Theory and Applications of Heat Transfer in Humans

An authoritative guide to theory and applications of heat transfer in humans 

Theory and Applications of Heat Transfer in Humans 2V Set offers a reference to the field of heating and cooling of tissue, and associated damage. The author—a noted expert in the field—presents, in this book, the fundamental physics and physiology related to the field, along with some of the recent applications, all in one place, in such a way as to enable and enrich both beginner and advanced readers.  The book provides a basic framework that can be used to obtain ‘decent’ estimates of tissue temperatures for various applications involving tissue heating and/or cooling, and also presents ways to further develop more complex methods, if needed, to obtain more accurate results.  The book is arranged in three sections: The first section, named ‘Physics’, presents fundamental mathematical frameworks that can be used as is or combined together forming more complex tools to determine tissue temperatures; the second section, named ‘Physiology’, presents ideas and data that provide the basis for the physiological assumptions needed to develop successful mathematical tools; and finally, the third section, named ‘Applications’, presents examples of how the marriage of the first two sections are used to solve problems of today and tomorrow.

This important text is the vital resource that:

• Offers a reference book in the field of heating and cooling of tissue, and associated damage.
• Provides a comprehensive theoretical and experimental basis with biomedical applications
• Shows how to develop and implement both, simple and complex mathematical models to predict tissue temperatures
• Includes simple examples and results so readers can use those results directly or adapt them for their applications

Designed for students, engineers, and other professionals, a comprehensive text to the field of heating and cooling of tissue that includes proven theories with applications. The author reveals how to develop simple and complex mathematical models, to predict tissue heating and/or cooling, and associated damage. 

• Language: English
• Publisher: Wiley
• Release date: Apr 16, 2018
• ISBN: 9781119127321

Book preview

List of Contributors to Volume I

Anant Agrawal

US Food and Drug Administration, Silver Spring, MD, USA

Jennifer K. Barton

Biomedical Engineering, The University of Arizona, Tucson, AZ, USA

John C. Bischof

Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA

Emad S. Ebbini

Department of Electrical and Computer Engineering, University of Minnesota Twin Cities, Minneapolis, MN, USA

Pradyumna Ghosh

Department of Mechanical Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi, UP, India

Anand Gopinath

Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, USA

Do-Hyun Kim

US Food and Drug Administration, Silver Spring, MD, USA

Harishankar Natesan

Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA

John Nyenhuis

School of Electrical and Computer Engineering, Purdue University,

West Lafayette, IN, USA

Henrik Odéen

Imaging & Neurosciences Center, Utah Center for Advanced Imaging Research, Department of Radiology, University of Utah, Salt Lake City, UT, USA

Dennis L. Parker

Imaging & Neurosciences Center, Utah Center for Advanced Imaging Research, Department of Radiology, University of Utah, Salt Lake City, UT, USA

Daniela Zavec Pavlinic

TITERA Ltd., Murska Sobota, Slovenia; and

Faculty of Mechanical Engineering University of Maribor, Maribor, Slovenia

T. Joshua Pfefer

US Food and Drug Administration, Silver Spring, MD, USA

David Porter

Minnesota Supercomputing Institute (MSI), University of Minnesota, Minneapolis, MN, USA

Elaine P. Scott

School of Science, Technology, Engineering and Mathematics, University of Washington Bothell, Bothell, WA, USA

Keith M. Sharp

Department of Mechanical Engineering, JB Speed School of Engineering, University of Louisville, Louisville, KY, United States

Devashish Shrivastava

US Food and Drug Administration, Silver Spring, MD, USA; and

In Vivo Temperatures, LLC, Burnsville, MN, USA

Joshua E. Soneson

US Food and Drug Administration, Silver Spring, MD, USA

Jinfeng Tian

Center for Magnetic Resonance Research, University of Minnesota, Minneapolis, MN, USA

Jonathan W. Valvano

Department of Electrical and Computer Engineering, University of Texas, Austin, TX, USA

Rachana Visaria

In Vivo Temperatures, LLC, Burnsville, MN, USA

William C. Vogt

US Food and Drug Administration, Silver Spring, MD, USA

Eugene H. Wissler

McKetta Department of Chemical Engineering, University of Texas at Austin, Austin, TX, USA

List of Contributors to Volume II

John P. Abraham

University of St. Thomas, School of Engineering, St. Paul, MN, USA

John G. Baust

Institute of Biomedical Technology, State University of New York at Binghamton, Binghamton, NY, USA; and

Department of Biological Sciences, Binghamton University, Binghamton, NY, USA

John M. Baust

Institute of Biomedical Technology, State University of New York at Binghamton, Binghamton, NY, USA; and

CPSI Biotech, Owego, NY, USA

Robert G. Van Buskirk

Institute of Biomedical Technology, State University of New York at Binghamton, Binghamton, NY, USA; and

Department of Biological Sciences, Binghamton University, Binghamton, NY, USA; and

CPSI Biotech, Owego, NY, USA

Zhongrong Chen

Department of Electronic Science and Technology, School of Information Science and Technology, University of Science and Technology of China, Hefei, P. R. China

Christopher J. Gordon

US Environmental Protection Agency, Research Triangle Park, NC, USA

John M. Gorman

Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA

Michael Hubig

Institute of Forensic Medicine, Jena University Hospital – Friedrich Schiller University Jena, Jena, Thuringia, Germany

Alexander LeBrun

Department of Mechanical Engineering, University of Maryland, Baltimore County, Baltimore, MD, USA

Gita Mall

Institute of Forensic Medicine, Jena University Hospital – Friedrich Schiller University Jena, Jena, Thuringia, Germany

Thomas L. Merrill

Mechanical Engineering, Rowan University, Glassboro, NJ, USA

Jennifer E. Mitchell

Mechanical Engineering, Rowan University, Glassboro, NJ, USA

Holger Muggenthaler

Institute of Forensic Medicine, Jena University Hospital – Friedrich Schiller University Jena, Jena, Thuringia, Germany

Brittany B. Nelson-Cheeseman

University of St. Thomas, School of Engineering, St. Paul, MN, USA

Senta Niederegger

Institute of Forensic Medicine, Jena University Hospital – Friedrich Schiller University Jena, Jena, Thuringia, Germany

Sean R. Notley

Centre for Human and Applied Physiology, School of Medicine, University of Wollongong, Wollongong, Australia; and

University of Ottawa, Faculty of Health Sciences, Human and Environmental Physiology Research Unit, Ottawa, Canada

Fazil Panhwar

Department of Electronic Science and Technology, School of Information Science and Technology, University of Science and Technology of China, Hefei, P. R. China

John Pearce

Department of Electrical and Computer Engineering, University of Texas, Guadalupe, Austin, TX, USA

Brian D. Plourde

University of St. Thomas, School of Engineering, St. Paul, MN, USA

Surya Prakash

Tej Bahadur Sapru Hospital, Allahabad, UP, India

Satish Ramadhyani

Galil Medical Inc., St Paul, MN, USA

Anthony T. Robilotto

Institute of Biomedical Technology, State University of New York at Binghamton, Binghamton, NY, USA; and

Department of Biological Sciences, Binghamton University, Binghamton, NY, USA; and

CPSI Biotech, Owego, NY, USA

Avraham Shitzer

Department of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa, Israel

Devashish Shrivastava

US Food and Drug Administration, Silver Spring, MD, USA; and

In Vivo Temperatures, LLC, Burnsville, MN, USA

Mayank Singh

Department of General Surgery, MLN Medical College, Allahabad, UP, India

Ephraim M. Sparrow

Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA

John R. Stark

Department of Mechanical Engineering, University of Kansas, Lawrence, KS, USA

Nigel A. S. Taylor

Centre for Human and Applied Physiology, School of Medicine, University of Wollongong, Wollongong, Australia

Jinfeng Tian

Center for Magnetic Resonance Research, University of Minnesota, Minneapolis, MN, USA

Lauren J. Vallez

University of St. Thomas, School of Engineering, St. Paul, MN, USA

Thad E. Wilson

Marian University, Indianapolis, IN, USA

Gang Zhao

Department of Electronic Science and Technology, School of Information Science and Technology, University of Science and Technology of China, Hefei, P. R. China

Liang Zhu

Department of Mechanical Engineering, University of Maryland, Baltimore County, Baltimore, MD, USA

Preface

The story of this book started with a phone call to my parents. I was complaining how there was a need for a comprehensive book to introduce the great width and breadth of the field of bioheat transfer, something like the impressive two-volume collection of Avraham Shitzer and Robert C. Eberhart, Heat Transfer in Medicine and Biology, that was published in 1985. My dad suggested that, maybe, I should take on the endeavor of producing such a work. The next thing I knew, I was seeking help from John Wiley & Sons, asking whether they were interested in publishing such a work, and from all of my friends and colleagues, asking them if they were willing to help produce such a book. The response was amazing – everyone most enthusiastically supported the idea, and collectively, we, as a community, were on our way to try to cover the vast ground that the field of bioheat transfer has grown into over the last 40 or so years.

The basic idea was to produce a book that presented the fundamental physics and physiology related to the field of bioheat transfer, along with some of the recent applications, all in one place, in such a way as to enable and enrich both a beginner and an advanced reader. It would provide a basic framework that could be used to obtain decent estimates, and also present ways to further develop more complex methods, if needed, to obtain more accurate results. To this end, the book is arranged in three sections. The first section, Physics (Chapters 1–20), presents the fundamental mathematical framework that can be used as is or combined together forming more complex tools to determine in vivo heating; the second section, Physiology (Chapters 21–27), presents ideas and data that provide the basis for the physiological assumptions needed to develop successful mathematical tools, and, finally, the third section, Applications (chapters 28–36), presents a few recent examples of how the marriage of the first two sections are used to solve problems of today and tomorrow.

More specifically, under Physics, Chapters 1 and 2 present the fundamentals of bioheat transfer modeling based on conservation of mass, momentum, and energy (i.e., first principles). The presented material is such that it can be used as is, or made more complex or simplified to build custom models for one's own applications and desired accuracy. Chapter 3 discusses the role of various blood vessels in transporting thermal energy and affecting temperature distribution, in order to help develop application-specific thermal models. Chapter 4 discusses how a physiologically realistic blood-vessel network may be generated to gain a better understanding of the thermal nature of the vascular bed. Chapter 5 discusses how whole-body models of humans can be built and used to investigate the effect of clothing on humans. Chapter 6 presents physical and computational models of the human cardiovascular system. Chapter 7 discusses lumped-parameter-based computational models of the human respiratory system. The models presented in Chapters 6 and 7 may be combined with the bioheat models presented in Chapters 1 and 2 to build more complex simulation models to better understand human physiology. Chapter 8 presents various techniques of inverse heat transfer that may be used to determine parameters of computational models for biomedical applications. Chapters 9–12 present techniques to determine the source-term distribution (or energy density) in various applications. Source-term distribution is needed to determine resultant in vivo temperature rise. Chapter 9 presents ways to quantify the source-term distribution due to the propagation of light in tissue. Chapter 10 presents ways to quantify the source-term distribution due to the propagation of ultrasound in tissue. Chapters 11 and 12 present ways to quantify the source-term distribution due to electromagnetic field propagation in tissues as experienced in magnetic resonance imaging (MRI) and other applications without and with very high conductivity material present in the body, respectively. Chapter 13 presents methods for fast computation and for evaluation of the computational performance. Chapters 14–16 present methods of thermometry using conventional methods, MRI, and ultrasound, respectively. Chapters 17–19 present methods for measuring thermal, optical, and dielectric properties of tissue, respectively. Chapter 20 presents methods of calorimetry for micro- and nanoscale biomedical applications.

Under Physiology, Chapter 21 presents cardiovascular and metabolic response to various thermal stimuli. Chapter 22 presents the role of morphological and physiological considerations in the modeling of human heat loss, mainly sweating. Chapter 23 presents changes in thermoregulatory response due to radiofrequency heating. Chapter 24, briefly, describes the clinical management of skin burns. It is hoped that presenting brief description of the current clinical-management methods in this book will prompt development of more effective ways to address clinical needs in the future. Chapter 25 describes thermoregulatory responses to toxic agents. Chapters 26 and 27 present methods for characterizing thermal damage due to high and low temperatures, respectively.

Under Applications, Chapter 28 presents the use of bioheat transfer models developed in Chapter 1 and techniques for determining the source term due to the electromagnetic field distribution in MRI developed in Chapter 11 in predicting heating during MRI. Chapter 29 presents uses of nanoparticles in cancer treatment. Chapter 30 presents how to use models of the blood-vessel network in planning hyperthermic and cryothermic treatments. Chapter 31 describes progress in thermal-imaging-assisted cryosurgery in cancer treatment and pain management. Chapter 32 presents methods for determining the role of blood flow on MRI-induced heating near stents. Chapter 33 presents ways to assess the damage associated with skin burns using mathematical models. Chapter 34 presents surface and endovascular cooling methods, models, and measurements. Chapter 35 presents methods for assessing the effects of wind chill on body temperature. The last chapter, Chapter 36, presents uses of the principles of bioheat transfer in determining time of death.

This book is intended for engineers, physicists, and everyone else (e.g., physiologists, physicians, garment designers, researchers working on performance enhancements and thermal safety in sports, defense, and challenging environments, etc.) who might be interested in the role of different ‘thermal’ stimuli (e.g., drugs, exercise, environmental conditions like in desert of Iraq or mountains of Afghanistan, clothing, electromagnetic fields, sound waves, etc.) on in vivo temperatures.

Writing this book has been a rewarding experience. I am deeply indebted to all my friends and colleagues who took time out from their busy schedules to contribute to this book and who responded gently to all my sobbing, prodding, and threats. I also appreciate the support of those who promised to contribute but for some reason failed to follow through. I promise all of them that I will not haunt them until it is time for the next edition. Next, I would like to acknowledge and appreciate the enthusiastic support of my PhD mentor, Professor Robert Roemer of the University of Utah, and Dr. Chris Gordon of the US EPA throughout this project. Their help was instrumental in finding the experts I needed every time I hit the wall. I am also immensely thankful to Charlie Lemaire of In Vivo Temperatures, LLC, who patiently read through several chapters of the book and provided suggestions on how to improve the message, and to the kind staff of John Wiley & Sons (Elsie Merlin, Tim Bettsworth, and Jenny Cossham) who were quick, efficient, and always ready to help. Finally, I would like to thank my friend, colleague, and worst critic of 19 years (and wife for 15 or so years – thank goodness she married me), Dr. Rachana Visaria, for making sure that I kept making progress and finished in time.

Devashish Shrivastava, PhD

Clarksville, MD, USA

Supplementary Material

To access supplementary materials for this book please use the download links shown below.

There you will find valuable material designed to enhance your learning.

The website includes supplementary material, organized in four folders, to facilitate readers with their bioheat work. More specifically, the first folder, TissueHeating, contains the program Tissue.Heating.Ver1.0 (and a readme file containing instructions on how to run the program) designed to compute the worst-case local in vivo temperature rise due to a given specific absorption rate (SAR) in normally perfused muscle tissue and associated thermal dose values based on cumulative equivalent minutes @ 43 °C (CEM43) calculations. The second folder, GBHTM, contains the program InVivoTemperatures.Ver1.2s.Student (and a readme file containing instructions on how to run the program) designed to compute in vivo temperatures, in three-dimensions and time, using user inputs and the generic bioheat transfer model (GBHTM)/Pennes BHTM. The program is limited to the matrix size of 100 × 100 × 250 (100 nodes each in X and Y direction and 250 nodes in Z direction). The third folder, bioheat_vasculature.tarz, contains the source code, along with examples, of how to build the application and generate the vasculature examples discussed in Chapter 4. Finally, the fourth folder, bioheat_performance_tests.tarz, includes the source code, along with examples, of how to build the applications and run benchmark tests to evaluate the performance of codes discussed in Chapter 13.

http://booksupport.wiley.com

Please enter the book title, author name or ISBN to access this material.

Section I

Theory: Physics

Chapter 1

A Generic Thermal Model for Perfused Tissues

Devashish Shrivastava¹,²

¹US Food and Drug Administration, Silver Spring, MD, USA

²In Vivo Temperatures, LLC, Burnsville, MN, USA

* Corresponding author: [e-mail]

1.1 Introduction

Many diagnostic and therapeutic procedures require a thermal model for perfused tissues for determining in vivo temperatures in order to better plan and implement those procedures (e.g., heating during MRI, burn management, etc.). However, it is extremely challenging to determine in vivo temperatures by solving the ‘exact’ thermal model, derived from first principles, and called the convective energy equation (CEE) [1]. This is so because it requires at least 20 linear computational nodes across the diameter of a blood vessel to obtain a numerically converged temperature solution of the CEE [2]. Blood vessel diameters range from ∼3 cm in large vessels (e.g., aorta, vena cava) to ∼3 µm in capillaries inside a human body. Thus, it requires a stupendous amount of computational power (∼3(10¹¹) nodes for every 1 mm³ assuming a uniform mesh resolution of 0.15 µm) to solve for the temperatures in perfused tissues if the CEE is used alone. This is in addition to the daunting challenge of knowing the blood velocity field in all the vessels down to every single capillary as a function of space and time since the blood velocity is a necessary input to the CEE. Therefore, to manage computational costs, temperatures in tissues embedded with ‘small’ (<1 mm in diameter) more frequent blood vessels are determined using ‘approximate’ thermal models known as bioheat transfer models (BHTMs) [3, 4], and temperatures in ‘large’ (vessel diameter ≥ 1 mm), less frequent blood vessels are determined using the ‘exact’ thermal model, the CEE [1].

BHTMs can be derived using first principles or proposed intuitively. The objective of this chapter is to present a general methodology to derive BHTMs from first principles. Deriving BHTMs from first principles is important since it helps relate the variables and parameters of the bioheat models to the underlying physiology, which provides better insight into the most fundamental mechanisms at play. The methodology is used to, first, derive a general, ‘two-compartment’ BHTM with very few assumptions – herein called the two-compartment generic bioheat transfer model (GBHTM) [5]. Later, more general forms of the model (i.e., a three-compartment GBHTM and an ‘N + 1’ compartment GBHTM) are derived using the same methodology. Next, the newly derived two-compartment GBHTM is compared with Pennes' intuitively proposed (and thus, empirical) ‘gold standard’ BHTM to better understand the implicit and explicit assumptions, and thereby application regime of Pennes' BHTM. Pennes' BHTM is chosen since it is a simple, empirical, and widely used BHTM. Lack of a formal derivation makes it difficult to relate the parameters and variables of this simple BHTM to the underlying physiology (e.g., blood flow, blood vasculature geometry, thermal properties of tissue and blood vessels), which in turn has, historically, made the implementation and interpretation of Pennes' BHTM and its results controversial and unreliable. Finally, in vivo temperature predictions of the two-compartment GBHTM and Pennes' BHTM are compared to the measured temperatures for magnetic resonance imaging (MRI) applications to further illustrate the usefulness of the bioheat models.

1.2 Derivation of Generic Bioheat Thermal Models (GBHTMs)

Above, we discuss the complexity and the impracticality of solving the exact thermal model, the CEE, alone in perfused tissues to obtain point-wise true tissue temperature distribution in space and time. That scenario forces us, as a compromise, to develop an approximate thermal model to determine the temperature of a ‘volume’ of tissue (i.e., a ‘volume averaged’ tissue temperature), rather than determining the temperature of each point in tissue (i.e., point-wise true tissue temperature).

Let's derive an approximate thermal model, a two-compartment GBHTM, by volume averaging the CEE. The presented methodology is general and is used to derive a three-compartment GBHTM and an ‘N + 1’ compartment GBHTM later in this chapter. For those of you who are not interested in the derivation, you may refer directly to the final differential form of the two-compartment GBHTM presented in Equations 1.10 and 1.11 below. The three-compartment GBHTM is presented in Equations 1.14. The ‘N + 1’ compartment GBHTM is presented in Equations 1.16. Simplifications used to obtain the GBHTMs are presented in Equations 1.9.

1.2.1 A Two-Compartment Generic Bioheat Transfer Model

Let's consider a finite, vascularized, heated tissue. Let's assume that the blood stays inside the vasculature and everything surrounding the blood is a non-moving solid tissue. Conserving energy at a point in the solid tissue and blood results in the following point-wise true, exact thermal model, the CEE [1]. Note that the velocity of solid tissue uT is zero in Equation 1.1 by our assumption.

1.1

equation

Also, note that solving Equation 1.1 for the point-wise temperatures in tissue perfused with smaller (<1 mm in diameter), more frequent blood vessels is impractical due to the tremendous cost of computation and the unavailability of the three-dimensional blood velocity field uBl. Next, let's imagine that our perfused tissue is made of several smaller volumes put together, herein called the averaging volume and each averaging volume V consists of two sub-volumes or compartments: a solid tissue sub-volume VT and blood sub-volume VBl. Integrating Equation 1.1 over the solid tissue and blood sub-volumes, separately, in an averaging volume results in Equation 1.2.

1.2

equation

Applying divergence theorem to the first terms on the left-hand side (LHS) and right-hand side (RHS) of Equation 1.2 results in Equation 1.3. Interested readers are encouraged to derive Equation 1.3 using Equation 1.2 for themselves.

1.3

equation

where, i and j = T, Bl and i ≠ j.

Assuming (a) constant density, (b) constant specific heat, and (c) incompressible blood and blood vessels in the averaging blood sub-volume, the second term on the LHS reduces to zero due to the principle of conservation of mass. (Note that this term is zero for solid tissue since uT = 0.) Thus, Equation 1.3 simplifies as follows.

1.4

equation

where, i = j = T, Bl and i ≠ j.

In Equation 1.4, the first term on the RHS represents the energy gained by a solid tissue (or blood) sub-volume from adjacent solid tissue (or blood) sub-volumes. The second term on the RHS represents the energy exchange due to the interaction between the solid tissue and blood sub-volumes inside an averaging volume. The third term on the RHS represents the energy gained by the solid tissue (or blood) sub-volume in an averaging volume due to source terms.

Next, normalizing Equation 1.4 by sub-volume Vi the following general integral form of the generic BHTM (Equation 1.5) is obtained. This form satisfies the energy equation and is valid for any unheated and heated tissue with no phase change. Note that Equation 1.5 represents two equations; one for solid tissue and another for blood.

1.5

equation

where, i = j = T, Bl and i ≠ j and

1.6

equation

1.2.2 Simplifications

The following three simplifications are made to obtain a differential form of the two-compartment GBHTM.

1.7

equation

1.8

equation

where, i = j = T, Bl and i≠ j, and

1.9

equation

The first simplification (Equation 1.7) relates the energy exchange among the tissue (or blood) sub-volume of an averaging volume and tissue (or blood) sub-volumes of surrounding averaging volumes to the average temperatures of tissue (or blood) sub-volumes. This simplification is similar to the simplifications used by many other BHTMs [e.g., [6–16]]. A new, to be determined, parameter Ci1 is introduced in Equation 1.7 to keep the simplification general.

The second simplification (Equation 1.8) defines the thermal interaction between a tissue and the embedded vasculature using the tissue and blood sub-volume averaged temperatures and a heat transfer coefficient (i.e., U). Note that this heat transfer coefficient is different from conventional heat transfer coefficients and must be evaluated to appropriately implement the GBHTM since the new heat transfer coefficient is defined based on the volume-averaged temperatures. Conventional heat transfer coefficients are defined using a tissue boundary temperature and a mixed mean blood temperature.

The third simplification (Equation 1.9) is similar to the simplification proposed by Equation 1.7 and defines a new, to be determined, perfusion-related parameter P. The simplification relates the energy transported through blood vessels of a blood sub-volume to surrounding blood sub-volumes to the gradient of blood sub-volume averaged temperature.

With the above simplifications substituted in Equation 1.5, the following differential form of the two-compartment GBHTM is obtained. Equations 1.10 and 1.11 are coupled equations. Equation 1.10 is valid for the tissue sub-volume, and Equation 1.11 is valid for the blood sub-volume in an averaging volume.

1.10

equation

1.11

equation

where, V = VT + VBl and c01-math-012

Note that in the above equations the local perfusion-related parameter P and the blood volume ratio ϵ are unknown. These values need to be determined for a given application and tissue using imaging methods (e.g., MRI, positron emission tomography, ultrasound, etc.). Alternatively, these values can be estimated by developing physiologically realistic geometric vascular network maps and assuming mean blood flow of ∼10 (diameter of vessel) m/s [17]. The tissue sub-volume VT can be estimated as the difference between the imaging voxel volume and blood sub-volume.

1.2.3 A Three-Compartment and ‘N + 1’ Compartment GBHTM

We derived above a two-compartment GBHTM by assuming that the averaging volume is composed of only two sub-volumes or compartments: a non-moving tissue sub-volume and a moving blood sub-volume. To derive a three-compartment GBHTM, let's assume, as an example, that the averaging volume is composed of the following three sub-volumes or compartments (instead of two): a non-moving tissue sub-volume, an arterial blood sub-volume, and a venous blood sub-volume. In this case, one can derive the following set of equations for a new three-compartment GBHTM using the methodology presented above and simplifications similar to the ones presented in Equations 1.9.

1.12

equation

1.13

equation

1.14

equation

where, V = VT + Var + Vvn , c01-math-016 and c01-math-017 . A point to note in this new GBHTM is that it is relatively more complex than the two-compartment GBHTM presented in Equations 1.10 and 1.11. In this new, three-compartment GBHTM, we have created a need to separately identify and account for the two blood sub-volumes (i.e., arterial and venous blood in this example) to solve for in vivo temperatures, which may or may not be useful and/or feasible for a given application. Also, the sub-volumes don't have to be arterial or venous sub-volumes. One can think about creating blood sub-volumes based on the relative thermal importance and frequency of blood vessels, as well. Generalizing the above result further, the following set of equations can be obtained for a GBHTM where the averaging volume is composed of ‘N + 1’ compartments: a non-moving tissue sub-volume and ‘N’ blood sub-volumes.

1.15

equation

1.16

equation

where, i, j = 1… N blood sub-volumes, i≠ j, c01-math-020 and c01-math-021 .

1.3 Comparing the Two-Compartment GBHTM with Pennes' BHTM

Pennes' BHTM is a simple and widely used BHTM [6] that was intuitively proposed by a clinician named Henry H. Pennes in 1948 to predict in vivo temperatures (Equation 1.17).

1.17

equation

Traditionally, Pennes' BHTM has been a point of contention and a source of confusion. This is so because the lack of a formal derivation of Pennes' BHTM makes it difficult to interpret Pennes' variables, parameters, and results in terms of the underlying physiology. For example, it is not clear what ‘blood temperature’ is in the Pennes' model. Thermodynamically, the blood temperature in Pennes' BHTM can be defined in at least four different ways: (1) local, point-wise true blood temperature, (2) blood velocity-weighted blood vessel area-averaged blood temperature, (3) blood vessel area-averaged blood temperature and (4) perfused tissue volume-averaged blood temperature. Assuming the blood temperature as the point-wise true temperature or the velocity-weighted temperature makes Pennes' BHTM physically invalid since it makes the model violate first principles. Assuming the blood temperature as the area-averaged or volume-averaged temperature renders physically invalid the traditional conception of the Pennes' w term as a blood flow term. It should be noted, however, that Pennes' BHTM has been shown to be capable of predicting reasonably accurate in vivo temperatures in ‘local’ tissue regions for ‘short’ durations by adjusting Pennes' perfusion-related parameter w(cp)Bl(1 − ζ) to minimize error between model predictions and measurements. Adjusting Pennes' perfusion related parameter w(cp)Bl(1 − ζ) is necessary because of the lack of a formal derivation of Pennes' BHTM, which makes it difficult to evaluate the parameter independently based on the underlying physiology. Simplicity and reasonable performance of Pennes' BHTM (once the blood-perfusion-related parameter w(cp)Bl(1 − ζ) is adjusted, of course) has always been puzzling for a ‘physically and theoretically invalid’ model.

Let's compare Pennes' BHTM with the two-compartment GBHTM to gain a better understanding of the implicit and explicit assumptions, and thus the application regimes of Pennes' model. Comparing Pennes' BHTM with the newly developed two-compartment GBHTM one notes the following:

1. The basic form of Equation 1.17 (i.e., Pennes' BHTM) and Equation 1.10 (i.e., the tissue sub-volume equation in the GBHTM) is similar (with CT1 = 1). Thus, Pennes' tissue temperature and blood temperature should be interpreted as a volume-averaged tissue temperature and a volume-averaged blood temperature, respectively, instead of the point-wise true temperatures. Tissue properties (e.g., density, specific heat, conductivity) and source term should also be interpreted as volume-averaged quantities.

2. Pennes' blood perfusion related parameter w(cp)Bl(1 − ζ) should be interpreted as equivalent to the blood-tissue heat transfer rate term c01-math-023 in the GBHTM. In practice, Pennes' blood perfusion related parameter w(cp)Bl(1 − ζ) is always determined by minimizing the error between the modeled and measured temperatures to obtain reasonable estimates of the in vivo temperatures. This is equivalent to obtaining c01-math-024 values by minimizing the error.

3. Pennes' BHTM assumes the blood temperature as constant. This is equivalent to assuming that the blood has infinite thermal capacity in the two-compartment GBHTM. Thus, Pennes' BHTM artificially forces tissue temperatures to stay close to the assumed blood temperature. The assumption results in the overestimation of the blood-tissue heat transfer rate and the underestimation of the tissue heating in deep tissues due to source terms. The lack of the ability to model the blood temperature variation further limits Pennes' BHTM to applications where the blood temperature does not vary ‘significantly’ from the assumed value in space and time.

4. Pennes introduced a blood equilibration parameter ζ (0 ≤ ζ ≤ 1) to quantify the blood-tissue heat transfer rate based on the local blood flow w. However, the spatial and temporal behavior of this parameter ζ is never studied in vivo. In the absence of a known spatial and temporal variation of ζ for an application and tissue type, substituting local blood flow value for w in Pennes' model estimates blood-tissue heat transfer rate incorrectly. Substituting local blood perfusion values for the term w (assuming ζ = 0) or w(1-ζ) (assuming ζ ≠ 0) and the absence of an equation for modeling the blood sub-volume averaged blood temperature variation in Pennes' BHTM have been shown to result in large deviations between the predictions of the model and the absolute temperature field in vascularized tissues [18].

1.4 Comparing the Predictions of the Two-Compartment GBHTM and Pennes' BHTM with Measured in vivo Temperature Changes during MRI

Figures 1.1–1.4, below compare the predictions of the two-compartment GBHTM and Pennes' BHTM with the temperature changes measured in swine during MRI. MRI scanners deposit radio frequency (RF) energy inside the body during MRI, which produces heating. Accurate prediction of this heating is necessary for the safety and efficacy of the scanners. Note that Pennes' BHTM is implemented by assuming infinite thermal capacity of the blood in the two-compartment GBHTM, as it should.

Graphical illustration of radiofrequency (RF) power deposition-induced brain heating, as measured with a fluoroptic probe placed 15 mm deep in the swine brain after the dura, in a 3T MR scanner.

Figure 1.1 RF power deposition-induced brain heating, as measured with a fluoroptic probe placed 15 mm deep in the swine brain after the dura, in a 3T (Larmor frequency = ∼123.2 MHz) MRI scanner. The RF power was delivered using a body coil with the swine head placed in the isocenter.

Graphical illustration of radiofrequency power deposition-induced whole-body heating, as measured with a fluoroptic probe placed 10 cm deep in the swine rectum, in a 3T MR scanner.

Figure 1.2 RF power deposition-induced whole-body heating, as measured with a fluoroptic probe placed 10 cm deep in the swine rectum, in a 3T (Larmor frequency = ∼123.2 MHz) MRI scanner. The RF power was delivered using a body coil with the swine head placed in the isocenter.

Graphical illustration of radiofrequency power deposition-induced brain heating, as measured with a fluoroptic probe placed 15 mm deep in the swine brain after the dura, in a 3T MR scanner.

Figure 1.3 RF power deposition-induced brain heating, as measured with a fluoroptic probe placed 15 mm deep in the swine brain after the dura, in a 3T (Larmor frequency = ∼123.2 MHz) MRI scanner. The RF power was delivered using a body coil with the center of the swine trunk placed in the isocenter.

Graphical illustration of radiofrequency power deposition-induced whole-body heating, as measured with a fluoroptic probe placed 10 cm deep in the swine rectum, in a 3T MR scanner.

Figure 1.4 RF power deposition-induced whole-body heating, as measured with a fluoroptic probe placed 10 cm deep in the swine rectum, in a 3T (Larmor frequency = ∼123.2 MHz) MRI scanner. The RF power was delivered using a body coil with the center of the swine trunk placed in the isocenter.

As discussed above, the figures clearly demonstrate that (1) the two-compartment GBHTM predicts in vivo tissue heating accurately in space and time, (2) appropriately implemented Pennes' BHTM predicts tissue heating accurately only until the blood temperature does not start varying appreciably in space and time from the baseline value, and (3) the assumption of constant blood temperature makes Pennes' BHTM overestimate the blood-tissue heat transfer rate, and thus underestimate the deep tissue heating. More details regarding the experimental evaluation of the two-compartment GBHTM and Pennes' BHTM can be found in [19, 20].

1.5 Summary

A general methodology is presented to derive BHTMs from first principles. The methodology was used to, first, derive a general ‘two-compartment’ BHTM for perfused tissues with very few assumptions – herein called the two-compartment generic bioheat transfer model (GBHTM). Later, the same methodology was used to derive more general forms of the model (i.e., a three-compartment GBHTM and an ‘N + 1’ compartment GBHTM). Finally, the new, two-compartment GBHTM was compared with the empirical, ‘gold standard’ Pennes' BHTM, theoretically as well as experimentally, to better understand the potential and limitations of the GBHTM and Pennes' BHTM. The comparison helped better understand the relation between the variables and parameters of Pennes' BHTM and the underlying physiology. It was demonstrated that the two-compartment GBHTM predicted accurate in vivo tissue heating in space and time while an appropriately implemented Pennes' BHTM predicted accurate in vivo tissue heating when the blood temperature did not change ‘appreciably’ from the baseline value.

Disclaimer

The subject matter, content, and views presented do not represent the views of the Department of Health and Human Services (HHS), US Food and Drug Administration (FDA), and/or the United States.

Nomenclature

Subscripts

Greek

References

1 Kays, W. M. & Crawford, M. E., 1993, Convective Heat and Mass Transfer, New York, McGraw-Hill, Inc.

2 White, J. A., Dutton, A. W., Schmidt, J. A., and Roemer, R. B., 2000, An accurate, convective energy equation based automated meshing technique for analysis of blood vessels and tissues, Int. J. Hyperthermia, 16(2): 145–158.

3 Roemer, R. B., 1999, Engineering aspects of hyperthermia therapy, Annu Rev Biomed Eng, 1: 347–376.

4 Craciunescu, O. I., Raaymakers, B. W., Kotte, A. N., et al., 2001, Discretizing large traceable vessels and using DE-MRI perfusion maps yields numerical temperature contours that match the MR noninvasive measurements, Med Phys, 28: 2289–2296.

5 Shrivastava, D. and Vaughan, J. T., 2009, A generic bioheat transfer thermal model for a perfused tissue, ASME: Journal of Biomechanical Engineering, 131(7): 074506.

6 Pennes, H. H., 1998, Analysis of tissue and arterial blood temperatures in the resting human forearm: 1948, J Appl Physiol, 85: 5–34.

7 Wulff, W., 1974, The energy conservation equation for living tissue, IEEE Trans Biomed Eng, BME-21: 494–495.

8 Klinger, H. G., 1974, Heat transfer in perfused biological tissue: I: General theory, Bull Math Biol, 36: 403–415.

9 Klinger, H. G., 1978, Heat transfer in perfused biological tissue: II: The macroscopic temperature distribution in tissue, Bull Math Biol, 40: 183–199.

10 Chen, M. M. and Holmes, K. R., 1980, Microvascular contributions in tissue heat transfer, Ann N Y Acad Sci, 335: 137–150.

11 Osman, M. M. and Afify, E. M., 1984, Thermal modeling of the normal woman's breast, J Biomech Eng, 106: 123–130.

12 Weinbaum, S. and Jiji, L. M., 1985, A new simplified bioheat equation for the effect of blood flow on local average tissue temperature, J Biomech Eng, 107: 131–139.

13 Brinck, H. and Werner, J., 1994, Efficiency function: Improvement of classical bioheat approach, J Appl Physiol, 77: 1617–1622.

14 Weinbaum, S., Xu, L. X., Zhu, L., and Ekpene, A., 1997, A new fundamental bioheat equation for muscle tissue: Part I: Blood perfusion term, J Biomech Eng, 119: 278–288.

15 Roemer, R. B. and Dutton, A. W., 1998, A generic tissue convective energy balance equation: Part I: Theory and derivation, J Biomech Eng, 120: 395–404.

16 Wren, J., Karlsson, M., and Loyd, D., 2001, A hybrid equation for simulation of perfused tissue during thermal treatment, Int J Hyperthermia, 17: 483–498.

17 Chato, J. C., 1980, Heat transfer to blood vessels, J Biomech Eng, 102: 110–118.

18 Brinck, H. and Werner, J., 1995, Use of vascular and non-vascular models for the assessment of temperature distribution during induced hyperthermia, Int J Hyperthermia, 11: 615–626.

19 Shrivastava, D., Utecht, L., Tian, J., et al. 2014, In vivo radiofrequency heating in swine in a 3T (123.2 MHz) birdcage whole-body coil, Magnetic Resonance in Medicine, 72(4): 1141–50.

20 Shrivastava, D., Hanson, T., Kulesa, J., et al., 2011, Radiofrequency heating in porcine models with a ‘large’ 32 cm internal diameter, 7 T (296 MHz) head coil, Magnetic Resonance in Medicine, 66(1): 255–263.

Chapter 2

Alternate Thermal Models to Predict in vivo Temperatures

Devashish Shrivastava¹,²

¹US Food and Drug Administration, Silver Spring, MD, United States

²In Vivo Temperatures, LLC, Burnsville, MN, United States

* Corresponding author: [e-mail]

2.1 Introduction

There are several applications where the length and time scales of the heat transfer processes and the resultant in vivo temperature changes are such that the use of typical volume-averaging based bioheat transfer models, such as the generic bioheat transfer model (GBHTM) or the GBHTM's simplified variant the Pennes bioheat transfer model (BHTM), is undesirable or inapplicable. Here, in this chapter, we develop alternate, first principles based, thermal models for a few of such applications.

2.2 Estimating Core Temperature

Estimating core temperature due to external and/or internal source terms, by solving a volume-averaged bioheat transfer model (e.g., GBHTM or Pennes BHTM) in whole-body human models with hundreds and thousands, if not millions, of computational nodes can be overly burdensome. Examples of external source terms are solar or other electromagnetic radiation, convective heating or cooling due to natural flow of fluids over the body, advective heating or cooling due to forced flow of fluids over (as in forced flow of fluids with a fan or pump) or inside the body (as in forced flow of fluids during induced hypothermia assisted surgeries), evaporation, conductive heat transfer due to body's external surface (e.g., skin or clothes) contacting a heated or cooled surface, etc. Examples of internal source terms are metabolism during exercise, fever, or induction of anesthesia; deposition of ultrasound, radio frequency (RF) or microwave energy during thermal intervention or diagnostic imaging; etc. For such applications, a relatively simple first principles based thermal model can be developed by averaging the convective energy equation (CEE) over the complete volume of the human body, as shown below. The resultant model is computationally inexpensive and easy to implement.

2.2.1 Thermal Model

Let's consider the CEE (Equation 2.1), derived using first principles.

2.1

equation

where, ρ is density, cp is specific heat, T is temperature, t is time, u is the three-dimensional velocity field, k is thermal conductivity, and Q is a source term.

Integrating Equation 2.1 over the whole body volume, Equation 2.2 is obtained.

2.2

equation

After applying the divergence theorem and some simplifications, Equation 2.2 can be rewritten as the following Equation 2.3 our alternate, simple thermal model to estimate the core temperature.

2.3

equation

where

2.4 equation

2.5 equation

Equation 2.3 relates core temperature (Tc) change over time to the net energy gain from the surface and internal source terms. The first term on the left-hand side (LHS) of Equation 2.3 represents the change in the energy content of the whole body. The second term on the LHS of Equation 2.3 represents the net energy loss due to bulk fluid motion like respiration, sweating, or forced cooling during induced hypothermia assisted surgeries. The first term on the right-hand side (RHS) of Equation 2.3 represents the net energy flux gain from the surface of the body due to external source terms. The second and last term on the RHS of Equation 2.3 represents the net energy gain by the body due to internal source terms. Also, Equation 2.4 defines the core temperature Tc) and Equation 2.5 defines the surface temperature Ts) and relates the net gain in the energy flux from the surface of the body S to a heat transfer coefficient h, the ambient temperature Tamb, and the surface temperature Ts.

2.2.2 Example: The Effect of Anesthetics on the Core Temperature Change

Several anesthetics (e.g., isoflurane) are known vasodilators and suppressors of metabolism. The use of these anesthetics, typically, results in a decrease of core temperature over time [1]. Let's use our model developed above in Equations 2.5 to estimate the effect of anesthesia-induced vasodilation and suppression of metabolism, as well as the ambient conditions, on the core temperature.

Using Equation 2.3, Equation 2.6 can be written as the governing equation for the core temperature (Tc) due to the internal source term of metabolism (Qmet).

2.6

equation

Neglecting energy loss due to respiration and sweating, the second term in Equation 2.6 goes to zero and Equation 2.6 reduces to Equation 2.7.

2.7

equation

Let's simplify Equation 2.7 further by expressing the surface temperature in terms of the core temperature:

2.8 equation

where the difference temperature δ, in general, is a function of time. Substituting Equation 2.8 in Equation 2.7 gives Equation 2.9.

2.9

equation

Rearranging the above Equation 2.9, we obtain Equation 2.10:

2.10 equation

where

equation

Equation 2.10 presents a simple, but powerful, tool to study the effect of changing surface temperature, internal source terms, and environmental conditions on the core temperature, when thermal energy loss due to respiration and sweating may be neglected.

Next, for simplicity, lets assume that δ is constant, instead of a number that varies with time. This is a reasonable assumption since prior experiments in 60–100 kg swine, anesthetized with isoflurane, have shown that the surface temperature follows the core temperature closely and is lower than the core temperature by a relatively constant amount (e.g., refer to Figure 2.1). Also, let's assume that the metabolism drops by a given amount, instantaneously, due to a given amount of anesthesia and that the metabolism, once dropped, does not vary with time, afterwards, as long as the amount of the anesthesia is not altered. With these two assumptions, Equation 2.10 can easily be solved analytically for the core temperature Tc and Equation 2.11 is obtained:

2.11

equationGraphical illustration of drop in body temperature as a function of time in anesthetized swine.

Figure 2.1 Drop in body temperature as a function of time in anesthetized swine.

Equation 2.11 presents a simple, analytical expression derived using first principles to study the effect of anesthesia-induced vasodilation and change in metabolism, as well as ambient conditions, on the core temperature in anesthetized subjects. The effect of vasodilation can be studied by changing the value of δ (i.e., the difference between the core temperature and the surface temperature) since vasodilation brings warmer blood from deeper parts of the body to near the subject's surface, resulting in changing values for δ (refer to Figure 2.2). The effect of change in metabolism can be studied by changing P (refer to Figure 2.3). The effect of ambient conditions can be studied by changing β (note that β includes convective heat transfer coefficient h) (Figure 2.4). The potential of the alternate thermal model, presented in Equations 2.5, and its simple variant, presented in Equation 2.11, in predicting accurate core temperatures with significant computational savings becomes obvious by noting that the trends in core temperature change presented in Figures 2.2–2.4 are comparable to the experimental observations made in anesthetized subjects [1].

Graphical illustration of the effect of the change in the difference between the core and surface temperatures due to anaesthesia-induced vasodilation on the core temperature as a function of time.

Figure 2.2 The effect of the change in the difference between the core and surface temperatures due to anesthesia-induced vasodilation on the core temperature as a function of time.

Graphical illustration of the effect of the anesthesia-induced suppression in energy production on the core temperature as a function of time.

Figure 2.3 The effect of the anesthesia-induced suppression in energy production on the core temperature as a function of time.

Graphical illustration of the effect of enhanced surface cooling on the core temperature change as a function of time.

Figure 2.4 The effect of enhanced surface cooling on the core temperature change as a function of time.

Relatively more complex, analytical solutions can be obtained by approximating the general, time-dependent behavior of δ and P, measured for a set of conditions, assuming the following general expressions and solving Equation 2.10. Interested readers are encouraged to develop such expressions to investigate the relative effect of the additional, time-dependent terms on the estimation of the core temperature as a function of time.

equation

Further, the presented methodology can also be expanded to situations where a biological system can be considered as a combination of multiple control volumes (CVs) with different source inputs. Again, interested readers are encouraged to generalize the presented methodology for biological systems made of multiple CVs to explore the potential of the presented method for such applications.

2.3 Estimating Worst-Case in vivo Temperature Change due to a ‘Regional’ Source Term

More often than not, we are only interested in estimating the worst-case transient in vivo temperature change over the baseline due to a source term in a local region inside the body to determine the safety and effectiveness of a device and/or procedure. For example, during magnetic resonance imaging (MRI), we are interested in constraining the maximum in vivo temperature to 40°C and the maximum in vivo temperature change to 1°C, per the current International Electrotechnical Commission (IEC) standard 60601-2-33 ed 3.0b. As another example, during therapeutic thermal interventions, we are interested in constraining maximum in vivo temperature to a given value to achieve intended therapeutic effect with minimum collateral tissue damage. In this section, we develop a simplified thermal model using the first of the two equations of the ‘two-compartment’ generic bioheat transfer model (GBHTM), developed earlier in Chapter 1, to determine the maximum in vivo temperature change due to a regional source term without solving for the complete GBHTM in segmented whole-body models.

2.3.1 Thermal Model

Let's consider the ‘two-compartment’ GBHTM, derived earlier in Chapter 1 from first principles (Equations 2.13). Please note that these two equations are coupled by the blood-tissue heat transfer rate term (i.e., the second term from the end on the RHS of the equations). Equation 2.12 solves for the volume-averaged tissue temperature, and Equation 2.13 solves for the volume averaged blood temperature.

2.12

equation

2.13

equation

Since we are interested in estimating the worst-case ‘tissue’ temperature change over the baseline tissue temperature due to a source term, let's consider Equation 2.12 alone. Substituting metabolic energy production as the source term in Equation 2.12, the baseline tissue temperature distribution can be presented using Equation 2.14.

2.14

equation

Using the same Equation 2.12, the tissue temperature due to the metabolic source and additional sources (e.g., RF or ultrasound energy) can be represented as Equation 2.15.

2.15

equation

Subtracting Equation 2.14 from Equation 2.15 gives us Equation 2.16, which represents the tissue temperature change over the baseline due to source terms alone.

2.16

equation

Please note that the worst-case tissue temperature change over the baseline due to source terms δTT,S must be higher than the local change in the blood temperature δTBl,S (since this is the hottest spot in the body and the blood will be carrying energy away from here, dumping the collected energy elsewhere along its flow path).Therefore, the magnitude of the second term on the RHS of Equation 2.16 (i.e., the blood-tissue heat transfer rate term) will be negative, representing a sink term – and not a source term – near the worst-case tissue temperature change. Further, note that this sink term will become maximum when δTBl,S = 0 (i.e., when the blood temperature does not change) and minimum when δTBl,S = δTT, S (i.e., when the blood temperature equilibrates with the local tissue temperature). Thus, the worst-case tissue temperature change over the baseline is bound by the following two Equations 2.17 and 2.18, which represent the upper and lower bound of the temperature, respectively.

2.17

equation

2.18

equation

One can imagine substituting different functions for the local change in the blood temperature δTBl,S in Equation 2.16 that vary between 0 and δTT,S (e.g., linear, exponential, etc.) to produce various estimates for the local tissue temperature change as a function of time between the upper and lower bounds. The appropriate choice of such function would depend on the desired spatio-temporal accuracy for a given application. One such obvious choice for the blood temperature δTBl,S would be to assume that the blood temperature δTBl,S varies as the change in core temperature. The expression for the core temperature change can be obtained by solving Equations 2.5 derived above. An example of the various tissue temperature change estimates is presented in Figure 2.5 for various functions of the blood temperature change. The tissue temperature estimates are compared to in vivo temperature measurements made in humanized swine exposed to RF power deposition during MRI (Figure 2.5) [2]. Once again, the potential of the alternate thermal model, presented in Equation 2.16, in predicting accurate worst-case tissue temperature with significant computational savings becomes obvious by noting that the trends in the temperature change presented in Figure 2.5 are comparable to the experimental observations made in swine, without explicitly solving for the full ‘two-compartment’ GBHTM in a segmented swine model with millions of computational nodes [2].

Graphical illustration of worst-case tissue temperature change for various approximations of blood temperature change in anesthetized swine exposed to the whole-body average specific absorption rate (SAR) of ~2.7 W/kg.

Figure 2.5 Worst-case tissue temperature change for various approximations of blood temperature change in anesthetized swine exposed to the whole-body average specific absorption rate (SAR) of ∼2.7 W/kg.

2.4 Estimating in vivo Temperature Change due to a ‘Local’ Source Term

An example, where the use of volume averaging based bioheat transfer models is inapplicable, is predicting in vivo temperature change in a ‘very small’ region due to local source term distribution with dimensions on the order of sub-millimeters, since there may not be enough ‘volume’ to develop appropriate averaging properties to model the thermal effects of thermally important blood vessels. Derived below is an alternate thermal model that may be applicable for such applications.

2.4.1 Thermal Model

Let's start with the CEE, derived from first principles, and presented in Equation 2.1. Equation 2.1 takes the following form due the source term of metabolism, alone.

2.19

equation

With additional source terms, Equation 2.1 can be rewritten as Equation 2.20.

2.20

equation

Subtracting Equation 2.19 from Equation 2.20 and assuming that the additional source term does not result in an appreciable change in thermal properties, flow, and baseline metabolism, results in Equation 2.21, an alternate thermal model to predict tissue temperatures in ‘local’ regions with ‘local’ source term distributions of sub-millimeter dimensions.

2.21

equation

Note that solving for Equation 2.21 requires one to know about thermally important blood vessels and their flow fields, among other parameters such as thermal properties and source term distribution. These flow fields might be obtained using imaging methods or estimated using artificially generated physiologically realistic vessel networks. In the absence of such flow fields or when implementing such flow fields is computationally very expensive, one may have to resort to determining suitable parameters using inverse heat transfer methods based of experimental observations in the region of interest.

2.5 Summary

In this chapter, we derived alternate bioheat transfer models, using first principles, for applications where implementing traditional volume averaged based bioheat transfer models may be computationally very expensive or inapplicable. Also, the potential of these new bioheat transfer models in predicting useful in vivo temperatures is demonstrated using the experimental observations made before.

Disclaimer

The subject matter, content, and views presented do not represent the views of the Department of Health and Human Services (HHS), US Food and Drug Administration (FDA), and/or the United States.

References

1 Ramchandra, V., Moore, C., Kaur, N., and Carli, F., 1989, Effect of halothane, enflurane, and isoflurane on body temperature during and after surgery, Br. J. Anaesth, 62, 409–414.

2 Shrivastava, D., Utecht, L., Tian, J., et al., In vivo radiofrequency heating in swine in a 3T (123.2 MHz) birdcage whole-body coil, Magn Reson Med, 2014 Oct; 72(4): 1141–1150.

Chapter 3

Thermal Effects of Blood Vessels

Devashish Shrivastava¹,²

¹US Food and Drug Administration, Silver Spring, MD, United States

²In Vivo Temperatures, LLC, Burnsville, MN, United States

* Corresponding author: [e-mail]

3.1 Introduction

The safe and effective implementation of several diagnostic and therapeutic applications, which produce intended or unintended tissue heating, depends upon the accurate modeling of the thermal effects of blood vessels for the range of physiological conditions pertinent to a given application and region of interest, in vivo. Thus, in this chapter, to better understand the thermal effects of blood vessels, we investigate the effect of the diameter of blood vessels on the volume-averaged tissue and blood temperatures, as well as on the amount of thermal energy transported away by the vessels for physiologically realistic mean flows. One should recall that volume-averaged tissue and blood temperatures are variables in bioheat transfer models. Additionally, the amount of thermal energy transported away by the blood vessels provides a basis to quantify the parameter related to the vessel-tissue heat transfer rate in bioheat transfer models.

3.2 Methods

Thermal effects of blood vessels are studied using simple geometries, herein called an elementary control volume (ECV). These ECVs are embedded with a manageable number of blood vessels since it is computationally very expensive to study the whole vasculature together. The temperature distribution in the ECV (comprising vessels and surrounding tissue) is obtained by solving the steady state convective energy equation (CEE) and relevant parameters are evaluated [1]. The study is typically conducted using a non-dimensional CEE since it reduces the number of parameters to be studied. For brevity and to gain fundamental understanding of the thermal behavior of blood vessels, we will, herein, investigate the thermal effects of a single blood vessel as a function of vessel diameter and flow.

Let's consider a cubic, uniform ECV, with a side of length L, perfused with a single cylindrical blood vessel, placed symmetrically in the center of the ECV. The diameter of the blood vessel may range from 0.05 L to 0.9 L. Considering a cubic ECV makes the results independent of the angular direction (i.e., x, y, or z) of the blood vessel. Considering only a single blood vessel in the ECV is a reasonable assumption for the ECVs with a side of length on the order of 1 mm (∼O(1 mm)). This is so since in vivo measurements have shown that in such ECVs only 1–3 blood vessels of diameter ≥ 50 µm (50 µm = 0.05 L for L = 1 mm) and 1–2 blood vessels of diameter ≥ 100 µm are expected to be present. Additionally, these measurements reveal that in such ECVs fewer than 31% of blood vessels of diameter ≥ 50 µm are found paired [2]. Varying the diameter of blood vessels from 0.05 L to 0.9 L in such an ECV helps us understand the thermal effects of blood vessels with diameter ranging from 50 to 900 µm. The ECV is assumed to be uniform for simplicity.

The following steady state (wherein the time-varying terms reduce to 0) CEE [1] Equation 3.1, with no source term, is solved in the ECV.

3.1 equation

where, k is thermal conductivity, T is temperature, ρ is density, cp is specific heat at constant pressure, and U is blood velocity. Assuming that (1) the blood flows in the z direction alone and (2) the flow is Laminar and fully developed, governing Equation 3.1 can be simplified as Equation 3.2. Equation 3.2 is presented in Cartesian coordinates.

3.2

equation

where, U = Uz (1 − r²/R²) is blood velocity, Uz is twice of mean blood flow, r = (x² + y²)⁰.⁵ ≤ R is radial distance from the center of the vessel inside the vessel, and R is the radius of the vessel. Assuming that the blood flow velocity is fully developed is a reasonable assumption since (1) the length of the vessel necessary for the development of the Laminar velocity profile is significantly small compared to the diameter of the vessel for vessels with diameter ≤ 0.5 L (i.e., 500 µm) and (2) for vessels with diameter > 500 µm, the Peclet number is significantly larger than 1 [1]. It is reasonable to assume that the blood flow is Laminar and non-pulsatile, since this makes the problem in hand relatively simpler and further, since the difference between the Laminar, non-pulsatile flow and pulsatile flow on the heat transfer around vessels has been shown to be insignificant [3]. Finally, we assume that the boundary temperature of the tissue and the inlet temperature of the blood are set to uniform temperatures of TT, boundary and TBl, inlet, respectively.

Governing Equation 3.2 and the boundary conditions are non-dimensionalized using the following variables.

equation

The non-dimensional governing equation and boundary equations are given below.

3.3 equation

with

3.4 equation

The non-dimensional governing Equation 3.3 is solved together with the boundary conditions presented in Equation 3.4 to quantify the non-dimensional volume-averaged tissue (⟨θT⟩) and blood (⟨θBl⟩) temperatures, as well as the non-dimensional vessel-tissue heat transfer rates (QT-Bl) as a function the vessel diameter and flow. These parameters are defined below.

equation

where, non-dimensional

c03-math-005

.

Next, the following parameter values are used in the simulations: density ρ = 1000 kg/(m³·s), specific heat at constant pressure cp = 3600 J/(kg·K), thermal conductivity k = 0.5 W/(m·K), L = 1 mm, mean physiologic blood flow = 20 R m/s, 50 R m/s [4]. As stated before, the diameter of the vessel is varied from 0.05 L to 0.9 L (i.e., 50–900 µm). The mean physiologic flow of 20 R m/s is used since normal mean physiologic flow in humans is ∼20 R m/s, where R is the radius of the vessel in meter [5–7]. The mean physiologic flow of 50 R m/s is used to understand the thermal effects of increased blood flow, as may be encountered under enhanced thermal load on the body (e.g., during exercise).

3.3 Results

Figure 3.1 presents the volume-averaged tissue temperature, volume-averaged blood temperature, and the difference between the volume-averaged tissue and blood temperaturesas a function of the vessel diameter for mean physiologic flow of 20 R m/s and 50 R m/s. The results show that the volume-averaged blood temperature approaches 1 (i.e., the non-dimensional tissue boundary temperature) for vessels with diameter ≤ 0.1 L (i.e., 100 µm). Additionally, the volume-averaged blood temperature does not approach 0 (i.e., the non-dimensional blood inlet temperature) for vessels with diameter ≥ 0.5 L (i.e., 500 µm).

Graphical illustration of non-dimensional volume-averaged tissue temperature, volume-averaged blood temperature, and the difference between the non-dimensional volume-averaged tissue and blood temperatures, for mean physiologic flow of 20R m/s and 50R m/s.

Figure 3.1 Non-dimensional volume-averaged tissue temperature, volume-averaged blood temperature, and the difference between the non-dimensional volume-averaged tissue and blood temperatures as a function of the non-dimensional vessel diameter for mean physiologic flow of 20 R m/s and 50 R m/s.

Figure 3.2 presents the tissue-blood heat transfer rates, normalized by 1, the difference between the volume-averaged tissue and blood temperatures, and the difference between the volume-averaged tissue and blood temperatures multiplied by vessel wall surface area (i.e., c03-math-006 ) as a function of the vessel diameter for mean physiologic flow of 20 R m/s and 50 R m/s. The results show that the tissue-blood heat transfer rate, when normalized by 1 or the difference between the volume-averaged tissue and blood temperatures, increases as the vessel diameter increases. However, the tissue-blood heat transfer rate per unit vessel wall surface area increases as the vessel diameter decreases.

Graphical illustration of non-dimensional tissue-blood heat transfer rates, normalized by 1, the difference between the non-dimensional volume-averaged tissue and blood temperatures, and the difference between the non-dimensional volume averaged tissue and blood temperatures multiplied with non-dimensional vessel wall surface area.

Figure 3.2 Non-dimensional tissue-blood heat transfer rates, normalized by 1, the difference between the non-dimensional volume-averaged tissue and