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IEEE JOURNAL ON SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 7, NO. 6, NOVEMBER/DECEMBER 2001
Thermal Analysis of Blood Undergoing
Laser Photocoagulation
Jennifer Kehlet Barton, Member, IEEE, Dan P. Popok, and John F. Black, Member, IEEE
Abstract—The temperature of blood undergoing laser-induced
photocoagulation during long-pulse (10 ms) 532 nm irradiation
was measured in a time- and spatially-resolved manner using a
novel technique. This method is based on the change in reflectivity
of a solid–liquid interface given a dynamically changing refractive
index in the liquid phase. In our case, the temperature-dependent
change in the refractive index of blood was utilized, and the reflectivity at a glass–blood interface was measured. Measurements
were compared to predictions from a finite-element model incorporating the effects of time-dependent changes in the absorption
coefficients of the blood, and phase changes representing coagulation and the liquid/vapor transition. Previous studies have linked
the onset of blood coagulation to a sharp rise in the 532-nm reflectance of the blood. Based on the thermal measurements and the
results of an Arrhenius analysis, we postulate that the reflectance
rise is a combination of protein denaturation and red blood cell
conformal changes.
Index Terms—Biomedical applications of radiation, blood,
lasers, modeling, photocoagulation, reflectometry.
I. INTRODUCTION
L
ASER-INDUCED photocoagulation of blood and other
biological tissues is a widely used technique in ophthalmology, surgery and dermatology. Applications range from
treatment of diabetic retinopathy to tumor thermotherapy to port
wine stains and other cutaneous vascular disorders. Despite the
widespread applicability and use of this technique, definitive
clinical endpoints to the procedure are often lacking, leading to
subjective treatment parameter spaces and an intrinsic barrier
to communication of good technique. Optical–thermal models
that accurately relate measurable parameters, such as surface
temperature or laser light remittance, to damage at the site of
interest could assist in the determination of quantifiable end
points. They may also allow tailoring of the photocoagulation
parameters to produce good clinical results with minimal
collateral damage.
Optical–thermal models have become increasingly complex in an effort to model complex tissue geometries with
time-varying optical and thermal properties. Typically, these
models employ Monte Carlo modeling of light propagation
[1], finite-difference [2] or finite-element [3] solutions of the
Fourier heat conduction equation, and the Arrhenius equation
Manuscript received May 21, 2001; revised October 29, 2001.
J. K. Barton is with the University of Arizona, Tucson, AZ 85721-0104 USA
(e-mail: barton@u.arizona.edu).
D. P. Popok is with Network Analysis, Inc., Tempe, AZ 85284 USA (e-mail:
dan@sinda.com).
J. F. Black is with Lightwave Electronics, Mountain View, CA 94043 USA
(e-mail: jblack@lwecorp.com).
Publisher Item Identifier S 1077-260X(01)11188-3.
for thermal damage [4]. Models have been developed that are
capable of accommodating any arbitrary tissue geometry [5] or
incorporate temperature-dependent optical properties [6].
An obstacle to the creation of truly useful models is a
lack of understanding about basic processes involved in light
propagation, heating, and tissue response to damage. Several
studies have been performed to examine changes that occur
during the heating of blood. Haldorsson [7] postulated that
optical and thermal property changes occur in blood irradiated
by an Nd : YAG laser, due to hemoglobin deoxygenation
and denaturation of blood constituents. Verkruysse et al. [8]
concluded that absorption increases at the beginning of a 586
nm, 0.5 ms laser pulse, then scattering increases due to blood
coagulation. Nilsson et al. [9] noted an abrupt increase then
decrease in the reduced scattering coefficient of slowly heated
blood at approximately 46 C and 49 C, respectively, due to
conformal changes of the red blood cells (RBCs). In previous
studies [10] the authors have concluded that, during 532 nm,
10 ms pulse duration laser irradiation, optical reflectance and
transmission of blood exhibit complex, wavelength-dependent
behavior. Three effects were noted: a time/temperature-dependent bathochromic shift of the oxy-hemoglobin chromophore
(a shift of the spectrum to longer wavelengths), two distinct
time- and length-scale coagulation events, and the creation of a
new chemical species, met-hemoglobin.
In this paper, we examine the temperature of laser-irradiated
blood in cuvettes. A novel, spatially and temporally resolved
method of measuring the temperature at the blood–glass interface is described and implemented. Temperatures are compared
to predictions from a thermal model that incorporates a temperature-dependent absorption coefficient as well as a unique
method of incorporating the latent heat of vaporization. The Arrhenius equation is then used to predict the time and temperature of the onset of the RBC damage noted in Nilsson [9]. This
experimental and theoretical analysis is then used to refine our
understanding of the nature of laser coagulation of blood.
II. METHODS
A. Experimental
We wish to make temperature measurements in a manner
which is
1) nonperturbative to the sample;
2) spatially selective in the axis (where is the propagation
direction of the pump laser);
3) time-resolved;
4) “real-time” in nature in that the results can be readily related to a temperature.
1077–260X/01$10.00 © 2001 IEEE
BARTON, et al.: THERMAL ANALYSIS OF BLOOD UNDERGOING LASER PHOTOCOAGULATION
Fig. 1. Total internal reflection thermal probe experimental schematic.
Thermocouples and high-speed IR cameras do not simultaneously meet all these requirements [11], [12]. We have chosen
instead to make time-domain measurements of the thermal
properties of the system using a technique similar to that of
Martinelli et al. [13] based on the change in reflectivity of a
solid–liquid interface given a dynamically changing refractive
index in the liquid phase. For -polarized light, the reflection
coefficient at the solid–liquid interface is given by
(1)
where is the angle of incidence, is the refractive index of the
is the refractive index of the liquid. The
solid substrate, and
is the square of this quantity. The
observable reflectivity
configuration used in this experiment is shown in Fig. 1. The
angle of incidence is 45 at the cuvette outer face and therefore
27.8 at the inner wall/blood interface given a cuvette refractive index of 1.516. This angle is less sensitive in terms of its
absolute response to changes in , but is also less sensitive to
systematic errors such as the intrinsic divergence of the laser
beam or in measuring the angle of incidence at the sample.
The sample holder is formed using a standard colorimeter cuvette (96-G-2.5, Starna Cells, Atascadero, CA), with a 2.5-mm
path length. As shown, the 532 nm pump beam may interact
with the sample at normal incidence from the top surface of the
cuvette while allowing the probe beam to interact with the interface at various angles of incidence. The thick (3-mm) walls
of the cuvette cause the first surface Fresnel reflection and the
internal interface reflection to be separated by several mm, allowing isolation of the relevant reflection using an iris. It is important that the first-surface cuvette reflection be completely
eliminated from view in the detector, as this reflection would
constitute a constant, more intense background on which the
desired signal is homodyned. Since the method requires an absolute measurement of the change in to be made, this background would decrease the sensitivity and lead also to an apparent decrease in the temperature measured at the interface. It is
important that the pump beam be parallel to the sample surface
normal to avoid asymmetric heating of the sample. A He-Ne
937
laser (05-LHP-151, Melles-Griot, Carlsbad, CA) impinges on
the cuvette at 45 0.5 (external) in -polarization with a 1
mm beam diameter. The reflected beam propagates for approximately 1 m and then impinges on a detector (Det 110, Thorlabs
Inc., Newton, NJ) placed behind an iris and a 632.8 nm interference filter. The signal from the diode is fed to an oscilloscope
(TDS210, Tektronix Inc., Beaverton, OR) and recorded contemporaneously with the 532-nm remittance from the sample.
Blood samples were obtained from two apparently healthy
Caucasian volunteers. The samples were kept refrigerated and
used within 72 h after extraction. The starting refractive index
of whole (centrifuged) blood plasma was measured using a refractometer (Atago R5000, Cole-Parmer Instrument Company,
Chicago, IL) and found to be 1.3500 0.0005 at 20 C room
temperature. The refractive index of isotonic saline was measured to be 1.3345 0.0005, consistent with book values [14].
The experiment was performed using 2 and 3 dilutions of
whole blood. Since the starting composition of blood plasma
is 91% water [12], and our dilution only increases this percentage, we have assumed that the functional form of
is the same as that for pure water [15]. The change in the refractive index of the cuvette substrate as a function of temperature is a factor of ten smaller than that for water and we have
neglected it in the analysis to follow [13]. By recording the
time-domain reflectivity at the interface, we therefore monitor
the time-domain refractive index of the sample. This can then be
inverted in a straightforward manner to give the temperature at
the sample/glass interface at any given moment. We measure the
0 , this
temperature in the sample with spatial selectivity
measurement then becoming the boundary condition for subsequent finite-element modeling on the sample. The only free
parameter in the inversion is the functional form of the refractive index of water versus temperature. This technique for temperature measurement has the advantage of being nonperturbative (unlike the insertion of a thermocouple), insensitive to the
presence of a large pump laser pulse (unlike a thermocouple),
spatially selective (unlike an IR camera or thermocouple) and
real-time (unlike an IR camera).
A commercial long-pulse green laser (VersaPulse Cosmetic
Laser, Coherent Medical Group, Santa Clara, CA) was used
to provide the 532-nm laser pulses. At a 3-mm-diameter spot,
the laser could operate at radiant exposures up to 16 J cm
with a 10-ms pulsewidth. Laser pulse energies were monitored using a power meter (PM30V1 volume-absorbing head,
EM5200 readout; Molectron Detector Inc., Portland, OR). The
long-pulse 532 nm laser was transmitted to the experiment
using a fiber-optic cable and the output recollimated using
a standard handpiece supplied with the laser. Incident pulse
temporal waveforms and energies were monitored using the
on-board photodiodes on the laser.
B. Thermal Modeling
A finite-element thermal model was developed iteratively in
parallel with experimental efforts. The model considers a simple
geometry, assuming a laser beam diameter of 3 mm, variable
(homogenous) sample thickness, and variable-thickness container walls. The container is assumed much wider than the
beam diameter, allowing a two-dimensional (2-D) axisymmetric
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IEEE JOURNAL ON SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 7, NO. 6, NOVEMBER/DECEMBER 2001
TABLE I
THERMAL PROPERTIES USED IN THE ANALYSIS
Fig. 2. Geometry of the thermal model. The cell wall and sample thicknesses
are exemplary.
modeling approach. Fig. 2 shows the assumed geometry in relation to the body of revolution represented by the model. The
-the time-varying temperature distrimodel computes
bution in the heated blood sample and container walls. A finite-element thermal model was created for the region of interest, using the SINDA/ATM thermal analysis system (Network Analysis, Inc., Tempe, AZ) [16]. SINDA/ATM consists
of a graphics-based finite-element preprocessor (FEMAP) combined with the SINDA/G advanced thermal analysis solver. This
modular approach has the advantage that embedded FORTRAN
routines can be easily added to SINDA/G models. This in turn
can be used to add any desired thermal effect, to produce custom
output, or to interact with the user. This analysis described here
made extensive use of this advanced capability.
The finite-element mesh for the model was constructed
following the sample interaction region dimensions described
above and given in Fig. 2. The mesh is locally refined within
the beam, near the interface between the blood sample and the
illuminated side of the container. Maximum sample heating occurs at the entrance face in the interaction region, although this
does not necessarily yield the maximum system temperature
due to proximity to the container wall acting as a heat sink.
Orthotropic material models were used to represent the sample
and the container walls. These materials allow different thermal
conductivity values to be specified for the and directions.
The laser energy deposition profile in the sample was modeled using a standard exponential decay (Beer–Lambert Law)
(2)
energy transmitted per unit area,
absorpwhere
incident energy per
tion coefficient for the blood sample,
distance in the sample from
unit area due to the laser, and
the cuvette wall interface. Monte-Carlo simulations of 532-nm
propagation in blood confirm that the Beer–Lambert form is an
excellent approximation to the photon flux in the sample. The
Monte Carlo program has been described previously [5]. The
optical properties of blood were assumed as follows: absorption
coefficient 266 cm , scattering coefficient 47.3 cm , refractive index 1.34, and anisotropy 0.995 [17]. A simple energy
balance on a differential element
shows
, the heat absorbed by the sample per unit volume, is given by the following
equation:
(3)
FEMAP was then used to create a candidate spatial distribution for these heating terms. Though not used directly in the
analysis, the distribution provided a scaling function generically
describing the variation as a function of depth, which was embedded in a more complete calculation. The program allows the
input of an absorption coefficient (in mm ) and a “degradation” factor to account for the bathochromic shift, which we
have identified previously [10]. The change in absorption with
respect to temperature is modeled as a linear function, based
on an extrapolation of data provided in the literature [18]–[20].
Sample dilution is accounted for by scaling the absorption coefficient.
The model also allows the thickness of the sample and container walls to be specified as parameters. The orthotropic material properties mentioned earlier make this possible, and embedded routines in the SINDA/G model compute the resulting
modifications to the mesh. Suppose a wall thickness of 1.6 mm,
not 3 mm, is to be considered. In this example, the modification
would consist of applying multiplying factors to all heat capacitance and thermal conductivity terms representing the container
wall. For heat capacitance and thermal conduction in the direction, the factor would be (1.6/3). For conduction in the
direction, the factor would be (3/1.6). Table I lists the thermal
properties for the various sample container materials considered
in the analysis.
The standard thermo-physical properties of water were assumed to apply for the blood sample, including constant density
and temperature-varying thermal conductivity [15]. It is known
from the work of Pfefer et al. [12] that under rapid laser-heating,
blood can exhibit signs of microvaporization at temperatures
lower than the boiling point of bulk water. We have chosen
to simulate this phase transition using a temperature-dependent
specific heat, as shown in Fig. 3.
The function superimposes two simpler functions: 1) the constant specific heat of water and 2) a “pulse” that simulates phase
change. If integrated as a function of temperature, the area added
by the pulse equals the latent heat of vaporization. This phase
change simulation technique is known as the “enthalpy method”
[21].
BARTON, et al.: THERMAL ANALYSIS OF BLOOD UNDERGOING LASER PHOTOCOAGULATION
939
C. Arrhenius Damage Modeling
The Arrhenius equation quantifies damage as a single parameter , which is the logarithm of the ratio of the original concentration of native state tissue to the remaining native state tissue
(6)
Fig. 3. Enthalpy method to model phase change.
The practical effects of this “gradual” onset of the transition
between liquid water and steam are twofold. The first effect is to
arrest the progress of the system to physically unrealistic temperatures seen for example in models that do not take the latent heat of vaporization into account. The second effect is to
“smooth” the transition from the abrupt step function (Heaviside function) intrinsic to a pure sample. In Fig. 3, the shape of
the curve is empirical, typically assuming a fairly broad temperature band. Classically with the enthalpy method, a narrow
band would be chosen for a pure substance. While this approach
is necessarily approximate, it is to the best of our knowledge
the first attempt to adjust for the transition between coagulating
tissue and boiling water in a simple but realistic manner. The
model allows the user to specify the temperature band and heat
of vaporization, thereby fully determining the curve in Fig. 3.
Finally, the interface dialog box also provides parametric
specifications describing the laser pulse. The overall fluence is
specified as well as the duration of the pulse. An arbitrary laser
pulse shape could also be incorporated into the model. The laser
used in the experiments described here is well approximated by
a rectangular shape. More complicated functional forms (e.g.,
Gaussian) could be used depending on the laser and target. At
its most complicated, the heating distribution in the sample is
a function of distance, time and temperature. Reflecting these
additional complications, (3) is expressed as follows:
(4)
For convenience, is handled nondimensionally, allowing the
candidate spatial distribution mentioned earlier to be re-scaled
to any sample thickness, and not limited to just the baseline case.
, where is the thickness of
For nondimensional , let
the blood sample
(5)
The standard parameters used to model the experimental results
are given in Table I (BK-7 values) with a water latent heat of
vaporization of 2257 kJ kg .
, is the total heating time,
where is a frequency factor
an activation energy J mole , the universal gas constant
mole
K ], and the absolute temperature
.
[8.32
Given the frequency factor and activation energy for a particular
damage process, the Arrhenius equation allows a damage estimate to be made for any tissue temperature history. The onset
1.
of damage is often chosen to be at a value of
In this study, we are interested in two damage processes noted
by Nilsson et al. [9] under slow linear heating of blood: the
transformation of erythrocytes to spherocytes, and cell fragmentation. From their data, it is seen that the first process occurred
at approximately 1800 s and a temperature of 46 C, and the
second at approximately 2100 s and a temperature of 49 C. In
the present study, it would be expected that the damage temperatures would be higher, since the heating occurs over a much
shorter time period. In order to estimate these temperatures, the
frequency factor A was assumed to be the same as a published
[22]. Then assuming the
value for hemoglobin 7.6 10
Nilsson temperature history, values for the activation energy that
1 at the correct time/temperature for erythrocyte
produced
shape change and cell fragmentation were determined by fitting
the data.
Given these activation energies, the time and temperature pro1 for each damage condition could be determined
ducing
for this study. The temperature history predicted from thermal
modeling at a depth of 50 m into the blood sample, along with
the same assumed frequency factor and the calculated activation energies, were input into the Arrhenius equation. Then the
damage parameter was calculated for each timestep.
III. RESULTS
A. Comparison of Experimental Data and Modeling
Fig. 4 shows a plot of the time-domain interface temperature
24.5
being illumiprofile for a 2 dilution of blood hct
nated by a 10 J cm , 10 ms laser pulse. The experimental data
is the mean of six shots at the same parameters. The error bars
are 1-sigma from these six shots. Also shown is the output of
the 532-nm reflectance photodiode. Superimposed on the experimental temperature profile is a representative prediction of the
interface and peak sample temperatures from the finite-element
model. The only parameters adjusted in performing this simulation are 1) the bathochromic-shift-dependent change in the
absorption coefficient for blood at 532 nm, and 2) the onset temperature for the phase change between liquid blood, coagulating
blood and steam. The specific parameters used in this simulation are given in Table II. The parameters represent a 2 dilution
of blood and a phase change onset of 95 C. The peak sample
temperature occurs approximately 30 m into the sample from
0.03 mm .
the “front” surface
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IEEE JOURNAL ON SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 7, NO. 6, NOVEMBER/DECEMBER 2001
Fig. 4. Measured and predicted (peak and surface) temperatures in laser-irradiated blood. The 532-nm reflectance signal is given for comparison. FEM:
finite-element model.
TABLE II
SPECIFIC PARAMETERS USED FOR THE SIMULATION SHOWN IN FIG. 4
Fig. 5. Predicted temperatures at various depths in laser-irradiated blood.
Outlined data points are times when damage to the erythrocytes is expected to
occur.
B. Arrhenius Results
Activation energies calculated from Nilsson data [9] are
422 500 and 427 000 J mole for the shape change of the
erythrocytes and the cell membrane disintegration, respectively.
When these values are used in the Arrhenius equation along
with the current study’s temperature history, it is predicted that
the shape change will occur at approximately 76 C and 3 ms
into the 10-ms laser pulse, and that cell fragmentation occurs
shortly thereafter at approximately 80 C and 3.5 ms after the
start of the laser pulse. It should be kept in mind that these
results are estimates at a specific depth (50 m) into the blood;
damage temperatures and times at other depths will vary. To
illustrate this fact, Fig. 5 shows the predicted temperature as
a function of time for various depths in the sample. The time
at which the erythrocytes are predicted to change shape is
outlined for each depth. At depths below approximately 150
m, there is insufficient heating to cause damage. The effect
of heat diffusion can be seen in the decrease in temperature
after the end of the laser pulse at shallow depths, and continued
gradual increase in temperature at greater depths.
IV. DISCUSSION
A. Comparison of Measured and Predicted Temperatures
We have identified previously three distinct temporal phases
involved in the long-pulse laser photocoagulation of blood [10].
BARTON, et al.: THERMAL ANALYSIS OF BLOOD UNDERGOING LASER PHOTOCOAGULATION
We have categorized these as
1) heating phase;
2) primary coagulation phase;
3) secondary coagulation phase;
shown schematically in Fig. 6, where the shaded area is the reflectance of the 532-nm laser light from the blood, and the curve
is the transmission of a 1064-nm probe laser.
The data presented here show that, at least up to the onset
of the primary coagulation phase, and perhaps beyond this, the
thermal dynamics of the system are quite reasonably modeled
assuming exponential decay of the pump pulse in the sample
and incorporating the effects of the water liquid/steam phase
transition into the model. The results become particularly noteworthy when it is remembered that there are no free parameters
(and only one assumption) in converting the experimental total
internal reflection (TIR) data to a temperature, and only two
free parameters in performing the simulations. Despite some
discrepancies which will be discussed subsequently, the overall
agreement between experiment and model is encouraging.
The model accurately confirms a naïve prediction, based on
Beer–Lambert law absorption and the heat subsequently deposited in the sample, that some parts of the system must reach
100 C after only a short time in the laser pulse. The fact that we
do not observe the visual and acoustic signatures of vaporization
(bubbles, “popping” noises) unless we increase the radiant exposure to beyond a critical threshold underscores the importance
of incorporating the latent heat of vaporization into the model.
The model correctly predicts temperatures >100 C at the end of
the laser pulse at supra-threshold radiant exposures, indicative
of the transition of some region of the sample to steam. Under
these conditions, experimental samples also exhibit evidence of
microvaporization.
B. Arrhenius Analysis and Comparison to Earlier
Observations
We can attempt to relate the observed/predicted system temperatures to some of the features in the optical/spectroscopic
data set shown in Fig. 6 and described previously [10], [23].
The primary coagulation phase is marked by the rapid rise in
remittance from the sample, which we have hypothesized previously is due to denaturation of proteins at a molecular level. At
the time our original analysis was undertaken, the system temperatures at which the changes in the optical properties of the
sample became apparent seemed to be too high to allow other
explanations for the rise in scattering, such as the phase transition in the RBCs from biconcave disc to spheroid identified by
Nilsson et al. [9], and the subsequent melting of the RBC membranes. Arguments against these explanations also included the
observations that the backscattering peak was present in
a) whole blood diluted with isotonic saline;
b) plasma-free suspensions of RBCs;
c) macerated suspensions of plasma-free RBCs;
d) solutions of oxy-hemoglobin.
Based on the above observations, we concluded that the
backscattering peak was independent of the state of the RBC,
but did require that molecular Hb or some other cytoplasmic
component of the RBC be present.
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Fig. 6. Schematic representation of the three phases of coagulation. The
shaded area is the 532-nm pump laser reflectance, and the curve is the 1064-nm
probe laser transmission, which continues to drop after the end of the pump
pulse.
However, the Arrhenius analysis predicts that the transition
from biconcave disc to spheroid will occur around 76 C and
that the RBC cell membranes will start to fragment at around
80 C under our conditions. These temperatures are much closer
to those observed in the current set of experiments around the
onset of increased remittance. We therefore now believe that the
situation in the primary coagulation phase is much more complicated than previously thought, with at least three “mechanical” processes simultaneously occurring in conjunction with
the chemical conversion of oxy-Hb to met-Hb [10], [23]. These
mechanical processes are
• “molecular level” protein denaturation previously identified;
• phase transition in the RBC morphology;
• RBC cell membrane disintegration.
The development of a scattering coagulum close to the cuvette wall will affect the transmission of the pump beam postonset. Increased scattering will in general lead to a pile-up of
photon trajectories close to the cuvette blood interface [24], resulting in an increased capture probability and concomitantly
an increased heating rate in this region. It is possible that this
contributes to the overshoot of the observed interface temperature compared to the calculation around the time the sample
coagulates. The model currently assumes a uniform exponential decay in fluence through the sample, with a linear change
in the absorption coefficient throughout the pulse (due to the
bathochromic shift as a function of temperature).
One of the notable aspects of our initial work with this
system was the observation that the optical properties of the
whole-blood system continued to evolve even after the drive
laser pulse was removed (Fig. 6). We have inferred from
the time-resolved optical and OCT [25] experiments that
this “secondary” phase of coagulation is dominated by the
gradual accretion of a macroscopic coagulum composed of the
denatured hemoglobin already formed, the cell membranes of
ruptured erythrocytes and thermally denatured plasma proteins
(fibrin etc.) [12]. We believe that the dominant influence on
the optical properties at this stage comes from the change in
scattering resulting from the emergence of this macroscopic
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IEEE JOURNAL ON SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 7, NO. 6, NOVEMBER/DECEMBER 2001
coagulum. When the time-domain thermal properties of the
system are examined both theoretically using the finite-element
model, and experimentally using the TIR technique, it becomes
clear that the heated “disc” of blood remains hot long after
the laser pulse terminates. The “front” of the sample remains
at temperatures over 95 C for several tens of ms (absent the
effects of convection), and a “wave” of heating spreads through
the sample from front to back by diffusion. Bearing this in
mind, it is then perhaps not surprising that the system continues
to evolve.
C. Finite-Element Model
In addition to the results described, the model allows a
number of important practical aspects of the experiment to
be investigated. As an example, early results using a high
speed infra-red camera and a cuvette with sapphire walls (to
allow 3–5- m transmission) showed remarkable oscillatory
behavior in the optical properties and temperature isotherms
[unpublished data]. The oscillations were not present when
a cuvette with fused silica or BK-7 windows was used. The
finite-element model described above correctly predicts that
in this case the sapphire windows act as highly effective heat
sinks, and significant perturbations to the sample result. The
TIR temperature probe described above would certainly not
give accurate results in this case. As a corollary, a window
material predicted to satisfy simultaneously the requirements
that 1) sample heating be adiabatic and 2) that it be transmissive
to 3–5- m radiation is magnesium fluoride.
The model also allows an analysis to be performed on the
system, investigating the effects of varying parameters (absorption coefficient, bathochromic shift, laser pulse shape, etc). As
an example, we have investigated the effects of variable wall
thickness, in the light of the observations made with the sapphire
cuvette above. The thermal model was run for a range of wall
thickness values, holding all the other parameters constant. For
each run, the temperature rise at the end of the pulse was noted
for a key location on the outside container wall; the center of
the beam on the illuminated side is this key location. The wall
thicknesses required to ensure adiabatic heating in the sample
are, for BK7, about 0.35 mm, and for sapphire about 1 mm. For
BK7, if the thickness of both walls exceeds 0.35 mm, the sample
thermal response is not a function of wall thickness. Further, the
model remains accurate for configurations with wall of unequal
thickness, as long as both wall are thicker than 0.35 mm (for
BK7). Thermal analysis is ongoing.
D. Sensitivity Analysis
We close the discussion with a short sensitivity analysis of the
experimental and theoretical techniques. From an experimental
point of view, the method is clearly sensitive to perturbations
at the interface caused by effects other than the refractive index
change of the liquid with temperature. For example, we can expect convection currents developing in the sample to perturb the
results. This may for example account for the progressive deterioration in the size of the errors toward the end of the pulse.
Acoustic transients inside the liquid, resulting from microvaporization for example, could also perturb the interface. A possible
mitigation for both convection and microvaporization is to use
a larger probe beam spot with a view to integrating out these
fluctuations. The finite-element model shows that the spot created by the pump beam is, radially, “top-hat” in nature for the
temperature isotherms for upwards of 90% of the pump beam
diameter, so an increase in probe beam diameter can certainly
be tolerated in this sense.
Other possible perturbations of the interface that could affect
the results include the development of a “lens” on the surface of
the cuvette through localized thermal expansion of the glass in
contact with the hot blood. This could be ameliorated through
the use of imaging optics to collect the probe beam after the
TIR bounce. Finally for the experiment, we have assumed that
the change in refractive index for the cuvette with respect to
can be ignored. This assumption should be investigated more
thoroughly since it is conceivable that a change in for the
cuvette wall material could give a phase lag, given sufficient
thermal “inertia”. This might in turn be an explanation for the
observed discrepancies between experiment and theory.
For the finite-element model, the major issues going forward
would seem to be 1) the correct treatment of convection in the
sample and 2) the correct treatment of the energy deposition
profile in a medium which becomes highly scattering during the
calculation. Both problems will present formidable challenges
involving computational fluid dynamics and diffusion.
V. CONCLUSION
We have demonstrated a technique to measure temperature,
with temporal resolution, at a solid–liquid interface in a manner
which does not perturb the sample, allows for straightforward
data work up, and offers excellent temporal and temperature resolution. We have also developed a finite-element thermal model
of a laser-heated liquid incorporating for the first time to the best
of our knowledge a “realistic” model of the onset of coagulation
and first order time-dependent optical properties. We have compared the experimental and theoretical time/temperature profiles
for laser-heated blood, and found highly encouraging agreement
between experiment and theory. This data has in turn permitted
the refinement of our understanding of blood photocoagulation
dynamics.
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Jennifer Kehlet Barton (S’95–M’98) received the
B.S. and M.S. degrees in electrical engineering from
the University of Texas (UT) at Austin and University
of California, Irvine, respectively. She received the
Ph.D. degree in biomedical engineering from UT in
1998
She worked for McDonnell Douglas on the Space
Station Program. She is now an Assistant Professor
of biomedical engineering, electrical and computer
engineering, and optical sciences at the University of
Arizona, Tucson. Her research interests include optical imaging of tissue and laser–blood vessel interaction.
Dan P. Popok received the B.S. degree in 1982 and
the M.S. degree in 1983 from North Carolina State
University, Raleigh, both in mechanical engineering.
After working in the aerospace industry for
much of his career, he joined Network Analysis,
Inc., Tempe, AZ, in 1996 where he designs and
develops software for the SINDA/G thermal analysis
system, including finite-element interfaces. His
work for Network Analysis also includes consulting
in the areas of thermal and mechanical engineering
analysis. His professional interests include any
aspect of the thermal sciences, programming, and numerical methods.
John F. Black (M’00) was born in Cheshire, U.K.,
in 1962. He received the B.Sc. degree in chemistry
and the Ph.D. degree in physical chemistry from Nottingham University, Nottingham, U.K., in 1984 and
1987, respectively.
He was awarded an SERC-NATO Fellowship
to pursue postdoctoral research in molecular
spectroscopy and dynamics at Stanford University,
Stanford, CA, from 1987 to 1990, and was a Postdoctoral Research Fellow at Columbia University,
New York, from 1991 to 1993. Since 1994, he has
been employed in the laser industry, first at Continuum and then for five
years at Coherent Medical Group. He is currently with Lightwave Electronics,
Mountain View, CA. His research interests are in the areas of laser–tissue
interaction, laser cavity design with application to diagnostic and therapeutic
medicine, and tissue optics and spectroscopy.
Dr. Black is a member of the Optical Society of America, the American Physical Society, SPIE, and Sigma Xi.