Ebook Physics of thermal therapy - Fundamentals and clinical applications: Part 1

Physics of Thermal Therapy
Fundamentals and Clinical Applications

Edited by

Eduardo G. Moros



Physics of Thermal Therapy


ImagIng In medIcal dIagnosIs and Therapy
William R. Hendee, Series Editor
Quality and safety in radiotherapy

Targeted molecular Imaging

Todd Pawlicki, Peter B. Dunscombe, Arno J. Mundt,
and Pierre Scalliet, Editors
ISBN: 978-1-4398-0436-0

Michael J. Welch and William C. Eckelman, Editors
ISBN: 978-1-4398-4195-0

adaptive radiation Therapy

C.-M. Charlie Ma and Tony Lomax, Editors
ISBN: 978-1-4398-1607-3

X. Allen Li, Editor

ISBN: 978-1-4398-1634-9

proton and carbon Ion Therapy

Quantitative mrI in cancer

comprehensive Brachytherapy:
physical and clinical aspects

Thomas E. Yankeelov, David R. Pickens, and
Ronald R. Price, Editors
ISBN: 978-1-4398-2057-5

Jack Venselaar, Dimos Baltas, Peter J. Hoskin,
and Ali Soleimani-Meigooni, Editors
ISBN: 978-1-4398-4498-4

Informatics in medical Imaging

physics of mammographic Imaging

George C. Kagadis and Steve G. Langer, Editors
ISBN: 978-1-4398-3124-3

Mia K. Markey, Editor
ISBN: 978-1-4398-7544-5

adaptive motion compensation in
radiotherapy

physics of Thermal Therapy:
Fundamentals and clinical applications

Martin J. Murphy, Editor
ISBN: 978-1-4398-2193-0

Eduardo Moros, Editor
ISBN: 978-1-4398-4890-6

Image-guided radiation Therapy

emerging Imaging Technologies in medicine

Daniel J. Bourland, Editor
ISBN: 978-1-4398-0273-1

Mark A. Anastasio and Patrick La Riviere, Editors
ISBN: 978-1-4398-8041-8

Forthcoming titles in the series
Informatics in radiation oncology

Image processing in radiation Therapy

Bruce H. Curran and George Starkschall, Editors
ISBN: 978-1-4398-2582-2

Kristy Kay Brock, Editor
ISBN: 978-1-4398-3017-8

cancer nanotechnology: principles and
applications in radiation oncology

stereotactic radiosurgery and radiotherapy

Sang Hyun Cho and Sunil Krishnan, Editors
ISBN: 978-1-4398-7875-0

Stanley H. Benedict, Brian D. Kavanagh, and
David J. Schlesinger, Editors
ISBN: 978-1-4398-4197-6

monte carlo Techniques in radiation Therapy

cone Beam computed Tomography

Joao Seco and Frank Verhaegen, Editors
ISBN: 978-1-4398-1875-6

Chris C. Shaw, Editor
ISBN: 978-1-4398-4626-1


Physics of Thermal Therapy
Fundamentals and Clinical Applications

Edited by

Eduardo G. Moros

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in
this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks
of a particular pedagogical approach or particular use of the MATLAB® software.

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To Kimberly,
your love and noble character strengthen me,
and
to our wonderful sons, Jonas and Ezra



Contents
Series Preface. . ............................................................................................................................................... ix
Preface........................................................................................................................................................... xi
Editor........................................................................................................................................................... xiii
Contributors..................................................................................................................................................xv

part I:  Foundations of Thermal Therapy Physics

1Fundamentals of Bioheat Transfer..........................................................................................................................3
Kenneth R. Diller

2Thermal Dose Models: Irreversible Alterations in Tissues............................................................................23
John A. Pearce

3Practical Clinical Thermometry............................................................................................................................ 41
R. Jason Stafford and Brian A. Taylor

4Physics of Electromagnetic Energy Sources........................................................................................................ 57
Jeffrey W. Hand

5The Physics of Ultrasound Energy Sources ........................................................................................................ 75
Victoria Bull and Gail R. ter Haar

6Numerical Modeling for Simulation and Treatment Planning of Thermal Therapy: Ultrasound...... 95
Robert J. McGough

7Numerical Modeling for Simulation and Treatment Planning of Thermal Therapy............................ 119
Esra Neufeld, Maarten M. Paulides, Gerard C. van Rhoon, and Niels Kuster

part II:  Clinical Thermal Therapy Systems

8External Electromagnetic Methods and Devices............................................................................................ 139
Gerard C. van Rhoon

9Interstitial Electromagnetic Devices for Thermal Ablation......................................................................... 159
Dieter Haemmerich and Chris Brace

10

Clinical External Ultrasonic Treatment Devices ............................................................................................ 177

11

Endocavity and Catheter-Based Ultrasound Devices .................................................................................... 189

Lili Chen, Faqi Li, Feng Wu, and Eduardo G. Moros
Chris J. Diederich

vii

viii

Contents

part III: Physical Aspects of Emerging Technology for Thermal Therapy

12
13

Evolving Tools for Navigated Image-Guided Thermal Cancer Therapy .................................................. 203
Kevin Cleary, Emmanuel Wilson, and Filip Banovac

Temperature Imaging Using Ultrasound.. ......................................................................................................... 219

R. Martin Arthur

14

Focused Ultrasound Applications for Brain Cancer ...................................................................................... 241

15

Extracorporeal Ultrasound-Guided High-Intensity Focused Ultrasound Ablation for Cancer
Patients . . ..................................................................................................................................................................... 255

Meaghan A. O’Reilly and Kullervo Hynynen

Feng Wu

16

Using Hyperthermia to Augment Drug Delivery . . .......................................................................................... 279

17

Magnetic Nanoparticles for Cancer Therapy ................................................................................................... 293

18

Application of Gold Nanoparticles (GNP) in Laser Thermal Therapy ..................................................... 319

19

Thermochemical Ablation ..................................................................................................................................... 339

Mark W. Dewhirst

Michael L. Etheridge, John C. Bischof, and Andreas Jordan
Zhenpeng Qin and John C. Bischof
Erik N. K. Cressman


Series Preface
Advances in the science and technology of medical imaging and radiation therapy are more profound and rapid than ever before,
since their inception over a century ago. Further, the disciplines are increasingly cross-linked as imaging methods become more
widely used to plan, guide, monitor, and assess treatments in radiation therapy. Today the technologies of medical imaging and
radiation therapy are so complex and so computer driven that it is difficult for the persons (physicians and technologists) responsible for their clinical use to know exactly what is happening at the point of care, when a patient is being examined or treated. The
professionals best equipped to understand the technologies and their applications are medical physicists, and these individuals are

assuming greater responsibilities in the clinical arena to ensure that what is intended for the patient is actually delivered in a safe
and effective manner.
The growing responsibilities of medical physicists in the clinical arenas of medical imaging and radiation therapy are not without
their challenges, however. Most medical physicists are knowledgeable in either radiation therapy or medical imaging and expert
in one or a small number of areas within their discipline. They sustain their expertise in these areas by reading scientific articles
and attending scientific meetings. In contrast, their responsibilities increasingly extend beyond their specific areas of expertise. To
meet these responsibilities, medical physicists periodically must refresh their knowledge of advances in medical imaging or radiation therapy, and they must be prepared to function at the intersection of these two fields. How to accomplish these objectives is a
challenge.
At the 2007 annual meeting in Minneapolis of the American Association of Physicists in Medicine, this challenge was the topic
of conversation during a lunch hosted by Taylor & Francis Publishers and involving a group of senior medical physicists (Arthur L.
Boyer, Joseph O. Deasy, C.-M. Charlie Ma, Todd A. Pawlicki, Ervin B. Podgorsak, Elke Reitzel, Anthony B. Wolbarst, and Ellen D.
Yorke). The conclusion of this discussion was that a book series should be launched under the Taylor & Francis banner, with each volume in the series addressing a rapidly advancing area of medical imaging or radiation therapy of importance to medical physicists.
The aim would be for each volume to provide medical physicists with the information needed to understand technologies driving a
rapid advance and their applications to safe and effective delivery of patient care.
Each volume in the series is edited by one or more individuals with recognized expertise in the technological area encompassed by
the book. The editors are responsible for selecting the authors of individual chapters and ensuring that the chapters are comprehensive and intelligible to someone without such expertise. The enthusiasm of the volume editors and chapter authors has been gratifying and reinforces the conclusion of the Minneapolis luncheon that this series of books addresses a major need of medical physicists.
Imaging in Medical Diagnosis and Therapy would not have been possible without the encouragement and support of the series
manager, Luna Han of Taylor & Francis Publishers. The editors and authors and, most of all, I are indebted to her steady guidance
of the entire project.
William Hendee
Series Editor
Rochester, Minnesota

ix



Preface
The field of thermal therapy has been growing tenaciously in the last few decades. The application of heat to living tissues, from
mild hyperthermia to high temperature thermal ablation, produces a host of well-documented genetic, cellular, and physiological

responses that are being intensely researched for medical applications, in particular for the treatment of solid cancerous tumors
using image guidance. The controlled application of thermal energy (heat) to living tissues has proven to be a most challenging
feat, and thus it has recruited expertise from multiple disciplines leading to the development of a great number of sophisticated
preclinical and clinical devices and treatment techniques. Among the multiple disciplines involved, physics plays a fundamental
role because controlled heating demands knowledge of acoustics, electromagnetics, thermodynamics, heat transfer, fluid mechanics, numerical modeling, imaging, and many other topics traditionally under the umbrella of physics. This book attempts to capture
this highly multidisciplinary field! Therefore, it is not surprising that when I was offered the honor of editing a book on the physics
of thermal therapy, I was faced with trepidation. After 25 years of research in thermal therapy physics and engineering and radiation
oncologic physics, I was keenly aware of the vastness of the field and my humbling ignorance. Even worse, the rapid growth of the
field makes it impossible, in my opinion, to do it justice in one tome. Consequently, tough decisions had to be made in choosing the
content of the book, and these were necessarily biased by my experience and the kindness of the contributing authors.
The book is divided into three parts. Part I covers the fundamental physics of thermal therapy. Since thermal therapies imply a
source of energy and the means for the controlled delivery of energy, Part I includes chapters on bio-heat transfer, thermal dose,
thermometry, electromagnetic and acoustic energy sources, and numerical modeling. This part of the book, although not exhaustive, can be thought of as an essential requirement for any person seriously seeking to learn thermal therapy physics.
Part II offers an overview of clinical systems (or those expected to be clinical in the near future) covering internally and externally
applied electromagnetic and acoustic energy sources. Despite the large number of devices and techniques presented, these must be
regarded as a sample of the current clinical state of the art. A future book on the same topic may have a similar Part I while the contents of Part II would be significantly different, as clinical technology experiences advances based on clinical practice and new needs.
The last section of the book, Part III, is composed of chapters describing the physical aspects of an emerging thermal therapy technology. The spectrum is wide, from new concepts relatively far from clinical application, such as thermochemical ablation, through
technologies at various stages in the translational continuum, such as nanoparticle-based heating and heat-augmented liposomal
drug delivery, to high-intensity-focused ultrasound interventions that are presently being investigated clinically. Imaging plays a
crucial role in thermal therapy, and many of the newer approaches are completely dependent on image guidance during treatment
administration. Therefore, Part III also covers both conventional as well as emerging imaging technologies and tools for imageguided therapies.
Although there are published books covering the physics and technology of hyperthermia, therapeutic ultrasound, radiofrequency ablation, and other related topics, to my knowledge this is the first book with the title Physics of Thermal Therapy. For this I
have to thank Dr. William Hendee, a medical physicist par excellence and the series editor, who had the original idea for the book.
In regard to the target audience, the book has been written for physicists, engineers, scientists, and clinicians. It will also be useful
to graduate students, residents, and technologists.
Finally, I must confess that it is extremely difficult to remain modest about the list of outstanding contributors. A well-established
expert, at times in collaboration with his/her colleagues, graduate student(s), or postdoctoral fellow(s), has authored each chapter. I
am profoundly grateful to all for the time and effort they invested in preparing their chapters. I would also like to thank Luna Han
and Amy Blalock from Taylor & Francis for their patience, assistance, and guidance during the entire process leading to this book.

xi


xii

Preface

MATLAB ® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy
of the text or exercises in this book. This book’s use or discussion of MATLAB ® software or related products does not constitute
endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB ® software.
MATLAB ® is a registered trademark of The MathWorks, Inc. For product information, please contact:
The MathWorks, Inc.
3 Apple Hill Drive
Natick, MA, 01760-2098 USA
Tel: 508-647-7000
Fax: 508-647-7001
E-mail:
Web: www.mathworks.com


Editor
Eduardo G. Moros earned a PhD in mechanical engineering from the University of Arizona,
Tucson, in 1990. His graduate studies were performed at the radiation oncology department
in the field of scanned focused ultrasound hyperthermia for cancer therapy. After a year as a
research associate at the University of Wisconsin, Madison in the human oncology department,
he joined the Mallinckrodt Institute of Radiology at Washington University School of Medicine,
St. Louis, Missouri, where he was the chief of hyperthermia physics (1991–2005) and the head of
the research physics section (2001–2005). He was promoted to associate professor with tenure in
1999 and to professor in 2005. In August 2005, Dr. Moros joined the University of Arkansas for
Medical Sciences as the director of the division of radiation physics and informatics. Currently,

he is the chief of medical physics for the departments of radiation oncology and diagnostic imaging at the H. Lee Moffitt Cancer Center and Research Institute in Tampa, Florida.
Dr. Moros served as president of the Society for Thermal Medicine (2004–2005), as associate
editor for the journal Medical Physics (2000–2007) and the International Journal of Hyperthermia
(2006–2009), and was a permanent member of the NIH Radiation Therapeutics and Biology
Study Section (2002–2005). He is an associate editor of the Journal of Clinical Applied Medical Physics and the Journal of Radiation
Research. He is an active member of several scientific and professional societies, such as the American Association for Physicists in
Medicine, the American Society for Therapeutic Radiology and Oncology, the Bioelectromagnetics Society, the Radiation Research
Society, the Society for Thermal Medicine, and the International Society for Therapeutic Ultrasound. Dr. Moros holds a certificate
from the American Board of Radiology in therapeutic radiologic physics.
Dr. Moros’s strength has been to collaborate with scientists and clinicians in the application of physics and engineering to facilitate biomedical research and promote translational studies. He has published more than one hundred peer-reviewed articles and has
been a principal investigator/coinvestigator on multiple research grants from the National Institutes of Health, other federal agencies, and industry. He was a recipient of an NIH Challenge Grant in Health and Science Research (RC1) in 2009.

xiii



Contributors
R. Martin Arthur
Department of Electrical and Systems Engineering
Washington University in St. Louis
St. Louis, Missouri

Mark W. Dewhirst
Department of Radiation Oncology
Duke University Medical Center
Durham, North Carolina

Filip Banovac
Department of Radiology
Georgetown University Medical Center

Washington, DC

Chris J. Diederich
Department of Radiation Oncology
University of California, San Francisco
San Francisco, California

John C. Bischof
Department of Mechanical Engineering
Department of Biomedical Engineering
Department of Urologic Surgery
University of Minnesota
Minneapolis, Minnesota
Chris Brace
Department of Radiology
Department of Biomedical Engineering
University of Wisconsin, Madison
Madison, Wisconsin
Victoria Bull
Division of Radiotherapy and Imaging
Institute of Cancer Research
Sutton, Surrey, United Kingdom
Lili Chen
Department of Radiation Oncology
Fox Chase Cancer Center
Philadelphia, Pennsylvania
Kevin Cleary
The Sheikh Zayed Institute for Pediatric Surgical Innovation
Children’s National Medical Center
Washington, DC

Erik N. K. Cressman
Department of Radiology
University of Minnesota Medical Center
Minneapolis, Minnesota

Kenneth R. Diller
Biomedical Engineering Department
University of Texas
Austin, Texas
Michael L. Etheridge
Department of Mechanical Engineering
Department of Biomedical Engineering
University of Minnesota
Minneapolis, Minnesota
Dieter Haemmerich
Department of Pediatrics
Medical University of South Carolina
Charleston, South Carolina
Jeffrey W. Hand
King’s College London
London, United Kingdom
Kullervo Hynynen
Sunnybrook Health Sciences Centre
Toronto, Ontario, Canada
Andreas Jordan
Department of Radiology
Charité-University Medicine
Berlin, Germany
Niels Kuster
Foundation for Research on Information Technologies in

Society (IT’IS)
and
Swiss Federal Institute of Technology (ETHZ)
Zurich, Switzerland
xv


xvi

Faqi Li
College of Biomedical Engineering
Chongqing Medical University
Chongqing, China
Robert J. McGough
Department of Electrical and Computer Engineering
Michigan State University
East Lansing, Michigan
Eduardo G. Moros
H. Lee Moffitt Cancer Center and Research Institute
Tampa, Florida
Esra Neufeld
Foundation for Research on Information Technologies in
Society (IT’IS)
and
Swiss Federal Institute of Technology (ETHZ)
Zurich, Switzerland
Meaghan A. O’Reilly
Sunnybrook Health Sciences Centre
Toronto, Ontario, Canada
Maarten M. Paulides

Department of Radiation Oncology
Erasmus MC Daniel den Hoed Cancer Center
Rotterdam, The Netherlands
John A. Pearce
Department of Electrical and Computer Engineering
University of Texas at Austin
Austin, Texas
Zhenpeng Qin
Department of Mechanical Engineering
University of Minnesota
Minneapolis, Minnesota

Contributors

R. Jason Stafford
Department of Imaging Physics
University of Texas MD Anderson Cancer Center
Houston, Texas
Brian A. Taylor
Department of Radiological Sciences
St. Jude Children’s Research Hospital
Memphis, Tennessee
Gail R. ter Haar
Division of Radiotherapy and Imaging
Institute of Cancer Research
Sutton, Surrey, United Kingdom
Gerard C. van Rhoon
Department of Radiation Oncology
Erasmus MC Daniel den Hoed Cancer Center
Rotterdam, The Netherlands

Emmanuel Wilson
The Sheikh Zayed Institute for Pediatric Surgical
Innovation
Children’s National Medical Center
Washington, DC
Feng Wu
Institute of Ultrasonic Engineering in
Medicine
Chongqing Medical University
Chongqing, China
and
Nuffield Department of Surgical Sciences
University of Oxford
Oxford, United Kingdom


Foundations of Thermal
Therapy Physics

I



1 Fundamentals of Bioheat Transfer  Kenneth R. Diller........................................................................................................... 3



2 Thermal Dose Models: Irreversible Alterations in Tissues  John A. Pearce.................................................................... 23

3 Practical Clinical Thermometry  R. Jason Stafford and Brian A. Taylor........................................................................... 41



4 Physics of Electromagnetic Energy Sources  Jeffrey W. Hand............................................................................................ 57



5 The Physics of Ultrasound Energy Sources  Victoria Bull and Gail R. ter Haar.............................................................. 75



6 Numerical Modeling for Simulation and Treatment Planning of Thermal
Therapy: Ultrasound  Robert J. McGough............................................................................................................................... 95

Introduction  •  Heat Transfer Principles  •  Special Features of Heat Transfer in Biomedical Systems

Introduction  •  Irreversible Thermal Alterations in Tissues  •  Physical Chemical Models: Arrhenius
Formulation  •  Comparative Measures for Thermal Histories: Thermal Dose Concept  •  Applications in Thermal
Models  •  Summary
Introduction  •  Invasive Thermometry  •  Noninvasive Thermometry  •  Summary

Introduction  •  Static Electric and Magnetic Fields  •  Time-Varying Electric and Magnetic Fields  •  Interaction of Electric
and Magnetic Fields with Tissues  •  Propagation of Electromagnetic Fields in Tissues  •  Principles of Electromagnetic
Heating Techniques  •  Invasive Heating Techniques  •  External Heating Techniques

Introduction  •  Ultrasound Transduction  •  Acoustic Field Propagation  •  Interactions of Ultrasound with
Tissue  •  Characterization and Calibration

Introduction  •  Models of Ultrasound Propagation  •  Thermal Modeling and Treatment Planning  •  Summary



7 Numerical Modeling for Simulation and Treatment Planning of Thermal Therapy  Esra Neufeld,
Maarten M. Paulides, Gerard C. van Rhoon, and Niels Kuster.............................................................................................. 119
Need for Treatment Planning in Thermal Therapy  •  Hyperthermia Treatment
Planning (HTP)  •  Segmentation  •  Electromagnetic Simulations  •  Thermal Simulations  •  Field
Optimization  •  Biological Effect Determination  •  Thermometry and Experimental Validation  •  Tissue
Parameters  •  Related Treatments  •  Challenges  •  Conclusions

1



1
Fundamentals of Bioheat Transfer
1.1Introduction..................................................................................................................................3
1.2 Heat Transfer Principles..............................................................................................................3
1.3

Kenneth R. Diller
University of Texas

Thermodynamics and the Energy Balance  •  Conduction Heat Transfer  •  Convection Heat
Transfer  •  Radiation Heat Transfer

Special Features of Heat Transfer in Biomedical Systems....................................................19
Blood Perfusion Effects  •  Thermal Properties of Living Tissues

Acknowledgments..................................................................................................................................21
References................................................................................................................................................21

1.1 Introduction
The science of heat transfer deals with the movement of thermal energy across a defined space under the action of a temperature gradient. Accordingly, a foundational consideration
in understanding a heat transfer process is that it must obey
the law of conservation of energy, or the first law of thermodynamics. Likewise, the process must also obey the second law
of thermodynamics, which, for most practical applications,
means that heat will flow only from a region of higher temperature to one of lower temperature. We make direct and repeated
use of thermodynamics in the study of heat transfer phenomena, although thermodynamics does not embody the tools to
tell us the details of how heat flows across a spatial temperature
gradient.
A more complete analysis of heat transfer depends on further information about the mechanisms by which energy is
driven from a higher to a lower temperature. Long experience has shown us that there are three primary mechanisms
of action: conduction, convection, and radiation. The study of
heat transfer involves developing a quantitative representation for each of the mechanisms that can be applied in the
context of the conservation of energy in order to reach an
overall description of how the movement of heat by all of the
relevant mechanisms influences changes in the thermal state
of a system.
Biological systems have special features beyond inanimate
systems that must be incorporated in the expressions for the
heat transfer mechanisms. Many of these features result in
effects that cause mathematical nonlinearities and render the
analytical description of bioheat transfer more complex than

more routine problems. For that reason, you will find numerical
methods applied for the solution of many bioheat transfer problems, including a large number in this book. The objective of this
chapter is to provide a simple introduction of bioheat transfer
principles without attempting to delve deeply into the details

of the very large number of specific applications that exist. The
following chapters will provide this particular analysis where
appropriate.

1.2  Heat Transfer Principles
In this section we will review the general principles of heat
transfer analysis without reference to the special characteristics
of biological tissues that influence heat transfer and the energy
balance. These matters will be addressed in the next section.
Here we will first consider the energy balance as it applies to all
types of heat transfer processes and then each of the three heat
transport mechanisms.

1.2.1 Thermodynamics and the Energy Balance
The starting point for understanding the movement of heat
within a material is to consider an energy balance for the system of interest. When an appropriate system has been identified
in conjunction with a heat transfer process, an energy balance
shows that the rate at which the internal energy storage within
the system changes is equal to the summation of all energy

3


4

Physics of Thermal Therapy

interactions the system experiences with its environment. This
relationship is expressed as the first law of thermodynamics, the
conservation of energy:

dE
=
dt

∑Q −W +∑m (h

in

− hout ) + Q gen (1.1)

where E is the energy of the system; ΣQ is the sum of all heat
flows, taken as positive into the system; W is the rate at which
work is performed on the environment; Σm(hin − hout ) is the sum
of all mass flows crossing the system boundary, with each having
a defined enthalpy, h, as it enters or leaves the system; and Q gen is
the rate at which energy generation and dissipation occur on the
interior of the system. These terms are illustrated in Figure 1.1,
depicting how the energy interactions with the environment
affect the system energy. In this case the system is represented on
a macroscopic scale, but there are alternative situations in which
it is of advantage to define the boundary as having microscopic
differential scale dimensions.
For the special case of a steady state process, all properties
of the system are constant in time, including the energy, and
the time derivative on the left side of Equation 1.1 is zero. For
these conditions, the net effects of all boundary interactions are
balanced.

Each term in Equation 1.1 may be expressed in terms of a specific constitutive relation, which describes the particular energy
flow as a function of the system temperature, difference between
the system and environmental temperatures, and/or spatial temperature gradients associated with the process as well as many
thermal properties of the system and environment. When the
constitutive relations are substituted for the individual terms
in the conservation of energy (Equation 1.1), the result is a partial differential equation that can be solved for the temperature
within the system during a heat transfer process as a function of
position and time. There are well-known solutions for many of
the classical problems of heat transfer (Carslaw and Jaeger 1959),
but numerous biomedical problems involve nonlinearities that
require a numerical solution method.
Development of the specific equations for the various constitutive relations constitutes a major component of heat transfer
analysis. We will review these relations briefly in the following
sections. The one constitutive equation we will discuss here is
that for system energy storage.

W
E (t)

mout
min

Qgen
Qconv

Qconv
Qrad

FIGURE 1.1  A thermodynamic system that interacts with its environment across its boundary by flows of heat, mass, and work that contribute to altering the stored internal energy.

Although there are a large number of energy storage mechanisms in various materials, those that are likely to be most
relevant to processes encountered in biomedical applications
include: mechanical, related to velocity (kinetic), relative position
in the gravity field (potential), and elastic stress; sensible, related
to a change in temperature; and latent, related to a change in
phase or molecular reconfiguration such as denaturation. Thus,


E = KE + PE + SE + U + L(1.2)

where substitution of a constitutive relation for each term
yields
KE = 1 2 mV 2 , PE = mgz, SE = 1 2 κx 2 , U = mc pT , L = mΛ(1.3)
with properties defined as: KE is kinetic energy; V is velocity; PE
is potential energy; g is the acceleration of gravity; z is position
along the gravity field; SE is the elastic energy; κ is the spring
constant; x is the elastic deformation; U is the internal energy; cp
is the specific heat; T is the temperature; L is the latent energy;
and Λ is the latent heat. The most commonly encountered mode
of energy storage is via temperature change.

1.2.2  Conduction Heat Transfer
Energy can be transmitted through materials via conduction under the action of an internal temperature gradient.
Conduction occurs in all phases of material: solid, liquid, and
gas, although the effectiveness of the different phases in transmitting thermal energy can vary dramatically as a function of
the freedom of their molecules to interact with nearest neighbors. The conductivity and temperature of a material are key
parameters used to describe the process by which a material may
be engaged in heat conduction.
The fundamental constitutive expression that describes the
conduction of heat is called Fourier’s law:

Qcond = −kA

dT
(1.4)
dr

where r is a coordinate along which a temperature gradient exists, and A is the area normal to the gradient and the
cross section through which the heat flows. The negative sign
accounts for the fact that heat must flow along a negative gradient from a higher to a lower temperature. This phenomenon is
described by the second law of thermodynamics and is illustrated in Figure 1.2.
For a process in which the only mechanism of heat transfer is
via conduction, a microscopic system may be defined as shown
in Figure 1.3. Equation 1.4 may be applied to the conservation
of energy (Equation 1.1) to obtain a partial differential equation for the temporal and spatial variations in temperature.
A microscopic system of dimensions dx, dy, and dz is defined
in the interior of the tissue as shown. The various properties
and boundary flows illustrated represent the individual terms


5

Fundamentals of Bioheat Transfer

The differentials in the conduction terms on opposing boundary faces are approximated via a Taylor series expansion with all
higher order terms dropped:

T

∂T
<0
∂r

Qx − Qx + dx = −

∂Qx
dx
∂x

Q y − Q y + dy = −

∂Qx
dy .
∂y
(1.7)

Qz − Qz + dzx = −

∂Qx
dz
∂z

Heat flux

r

FIGURE 1.2  A positive flow of heat along a coordinate occurs by
application of a negative gradient in temperature along the direction
of flow.

When Fourier’s law is substituted for the heat flows, the
boundary interactions are written in terms of the temperature
gradients

to be accounted for in applying conservation of energy to this
elemental volume.
The time rate of change of energy stored in the elemental system is described as



Est = ρc p

dT
dx ⋅ dy ⋅ dz. (1.5)
dt

The individual conduction exchanges across the system boundary are written in terms of the Fourier law:



Qx = − k ⋅ dy ⋅ dz

∂T
∂x

Q y = − k ⋅ dx ⋅ dz

∂T
.
∂ y (1.6)

Qz = − k ⋅ dx ⋅ dy

∂T
∂z



Qx − Qx + dx =

∂  ∂T 

 dx ⋅ dy ⋅ dz
∂x  ∂x 

Q y − Q y + dy =

∂  ∂T 
dx ⋅ dy ⋅ dz .
(1.8)
∂ y  ∂ y 

Qz − Qz + dzx =

∂  ∂T 

 dx ⋅ dy ⋅ dz

∂z  ∂z 

The constitutive Equations 1.5 and 1.8 may be substituted in
the conservation of energy Equation 1.1 for the limited boundary interactions assumed in this analysis, noting that each
resulting term contains the system volume, dx⋅dy⋅dz, which can
be divided out:
ρc p



∂T
∂  ∂T  ∂  ∂T  ∂  ∂T 
=
+  k  . (1.9)
k
k  +
∂t ∂ x  ∂ x  ∂ y  ∂ y  ∂ x  ∂z 

Qz+dz
Qy+dy
z
Qmet

Qx

Est
Qy

dz

y
x

Qx+dx

dy
dx
Qz

FIGURE 1.3.  A small interior elemental system for analysis of heat conduction consisting of differential lengths dx, dy, and dz in Cartesian coordinates as identified within a larger overall system.


6

Physics of Thermal Therapy

This expression is known as Fourier’s equation, and it has units
of W/m3. Although Equation 1.9 was derived in Cartesian
coordinates, it can be generalized to be applicable for alternate
coordinate systems:



ρc p

∂T
=
∂t

(k

T ) . (1.10)

The foregoing equation may be divided by the product ρcp to isolate the temperature term on the left side. The resulting thermal
property is the thermal diffusivity, α = k/ρc:



∂T
=
∂t

(α

T ) . (1.11)

Applications involving therapeutic hyperthermia generally
involve the deposition of a temporally and spatially distributed
internal energy source to elevate the temperature within a target
tissue. In this case, the energy generation term must be included
in the conservation of energy equation, resulting in



Q gen
∂T
= (α T ) +
.
∂t
ρc p (1.12)

The complete solution of Equation 1.11 requires the specification of one (initial) boundary condition in time and two spatial boundary conditions for each coordinate dimension along
which the temperature may vary independently. These boundary conditions are used to evaluate the constants of integration
that result from solution of the partial differential equation.
They are determined according to: (a) the geometric shape of
the system, including whether there is a composite structure
with component volumes having distinct material properties;
(b) what the temperature field interior to the system is like at
the beginning of the process; (c) the geometry of imposed heat
transfer interactions with the environment, such as radiation
and/or convection; and (d) how these environmental interactions may change over time. As an aggregate, these four types
of conditions dictate the form and complexity of the mathematical solution to Equation 1.11, and there are many different outcomes that may be encountered. Mathematical methods
for solving this equation have been available for many decades,
and some of the most comprehensive and still useful texts are
true classics in the field (Morse and Feshback 1953; Carslaw and
Jaeger 1959).
The temporal boundary condition is generally defined in
terms of a known temperature distribution within the system at
a specific time, usually at the beginning of a process of interest.
However, definition of the spatial boundary conditions is not so
straightforward. There are three primary classes of spatial boundary conditions that are encountered most frequently. The thermal
interaction with the environment at the physical boundary of the

system may be described in terms of a defined temperature, heat
flux, or convective process. The energy source applied to create a
hyperthermia state in tissue nearly always results in a geometrically complex internal temperature field imposed onto the system
of analysis. The source can be viewed as a type of internal boundary condition. The solution of the Fourier equation issues in an
understanding of the spatial and temporal variations in temperature, T(x,y,z,t), which can then be applied to predict the therapeutic outcome of a procedure. This analysis is covered in Chapter
2, this book. The solution for the temperature field in tissue may
also be incorporated into feedback control algorithms to achieve

specific therapeutic outcomes.
Several classes of boundary conditions will be discussed to
illustrate how different environmental interactions influence
the flavor of the solution for the temperature field. We will first
consider semi-infinite geometries for which there is an exposed
surface of the tissue and an elevated temperature develops over
space and time in the interior. The overall tissue dimensions are
assumed to be large enough so that the effects of the free surface on the opposing side of the body are not encountered. This
geometry simplifies to a one-dimensional Cartesian coordinate
system, which we will represent in the coordinate x. The three
classes of boundary conditions we will consider for semi-infinite
geometry are: (a) constant temperature, (b) convection, and (c)
specified heat flux.
1.2.2.1 Semi-Infinite Geometry—Constant
Surface Temperature: T (0, t ) = Ts
A temperature Ts is assumed to be applied instantaneously to the
surface of a solid and then to be held constant for the duration of
the process. The solution for this problem is the Gaussian error
function, erfφ, where



erf = 2

π

∫

0

exp(−ξ 2 )d ξ.

For a uniform initial temperature, Ti, throughout the material, the solution is expressed as a dimensionless ratio as



T ( x , t ) − Ts
 x 
= erf 
.
 2 αt  (1.13)
Ti − Ts

1.2.2.2 Semi-Infinite Geometry—Convection:
∂T (0, t )
−k
= h[T∞ − T (0, t )]
∂t
Here, the symbol h is the convective heat transfer coefficient (in
other contexts it may be used for specific enthalpy (Equation 1.1)
or for the Planck constant (Equation 1.71)), which is a function
of the boundary interaction between a solid substrate and the
environmental fluid that is at a temperature T∞. Convective heat
transfer analysis is focused primarily on determining the value
for h to be applied as the boundary condition for a conduction


7

Fundamentals of Bioheat Transfer

process within a solid immersed in a fluid environment. The
solution for the internal temperature field is
2
T ( x ,t ) − Ts
 x    hx h αt  
= erfc 
− exp 
+ 2 

 2 αt    k
Ti − Ts
k 


 x
h αt  
× erfc 
+

 2 αt
kw  




where Fo is called the Fourier number, representing a dimensionless time. It is the ratio of the actual process time compared
to the thermal diffusion time constant for the system.
The Fourier equation (Equation 1.11) in one dimension can be
written in terms of these dimensionless variables as

∂θ∗ ∂2 θ∗ (1.19)
=
∂Fo ∂x ∗2

(1.14)


where erfcφ is the complementary error function defined as
erfcφ = 1 − erfφ.

for which the initial and boundary conditions are written as

1.2.2.3 Semi-Infinite Geometry—Defined
∂T (0, t ) Qs
Surface Heat Flux: − k
=
= qs
∂t
A
A heat flow per unit area of the surface is assumed to be applied
instantaneously and then maintained continuously for the duration of the process. Typical causes of this boundary condition
are an external noncontact energy source that is in communication with the surface of a solid via electromagnetic radiation.
The solution of this problem is



θ∗ ( x ∗ ,0) = 1

∂θ∗ 
= 0 (1.21)

∂ x ∗ x ∗ = 0



T ( x ,t ) − Ti =

2qs

αt

k

π

 − x 2  qs x
 x 
erfc 
(1.15)
exp 
−
 4αt  k
 2 αt 

1.2.2.4 Finite Dimensioned System with
Geometric and Thermal Symmetry
Another boundary condition encountered frequently occurs
when a finite-sized solid is exposed to a new convective environment in a stepwise manner. If the system and process both

exhibit geometric and thermal symmetry, an explicit mathematical solution exists for one-dimensional Cartesian, cylindrical,
and spherical coordinates in the form of an infinite series. As
will become apparent, it is advantageous to write the problem
statement and solution in terms of dimensionless variables.
The temperature is scaled to the environmental value as
θ = T − T∞ and is normalized to the initial value:



θ T − T∞
θ∗ = =
. (1.16)
θi Ti − T∞

Likewise, the independent variables for position and time are
normalized to the size and thermal time constant of the system,



x∗ =

x
(1.17)
L

where L is the half width of the system along the primary thermal diffusion vector,



t ∗ = Fo =

αt (1.18)
L2

(1.20)

which is a result of thermal and geometric symmetry,
−


∂θ∗ 
= Biθ∗ (1, Fo) (1.22)

∂x ∗ x ∗ =1

where Bi is defined as the Biot number, which represents the
ratio of thermal resistances by condition on the interior of the
solid and by convection at the surface interface with a fluid
environment:
Bi =



hL
=
k

L
1

kA
hA

. (1.23)

The solution for this problem is in the form of an infinite
series,
∞

θ∗ =


∑C e
n

−ζn2 Fo

(

)

cos ζn x ∗ (1.24)

n =1

where Cn satisfies for each value of n,



Cn =

4sin ζn
2ζn + sin(2ζn ) (1.25)

and the eigenvalues ζn are defined as the positive roots of the
transcendental equation


ζn tan ζn = Bi . (1.26)

Thus, there are unique values of Cn and ζn for each value of Bi.
The first six roots of this expression have been compiled as a
function of discrete values for Bi between 0 and ∞, and are available widely (Carslaw and Jaeger 1959).