness (9-12), L species activity (20-2), ML 31moIe amplitude coefficient (7-16b) endurance time parameter (13-67) magnetic field (16-16), (Mohmlt)I/2 coefficient in general solution of bioheat equation (15-7) heat convection parameter (A3-20), MI Lt3 T basal metabolic rate (22-41), Mlt3

Biot number (A1-Table 4) tissue thickness (14-47) or blood vessel spacing (4.5-28), L cQefficient determining effect oE species concentration on per-

fusion rate (13-46), L 31moIe speed oE sound (16-10), Llt thermal equivalent O 2 consumption (12-71), MI L 3 t 2

wminlwmax (12-34) thermal capacitance matrix in finite element formulation of

bioheat equation (A3-33), MI LtzT cardiac index (22-30), L 2 I t species concentration, moleiL 3

shear velocity (8-37), LI t coefficients oE Planck radiation equation (Al-5)

xvii

xviii / Nomenclature

C, cp

D D D D D,d Die d E I1E Er F F F I1F FBF F,f Fe> Fw

fel fi fw C, Cl> C 2

C,g Gz

C .. ,i Ce g g .. ,o

H HR H,h Hf h h hfg

hr

I

heat capacity at constant pressure (6-3), L 2 I Tt2

diffusion coefficient (14-17), L 21t vasodilator outßow signal intensity (13-53), t- 1

heat generation and convection parameter (A3-20), MI Lt3

coefficient in general solution of bioheat equation (15-8) diameter, L weighting factor for vasodilator outßow (13-56) thickness, L blood ßow weighting function (12-37) activation energy (17-1), ML21t2

emissive power (8-2), M It3

vessel ßow (6-9), L 31t radiation shape factor (6-14) heat source function (10-15), LT net force (8-31), MLI t2

fore arm blood ßow (3-2), elt skin heat ßux (14-37), ML2/r' weighting functions for thermoreceptor afferent error signals

(13-41), (Tt)-l shivering thermogenesis parameter (13-62), t- 1

frequency (16-10), t- 1

probe heat generation rate function (9-12), r 1/ 2

number of independent intensive properties (20-1) heat convection and surface exchange matrix in finite element

formulation (A3-33), ML 2 I t3

percentage of surface area through wh ich convective exchange occurs (6-11)

ratio of surface area of clothed body to that of nude body (6-25) solution function for generalized bioheat equation (10-8) surface wetting coefficient (6-20) gain factors in shivering thermogenesis (13-62,63) generalized initial temperature distribution (10-7,14) Graetz number (AI-Table 4) temperature weighting function (14-51), T Green's function (11-8) gravitation constant (Al-Table 4), LI t2

temperature weighting function (14-52), T heat generation rate (12-71), ML 21t3

heart rate (3), r 1

heat transfer coefficient (6-11), MI Tt3

heat loss (3-1), ML 21t2

Planck constant = 6.625 X 10-34 W S2 (19-1) height, L latent heat of vaporization (6-17), L 2 I t2

radiative heat exchange coefficient (6-16), MI Tt3

radiation intensity (18-17) Mir'

Ii(x)

f fi(X)

fw j K K K

k

k' kij

L L L, I Lf M M NI M,N

M,m M,h m m m m fit N N Ni Nu

p

Nomenclature / xix

total thermal resistance offered by clothing ensemble (6-26), Tt 3 /M

modified Bessel function of the first kind, of order i (12-14) surface-absorbed thermal radiant heat flux (18-16), MI t3

Bessel function of the first kind, of order i (12-15) volume flux (20-31), elt current density (16-15), amperes I L 2

skin temperature gradient (18-12), TI L kernel of finite integral transform (10-21) thermal conduction matrix in finite element formulation of

bioheat equation (A3-33), ML 21 Tt] thermal conductivity parameter between body layers (2.2-1),

(22-34), MLI{3T modified Bessel function of the second kind, of order i (14-36) mass transfer coefficient (23-11), LI t thermal conductivity (6-4), MLI Tt] mass transfer coefficient (14-17), L 3 I t Boltzmann constant = 1.38 x 10-23 W s/K (27-1)

{COefficient in vasodilator equation (13-53); in glycogen depletion equation (13-71); in sweating rate equation (13-58)

chemical reaction rate constant (13-32), moleiL 3 t thermal conductivity tensor (11-56), MLITt3

work load (13-66) flow rate function (10-1), L-2

length, L latent heat of fusion (20-2), L 2 I t 2

molecular weight (23-9), MImoIe kernel of finite integral transform (10-39) molar flux rate of species (13-16), molel L 3 t

number of capillaries in average cube in x, y directions (11-44), L-3

Mass, M rate of shivering thermogenesis (13-59), ML 2 I t 3

water vapor permeation coefficient of skin (6-17), tl L ratio of electrode to tissue thermal inertia (17-10) mass fraction (8-9) concentration (20-4), moleiL 3

mass flow rate (6-21), Mit (M I L 3 t in other usage) number of heat transfer units (12-33) mass transfer rate (13-19), moleiL 2 t thermal conductance (12-46), L 3

Nusselt number(AI-Table 4) molecular concentration (20-21), L-] outward unit vector normal to surface element perimeter, L

xx / Nomenclature

P P Pe P,Po Pr Pi, Pi P P P Pw Q Q

R, r R R,Ro RQ R, r Rs

Re r r,R 5 5

5, s 5, s Sc Sh 5 b 52, 53 S

T tlT To t t tc

tf U U,u

reduced temperature function (10-l4), T power (8-31), ML 2 /t 3

Peclet number (A1-Table 4) power deposition in tissue (17-2), M / Lt3

Prandtl number (A1-Table 4) partial pressure of species i (6-17), M / Lt2

fluid pressure (20-15), M / Lt2

wave number (7-25) number of phases (20-1) water permeability (20-32), L 2 t / M heat input (16-13), ML 2 /t 2

heat generation rate, cooling rate, heat storage rate (6-1), ML 2 / r3

temperature coefficient of metabolism (22-40) heat generation rate per unit volume, M / Lt3

heat flux (6-6), M /t 3

heat source strength per unit length (8-26), ML/t3

electrical resistance (9-33), ohm ratio of heat loss via coronary arteries to myocardial heat pro-

duction (12-Fig. 36) molar chemical reaction rate (13-17), mole/ L 3 t stretching ratio for finite difference grid (18-44) universal gas constant, 8.317 W s/K mol (17-1) respiratory quotient, VC02 / V02 (22-41) radial coordinate; L or dimensionless real part of shear acoustic impedance (8-35), M / L 2 t Reynolds number (A1-Table 4) tissue: blood solubility ratio (13-33) resistance to heat exchange (22-Fig. 5), Tt3 / M body surface area (22-29), L 2

signal intensity in the autonomie nervous system (13-46), TO. 25

(t, t- 1 in other usage) heat source (26-4), M / Lt3

sweat generation rate (14-11), L 3 /t, M/L2 t Schmidt number (A1-Table 4) Sherwood number (A1-Table 4) contributions to shivering thermogenesis (13-59), ML 2/t 2

sensitivity of tissue damage to temperature (19-49) temperature, T temperature increment, T thermal parameter (12-13), T time, t thickness (22-6), L characteristic time for heat conduction (11-4), t endurance time (13-65), t overall heat transfer coefficient (12-31), M / Tt J

temperature difference (9-5), T

U,IIII(, .p

(1,1111(,0

V V V V;v Vw

V

V

V, V V

W W

x X, X

x. X

Y Y, Y Yi(X) Z,Z Zc Z

Z

Nomenclature I xxi

temperature inerement, heated, perfused tissue (9-4), T temperature inerement, heated, unperfused tissue (9-4), T voltage (9-33), V temperature differenee (12-13), T volume, L 3

volumetrie gas ßow rate (13-38), L 3 I t partial molar volume of water (20-42), L 3 /mol vapor solution funetion for generalized bioheat equation (10-8), T velocity, Llt ßow distribution funetion (28-24) weight (6-12), M total ßow rate (22-1), L 3 I t speeifie humidity (6-21) blood ßow rate in tissue, per unit volume (6-6), r l

weighting faetor for perfusion response to thermally indueed vasoeonstrietor outßow (13-46), T-O,25

eoneentration, one eompartment model (28-23), MI L 3

rectangular eoord~nate; L or dimensionless imaginary eomponent of shear aeoustic impedanee (8-36),

MIL2 t mole fraetion (20-19) eoneentration, two eompartment model (28-24), MI L 3

rectangular eoordinate; L or dimensionless Bessel funetion of the seeond kind, of order i (14-42) rectangular or axial eylindrieal eoordinate; L or dimensionless vasoeonstrietion faetor (13-43) body height (6-12), L perfusion parameter (9-36)

GREEK SYMBOLS

01

01

ß ß ß

ß

r

thermal diffusivity (8-20), L 2 I t radiation absorptivity (AI-9) radiation absorption eoefficient (18-18), L -I aeoustie absorption eoeffieient (8-34), L -I heat ßux funetion (14-38), TI L sweating eoeffieient (14-11), MI L 2 Tt parameter for transient probe heating rate (9-12), MI Lt2.5

perfusion parameter (AI-Table 4) eoefficients determining intensities of autonomie responses

(13-46, 50, 51, 58, 60), r l

thermal eoefficient of expansion =; (:;) p' T- 1

steady state heat generation rate (9-12), MI Lt3

xxii / Nomenclature

Y y' y, Yi

Yi

8 8 8A 8V e e e

11 11 0,8 8 8 K

K

A A A A A

Ai /L

/L /Li

jI

jI

jI

t TI p p (T

(T

(T

(Ti

(Ti

T

heat generation parameter (AI-Table 4) metabolie heat generation parameter (12-75) vessel spacing parameters (14-48), L- 1

coefficients determining intensities of autonomie responses (13-42,45)

thermal inertia parameter (17-25) depth of layer with varying temperature (26-18), L control element surface area (7-Fig. 2), L 2

control element volume (7-Fig. 2), L dielectric constant (14-12), (ohm)-1 radiative emission coefficient (6-14) heat transfer effectiveness (12-25) equivalent length parameter (14-40), L- 1

tube diameter to tube spacing ratio (14-31) coefficients determining intensities of autonomie responses

(13-53,54) heat transfer effectiveness parameter (16-3) dimensionless radial distance (19-11) dimensionless or reduced temperature angle (7-Fig. 2) freezing point depression (20-3), T reaction rate constant, Arrhenius Equation (18-4), r 1

perfusion ratio (28-13) thermal equilibration length coefficient (7-7), MI Lt2 T surface heat transfer parameter (10-13) ratio of tissue to blood heat capacity (9-8) wavelength (16-10), L depth of embedded heat source (26-21). area fraction of the ith vessel (7-22) viscosity (12), MI Lt chemieal potential (20-5), ML 2 I t2 mol dynamic shear stiffness j = 1 (8-35), MI Lt2 ; shear viscosity

j = 2 (8-76), MILt kinematic viscosity (23-4), L 2 I t dissociation constant (20-21) configuration parameter for general solution of bioheat

equation (10-1) dimensionless length (12-47) osmotie pressure (20-6), MI Lt2

density (6-3), MI L 3

radiation reßectivity (AI-9) Stefan-Boltzmann constant = 5.67 X 10-8 W 1m2 K4 (6-14) electrieal conductivity (16-12), (ohmt1

image radius (19-8), L spacing parameter (14-49) heat generation parameter (28-13) dimensionless time, Fourier number (Al-Table 4)

w w

radiation transmissivity (AI-9) dimensionless freezing time (21-43) dimensionless or reduced temperature efHux of a flow path (11-36), L 3 / t solution osmotic coefficient (20-18)

Nomenclature / xxiii

basis function in finite element formulation of bioheat equation (A3-28a)

heat source distribution and perfusion parameter (26-24) equivalent tissue heat production (9-14), M / Lt3

combination of modified Bessel functions (15-11) surface heat transfer parameter (10-3) axial temperature distribution function (10-60), T tissue damage function (17-1) solution osmolality (20-3), M / L 3

volume element (11-21), L 3

frequency (17-4), Cl

SUPERSCRIPTS

+

* * * (B) (F) (i) (i), (n) (8) (T) (0)

SUBSCRIPTS

a

a a, art amb avl A B B B,b

dimensionless or reduced variable transient dimensionless quantity limit of discrete blood vessels setpoint, reference dimensionless quantity bound free phase ith, nth iteration steady state total first spectral component

air afferent artery ambient available alveolus, airway body surface blood

xxiv / Nomenclature

br c c c C, conv CO2

ehern cl d d, diff e e e e e e, env e,ex eff es

f f f f, fr fg G,g g h, hy h

j k L I I M M,m m m m m max mbf min N

brain conduction core coolant convection carbon dioxide chemical reaction clothing dentin diffusion equilibrium enamel equivalent electrical evaporation environment expired, exhaled effective esophageal fabric fat length of exposure frozen fluid to gas glycogen generation hypothalamus heating tissue element in finite difference schemes initial blood vessel generation inspired blood vessel generation conduction lung lactic acid liquid mucus-air interface node in finite difference mesh metabolie mean muscle tissue, intrinsic maximum myocardial blood flow minimum necrotic

O2

0

p pe r r r, re ra ref res s s s, sw set sh sk ss st t ty u v v W,W 0 0 1 2 A 00

OVERLINES

oxygen outer probe, wave number phase change radiation resting, basal reetal right atrial referenee respiratory surfaee, skin solute sweating set point shivering skin steady state storage tissue tympanie uniform vein volumetrie water referenee, ambient initial prior to oeclusion, inner following oeclusion, outer wave length ambient

d dt

time-weighted funetion normalized parameter

Nomenclature / xxv

transformed function, averaged parameter unit vector

UNDERLINES

matrix

xxvi / Nomenclature

OPERATORS

V grad V2 div' grad {) Dirae delta funetion

ljJ 2 A V ---

At

J integral

2: summation

n produet

<> spatial average