The most wonderful skaters of Learn to Skate and Figure Skating Illini. It was truly ...... cancer tissue, and release toxic-singlet oxygen upon optical excitation.12 ...... 4000. 6000. 8000. 10000. 12000. 14000. 0. 500. 1000. 1500. 2000 excitation.
QUANTITATIVE FREQUENCY-DOMAIN FLUORESCENCE SPECTROSCOPY IN TISSUES AND TISSUE-LIKE MEDIA
BY ALBERT EDWARD CERUSSI B.S., Rensselaer Polytechnic Institute, 1991
THESIS Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 1999
Urbana, Illinois
iii QUANTITATIVE FREQUENCY-DOMAIN FLUORESCENCE SPECTROSCOPY IN TISSUES AND TISSUE-LIKE PHANTOMS
Albert Edward Cerussi, Ph.D. Department of Physics University of Illinois at Urbana-Champaign, 1999 Enrico Gratton, Advisor In the never-ending quest for improved medical technology at lower cost, modern nearinfrared optical spectroscopy offers the possibility of inexpensive technology for quantitative and non-invasive diagnoses. Hemoglobin is the dominant chromophore in the 700-900 nm spectral region and as such it allows for the optical assessment of hemoglobin concentration and tissue oxygenation by absorption spectroscopy. However, there are many other important physiologically relevant compounds or physiological states that cannot be effectively sensed via optical methods because of poor optical contrast. In such cases, contrast enhancements are required. Fluorescence spectroscopy is an attractive component of optical tissue spectroscopy. Exogenous fluorophores, as well as some endogenous ones, may furnish the desperately needed sensitivity and specificity that is lacking in near-infrared optical tissue spectroscopy.
The main focus of this thesis was to investigate the generation and
propagation of fluorescence photons inside tissues and tissue-like media (i.e., scattering dominated media).
The standard concepts of fluorescence spectroscopy have been
incorporated into a diffusion-based picture that is sometimes referred to as photon migration.
The novelty of this work lies in the successful quantitative recovery of
fluorescence lifetimes, absolute fluorescence quantum yields, fluorophore concentrations, emission spectra, and both scattering and absorption coefficients at the emission
iv wavelength from a tissue-like medium. All of these parameters are sensitive to the fluorophore local environment and hence are indicators of the tissue’s physiological state. One
application
demonstrating
the
capabilities
of
frequency-domain
lifetime
spectroscopy in tissue-like media is a study of the binding of ethidium bromide to bovine leukocytes in fresh milk. Ethidium bromide is a fluorescent dye that is commonly used to label DNA, and hence visualize chromosomes in cells. The lifetime of ethidium bromide increases by an order of magnitude upon binding to DNA. In this thesis, I demonstrated that the fluorescence photon migration model is capable of accurately determining the somatic cell count (SCC) in a milk sample. Although meant as a demonstration of fluorescence tissue spectroscopy, this specific problem has important implications for the dairy industry’s warfare against subclinical mastitis (i.e., mammary gland inflammation), since the SCC is often used as an indication of bovine infection.
v
DEDICATION
to my precious parents …
who have demonstrated the true meaning of the word 'love' by their endless sacrifices, bountiful care, and unshakable determination for my well being throughout every day of their lives
vi
ACKNOWLEDGEMENTS
I would like to thank everyone who ever invested any time into my life, because all of them have in some way or another helped me. The people at the LFD have been very patient, kind, and a lot of fun (did I forget to say knowledgeable too?). In particular, I would like to thank …
My Lord, Jesus Christ. My Savior, Yeshua who has been the shepherd of my soul, my trust, my help and my fortress. Without Him I am nothing. He sent me to Urbana, and it has been a wonderful time.
My Family. They have always supported me, prayed for me, and helped me as best they could. All of them have my gratitude and love forever.
Enrico Gratton, my advisor.
Enrico has orchestrated a most incredible medley of
personalities, talents, and characters.
The LFD is certainly a unique working
environment! The atmosphere here as has been a sweet song in my ears ever since I got here. I am very grateful to Enrico for giving a clueless graduate student a chance to do some exciting science. I am equally grateful that I escaped his notice as a target for one of his infamous practical jokes…
vii Sergio Fantini and Maria-Angela Franceschini. These are two are the 'dynamic duo' of the photon migration group. Their help has extended to every level of the research process. I have profited greatly from their wisdom, intuition, experimental prowess, and sense of humor. They taught me the difference between doing science and doing good science. Mi fei sentire orgoglioso di avere origini Italiane!
Bill Mantulin and Chip Hazlett.
These gentlemen have had to endure the helpless
wanderings of a physicist through their biochemistry laboratory. They taught me many useful things about chemistry, ranging from how not to pipette, to the finer points of making a buffer. I would still be figuring out how to turn on the electronic balance if it had not been for their expertise.
Joshua Fishkin, who took me under his wing when I first arrived at the LFD. He has been a great collaborator, a source of endless amusement, and a good friend.
Todd French and Brett Feddersen. These two have answered more questions than you can imagine about electronic circuits, computers, and programming. Who else could help me survive a battle against the evil forces of Microsoft and retain my sanity?
John Maier. From making toxic-rhodamine soup to combating the unspeakable evils of Windows NT, John has been a tremendous blessing and a great collaborator.
viii The whole Photon Migration Group. They are all truly great people. I thank all of them who I haven't mentioned already: Scott, Adelina, Vlad, Mattia, and Claudia.
They
represent more brain power than all of congress!
Julie and Dawn. They are sort of like system administrators. When all is well, you never think of them, but when something goes wrong, they are the first people you find. They have helped me in numerous ways, many of which I probably don't even know about.
Yan and Jochen. They have been good friends to me for many years. From late night sessions in the lab, to discussions about the history of pasta (noodles), it has been an honor to serve in the LFD with them. It is only fitting that Yan and I graduate on the same day!
Peter So. Peter truly embodied the 'dynamics' in the LFD. He was a great help to many graduate students in the LFD, and I was one of them. Peter the 'most awesome man' will live on in LFD folklore!
Martin vandeVen, who has been an endless fountain of useful information for a good many years.
The whole gang at the LFD, past and present. Everyone at the LFD has been a great source of knowledge, culture, and humor.
ix My many brothers and sisters who have lifted me up many a time over the years. In particular, I would like to thank Bill Barber, Joe Pollard, Bernie Ranchero, Emily Statland, Tim Knox, and the Jahde family. All of my 'kids' have been a truly great blessing, from age 3 to 19. I want to be just like them when I grow up!
The most wonderful skaters of Learn to Skate and Figure Skating Illini. It was truly an honor to have skated on the same ice as they did.
And or course, the people who shelled out the money for me to complete this research. This work was performed in the Laboratory for Fluorescence Dynamics, and it was supported by the NIH (RR03155 and CA57032) and the University of Illinois.
x
PREFACE
There are a few notes I would like to make about the style I adopted in this work:
•
Reference numbers in bold type signify a note in the "Reference and Notes"
section, as opposed to purely a literature reference. •
Although I am a die-hard experimentalist, I have adopted the theoretician's
method of recording errors.
I wrote all experimental values with their errors in
parentheses. As an example, 32 (1) means the same thing as 32 ± 1. •
Appendix A contains a list of all the important abbreviations I have used.
•
Appendix A also contains a list of variables I have used throughout this work.
•
I wrote large numbers in a consistent way. For example, I often wrote 400
103 M-1×cm-1 instead of the number 400,000 M-1×cm-1.
Most of all, I hope you enjoy reading this thesis as much as I enjoyed working on it (I actually did enjoy it, for the record).
xi
TABLE OF CONTENTS
I
INTRODUCTION: FLUORESCENCE MEETS NEARINFRARED TISSUE SPECTROSCOPY ......................... 1 I.A BIOMEDICAL OPTICS: A NEW FRONTIER OF MODERN MEDICINE ....1 I.B SOME MACROSCOPIC OPTICAL CHARACTERISTICS OF TISSUES ........3 I.B.1 I.B.2 I.B.3 I.B.4
OPTICAL PROPERTIES OF INTEREST: THE PHYSICIST’S MODEL OF TISSUE....................................................... 3 THE OPTICAL PROPERTIES OF TISSUES IN THE NEAR-INFRARED............ 4 WHO IS RESPONSIBLE FOR BIOLOGICAL SCATTERING?.......................... 9 CONTRAST NEEDED TO EXTEND NEAR-INFRARED TISSUE SPECTROSCOPY ......................................................................... 11
I.C FLUORESCENCE SPECTROSCOPY IN TISSUES ....................................11 I.C.1 I.C.2 I.C.3
FLUORESCENCE PARAMETERS OF INTEREST ......................................... 12 FLUORESCENCE PROPERTIES OF TISSUE ............................................... 14 FLUORESCENCE CONTRAST: SENSITIVITY AND SPECIFICITY ENHANCEMENT .................................... 16 I.C.4 USES OF FLUORESCENCE PARAMETERS IN TISSUE SPECTROSCOPY ...... 18 I.C.4.1 INTENSITY-BASED-FLUORESCENCE MEASUREMENTS ............................. 18 I.C.4.2 LIFETIME-BASED-FLUORESCENCE MEASUREMENTS .............................. 19 I.C.4.3 FLUORESCENCE MEASUREMENTS IN TISSUES ......................................... 20 I.C.5 FLUORESCENCE SPECTROSCOPY: MOVING FROM CUVETTES TO TISSUES ................................................. 22
I.D CHAPTER SUMMARY.........................................................................23
II OVERVIEW OF MODELS USED IN TISSUE OPTICS ..... 26 II.A II.A.1 II.A.2 II.A.3 II.A.4
MINIMALLY-SCATTERING MODELS ............................................26 BALLISTIC PHOTONS: NON-SCATTERED ............................................... 28 SINGLY-SCATTERED PHOTONS .............................................................. 29 SNAKE PHOTONS: ZIGZAG MASTERS .................................................... 29 USEFUL TISSUE OPTICS UNDER MINIMALLY SCATTERING CONDITIONS ................................................ 30
II.BHEURISTIC MODELS .........................................................................31 II.B.1
BEER-LAMBERT CORRECTION: CHANGE THE PATH LENGTH............... 31
xii II.B.2 II.B.3
II.C
MONTE CARLO: A USEFUL SIMULATION TOOL .................................... 32 KUBELKA-MUNK: A PIONEERING MODEL ............................................ 33
EXACT MODELS ..........................................................................35
II.C.1 TRANSPORT THEORY: A GENERIC PARTICLE TRACKER ...................... 36 II.C.1.1 THE DIFFUSION THEORY APPROXIMATION TO TRANSPORT THEORY ..................................................................... 36 II.C.1.2 DIFFUSION THEORY: RANDOM WALK MODEL .................................... 40 II.C.2 MIE THEORY: A CLASSIC PHYSICS PROBLEM ...................................... 40 II.C.3 EXACT E & M: USEFUL WITH A BIG COMPUTER .................................. 42
II.D
CHAPTER SUMMARY....................................................................43
III THE DIFFUSION EQUATION IN FREQUENCY-DOMAIN TISSUE SPECTROSCOPY ......... 45 III.A FREQUENCY-DOMAIN SOLUTION TO THE DIFFUSION EQUATION IN THE INFINITE-MEDIUM GEOMETRY ......................45 III.A.1 THE BASICS OF FREQUENCY-DOMAIN METHODS.................................. 45 III.A.2 THE STANDARD DIFFUSION EQUATION ................................................. 47 III.A.3 THE FREQUENCY-DOMAIN SOLUTION TO THE DIFFUSION EQUATION .. 49 III.A.3.1 MODEL FOR A FREQUENCY-DOMAIN ISOTROPIC-POINT SOURCE ........ 49 III.A.3.2 THE SOLUTION: SPHERICAL WAVES OF PHOTON DENSITY ................. 49 III.A.3.3 PHOTON DENSITY FREQUENCY-DOMAIN CHARACTERISTICS .............. 51
III.B LIMITATIONS OF THE DIFFUSION MODEL ...................................56 III.B.1 III.B.2 III.B.3
RULE #1: STAY AWAY FROM BOUNDARIES AND SOURCES .................... 56 RULE #2: ONLY SCATTERING-DOMINATED MEDIA .............................. 56 RULE #3: KEEP THE MODULATION FREQUENCY BELOW 1 GHZ ........... 57
III.C THE DIFFUSION EQUATION IN OTHER GEOMETRICAL CONFIGURATIONS...............................................57 III.C.1 III.C.2 III.C.3 III.C.4
SOURCE DISTRIBUTIONS........................................................................ 58 THE EXTRAPOLATED-BOUNDARY CONDITION ...................................... 60 SOLUTION FOR THE PHOTON DENSITY IN A SEMI-INFINITE GEOMETRY ................................................................... 63 SOLUTION FOR THE PHOTON DENSITY IN A SLAB GEOMETRY .............. 64
III.D MEASUREMENT OF THE OPTICAL COEFFICIENTS OF HOMOGENEOUS-TURBID MEDIA ............................................65 III.D.1
THE MULTI-DISTANCE APPROACH IN THE INFINITE-MEDIUM GEOMETRY ............................................................. 65
xiii III.D.2
A WORD ABOUT THE MULTI-DISTANCE METHOD IN OTHER GEOMETRICAL ARRANGEMENTS........................................................... 67
III.E CHAPTER SUMMARY....................................................................68
IV THE FLUORESCENCE PHOTON DIFFUSION MODEL .... 69 IV.A NOTATION CONVENTIONS ...........................................................69 IV.B FREQUENCY-DOMAIN DIFFUSION THEORY: EXCITATION PHOTONS ................................................................72 IV.C FREQUENCY-DOMAIN DIFFUSION THEORY: EMISSION PHOTONS ....................................................................73 IV.C.1 IV.C.2 IV.C.3
EFFECTIVE EMISSION SOURCE DISTRIBUTION ...................................... 73 SOLUTION OF THE FLUORESCENCE-DIFFUSION EQUATION: SINGLE SPECIES .................................................................................... 76 SOLUTION OF THE FLUORESCENCE-DIFFUSION EQUATION: N SPECIES.............................................................................................. 77
IV.D MEASURABLE PARAMETER .........................................................78 IV.E ANALYSIS OF THE EMISSION PHOTON DENSITY ..........................80 IV.F
V
CHAPTER SUMMARY....................................................................83
INSTRUMENTATION ........................................... 84 V.AWORKING IN THE FREQUENCY DOMAIN ..........................................84 V.A.1 V.A.2 V.A.3
V.B
WHY USE TIME-RESOLVED OR PHASE-SENSITIVE METHODS AT ALL? .................................................. 84 THE BASICS OF THE TIME-DOMAIN METHOD ....................................... 85 WHY CHOOSE THE FREQUENCY-DOMAIN METHOD?............................ 86
A GUIDED TOUR OF THE INSTRUMENTATION..............................88
V.B.1 V.B.2
GENERAL TOUR OF THE INSTRUMENTATION ......................................... 88 THE LIFELINE OF FREQUENCY-DOMAIN INSTRUMENTS: PHASE LOCKING .................................................................................... 88 V.B.2.1 THE MEANING OF A PHASE DIFFERENCE ........................................... 88 V.B.2.2 PHASE REFERENCES AND PHASE ARTIFACTS ..................................... 89
V.CFREQUENCY-DOMAIN LIGHT SOURCES ............................................91 V.C.1
METHODS FOR INTENSITY MODULATING LIGHT SOURCES ................... 91
xiv V.C.1.1 V.C.1.2 V.C.1.3
DIRECT OR INTERNAL MODULATION .................................................. 91 MODE LOCKING ................................................................................. 92 THE ACOUSTO-OPTICAL EFFECT: AN EXTERNAL MODULATION METHOD............................................... 92 V.C.1.4 THE POCKELS EFFECT: ANOTHER EXTERNAL MODULATION METHOD .................................................... 93 V.C.2 SEMICONDUCTOR LIGHT SOURCES ....................................................... 94 V.C.3 ND:YAG LASER .................................................................................... 96 V.C.4 ARGON-ION LASER................................................................................ 98 V.C.5 DYE LASERS .......................................................................................... 99 V.C.6 TITANIUM-SAPPHIRE LASERS.............................................................. 100 V.C.7 LAMPS ................................................................................................. 101
V.DLIGHT DELIVERY AND DETECTION ................................................101 V.D.1 OPTICAL FIBERS.................................................................................. 101 V.D.2 SYNTHESIZERS..................................................................................... 105 V.D.3 SCANNER ............................................................................................. 106 V.D.4 AMPLIFIERS......................................................................................... 106 V.D.5 WAVELENGTH SELECTION .................................................................. 107 V.D.5.1 MONOCHROMATORS......................................................................... 108 V.D.5.2 OPTICAL FILTERS ............................................................................ 109 V.D.6 PHOTO-MULTIPLIER TUBE.................................................................. 110
V.E
SIGNAL PROCESSING .................................................................112
V.E.1 V.E.2 V.E.3 V.E.4 V.E.5
V.F
HETERODYNING: FREQUENCY DOWN-CONVERSION........................... 112 DIGITIZATION OF THE WAVE .............................................................. 115 WAVE FOLDING: AVERAGE OUT THE NOISE ....................................... 116 FAST FOURIER TRANSFORM ................................................................ 116 STATISTICS .......................................................................................... 117
CHAPTER SUMMARY..................................................................117
VI QUANTITATIVE FLUORESCENCE IN TURBID MEDIA: SINGLE SPECIES ....................... 119 VI.A OUTLINE: SINGLE FLUORESCENT SPECIES MEASUREMENTS....119 VI.B THE MULTI-DISTANCE EMISSION MEASUREMENT ...................122 VI.B.1 VI.B.2 VI.B.3 VI.B.4 VI.B.5
SPECIFIC INSTRUMENTATION .............................................................. 122 THE PHANTOM: OPTICAL COEFFICIENTS ........................................... 122 THE PHANTOM: FLUORESCENCE PROPERTIES .................................... 124 THE SOURCE TERMS AND THE DETECTOR RESPONSE ......................... 126 EXPERIMENTAL RESULTS .................................................................... 127
xv VI.B.6 VI.B.7
DISCUSSION OF RESULTS ..................................................................... 131 SECONDARY EMISSION: SMALL BUT POTENT ...................................... 133
VI.C THE MULTI-FREQUENCY MEASUREMENT ................................135 VI.C.1 VI.C.2
BASICS OF THE MEASUREMENT ........................................................... 136 EXPERIMENTAL RESULTS & DISCUSSION............................................ 136
VI.D MEASUREMENTS AS FUNCTIONS OF THE FLUOROPHORE CONCENTRATION .....................................139 VI.D.1 VI.D.2 VI.D.3 VI.D.4 VI.D.5
EMISSION INTENSITY AS FUNCTIONS OF FLUOROPHORE CONCENTRATION ....................................................... 140 SUB-NANOMOLAR FLUOROPHORE CONCENTRATION DETECTION ...... 144 SCATTERING CAN ENHANCE THE EMISSION SIGNAL ........................... 147 SENSITIVITY TO CHANGES IN CHROMOPHORE CONCENTRATION: ABSORPTION VS. FLUORESCENCE ....................................................... 148 SENSITIVITY TO CHANGES IN CHROMOPHORE CONCENTRATION: THE EMISSION PHASE ......................................................................... 151
VI.E CHAPTER SUMMARY..................................................................152
VII QUANTITATIVE FLUORESCENCE IN TURBID MEDIA: TWO SPECIES ................................................ 154 VII.A EXPERIMENTAL DETAILS ..........................................................154 VII.A.1 VII.A.2 VII.A.3 VII.A.4 VII.A.5
OUTLINE OF THE EXPERIMENT............................................................ 154 SPECIFIC INSTRUMENTATION .............................................................. 154 THE PHANTOM: OPTICAL COEFFICIENTS ........................................... 157 THE PHANTOM: FLUORESCENCE PROPERTIES .................................... 158 THE SOURCE TERMS AND THE INSTRUMENT RESPONSE ...................... 159
VII.B EXPERIMENTAL RESULTS: TWO SPECIES IN A TURBID MEDIUM .........................................161 VII.B.1 THE MULTI-DISTANCE MEASUREMENT .............................................. 161 VII.B.2 THE MULTI-FREQUENCY MEASUREMENT........................................... 161 VII.B.3 DISCUSSION OF EXPERIMENTAL RESULTS ........................................... 165 VII.B.3.1 GENERAL NOTES ON THE FITS ......................................................... 165 VII.B.3.2 THE PROBLEM OF AUTOFLUORESCENCE .......................................... 166
VII.C THE GENERAL PROBLEM OF LIFETIME CONTRAST ..................167 VII.C.1 TWO LIFETIMES: THE NON-SCATTERING CASE ................................. 167 VII.C.2 TWO LIFETIMES: THE MULTIPLE-SCATTERING CASE ........................ 169
VII.D CHAPTER SUMMARY..................................................................170
xvi
VIII
EXPERIMENTS WITH CELLS IN A TURBID MEDIUM ................................. 172
VIII.A BACKGROUND OF THE PROBLEM...............................................172 VIII.A.1 MASTITIS: THE DAIRY FARMER’S WORST ECONOMIC NIGHTMARE ......................................................... 172 VIII.A.2 A BRIEF DESCRIPTION OF THE CAUSES OF MASTITIS ......................... 173 VIII.A.3 THE ECONOMIC EFFECTS OF MASTITIS: LOST PRODUCTION ............. 174 VIII.A.4 CLUES FOR DETECTING MASTITIS: THE SOMATIC CELL COUNT................................................................ 174 VIII.A.5 TOWARDS AN OPTICAL MONITOR OF THE SCC IN MILK .................... 176 VIII.A.6 WHY DISCUSS THIS PROBLEM ............................................................ 177
VIII.B THE PHYSICS OF THE PROBLEM ................................................177 VIII.B.1 CONTRAST FOR DETECTING CELLS: THE FLUORESCENCE LIFETIME........................................................... 178 VIII.B.2 HOW MANY BINDING SITES ARE AVAILABLE? ................................... 178 VIII.B.3 AFFINITY FOR BINDING ....................................................................... 179 VIII.B.4 MONITORING ETHIDIUM BROMIDE BINDING IN THE CELLULAR ENVIRONMENT VIA FREQUENCY-DOMAIN LIFETIME SPECTROSCOPY ............................... 182
VIII.C EXPERIMENTAL ARRANGEMENT ...............................................184 VIII.C.1 VIII.C.2 VIII.C.3 VIII.C.4
IDEA OF THE MEASUREMENT .............................................................. 184 INSTRUMENTATION ............................................................................. 185 THE SAMPLE ....................................................................................... 185 PROTOCOL .......................................................................................... 186
VIII.D EXPERIMENTAL RESULTS AND DISCUSSION ..............................188 VIII.E CHAPTER SUMMARY..................................................................191
IX CONCLUDING REMARKS ................................... 192 IX.A THE PAST AND THE PRESENT ....................................................192 IX.B THE FUTURE?............................................................................193 IX.B.1 IX.B.2 IX.B.3 IX.B.4
NON-INVASIVE BLOOD-LEAD INDICATOR ........................................... 193 PORTABLE CELL COUNTER FOR MASTITIS DETECTION ...................... 194 IN-VIVO DRUG KINETICS MEASUREMENT ........................................... 195 THE “DREAM MACHINE” FLUOROMETER .......................................... 195
APPENDIX A
ABBREVIATIONS AND VARIABLES ......... 197
xvii
A.ATABLE OF ABBREVIATIONS.............................................................197 A.B
TABLE OF VARIABLES ...............................................................200
APPENDIX B B.A B.A.1 B.A.2 B.A.3
B.B B.B.1 B.B.2 B.B.3
DERIVATION OF THE DIFFUSION EQUATION FROM TRANSPORT THEORY.. 205
THE ONE-SPEED BOLTZMANN TRANSPORT EQUATION ............205 DEFINING OUR VOCABULARY: THREE SIMPLE DEFINITIONS ............. 205 ASSEMBLING THE ONE-SPEED BOLTZMANN TRANSPORT EQUATION .................................................. 207 DO GENERAL SOLUTIONS TO THE ONE-SPEED BOLTZMANN TRANSPORT EQUATION EXIST?...................................... 210
THE STANDARD DIFFUSION EQUATION .....................................213 THE P1-APPROXIMATION: RESTRICT THE ANGULAR DEPENDENCE TO FIRST ORDER .......................................... 214 THE DIFFUSION APPROXIMATION: ASSUME SMALL TIME VARIATIONS IN THE PHOTON CURRENT........... 218 THE DIFFUSION EQUATION ................................................................. 219
APPENDIX C
MATHEMATICAL BASIS FOR THE DIFFUSION APPROXIMATION ............... 220
C.ATHE DIFFUSION APPROXIMATION ..................................................220 C.A.1 C.A.2
C.B
THE FIRST CONDITION: 3ωDV-1 > µA.................................................... 221
RANGES OF APPLICABILITY ......................................................223
APPENDIX D
THE EFFECT OF SECONDARY EMISSION ....................................... 226
D.ASTATEMENT OF THE PROBLEM .......................................................226 D.B
THE SECONDARY-EMISSION SOURCE DISTRIBUTION ................228
D.CTHE EMISSION PROBABILITIES ......................................................229 D.DTHE SECONDARY-EMISSION PHOTON DENSITY .............................230
xviii
D.E
SECONDARY EMISSION WITH MULTIPLE FLUOROPHORES ........232
REFERENCES AND NOTES ...................................... 234
VITA… … … … … … … … … … … … … … ..… … … … … ...… … 250
1
I
INTRODUCTION: FLUORESCENCE MEETS NEARINFRARED TISSUE SPECTROSCOPY
This chapter serves as background material for many topics.
The beginning of the
chapter is an attempt to describe the impact that optics is delivering to medicine. I will proceed by characterizing the optical properties of tissues in the near-infrared (near-IR) wavelength region. This characterization will naturally lure our attention to two of the major considerations in near-IR optical tissue spectroscopy: multiple scattering and contrast. The discussion will then shift to characterize the fluorescence properties of tissues. Some interesting examples of fluorescence spectroscopy as a diagnostic tool in medicine also find their way into this chapter. Our discussion concludes with a peek at how multiple scattering alters the way we must approach fluorescence spectroscopy.
I.A
BIOMEDICAL OPTICS: A NEW FRONTIER OF MODERN MEDICINE
Medicine has profited enormously from the ingenious application of physical principles to problems that often started out as non-medical in nature. For example, the discovery of x-rays by Roentgen in 1895 furnished a non-invasive method for exploring inside the human body. Isotope scanning methods made their medical debut in the 1950’s, and ultrasound burst upon the medical scene in the 1960’s. Computerized Tomography (CT) was a big hit in the 1970’s. Nuclear Magnetic Resonance, pioneered by many solid-state physicists in the 1940’s, paved the way towards the marvel of Magnetic Resonance Imaging in the 1980’s. Although there are myriad applications of the physical sciences to
2 medicine and biology, optics is one of the more explosive contestants emerging on the medical playing field in the 1990’s. Many researchers are developing wonderful techniques, featuring visible and near-IR light as a probe for investigating tissues. For example, there has been great success in the non-invasive optical sensing of hemoglobin saturation (i.e., the ratio of [HbO2] to the total hemoglobin concentration).1-4 As a tool in the hands of a physician, non-invasivehemoglobin sensing may aid in the analysis of diseases such as hematomas, strokes, and vascular deficiencies, as well as provide new means to investigate muscle physiology. Time will fail me if I mention of other potential uses of light in medicine, since new optical-diagnostic tools are emerging at a torrid pace. Powerful examples include noninvasive optical biopsies (i.e., non-tissue removal)5-8 and non-ionizing radiation mammograms. 9-11 The advantages of these techniques are crystal clear, especially in an age of ridiculously-increasing expenses in health care.
Optical methods are also
graduating into the disease-treatment business. For example, the field of photodynamic therapy concerns itself with photosensitizing drugs that have a preferential affinity for cancer tissue, and release toxic-singlet oxygen upon optical excitation.12 Optical methods may further contribute the exciting possibility of rapid, low-resolution, functional images, as opposed to the purely anatomical images of conventional imaging devices.13 One exciting example might be a real-time and low-resolution map of the oxygen consumption in a tissue. Additionally, optical instrumentation is easy to make both portable and inexpensive; an example of this is a portable tissue oximeter originally developed in our laboratory that is already undergoing some clinical trials as you read this document.14
3 At the heart of all of these medical innovations lies one major physical issue: photontissue interactions. The field of tissue optics concerns itself with topics such as "how many photons penetrate a given distance below the tissue surface and strike a tumor," or "how many degrees will the temperature of the tissue increase after absorbing a laser pulse?" Most importantly, the answers must be quantitative, which is a lofty requirement that further complicates an already challenging problem.
I.B
SOME MACROSCOPIC OPTICAL CHARACTERISTICS OF TISSUES
11
Tissues are an impressively complex creation comprised of a vast assortment of molecules, structures, and functional units. Despite this overwhelming complexity, we may still discuss average optical properties as long as we realize the limitations involved. The goal of this section is to provide a physical feel for the scattering and the absorption of near-IR radiation by tissues both in terms of (1) the origin of these processes, and (2) the typical values of the macroscopic coefficients that describe these processes.
I.B.1
OPTICAL PROPERTIES OF INTEREST: THE PHYSICIST’S MODEL OF TISSUE
There are four independent macroscopic parameters that are believed to characterize light propagation in tissue: the scattering anisotropy (g), the absorption coefficient (µa), the scattering coefficient (µs), and the index of refraction (n). These parameters are defined in Table (1.1). Mathematics defines the scattering anisotropy as:
4 g ≡ ∫dΩˆ′ f (Ωˆ′ → Ωˆ) Ωˆ′ ⋅Ωˆ ,
(1.1)
4π
where f (Ωˆ′ → Ωˆ) is the scattering-phase function (i.e., the probability to scatter from the direction Ωˆ′ into the direction Ωˆ , normalized such that the integral over the whole solid angle is unity). These optical parameters should communicate something about both the biochemical properties as well as the morphological and structural configurations of the tissue. Inverses of a mean-free path (mfp) are a convenient manner in which to describe the scattering and the absorption coefficients.
The reduced-
scattering coefficient (µs′ ) is not independent from the other parameters in Table (1.1), and we shall define it as: µ s ' ≡ µ s (1 − g ) .
(1.2)
I am presenting it here because the model I use cannot determine µs independently from g. The reduced scattering represents a length scale where the photon looses all memory of its initial direction; as such, µs′provides a scale for isotropic-scattering events.15 Generally one is forced to assume that the parameters of interest in Table (1.1) are macroscopically homogeneous throughout the tissue volume, although this is not a rigorous description of tissue. Despite this simplified vision of tissue, for many cases of interest it will be sufficient.
I.B.2
THE OPTICAL PROPERTIES OF TISSUES IN THE NEAR-INFRARED
Figure (1.1) convincingly demonstrates that near-IR photons achieve the largest photon penetration depth into tissues, where 1/(µs′+ µa) quantifies the penetration depth
5 (Ref. 16). On the IR end, water is highly absorbing. On the visible to ultraviolet end, water and proteins dominate the absorption. You may witness for yourself the deep penetration of red photons into tissues by holding a white light to your hand; only the red light emerges on the other side of your fingers.
Table (1.1) - Physical Quantities of Interest in Tissue Optics PARAMETER
SYMBOL
DEFINITION
absorption coefficient (cm-1)
µa
absorption mfp ≡ µa-1
scattering coefficient (cm-1)
µs
single-scattering mfp ≡ µs-1
scattering anisotropy
g
average cosine of the scattering angle (Eq. (1.1))
index of refraction
n
ratio of speed of light in vacuum to that in medium
reduced scattering coefficient (cm-1)
µs′
approximate inverse scale of isotropic scattering (Eq. (1.2))
The major absorbing components (save perhaps for melanin) of many soft tissues in the near-IR are the stars of Figure (1.2). This figure assumes a 100 µM concentration of hemoglobin saturated at 70%, and uses published values for the extinction coefficients of water
17
and hemoglobin.18 Hemoglobin is the strongest absorber of photons within the
700 to 900 nm range. Both oxy- and deoxy- forms of hemoglobin constitute about 90% of the absorption of near-IR radiation in muscle.19
This distinctive absorption by
hemoglobin provides excellent contrast for hemoglobin detection via absorption spectroscopy.
6
Figure (1.1) - Penetration depths in tissues for common laser wavelengths. The penetration depth is 1/(µs′+ µa). I adapted this figure from page 33 of Ref. 16. The nearIR has by far the deepest penetration depth into tissues for these selected wavelengths.
7
0.5
0.4
Hb
H2O
-1
µa (cm )
0.3
0.2
0.1
0.0
Hb2O
600
700
800
900
1000
1100
WAVELENGTH (nm)
Figure (1.2) - Absorption spectra for the dominant chromophores in bulk tissue within the near-IR spectral region. This figure assumes a 100 µM concentration of hemoglobin saturated at 70%, and uses published values for the extinction coefficients.17,18 Between 700-1100 nm, the total absorption is small compared to other wavelengths. Additionally, hemoglobin absorbs more strongly than water in the 700-900 nm region.
8 An extensive list of measured tissue optical properties exists.20 It is easy to notice in this list that for a given tissue type that these reported optical coefficients often differ by substantial amounts. One scapegoat for many of these discrepancies is the varied problems associated with in vivo measurements. However, there are also many glaring discrepancies reported in these measured coefficients because of different sample preparation protocols and varied measurement techniques.
Accurate and precise
measurements are an arduous task to perform in vivo, and nobody has proven that in vitro measurements compare realistically with in vivo measurements (for example, excised tissue samples are devoid of blood). Remaining cognizant of these many difficulties, accepted coefficient values for many soft tissues fall within the ranges of µa ~ 0.05 to 1 cm-1 and µs ~ 100 to 1000 cm-1 (Ref. 21).
Scattering in tissue is highly forwardly
directed; many reported values of g are 0.9 or greater, (although values as small as 0.6 have been reported)20 and at least in liver tissue, g is relatively wavelength independent.22 Typical values for µs′therefore fall into the range of 3 to 20 cm-1. An average value of 1.4 has been measured in various tissues for the index of refraction in the near-IR.23 Another way to think about the scattering properties of tissue is to consider the average propagation time of photons traversing a gap of 1 cm between the source and the detector. A photon will take only about 0.04 ns to cross the 1 cm gap in water (i.e., devoid of scattering). On the other hand, in a tissue such as muscle, the average photon time transit for a 1 cm source-detector separation falls in the range of a few nanoseconds. This transit time is at least 25 times longer than the transit time of the ballistic case. This greatly elongated photon path length, which is much greater than 1 cm, clearly demonstrates the highly-scattering nature of tissues.
9 I.B.3
WHO IS RESPONSIBLE FOR BIOLOGICAL SCATTERING?
The predominant factors that influence the optical scattering properties of tissues are the size, the shape, and the density of the scattering centers, as well as the index of refraction mismatches. Scattering in tissues most likely occurs from cells, cellular organelles, and proteins.24 Mammalian cells range in size from about 10 to 30 µm, and their internal organelles appear in a broad assortment of shapes and sizes. Table (1.2) lists some of the most common cellular organelles inside mammalian cells. 25 Proteins are often loosely denoted as “spherical” with diameters less than 7 nm (Ref. 26). Additionally, there are many variations in the tissue index of refraction. Lipids (n ≈1.46, Ref. 27) and proteins (n ≈1.51, Ref. 28) have relatively high indices of refraction, whereas components such as the extra-cellular fluid have lower indices of refraction (n ≈1.35, Ref. 29). Glucose also changes the index of refraction of water. Specific structures in tissue such as collagen scatter light in very different ways. Different classes of cells also contain different fractions of cellular components, and have distinctive surroundings (for example, epithelial cells in skin and osteoblasts in bone). The conclusion of the matter is that tissues possess a wide range of scatterers as evidenced by the many sizes, shapes, densities, and index of refraction mismatches found in them. The scattering coefficient should be able inform us about the size of the scattering centers in the tissue.
Saidi et al. demonstrated that scattering in neonatal skin is
dominated by Mie scattering from cylindrical collagen fibers (1-8 µm in diameter) that tend to have a general orientation. Rayleigh scattering was also present, emerging mostly from collagen fibrils, which are much smaller than the collagen fibers and do not bundle.30 Flock et al. measured the attenuation coefficient of various thin tissue samples
10 to be on the order of 100 to 1000 cm-1 (Ref. 21). These values imply a near-IR photon mfp of about 10 to 100 µm, which roughly compares with the typical mammalian cell size.
Similar results were found by Parsa et al. in liver tissue, which is about as
homogeneous as tissues get (Ref. 22). Beauvoit et al. found that the scattering properties of isolated mitochondria behaved very similarly to those of whole liver, and therefore suggested that mitochondria are responsible for most of the optical scattering in the liver.31 Given that mitochondria are about 1 µm in size and are present in great numbers in cells (about 20% of the cell volume in liver cells), mitochondria should certainly play a role in near-IR photon scattering. Jacques attributed the near-IR scattering of soft tissues to membranous structures that he modeled as spherical Mie scatterers in the 0.2 to 2 µm diameter range.32 Mitochondria also have an extensive membrane structure.
Table (1.2) – Approximate Shapes and Sizes of Common Cellular Organelles SCATTERER
SIZE PARAMETER
APPROXIMATE SHAPE
plasma membrane
0.005 µm (thickness)
sheet
cytoskeletal filaments
0.01 µm (diameter)
cylindrical
ribosomes
0.02 µm (diameter)
spherical
lysosomes
0.4 µm (diameter)
spherical
mitochondria
1 µm (diameter)
cylindrical
golgi apparatus
1 µm (length)
folded sheets
endoplasmic reticulum
few µm (length)
folded sheets
nucleus
7 µm (diameter)
spherical
11 I.B.4 CONTRAST NEEDED TO EXTEND NEAR-INFRARED TISSUE SPECTROSCOPY The amount of available contrast sets the amount of usefulness of an imaging or a spectroscopic technique. As I mentioned previously, in the specific case of the near-IR, both oxygenated and deoxygenated forms of hemoglobin provide superior absorption contrast. In using the term contrast, I imply the ability to detect the substance under study separate from the background. Few other important tissue chromophores can match the level of absorption contrast of hemoglobin in the near-IR. We could try to find wavelengths where other chromophores of interest absorb strongly, but these wavelengths dwell in regions where penetration into the tissue is low.
Besides this setback, the absorption of the
chromophore of interest must stand out from amongst the rest of the tissue. Thus, nearIR absorption spectroscopy provides little contrast for chromophores other than hemoglobin.
In the case of a tumor, where we expect a large concentration of
hemoglobin, contrast is still an open question. Some have reported little contrast,33 while others have reported measurable changes in the optical properties of tumors relative to healthy tissue.34,35 Regardless of these findings, contrast enhancements are necessary for many other important cases of medical interest. This is where fluorescence enters the scene.
I.C
FLUORESCENCE SPECTROSCOPY IN TISSUES
I will now follow a similar course with fluorescence as I did in the previous section. After starting with some basic definitions, I will proceed to describe the fluorescence
12 properties of tissues. Finally, the remainder of the chapter will focus upon two main points: (1) how fluorescence spectroscopy provides enhanced contrast to tissue spectroscopy, and (2) how multiple scattering dramatically influences fluorescence spectroscopy.
I.C.1
FLUORESCENCE PARAMETERS OF INTEREST
There are four fluorescence parameters of interest that we may recover from multiplescattering media: the fluorescence lifetime (τ), the fluorescence quantum yield (Λ), the fluorescence emission spectral efficiency (ϕ(λ)), and the fluorophore concentration ([F]). Table (1.3) provides definitions for each of these measurable parameters. Fluorescence polarization measurements such as anisotropy are quite valuable in non-scattering media; they are not included here because the incident excitation depolarizes after a few scattering events. Polarization measurements (both fluorescence and otherwise) in turbid media are not possible, save for surface reflectance measurements. A few notes about these parameters are in order. With regards to the fluorescence lifetime, the average time the molecule spends in the excited state:
∞
t ≡
∫dt t M (t ) 0 ∞
,
(1.3)
∫dt M (t ) 0
where M(t) is the average number of molecules in the excited state. Only in the case of a single decay rate (i.e., M(t) ~ exp[-t]) is the lifetime identically equal to the average
13 amount of time that the molecule spends in the excited state, (i.e., t = τ ). With regard to the fluorophore concentration, I need to clarify two items. First of all, there is a difference between absorption by the fluorophore and absorption by other compounds. I will designate the coefficient µaf as the absorption due to the fluorophore of interest, and µab as the absorption due to everything non-fluorescent (termed the background). If ε is the fluorophore molar-extinction coefficient (M-1×cm-1), then µaf becomes:
µ af = 2.303 ε [ F ] .
(1.4)
The 2.303 is a conversion factor from base 10 to base e. The second item I need to clarify is whether we are discussing endogenous (i.e., native) or exogenous (i.e., foreign) fluorophores. I will state which is the case as the need arises.
Table (1.3) - Parameters of Interest in Fluorescence Tissue Spectroscopy PARAMETER
SYMBOL
DEFINITION
fluorescence lifetime (ns)
τ
related to the average time a molecule spends in the excited state (see Eq. (1.3))
fluorescence emission spectral efficiency (nm-1)
ϕ(λ)
ϕ(λ)dλ≡ probability for emission into wavelength range dλabout λ
fluorescence quantum yield (%)
Λ
ratio of radiative de-excitation rate to all other de-excitation rates
fluorophore concentration (M)
[F]
moles of fluorophore per liter of solution
I.C.2 FLUORESCENCE PROPERTIES OF TISSUE The absorption and the emission maxima of some endogenous tissue fluorophores are provided in Table (1.4).36-39 Tryptophan and tyrosine are amino acids, which are the
14 building blocks of proteins. Tryptophan is responsible for about 90% of the intrinsic fluorescence of proteins. It is sometimes inserted into proteins, acting as a sensitive marker whose fluorescent properties change with the state of the protein. Collagen is a tough structural protein that one finds abundantly in mammals. The pyridine nucleotides NADH and NAD(P)H fluoresce strongly at 470 nm (nicotinamide adenine dinucleotide and nicotinamide adenine dinucleotide phosphate, respectively). The oxidized pyridine nucleotides, NAD and NADP, fluoresce but at three orders of magnitude weaker than their reduced counterparts. NADH is an important player in forming energy-rich ATP (adenosine triphosphate) and is therefore an indication of cellular metabolism.40 Flavins are better known by generic name of Vitamin B2, and the different varieties of flavins have very different quantum yields. The strongest of the lot are flavin mononucleotide (FMN) and riboflavin.41
Flavins can also bind to proteins (i.e., flavoproteins), but
flavoproteins are usually weak in fluorescence. Porphyrins are intermediate products in the biosynthesis of heme. Normal accumulations of porphyrins are low in tissues, but abnormalities in heme synthesis can cause a high accumulation of metal-free porphyrins such as protoporphyrin IX (PP IX). Abnormal accumulations of porphyrins occur in some hemolytic diseases, anemia, and alcoholism.36 Specifically, zinc protoporphyrin (ZPP) fluorescence is as an indicator of blood lead toxicity.42,43 Other fluorophores such as melanin exist in tissues but with very low quantum yields.44
15
Table (1.4) – Some Common Endogenous Chromophores in Tissues FLUOROPHORE
SOLVENT
ABSORPTION MAXIMA (nm)
EMISSION MAXIMA (nm)
LIFETIME(S) (ns)
COMMENTS
tryptophan
H2O
220, 280
348
2.6
1-6 ns in proteins
tyrosine
H2O
274
303
3.6
very low Λ in proteins
collagen
powder
300-340
420-460
2.7 & 8.9
elastin
powder
300-340
420-460
2 & 6.7
NADH
H2O
260, 340
470
0.4
H2O
260, 370
470
~ 0.5, ~ 1.8
human artery wall
320
400
0.3, 2, 7
H2O
260, 370, 450
530
rat kidney
488
500-550
0.36, 1.22
rat kidney (cancer)
488
500-550
0.22, 1.97
FMN
H2O
~ 450
~ 515
4.7
quenched when reduced
FAD
H2O
~ 450
~ 515
2.3
quenched when reduced
PP IX
DMSO
400
650-670
17, 3
long lifetime
ZPP
DMSO
400
650-670
2
NAD(P)H
flavins
quenched when reduced quenched when reduced
flavoproteins generally not fluorescent
16 A quick glance at the absorption maxima column in Table (1.4) reveals a limitation of endogenous tissue fluorescence sensing; these native fluorophores are excitable only in wavelength regions where tissue absorption is brutally strong. Since these excitation photons can only penetrate less than a few hundred microns into the tissue, native fluorescence measurements are restricted usually to surfaces, thin excised samples, endoscopy, or invasive measurements. In addition, notice that a single exponential often does not describe the fluorescence decays of many of these fluorophores. Table (1.4) lists the strongest of the lifetime components is listed first.
I.C.3 FLUORESCENCE CONTRAST: SENSITIVITY AND SPECIFICITY ENHANCEMENT Traditionally both the high sensitivity and specificity of fluorescence spectroscopy have made it an attractive tool for detecting traces of specific chemical compounds. Fluorescence spectroscopy enjoys a superior signal-to-noise ratio (SNR) over absorption spectroscopy in the low chromophore concentration regime. The reverse is true for higher concentrations.
A better SNR improves our sensitivity to detect the target
compound. The fluorescence parameters of Table (1.3) are sensitive indicators of the local environment of the fluorophore. The lifetime, the emission spectrum, and the quantum yield are all solvent dependent. As an example, consider how the emission and the absorption properties of rhodamine 3B+ change when it is dissolved in two different solvents in Table (1.5) (Ref. 45).
Modern instrumentation can easily resolve both
lifetimes and intensities with a precision of 0.2 ns and 1%, respectively. This enhanced sensitivity may be used as a contrast enhancement for tissue spectroscopy, if the
17 chromophore is fluorescent. This enhancement would allow the detection of compounds other than hemoglobin in near-IR tissue spectroscopy. However, there is more to contrast than just sensitivity. In an ideal case, there would be a hefty signal from the target compound, and zero signal from the background (i.e., everything else). Specific labeling of target compounds may be achieved in conjunction with fluorescence. For example, one may fluorescently label an antibody specific for a given tumor, and thus be able to tag fluorescently the tumor rather than the background tissue. This would be called “selective staining” in microscopy. Fluorophores may also be engineered to bind to a specific target, so that in principle one can coerce many compounds into becoming fluorescent.
Table (1.5) - Emission and Absorption Properties of Rhodamine 3B+ in Two Different Solvents PARAMETER
WATER
ETHANOL
absorption peak (nm)
559.0
556.0
emission peak (nm)
578.1
574.2
extinction (105 M-1×cm-1)
0.98
1.15
quantum yield (%)
19
41
lifetime (ns)
1.53
2.27
Exogenous chromophores are therefore required if we wish to monitor non-invasively physiologically relevant compounds other than hemoglobin in tissues. Such monitoring capabilities have been long available on the microscope slide. For in vivo use, the trick is
18 to come up with fluorophores that are both non-toxic and excitable in the near-IR; this search is currently an active area of research. To date, the fluorophore indocyanine green is the only Food and Drug Administration (FDA) approved fluorophore that absorbs and emits above 700 nm, suggesting that it is a leading candidate for in vivo tissue fluorescence spectroscopy. The background fluorescence of the tissue is substantially lower in the near-IR than in the ultra-violet (UV) and visible regions (see Table (1.4)). Thompson has provided a brief summary of currently available red and near-IR fluorophores.46
I.C.4
USES OF FLUORESCENCE PARAMETERS IN TISSUE SPECTROSCOPY
It is an insurmountable task to make due mention of all of the past applications of fluorescence spectroscopy to tissues. These varied techniques cover both exogenous and endogenous fluorophores. Rather than provide a comprehensive list, I will outline two interesting applications of fluorescence spectroscopy in tissues, and discuss the differences between intensity-based and lifetime-based measurements.
I.C.4.1 INTENSITY-BASED-FLUORESCENCE MEASUREMENTS The most common fluorescence measurements in tissues have been based upon measurements of the emission intensity. Although these measurements are relatively simple to perform, it is very difficult to be quantitative with intensity-based measurements. The emission intensity may change over time because of many unrelated factors, and it is generally impossible to account for all of them. Changes in the emission
19 intensity arise from drifts in the excitation source; we may fix this problem by referencing the output of the light source. However, the emission intensity also changes because the fluorophore may photobleach, diffuse in or out, or change in a chemical reaction.
The emission intensity depends upon the fluorophore concentration, and
usually it is impossible to know the fluorophore concentration, especially in tissues. In terms of calculations, the quantum yields are often unknown, especially in situations where the fluorophore binds to an unknown target. On top of all this, the photon path length is unknown, except in the case of a completely dilute solution (a case that does not exist in tissues). A common way to eliminate unknown factors is to measure a ratio of intensities. Usually a normalized emission spectrum provides information about the sample. If one knows which compounds are the major contributors to a given emission spectrum, one can monitor changes in the concentrations or states of these compounds by monitoring the emission spectrum. Additional information is required to discover the reasons behind the changes in this compound.
I.C.4.2 LIFETIME-BASED-FLUORESCENCE MEASUREMENTS An altogether different approach that has received much attention in the literature is the measurement of the fluorescence lifetime. Lifetime-based sensing does not suffer from the many shortcomings of intensity-based sensing, since the lifetime is independent of the fluorophore concentration. In principle, frequency-domain-lifetime measurements suffer from fewer artifacts than intensity-based measurements since the phase is inherently a
20 relative quantity so that drifts and certain systematic errors are usually not a problem. Additionally, lifetimes may be measured with very good precision (~ 0.1 ns or less). The fluorescence lifetime characterizes the molecular state of the sample, so that in a given environment, repeated measurements of the lifetime should be very accurate. Of course, we pay for these advantages with more complex instruments (i.e., it is more expensive!).
I.C.4.3 FLUORESCENCE MEASUREMENTS IN TISSUES Two prominent examples of fluorescence spectroscopy in tissues present in the literature are detecting cancer and monitoring metabolism. There have been numerous applications of native fluorescence to the study of healthy and diseased tissues using both intensitybased and lifetime-based measurements. Simple qualitative changes in the emission spectrum have been the traditional marker for identifying diseases in tissues. Typically the diseased and the normal emission spectra are independently normalized, and the disease is assumed responsible for the changes to the normalized spectrum.5-7,36,47 Multicomponent decays frequently confront lifetime-based measurements, making dataanalysis a trifle trickier. In some cases, the lifetimes remain constant but the weighting of the components indicates cancerous changes in the tissue.38 Porphyrins have also been a common target for fluorescence spectroscopy. Characterized by their relatively long lifetimes in the 10-20 ns range, porphyrins possess some distinction from most other endogenous fluorophores. In addition, porphyrins are weakly excitable in the red (610-690 nm), which allows reasonable penetration of the exciting light into the tissue. Porphyrins are endogenous to tissues but there are many different types of them. However, another approach has been to use endogenous agents such as 5-
21 aminolaevulinic acid (ALA) as a means to induce cells to increase their production of PP IX. Rapidly growing cancer cells will now be preferentially laced with PPIX, making them stand apart from normal tissue. Richards-Kortum and Sevick-Muraca provided a nice summary of references on the topic of exogenous porphyrin fluorescence.38 Fluorescence has also been used to determine the physiological state of a tissue. Two of the more popular targets have been NAD and its reduced form NADH. The critical factor is that NADH provides an intrinsically fluorescent indicator of cellular metabolism.48 Some in vivo studies have tried to provide quantitative vales of NADH fluorescence in muscle during low oxygen conditions such as ischemia and exercise.49 NADH may also serve as an indicator of the intracellular oxygen concentration.48 In addition to NADH, some researchers have tried to use the emission intensity to monitor the progression of atherosclerosis47,50-52 as well as oxygen concentration.53,54 All of these aforementioned studies used intensity-based methods for monitoring the light emitted from tissues. The fluorescence lifetime is another common indicator used for monitoring metabolism in biological media. As a tool in sensing, the monitoring of pO2 (Ref. 55), pH (Ref. 56, 57), Ca++ (Ref. 58), and glucose,58 to name a few, have all been suggested. The main problem with monitoring these states and analytes in vivo is that a non-toxic fluorophore must be used for in these applications. Given the general successes of lifetime-based methods, this is an area of research holding great promise for the future.
22 I.C.5
FLUORESCENCE SPECTROSCOPY: MOVING FROM CUVETTES TO TISSUES
Fluorescence spectroscopy is usually performed in non-scattering media. The intensity I transmitted through a purely absorbing medium (where scattering may be neglected) is related to the incident intensity I0 via the Beer-Lambert law:
I = I 0 exp( − ε[ F ]L) ,
(1.5)
where L is the photon path length (cm). Without scattering, L is equal to the width of the sample holder (called a cuvette).
The light absorbed by the medium is therefore
Io - Ioexp(-ε[F]L). The emitted fluorescence intensity is then proportional to the light absorbed by the medium via the fluorescence quantum yield (Λ). If we take the same cuvette and add many scattering particles, Eq. (1.5) is no longer valid for this medium. The effects of multiple scattering on the fluorescence emission are threefold:
(1)
The excitation light will now be spatially distributed within the medium, even in
the axial direction (Figure (1.3a)). This distribution in turn yields a position-dependent emission signal. All of the measured parameters become position dependent, which is usually not the case in a cuvette. (2)
The scattering increases the photon path length beyond the geometrical path of
the light (Figure (1.3a)). This distorts measurements of the fluorescence lifetime in either time- (the time of flight), or frequency-domain (the phase delay) methods. (3)
The scattering introduces an additional wavelength-dependent attenuation
(Figure (1.3b)), which causes the attenuation of the medium to differ from that of the absorption spectrum. In particular, the blue part of the spectrum will be attenuated more
23 than the red part, thus distorting the spectrum of light propagating through the medium. Red shifts in the emission spectrum will occur because of these differing optical properties.59
In addition, any background absorption in the medium will further
complicate matters.
Any model that we use must be able to separate the effects of fluorescence from the effects of multiple scattering. This realization now sets the table for the next chapter.
I.D
CHAPTER SUMMARY
Optical tissue spectroscopy is rapidly emerging as a cogent force in the growing arsenal of medical techniques by supplying portable and low cost instrumentation capable of non- or minimally invasive quantitative diagnoses. Although the near-IR spectral region provides excellent light penetration into tissues, the lack of optical contrast for compounds other than hemoglobin restricts the opportunity to investigate many interesting physiological processes in tissues. Exogenous fluorophores offer enhanced sensitivity and specificity as a viable solution to the contrast problem of near-IR tissue spectroscopy. Quantitative parameters of interest include optical properties (µa and µs′ ), as well as fluorescence characteristics such as Λ and τ. Typical values of µa and µs′for tissues exposed to near-IR photons are 0.1 and 10 cm-1, respectively.
The accurate
recovery of these many parameters provides a crucial step towards realizing quantitative tissue spectroscopy. The multiple-scattering nature of tissues affects the recovery of
24 these optical parameters because multiple scattering complicates the trajectories of both the emission and the excitation photons according to the wavelength-dependent scattering properties of the tissue.
25
OPTICALLY CLEAR
OPTICALLY TURBID
L2 = ?
ABSORPTION
WAVELENGTH
(b)
INTENSITY
SCATTERING
L1
(a)
WAVELENGTH
WAVELENGTH
Figure (1.3) - The effect of multiple scattering upon fluorescence spectroscopy. (a) Multiple scattering increases the path length and thus the photon's propagation time through the medium (L1 > µa
3 ω D/v µa. If the photon density is to be essentially isotropic, then for any point inside the medium photons must converge upon that point from all directions in equal numbers. Only after many scattering events will there be a sufficient randomization of photon direction vectors to establish an isotropic photon density. This randomization must occur before the photons are heavily attenuated
57 by absorption; thus, the absorption cannot be on the same scale as the scattering. This condition is sometimes called the “high albedo” condition, where the albedo is defined as µs '
( µ s ' + µa )
.
III.B.3 RULE #3: KEEP THE MODULATION FREQUENCY BELOW 1 GHZ First, the obscure term 3ωDv-1 has a lot more physical meaning if we re-write it another way.
The effective transport mfp (or diffusion mfp) of the photons is 1/(µs′+ µa).
However, in the diffusion approximation µs′>> µa so that vµ′ s represents the isotropiccollision frequency of the photons. We can easily write down the period between these isotropic interactions as τcoll. Not to be outdone, the intensity-modulated source has its own period: τmod = 1/f =2π/ω. The condition 3ωDv-1 z (Ref 110).
III.C.4 SOLUTION FOR THE PHOTON DENSITY IN A SLAB GEOMETRY The slab geometry is the next step up in complexity from a semi-infinite geometry. The solution again relies upon the method of images. Starting with the semi-infinite model and one image source, we now place an image source below the second surface to force the photon density to vanish on this second extrapolated boundary. However, this second image source changes the previous boundary condition on the first extrapolated boundary.
In retaliation, we could place another image source above the first
extrapolated boundary to cancel the effect. This of course, changes the photon density at the second extrapolated boundary. The only way to balance these competing boundary conditions is to use an infinite series of image sources, each one having the form of
65 Eq. (3.6), but using very different r’s. Note that the heavily damped nature of the photondensity wave assures us that only a few terms would be required to achieve the delicate balance.
Patterson et al. have calculated expressions for the reflectance and the
transmittance of a slab in the time domain using this approach.109
III.D
MEASUREMENT OF THE OPTICAL COEFFICIENTS OF HOMOGENEOUS-TURBID MEDIA
III.D.1 THE MULTI-DISTANCE APPROACH IN THE INFINITE-MEDIUM GEOMETRY The relationships expressed in the infinite-medium solution (i.e., Eq. (3.7)) represent the measurable frequency-domain parameters in terms of the optical coefficients of the medium. Now we would like to reverse this by finding expressions for µa and µs′in terms of the measured frequency-domain parameters. In the infinite-medium solution, the equations for the AC and DC intensities take on a new form if we multiply each side by r and take the natural log:
ln[ r AC ] = − r V+ ( ω )
ln[ r DC ] = − r
P (ω ) µa + ln = − r M AC (µ a , µ s ' , ω ) + BAC (µ s ' , P (ω )) 2D 4 πv D P ( 0) µa + ln = − r M DC (µ a , µ s ' ) + BDC (µ s ' , P (0)) D 4πv D
φ= r V− ( ω )
(3.18)
µa + φ0 (ω ) = r M PH (µa , µ s ' , ω ) + φ0 (ω ) 2D
There are three important facts about the three equations of Eq. (3.18) that we should notice:
66 1)
Each of the measurable frequency-domain parameters (ln(rAC), ln(rDC), and φ)
are linear functions of r. 2)
The slopes of these lines (MAC, MDC, and MPH) are functions of the desired optical
coefficients µa and µs′ . 3)
The intercepts of these lines (BAC, BDC, and φ0(ω)) contain the only appearances
of the source terms P(ω) and φ0(ω).
All we need to do is measure the slopes of the quantities ln(rAC), ln(rDC) and φ as functions of r. Since we have two unknowns and three equations, in reality we need to measure only two out of the three frequency-domain parameters. Typically, I have chosen to use the AC and the φas my parameters since the AC intensity and the φare less sensitive to ambient light than the DC intensity. Using these two slopes MAC and MPH, we may easily write down explicit expressions for µa and µs′(Ref. 115):
µa =
ω M PH M AC − 2v M AC M PH
µs ' =
( M AC ) 2 − ( M PH ) 2 3 µa
(3.19)
Optical coefficients have been recovered from a multiple-scattering medium by varying not only r, but also by changing the modulation frequency,116 and by changing both r and the modulation frequency.107 One very important feature of these equations is that we are able to obtain absolute information from a relative measurement. The importance of this observation cannot be overemphasized. For example, say that we place some optical fibers on the surface of a tissue. The amount of light entering the tissue depends upon many things, but one of
67 them is the coupling between fiber and tissue. The light intensity will increase if we impolitely shove the fibers harder into the skin. However, since we are only interested in the slope of the intensity vs. distance, as long as each distance is affected the same way (say with uniform pressure of the fibers onto the skin), we should recover the same optical properties. This fact has been experimentally demonstrated.14 Any method that pursues absolute optical coefficients by relying upon the absolute intensity signs its own death warrant, since there are countless ways in which the intensity may be affected, and it is unfeasible to account accurately for all of them.
III.D.2
A WORD ABOUT THE MULTI-DISTANCE METHOD IN OTHER GEOMETRICAL
ARRANGEMENTS I have tried to emphasize that the multi-distance method is only concerned with slopes and that we may use these slopes to determine absolute optical coefficients.
An
unexpected treat is that the spirit of the multi-distance measurement is also advantageous in geometrical configurations more complicated than the infinite-medium geometry. Fantini et al. demonstrated that the multi-distance measurement protocol works equally well in the semi-infinite geometry.110 Non-essential parameters such as the extrapolated boundary distance do not even play a role in the procurement of the optical coefficients. Although the equations are a bit more complex in the semi-infinite medium than those for the infinite medium, they still follow the same guidelines listed above. In addition, Cerussi et al. showed that under certain conditions, curved boundaries likewise have little effect upon the multi-distance slopes.117 Although the absolute vales of the intensity and
68 phase may be dramatically altered by a semi-infinite or curved boundary, the values relative to r do not change significantly.
III.E
CHAPTER SUMMARY
In the frequency-domain approach, an intensity-modulated light source injects photons into a multiple-scattering medium, generating a heavily-damped-spherical-photon-density wave. Measurements sample distances much less than one wavelength of the PDW. The quantities of interest for this wave are its amplitude (AC), average intensity (DC), and phase (with respect to the source). The characteristics of this wave are sensitive to the optical properties of the medium, µs′and µa.
Sampling this wave over a range of
distances permits accurate absolute measurements of µs′and µa. The limitations of this model are threefold. First, it is only valid in scattering-dominant media (µs′>> µa). Second, the model breaks down inside regions within a transport mfp of sources and boundaries. Finally, the modulation frequency must be below about 1 GHz when applied to media with optical properties similar to tissues.
69
IV THE FLUORESCENCE PHOTON DIFFUSION MODEL In the previous chapters, we assumed that photons migrate diffusely inside strongly multiple-scattering media such as tissues. The goal of this section is to incorporate the concepts of fluorescence spectroscopy into an optical-diffusion framework to model the generation and the propagation of emission photons traveling throughout a homogeneous multiple-scattering medium containing N uniformly-distributed fluorophores. We may safely use the standard diffusion equation to model both excitation and emission photons provided that we are attentive to the fact that the source distributions Q(r,t) will be very different for excitation and emission photons.
In addition, the absorption and the
scattering will differ at the excitation and the emission wavelengths. A graphical outline of the problem is provided by Figure (4.1). As an excitation photon enters the multiple-scattering medium, it undertakes a drunken random walk. As the scattering becomes increasingly vehement, the average photon time of flight will correspondingly increase. The circles in the figure represent a time clock for the photon, which ‘ticks’ as the photon travels. After absorbing an excitation photon, a fluorophore may emit an emission photon. The clock in the figure registers the average delay in time τ due to the fluorophore lifetime.
Finally, the new emission photon scatters as it
propagates which further augments the photon time of flight.
IV.A
NOTATION CONVENTIONS
In order to keep the record straight between excitation and emission photons, we require a consistent nomenclature for all of our many different optical coefficients. There are
70 three specific designations that we need to make in order to organize all of the categories of optical coefficients:
(1)
The subscript conventions of x and m denote the excitation and the emission As an example, the excitation wavelength is λx and the
wavelengths, respectively. emission wavelength is λm. (2)
The background chromophores are designated by the subscript b. In this thesis,
the term ‘background’ is employed to mean anything that is not fluorescent. The N independent fluorescent chromophores are designated by the subscript f. (3)
Finally, the index j is a species label for each of the N fluorescent species.
Table (4.1) provides a list of all the optical properties used in this formalism. Notice that we are assuming that the fluorophores themselves contribute negligible scattering. So that there is no misunderstanding, consider the total absorption of the medium at both excitation and emission wavelengths:
µ ax = µ abx +
N
∑
j =1
µafx j
µam = µabm +
N
∑µ j =1
afm j
.
(4.1)
µam represents the absorption of the medium as a whole at the emission wavelength, µafmj represents the absorption of only the fluorescent species j at λm, and µabm represents the absorption of the medium in the absence of any of the N fluorophores.
SOURCE EXCITATION LIGHT
DETECTED EMISSION LIGHT
r
TIME EXCITATION PHOTON
HOMOGENEOUS INFINITE MEDIUM FLUOROPHORE
τ
EMISSION PHOTON
FLUORESCENCE EVENT
Figure (4.1) - The propagation of excitation and emission photons inside a turbid medium containing a homogeneous distribution of fluorophores. When a fluorophore absorbs an excitation photon, it may decide to release this energy with an emission photon. The clocks in the figure represent the passage of time owing to multiple scattering and to fluorescence lifetime. Note that the scattering and absorption properties of the medium differ at the excitation and the emission wavelengths.
72
Table (4.1) – Optical Property Nomenclature
IV.B
COEFFICIENT
DEFINITION
µabx
absorption at λx by the background
µafxj
absorption at λx by the fluorophore species j
µabm
absorption at λm by the background
µafmj
absorption at λm by the fluorophore species j
µsx′
reduced scattering at λx
µsm′
reduced scattering at λm
FREQUENCY-DOMAIN DIFFUSION THEORY: EXCITATION PHOTONS
In similar fashion to the previous chapter, we will start with an intensity-modulated isotropic point source for the excitation source. In the spirit of the infinite medium, the excitation source is located inside a homogeneous multiple-scattering medium. This is a very simple problem to solve since we have already solved it in the previous chapter. The fact that the medium contains fluorophores is of no consequence here, except for the added absorption that these fluorophores present. All we need to do is write down the solution to the diffusion equation again; however this time we will adopt the notation conventions from the previous section to avoid any confusion:
U x ( r, ω ) =
Px ( ω ) exp[ − i φx 0 (ω )]1 exp[ − k x (ω ) r ] , 4 πvD x r
(4.2)
73 where the excitation photon-density-wave vector ikx is given by:
k x (ω ) ≡ 2
µ ax Dx
ω 1 − i v µ ax
.
(4.3)
Note that the source terms Px(ω) and φx0(ω) reflect the characteristics of the excitation source. The x subscript reminds us of this fact.
IV.C
FREQUENCY-DOMAIN DIFFUSION THEORY: EMISSION PHOTONS
This section proceeds along just as the previous section did. First we will determine the effective source distribution for the emission photons, and then we will solve the diffusion equation using this new source distribution.
IV.C.1 EFFECTIVE EMISSION SOURCE DISTRIBUTION Suppose that we distribute a fluorophore uniformly throughout a homogeneous multiplescattering medium.
With regards to this fluorophore, we shall make the following
assumptions:
(1)
The probability that the fluorophore will absorb emission light and subsequently
re-fluoresce (i.e., secondary fluorescence) is negligible. (Appendix D treats the case of secondary fluorescence). (2)
The fluorescence intensity decays at a single rate (i.e., a single exponential
decay). (3)
The photo-destruction of the fluorophore is negligible (i.e., photobleaching).
74 The emission source distribution for a given species of fluorophore (i.e., Qmj(r,ω,t)) must be proportional to the excitation photon density at r and at t (i.e., Ux(r,ω,t)), and it also must be proportional to the fluorophore quantum yield (Λj). In the linear regime, the probability per unit time for the absorption of an excitation photon is given by vµafxj. The strength of the emission of species j must therefore scale with (vµafxj)ΛjUx(r,ω,t). The fluorescence signal strength at λm also influences the emission spectrum of the fluorophore species. We will need to introduce the emission spectral efficiency as a probability density ϕ mj(λ). The probability density ϕ mj(λ) must be normalized to unity, since if the fluorophore radiates at all, it must radiate somewhere:
+∞
∫ϕ
mj
(λ) dλ= 1 .
(4.4)
0
The integral effectively takes place from λx to + ∞ in most cases since the fluorophore cannot emit more energy than the energy it has absorbed. This parameter ϕ mj(λ) is nothing more than the ideal emission spectrum of the fluorophore; i.e., without any contributions from the detection system. The lifetime τj of the fluorophore induces an average temporal delay in the fluorescence emission. This said delay manifests itself as a convolution between all possible decay times and the actual time at which the excitation photon is absorbed. The emission photon source conceived from absorption at λx that subsequently radiates into a wavelength range dλabout λtakes the form:
75 ∞
dQm j (r , ω, t ) = dλϕ m j (λ) ∫d (t − t ' ) Λj vµ afx j U x ( r, ω, t ' ) 0
t − t' 1 exp − , τ τj j
(4.5)
where t′is the time of absorption by the fluorophore. The fluorophore may emit light at any time after t-t′> 0. The dλ represents the fact that the fluorescence emission is not monochromatic. Although not written explicitly, the optical coefficients are likewise functions of the wavelength. We may separate the time dependence from Ux(r,ω,t′ ), and use the change of variables ∆t = t-t′to simplify the integral in Eq. (4.5):
dQm j ( r , ω , t ) = dλϕ m j (λ)
Λj vµ afx j τj
t − t' U x ( r, ω ) ∫d (t − t ' ) exp[ − iω t ' ]exp − τj 0 ∞
(4.6) = dλϕ m j (λ)
Λj vµ afx j τj
1 . U x ( r , ω ) exp[ − iω t ]∫d ( ∆t ) exp − ∆t i ω + τj 0 ∞
The integral presented in the final line of Eq. (4.6) may now be evaluated easily via a simple substitution. Since in the frequency domain the time dependence is given by exp(-iωt), the source distribution becomes dQmj(r,ω)exp(-iωt), where
dQm ( r, ω ) = dλϕ m j (λ) Λj vµ afx j U x ( r, ω ) j
1 + iωτ j 1 + (ωτ j ) 2
.
(4.7)
Equation (4.7) is the fluorescence analogue of Eq. (3.3) since it provides the effective source distribution for the generation of fluorescence light resulting from an isotropic point source excitation. This source distribution uncovers the origin of the emission signal. It is clear that the strongest emission will take place in the region of the strongest
76 excitation. However, as for which region supplies the emission that is most readily detected, this is a different matter.
IV.C.2 SOLUTION OF THE FLUORESCENCE-DIFFUSION EQUATION: SINGLE SPECIES In the case of the excitation, the source was localized to a single point; this is no longer true in the case of the emission. Fluorophores may emit wherever excitation photons may decide to travel. For this reason, every fluorophore has a finite probability to become a tiny emission source, each with the form of Eq. (4.7). We may evaluate the total emission photon density for a given fluorescent species through a spatial convolution of the fluorescence source distribution with the Green’s function solution of the diffusion equation (i.e., Eq. (4.2) with unit source strength and all emission optical coefficients):
dUmj(r,ω) =∫d 3 r ' U m j
GREEN ' S
(r' , ω ) dQm j (r − r' , ω )
V
exp[ 1 + iωτ j − r'⋅k m ( ω )] ′ − ω r r =∫d 3 r ' U ( , ) dλϕ m j (λ)Λj vµ afx j x 2 1 + (ωτ j ) V 4πvDm r' ∝ dλ∫d 3 r ' V
(4.8)
exp[ − r ′k m (ω )] exp[ − r − r ′k x ( ω )] . r′ r − r′
All we have done in the second line is substitute in the Green’s function and the emission source distributions in all their glory. In the last line of Eq. (4.8) all of the proportionality factors have been removed so that the integral looks a bit less formidable. We can evaluate the integral with the help of a sneaky trick, the geometry of which is outlined in
77 Figure (4.2). Using the substitution ψ ≡ r + (r ' ) 2 − 2rr ' cos(θ) , we may re-write the convolution integral using the spherical coordinate system:
2π
∞
π
exp[ − r ′ k m ( ω )] exp[ − ψ k x ( ω )] . dU m j ( r , ω ) ∝ dλ ∫dφ ∫dr ' r ' dθ sin( θ) ∫ r′ ψ 0 0 0 2
(4.9)
Please note that in the above equation, φ refers in standard fashion to the azimuthal angle and not the phase. Evaluation of Eq. (4.9) unveils an expression for the emission photon density inside the medium for the fluorescent species j:
dU m j (r , ω ) = dλΛj µ afx j ϕ m j (λ)
1 + iωτ j Px (ω ) exp[− i φx 0 (ω )] × 1 + (ωτ ) 2 4πv Dm Dx j − k x (ω ) r ]− exp[ − k m (ω ) r ] 1 exp[ 2 2 r k m (ω ) − k x (ω )
.
(4.10)
zˆ θ
r
r' yˆ
Figure (4.2) – Geometry for the calculation of the convolution integral.
φ
xˆ
IV.C.3 SOLUTION OF THE FLUORESCENCE-DIFFUSION EQUATION: N SPECIES Now that we have the solution for a single species, it is quite simple to generalize this result to an arbitrary number of species. If we assume that each of the N species do not interact with each other, then everything we have done simply adds together.
For
78 example, the total emission source distribution is now a linear sum of all the source distributions from the N independent fluorescent species. Notice that in Eq. (4.10), there are no species indices in the r dependent factor on the far right. The wave vectors km and kx depend upon the total absorption of the medium at their respective wavelengths, and are hence not specific to any individual fluorophore species (see Eqs. (4.1) and (4.3)). Therefore, we may safely add together the photon-density waves emerging from each fluorescent species as a linear sum:
N
dU m ( r , ω ) = ∑ dU m j ( r , ω )
(4.11)
j =1
IV.D
MEASURABLE PARAMETER
There has been some discussion in the literature as to what transport parameter is actually being measured in an experiment.
Researchers have contested that the measured
parameter is the photon current density,118 the boundary flux,109 and also the photon density. 111,119,120 The angular-photon density should be related to the quantity of interest, but in the diffusion approximation, the angular-photon density is essentially just the photon density (see Appendix B). Fantini et al. have performed extensive measurements using optical fibers to demonstrate that the quantity of interest is indeed proportional to the photon density.120 Since the ultimate goal of this effort is to perform experimental measurements, the final step in this derivation will be to evaluate the detected photon density. A detector has a spectral-intensity response denoted by γ (λ).
The detector also has a finite spectral
79 bandwidth that passes a range of wavelengths ∆λ centered about λ; within the context of this thesis the bandwidth is limited by either a monochromator or an optical filter. The measured photon density is therefore really:
λm +
∆λ 2
∫dU
Um (r, ω) =
m
(r, ω) γ (λ)
(4.12)
∆λ λm − 2
If we carry out this integration in Eq. (4.12), we will finally obtain the total detected emission photon density:
Um (r, ω) =
Px (ω) exp[ − iφx0 (ω)]1 exp[ − k x (ω) r]− exp [ − km (ω) r] × 2 2 4πvDx Dm r km (ω) − k x (ω) N
∑
j =1
1 + iω τj Λ µ Φ j afx m 2 j j 1 + (ω τ ) j
.
(4.13)
where the emission detection factor Φ mj is defined as:
λm +
Φ mj ≡
∆λ 2
∫dλ '
m
λm −
∆λ 2
ϕ m j (λ' m ) γ (λ' m ) .
(4.14)
The results of this formalism agree with those of others in the literature.83,85,90 Note that we have assumed that the optical coefficients are now averages over the spectral bandwidth of the detector. After some tedious algebra, Eq (4.13) can be more easily broken down into real and imaginary parts if we re-write it another way:
80
Um (r, ω) =
− iφx 0 (ω)] Px (ω) exp[ 1 × 2 r (β + ∆2 (ω)) 4πvDx Dm
exp − r (V+ x (ω) − iV+ x (ω)) µax − exp − r (V+ m (ω) − iV+ m (ω)) µam × Dx Dm 2
2
N N ∑ Λjµafxj Φ j B j (τ j , ω) + ∑ Λjµafx j Φ j Aj (τ j , ω) × j=1 j=1
(4.15) ,
N ∑ Λjµafxj Φ j B j (τ j , ω) j=1 exp i ArcTan N ∑ Λjµafx j Φ j Aj (τ j , ω) j=1
where the definitions of Eq. (3.8) along with the following definitions have been used:
∆(ω ) =
3ω ( µ sm ' − µ sx ' ) v
A j ( τ, ω ) ≡
IV.E
∆( ω ) + βωτ j 1 + (ωτ j ) 2
β = 3 µ am µ sm ' − 3 µ ax µ sx ' B j ( τ, ω ) ≡
β − ∆( ω )ωτ j 1 + (ωτ j ) 2
(4.16) .
ANALYSIS OF THE EMISSION PHOTON DENSITY
Equation (4.13) segments nicely into three natural factors: a spatial factor that contains the r dependence, a lifetime factor that contains all appearances of τ, and a yield & source factor that contains Λ as well as the source terms. The emission photon density is proportional to µafx only when µafx > µa, which
219 is a more strict requirement than the µs >> µa condition enforced by the P1 approximation.
B.B.3 THE DIFFUSION EQUATION Having obtained a relationship between U(r,t) and J(r,t), we may finally insert Eq. (B.22) into Eq. (B.5) in order to reveal the standard diffusion equation:102,109,112,165-167
∂U (r, t ) − vD∇ 2U (r, t ) + vµ aU (r, t ) = Q (r, t ) . ∂t
(B.23)
Thus, photon scattering in tissues within the near-IR wavelength region may be thought of as undergoing a random walk (i.e., a drunken photon) within the tissue. The diffusion equation relates the photon density to the optical coefficients of the medium, µa and µs′ (through D). One final note is that since in the diffusion approximation µs′>> µa, then we may safely write D ≈(3µs′ )-1 (Ref. 175).
220
APPENDIX C
MATHEMATICAL BASIS FOR THE DIFFUSION APPROXIMATION
In this appendix I would like to show how the approximations listed in Chapter III come about, mathematically speaking.
C.A
THE DIFFUSION APPROXIMATION
C.A.1 THE FIRST CONDITION: 3ωDV-1 > 3µaD will the wave vectors of Eqs. (C.8) and (3.4) match
identically. Thus, another constraint on the diffusion approximation is that we must have:
1 > > 3µ a D =
µa µ s ' + µa
(C.9)
Now, this expression is hauntingly familiar to the ‘high albedo’ constraint that we discussed in the P1 approximation section just moments ago. There is one subtle, yet
223 important, difference: the condition now reads that µs′>> µa rather than µs >> µa. The diffusion approximation therefore requires many more scattering events in order to keep the time between collisions far lower than the modulation time (recall that µs′is typically an order of magnitude greater than µs in biological tissues). Furutsu and Yamada have observed that since µs′>> µa, it is consistent to rewrite the diffusion coefficient as -1 vD≈v(3µ′ s) (Ref. 175). We may always check that the diffusion approximation is valid
by checking the quasi-isotropic condition of Eq. (C.4):
v U ( r, ω , t ) U (r, ω, t ) = >> 1 . 3 J ( r , ω, t ) ⋅Ωˆ 3 vD ∇ U ( r, ω, t )
(C.10)
This value is typically about 10 for tissue-like media.
C.B
RANGES OF APPLICABILITY
Figure (C.1) presents a crude map of the ranges of applicability of the Boltzmann Transport Equation (BTE), the P1 Approximation, and the standard diffusion equation (SDE).122 In Figure (C.1a) I have plotted the following conditions: µs′> 10 µa as the minimum condition for the SDE, and µs = µs′ (1-g)-1 > 10 µa as the minimum condition for the P1 approximation (assuming g = 0.9). The criteria of the symbol ">>" meaning a factor of ten is purely arbitrary.
Figure (C.1a) demonstrates the limited range of
applicability of the SDE, but thankfully, this range covers the values for biological tissues exposed to near-IR light (the gray-shaded area).
Figure (C.1b) explores the
-1 approximation of rµ′ s = rµs(1-g) > 10 as the minimum condition for the P1
approximation, and 3ωD/v λ′
(λ)
λ′
ϕ′ m ( λ) = 0
D.D
(D.4) λ ≤λ′
THE SECONDARY-EMISSION PHOTON DENSITY
As in the case of the primary-emission photon density, the secondary-emission photon density may originate from many points within the medium. Mathematically, we must convolve a photon-density wave at the emission wavelength λm with the secondaryemission source term:
d 2U ′ ( r , ω ) = ∫d 3 r ' U mGREEN ' S (r' , ω ) d 2Q ′ (r − r' , ω ) x′ x′
(D.5)
V
Using the mathematical results of Chapter IV, we can easily evaluate this integral. For general scalars A and B,
231
∫d
V
3
r'
exp[ − r ′A] exp[ − r − r ′B ] 4π exp[ − Br ] − exp[ − Ar ] = r′ r − r′ r A2 − B 2
(D.6)
This mathematical result allows us to find the secondary-emission photon density in the medium:
1 + iωτ (r , ω ) = dλdλ′ d 2U ′ x′ 2 1 + (ωτ)
Λ2 Px (ω ) exp[ − iφx 0 (ω )] µ afx µ afx′ ϕ m (λ′ ) ϕ′ m (λ) × 2 4 π D x Dm r D x′ k x′(ω ) − k x2 (ω ) 2
exp[ − k x (ω ) r ]− exp[ − k m (ω ) r ] exp[ − k x′(ω ) r ]− exp[ − k m (ω ) r ] − 2 2 2 2 k m (ω ) − k x (ω ) k m (ω ) − k x′(ω )
(D.7)
Since there is usually a range of wavelengths where the secondary-emission process may thrive, we must add up all of the contributions from these wavelengths. Assuming that the first line in (D.1) is the valid case, then the total secondary-emission contribution now becomes:
λ2
dU m′( r , ω ) = ∫d 2U m′( r , ω ) ,
(D.8)
λ1
where the integral is over the wavelength λ′ . All we are doing is summing up all of the emission photon-density waves resulting from re-absorption at each possible reabsorption wavelength. The result of this sum is
232
λ
1 + iωτ Λ2 Px (ω) exp[− iφx 0 (ω )] µ afx 2 µ afx′ ϕ m (λ′ )ϕ′ m (λ) ′ × ∫dλ′ dU m ( r, ω ) = dλ × 2 2 4 π D x Dm r λ1 Dx′ k x′(ω ) − k x2 (ω ) 1 + (ωτ) 2
(D.9) exp[ − k x ( ω) r ]− exp[ − k m (ω) r ] exp[ − k x′(ω) r ]− exp[ − k m (ω ) r ] − 2 2 2 2 k m (ω ) − k x (ω ) k m ( ω) − k x′(ω )
If we want to determine the measured secondary-emission-photon density, we must integrate dU′ m(r,ω) over the bandpass of the detector, just as we did in Chapter IV. The result of this integration yields the final form for the secondary-emission photon density (for a single species):
λ ′ 1 + iωτ Px (ω) exp[− i φx 0 (ω)] µafx 2 µ afx′ Φ ′ m (λ, λ) × U m′(r, ω) = Λ dλ′ 2 2 1 + (ωτ) 2 ∫ 4π Dx Dm r λ1 Dx′ k x′(ω) − k x (ω) 2
2
(D.10) exp[ − k x (ω ) r ]− exp[ − k m (ω ) r ] exp[ − k x′(ω) r ]− exp[ − k m (ω) r ] − 2 2 2 2 k m (ω ) − k x ( ω ) k m (ω ) − k x′(ω)
where the probability Φ ′ ) is defined as: m(λ,λ′
λm +
Φ′ )= m ( λ, λ′
∆λ 2
∫dλγ(λ) ϕ
m
(λ′ ) ϕ′ m ( λ) .
∆λ λm − 2
D.E
SECONDARY EMISSION WITH MULTIPLE FLUOROPHORES
If we start to add more fluorophores, we will need more terms to describe the secondaryemission effect.
Equation (D.10) accounts for only intra-species secondary absorption;
that is, re-absorption by species j and subsequent re-emission by the same species j.
(D.11)
233 However, there is also the case where the primary-emission-photon density of one species acts as the source for the secondary emission of another species. Thus, Λ, Φ ′ m, τ, µafx, and µafx′all depend upon the order of the absorption and the subsequent re-emission. As before, the photon-density-wave vectors are not species-dependent since they depend upon the total absorption of the medium. First there must be a Λ, τ, µafx, and ϕ m(λ′ ) for the driving species (that is, the source of the emission in Eq. (D.3)), which we will denote by the subscript α. The (secondary) emission species must also have its own Λ, τ, µafx′, and ϕ′ (λ), which we x′ keep track of with the subscript β. If we properly delegate where each of these variables goes, labeled by the appropriate species index, we will obtain the following result:
U m′( r , ω ) =
Px (ω ) exp[ − i φx 0 (ω )] 1 × r 4 π D x Dm
− k x ( ω ) r ]− exp [ k m2 (ω ) − 1 1 ∫dλ′k x2′(ω ) − k x2 (ω ) D x′exp[− k x ′( ω ) r ]− λ1 k m2 (ω ) −
λ2
− km (ω) r ] exp [ − 2 k x (ω ) × − k m (ω ) r ] exp [ k x2′(ω )
(D.12)
1 + iωτ α N 1 + iωτ β Λα µ afx α ) αβ ( λ, λ′ 2 ∑ 1 + (ωτ ) 2 Λβµ afx′βΦ ′ (ωτ α ) α =1 β=1 β N
∑ 1 +
where the probability Φ ′ ) is defined as αβ(λ,λ′
λm +
Φ′ αβ ( λ) =
∆λ 2
∫dλγ(λ) ϕ
∆λ λm − 2
mα
(λ′ ) ϕ′ mβ (λ) .
Note that for the case of N=1, Eq. (D.12) reduces to Eq. (D.10).
(D.13)
234
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[129] E. Gratton and M. Limkeman, “A Continuously Variable Frequency CrossCorrelation Phase Fluorometer with Picosecond Resolution,” Biophys. J. 44, 315324 (1983). [130] J. Wilson and J. F. B. Hawkes, Optoelectronics: An Introduction , 278-331 (Prentice-Hall, London, 1983). [131] R. Boyd, Radiometry and the Detection of Optical Radiation , 106-157 (John Wiley & Sons, New York, 1983). [132] B. A. Feddersen, "Digital Frequency Domain Fluorometry and the Study of Hoechst 33258 Dye-DNA Interactions," Ph. D Thesis, University of Illinois at Urbana-Champaign, 28-35 (1993). [133] Noting, of course, that the fluorophore absorption coefficients depend upon [F]. [134] F. López Arbeloa, T. López Arbeloa, M. J. Tapia Estévez, and I. López Arbeloa “Photophysics of Rhodamines: Molecular Structure and Solvent Effects,” J. Phys. Chem. 95, 2203-2208 (1991). [135] R. A. Velapoldi and K. D. Mielenz, A Fluorescence Standard Reference Material: Quinine Sulfate Dihydrate, National Bureau of Standards Special Publication 260264, U. S. Government Printing Office, Washington (1980). [136] I used the commerical program Origin for the global fits in this thesis (Microcal, Northampton, MA). [137] Ref.134 did not provide errors for the quantum yield. [138] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, 106-107 (John Wiley & Sons, New York, 1983). [139] J. N. Demas and G. A. Crosby, “The Measurement of Photoluminescence Quantum Yields: A Review,” J. Phys. Chem. 75, 991-1024 (1971). [140] The best of the absorption spectrometers are reliable in the 10-6 OD range on a good day. [141] I determined the slope of this line from the data presented in the thesis of Dr. Joshua Fishkin (Ref. 107). The recovery of the slope from the figure resulted in the seemingly large error in the emission reduced-scattering coefficient. [142] Molecular Probes provided the spectral data parameters for the sample lot number, but did not provide measurement errors for these parameters.
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[157] Spectroscopic data was provided by the manufacturer (Molecular Probes; Eugene, Oregon). [158] I measured the lifetime free ethidium bromide, and used the ratio of 12.5 provided by Ref. 156. [159] L. Stryer, Biochemistry, 4th Edition, 87 (W. H. Freeman, New York, 1995). [160] The Center for Biological Sequence Analysis at the Technical University of Denmark maintains a website that lists the genome sizes of a variety of organisms. This useful site is called DOGS (Database of Genome Sizes) and it has the address http://www.cbs.dtu.dk/databases . [161] M. J. Waring, "Complex Formation between Ethidium Bromide and Nucleic Acids," J. Mol. Biol. 13, 269-282 (1965). [162] P. T. C. So, T. French, W. M. Yu, K. M. Berland, C. Y. Dong, and E. Gratton, “Time-Resolved Fluorescence Microscopy Using Two-Photon Excitation,” Bioimaging 3, 49-63 (1995). [163] L. M. Angerer and E. N. Moudrianakis, "Interaction of Ethidium Bromide with Whole and Selectively Deproteinized Deoxynucleoproteins from Calf Thymus," J. Mol. Biol. 63(3), 505-521 (1972). [164] Raman scattering, though weak, constitutes a larger signal than you might think in the turbid medium. See VI.D.2 as an example. [165] K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading Mass., 1967). [166] A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978). [167] W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, “Light Dosimetry in Optical Phantoms and in Tissues: I. Multiple Flux and Transport Theory,” Phys. Med. Biol. 33, 437-454 (1988). [168] L. G. Henyey and J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70-83 (1941). [169] H. C. van de Hulst, Multiple Light Scattering, Vol 2 (Academic, New York, 1980). [170] J. H. Joseph, W. J. Wiscombe, and J. H. Weinman, “The Delta-Eddington Approximation for Radiative Flux Transfer,” J. Atmos. Sci. 33, 2452-2459 (1976).
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[171] L. Reynolds, C. Johnson, and A. Ishimaru, “Diffuse Reflectance from a Finite Blood Medium: Applications to the Modeling of Fiber Optic Catheters,” Appl. Opt. 15, 2059-2067 (1976). [172] J. J. Duderstadt and W. R. Martin, Transport Theory , 225-235 (Wiley, New York, 1980). [173] D. A. Boas, H. Lui, M. A O’Leary, B. Chance, and A. G. Yodh, “Photon Migration Within the P3 Approximation,” SPIE 2389, 240-247 (1995). [174] G. Yoon, S. A. Prahl, and A. J. Welch, “Accuracies of the Diffusion Approximation and Its Similarity Relations for Laser Irradiated Media,” Appl. Opt. 28, 2250-2255 (1989). [175] K. Furutsu and Y. Yamada, “Diffusion Approximation for a Dissipative Random Medium and the Applications,” Phys. Rev. E 50, 3634-3640 (1994). [176] This treatment excludes the cases of multi-photon excitation.
