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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38. NO. I . JANUARY 1991
Combined Microwave Heating and Surface Cooling of the Cornea B. Stuart Trembly,
Member, IEEE,
Abstract-We investigated a nonsurgical means of reshaping the cornea to correct hyperopia, keratoconus, or myopia. The object was to heat the central stroma of the cornea to the shrinkage temperature of collagen, 55-58°C. The heating device was an open-ended, coaxial, near-field applicator driven at 2450 MHz; it incorporates cooling of the cornea surface by flow of saline. We investigated the system theoretically by computing the 2-D, axisymmetric temperature distribution with the finite element method. We investigated the system experimentally by heating excised steer corneas. Histology showed the system could shrink the stroma to a depth of 0.6 mm while sparing the epithelium in 75%of cases; the diameter of shrinkage was 1.3 mm. Theory predicted a significantly deeper and narrower region of shrinkage than was observed.
INTRODUCTION
I?
can be useful to change the shape of the cornea to correct disorders such as myopia, keratoconus, and hyperopia. In myopia, the dioptric power of the eye is too great; this can be corrected by flattening the cornea. In keratoconus, the cornea has an abnormal cone-shaped projection at its center; it can also be corrected by flattening. Hyperopia can be corrected by increasing the curvature of, the cornea to increase the dioptric power of the eye. Radial keratotomy is a surgical procedure that can correct myopia by flattening the center portion of the cornea [ 11. Up to 16 nonpenetrating inscisions are made in the cornea, from near the center outward toward the rim. The internal pressure of the eye causes the cornea to bulge where cuts have been made; the uncut center of the cornea flattens as a result. The optical correction can be unpredictable; 10% of procedures yield a correction varying two diopters or more from that intended [6]. Complications include perforation of the cornea, glare, fluctuating vision, infectious keratitis, and loss of endothelial cells [2]-[6]. However, Salz [7] finds that proper technique can reduce incidence of infectious keratitis to 0.2%, and disputes the loss of endothelial cells, based on other studies [8], [9]. Thermal techniques for reshaping the cornea are based on the fact that the cornea stroma shrinks permanently when raised to 55-58OC [lo]. The stroma is the central, thickest layer, consisting of collagen fibers. If the pattern of shrinkage in the stroma were chosen properly, the resulting stresses could be used to reshape the cornea. The cornea has been flattened to correct keratoconus by applying a heated rod to it [ 111. Since the rod heats by conduction, however, the maximum temperaManuscript received December 18, 1987; revised March 29, 1990. This work was supported in part by Frigitronics Corp., Shelton, CT, and NSF Grant ES-8352580. B. S . Trembly is with the Thayer School of Engineering, Dartmouth College, Hanover, NH 03755. R. H . Keates is with the University Hospitals Clinic, The Ohio State University, Columbus, OH 43210. IEEE Log Number 9040627.
and Richard H. Keates
ture must occur at the cornea surface; consequently, the shrinkage effect may not obtain at depth, and the epithelium (the outer layer) may be destroyed. The corneal stroma of excised eyes has been heated by radio-frequency techniques combined with surface cooling [ 121. This produces a local maximum of temperature below the epithelium. However, the system requires two conductors between which the RF current flows. Since one conductor is a groundplane behind the eyeball, the technique may be difficult to apply in the clinic. At microwave frequencies a single applicator of small dimensions can deposit power directly at depth. In this paper, we describe a microwave heating system combined with surface cooling, and report its performance. Our intention was to cause shrinkage in a small dot-shaped region with the tip of a small cylindrical applicator; if the concept were feasible, then other radiators would be investigated to produce extended regions of shrinkage. For example, radial or circumferential lines of shrinkage could be created by an applicator with the diameter of the cornea, in which aperatures with the appropriate pattern are cut.
METHODS We investigated experimentally and theoretically the temperature distribution in the cornea produced by an open-ended, coaxial, near-field applicator in combination with cooling by forced convection of fluid. The microwave applicator and cooling channel are shown in Fig. 1. The microwave applicator is made of a 7.5 cm length of semi-rigid, copper, coaxial cable with an inner conductor OD of 0.91 mm, outer conductor ID of 2.98 mm, and outer conductor OD of 3.6 mm (Micro-Coax Components UT-141A). Its dielectric is PTFE (Teflon). An important feature is the gap between the tip of the microwave applicator and the cornea, through which flowing saline is guided by a channel. The apparatus holds the microwave applicator tip away from the cornea surface by a distance adjustable with a set screw. The cornea tends to bulge upwards when the applicator is applied firmly enough to the surface to form a fluid seal; to prevent this, three nylon threads cross the applicator centerline at the level of the cornea surface, as shown in Fig. 1. The microwave components were a Hewlett Packard 8620C sweep oscillator with 86222A RF Plug-In, Hughes 1277H traveling-wave-tube amplifier, circulator and load, dual directional coupler, Hewlett Packard 435B power meter, double stub tuner, and cables. The cooling system consisted of a reservoir, a positive-displacement pump, a flow integrator to smooth flow pulses, a basin, and tubing. We chose saline as the cooling fluid since it would be innocuous to a living eye. We heated the corneas of excised eyes of steer with this system. This was done within 8 h of excision; in the interval they
0018-9294/91/0100-0085$01.OO
0 1991 IEEE
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. I . JANUARY 1991
Stand-OHS
SMA
stained with hematoxylin and eosin, sliced with 10 pm thickness through the center of the treated spot, mounted in a slide, and photographed. We noted the treated spot was difficult to locate for histological sectioning in cases where little shrinkage was obtained. The results below show the effect on the region of shrinkage of varying gap size, microwave power, and saline flow rate. Swicord and Davis have derived a theoretical expression for the near field of an open-ended coaxial antenna using a magnetic current sheet as the source [ 131. We programed a form of their equations (22) and (23) on a Vax 11/785, and verified the program by computing the near electric field for a case they computed. In our work, the driving frequency was 2450 MHz; for cornea tissue, relative permittivity E ‘ is then 50.4, and the loss tangent, approximately equal to E ” / E ’ , is 0.343 ( E “ is the relative loss factor) [ 141. The electric field distribution is very nonuniform axially and radially (a typical distribution is given in [13, Fig. 51). The square of the electric field was 1/10 the value at the cornea surface at a depth 1.5 mm in the cornea. In our apparatus, a layer of cooling fluid is present between the microwave applicator and the comea surface. We did not model this separate layer electrically, but treated the medium loading the applicator as all cornea. Although saline relative permittivity is greater than that of cornea (80 versus 50), we assumed the discontinuity would not affect the fields greatly since the largest component of electric field near the end of the coaxial cable is parallel to the interface. The loss tangent of saline is close to that of tissue. The square of the magnitude of the electric field multiplied by ( U W E ~ E )” 2p gives specific absorption rate (SAR, W/kg). The quantity U is electrical conductivity, w is radian driving frequency, is permittivity of a vacuum, and p is mass density. This varying function of position forms a distributed heat source in the thermal analysis of the problem. We used a finite-element-method heat-transfer analysis program [ 151 to compute steady-state and transient temperatures on a 2-D axisymmetric grid representing the cornea and aqueous humour [Fig. 2(a)]. We used a domain of 3 mm depth (z-axis) and 6 mm width ( p axis) made up of 600 rectangular elements with a maximum aspect ratio of 6. The boundary conditions were: a constant temperature at the two edges farthest from the applicator (z = 3 mm and p = 6 mm); no heat flux crossing the domain axis ( p = 0); and a known convection coefficient and cooling fluid temperature at z = 0 [Fig. 2(b)]. The constant temperature boundary condition and cooling fluid temperature were set to 17.5”C, so that results could be compared with experiments conducted at room temperature. The convection coefficient is computed from the empirical correlation for the Nusselt number for laminar flow over a flat plate [ 161. The local Nusselt number is
+
Holding Screw ...
Side View Assembled
(C) Fig. 1. (a) Close-up of coaxial applicator with standoffs to maintain gap between applicator tip and comea. (b) Exploded view of applicator and saline flow channel. (c) Side view of assembly on cornea
were kept in a sealed, humidified container at room temperature. To heat, we fit the curved applicator to the eyeball to make a good seal with the channel, turned on the saline pump, adjusted its flowrate, then applied microwave power for 10 s. We made more than 50 heatings in total. As a practical matter, we found that leakage between the cornea and the sides of the applicator became significant at about 800 mL/min. We found in preliminary studies that a 5 s heating produced little shrinkage effect, and that 20 s was little different from 10 s. After heating, the cornea was excised from the eye, preserved in formalin,
Nu, = h , ( x / k ) = 0.332 Rei/2 Pr’’3, where h, = film coefficient of convective heat transfer at point x (W/m2 . K ) . x = distance from the beginning of plate to the point of interest (m). (The beginning of the plate is the edge of the coaxial applicator, and the point of interest is the center of the applicator).
k = thermal conductivity of cooling fluid (W/m K ) , Pr = Prantl number of fluid at given temperature (dimensionless), Re, = V x / v = Reynolds number at point x (dimensionless),
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TREMBLY AND KEATES: MICROWAVE HEATING AND COOLING OF CORNEA
87
and 17 800 W/m2 . K for the same two flow rates. We used the lower values from the flat plate correlation in our calculations. We assumed the temperature of the cooling fluid remains constant along the flow path, since a conservative calculation showed the temperature rise is only 0.6"C. We did not model any variations in thermal properties of the layers of the comea, because we obtained only averaged values for the cornea as a whole. We measured the thermal conductivity as 0.556 W/m . K (five measurements, s.d. = 0.0016 W/m . K) and thermal diffusivity as 1.45 X lo-' m2/s (five meam2/s) using the apparatus of surements, SD = 0.45 x Dr. F. Bowman at M.I.T. described in 1171. Assuming mass density is 1000 kg/m3, this yields 3830 J/kg . K for specific heat. In aqueous humour, thermal conductivity is 0.578 W/m . K and specific heat is 4180 J/kg * K (error not reported in source) [ 181. For simplicity, we used property values for cornea throughout the domain. In the steady-state, only conductivity values affect the temperature distribution, and these values differ by only 4%. We checked for numerical artifacts in the FEM model by making the grid size coarser and by increasing the depth at which the temperature boundary condition was enforced. A grid with half as many elements yielded a temperature profile with maximum temperature only 0.1 mm deeper. Moving the boundary condition T = 17.5"C from z = 3 mm to z = 6 mm changed the location of the temperature maximum by only 0.1 mm. Therefore, we concluded the grid in Fig. 2 was adequate. We used this theoretical model to predict the effect of varying surface cooling, and compared these results to experiment.
....a:i ......
2
3.0 mm'
RESULTS
(b) Fig. 2 . (a) The cornea and aqueous humour modeled as a two-dimensional axisymmetric domain. (b) The boundary conditions for calculation of temperature.
V = average velocity of fluid (m/s), and U = kinematic viscosity of fluid at given temperature (m2/s). With a cross-sectional area for saline flow of 2.4 mm2 for a 0.67 mm applicator-cornea gap, and fluid properties of water computed at 17.5"C, we obtain this relationship between vol; ume flow rate of the cooling fluid and heat transfer coefficient at the cornea surface below the applicator center: h, = [216W/m2
. K][(6.92
X
lo8 s/m')
. q]'/2
or h,
=
[216 W/m2
*
K][(11.6 min/mL)
. q]'I2.
For the two flow rates of interest, 100 mL/min and 400 mL/min, this expression yields h, equal to 7360 W/m2 . K and 14,700 W/m2 . K. These large values are due to the very small flow area. We also computed the convection coefficient using an average Nusselt number correlation for flow in a duct with combined thermal and hydrodynamic entrance effects 119, equation (8.55)]. This correlation yielded 11 200 W/m2 . K
Figs. 3-8 show experimental results. Fig. 3 shows the effect of the microwave source power on the diameter of the region of shrinkage on the cornea. The applicator-cornea gap is 0.67 mm, and the saline flow rate is 200 mL/min. The shrinkage diameters are classified as small ( < 1 mm), medium (1-2.5 mm), or large ( > 2.5 mm). Three heatings were made at each power level, and the number of samples in each size category is plotted. Based on these results, we chose an intermediate power level of 25 W was chosen for following heatings. Note that only a small fraction of the source power was absorbed in the comea since most is absorbed in the feedline before or after reflection at the interface between applicator and saline, or absorbed in the saline. Fig. 4 shows the effect of applicator-cornea gap on shrinkage diameter. Microwave source power is 25 W, and saline flow rate varies between 100 and 300 mL/min to keep the estimated convection coefficient equal to about lo4 W/m2 K. Three heatings were made at each power level. We found the shrinkage diameter is extremely sensitive to gap size. We chose an intermediate gap size of 0.67 mm for following heatings. Fig. 5 shows the effect of saline flow rate on shrinkage diameter. Microwave source power is 25 W, and gap size is 0.67 mm. Estimated convection coefficient varies from 7360 to 20 810 W/m2 . K for these flow rates. Four heatings were made at each flow rate, except the highest value, for which two were made. Shrinkage diameter varies more slowly with flow rate than with microwave source power or gap size. We present the cases of 100 and 400 mL/min flow rate in more detail below. Fig. 6 shows the effect of saline flow rate on the sparing of the corneal epithelium. For each flow rate of Fig. 5, the fraction of samples with an intact epithelium is plotted. We found that
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 38, NO. I , JANUARY 1991
88
Size categories.
0 Nothing Visible
0 Small (cl mm d i d Medium (1-2.5 mm dial
W Large b2.5"
15
20 30 Source Power (W)
dia)
35
Fig. 3. Effect of microwave source power on diameter of shrinkage region. Applicator-comea gap is 0.67 mm, and saline flow rate is 200 mL/min. Three measurements made at each power level
Size categories:
IJ
Nothing Visible
0 Small (4 mm dia) Medium (1-2 5mm dia) Large (>2 5 mm d i d
0.25
0.5
0.8
1
Gap size (mm)
Fig. 4. Effect of applicator-cornea gap on diameter of shrinkage region. Microwave source power is 25 W, and estimated convection coefficient is lo4 W/m2 . K (saline flow rate between 100 and 300 mL/min). Three measurements made at each gap size.
4 .-c
Size categories.
n
2 >, 0.L
2 in 'a
0 Nothing Visible
3 0
% E
t !.
S I
Small (clmm d i d
Y
W Medium (1-2 5mm dial
2
'A
W Large (>2 Smm dial
E L 1
100
200
300
400
600
800
Flow Rate, ml/min
Fig. 5 . Effect of saline flow rate on diameter of shrinkage region. Microwave source power is 25 W, and applicator-comea gap size is 0.67 mm. Four measurements made at each flow rate, except the highest, with two.
~
I'
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TREMBLY AND KEATES: MICROWAVE HEATING AND COOLING OF CORNEA
0
20 0
200
400
600
800
1000
Flow Rate, ml/min Fig. 6. Effect of saline flow rate on sparing of epithelium. Microwave source power is 25 W, and applicator-cornea gap size is 0.67 mm. Four data points at each flow rate, except the highest, with two. Linear regression line shown: r = 0.51.
Fig. 8. Cross-section of a cornea heated with 25 W source power, 400 mL/min saline flow at 17.5"C, and a antenna-cornea gap of 0.67 mm. Scale indicated.
T = 17 5 deg C
\
\
I
\
5 deg C
- P, mm
2
4
6
(cornea surface)
Fig. 7. Cross-section of a cornea heated with 25 W source power, 100 mL/min saline flow at 17.5"C, and a antenna-cornea gap of 0.67 mm. Scale indicated.
a high flow rate tends to spare the epithelium, but note the large scatter from the linear regression line (correlation coefficient r equals 0.51). Fig. 7 shows the cross-section of a steer cornea heated with 25 W of microwave source power, flow rate equal to 100 mL/min, (estimated convection coefficient is 7360 W/m2 * K), and a microwave applicator-cornea gap of 0.67 mm. The epithelium is missing. Shrinkage extends to 0.6 mm depth, excluding the missing epithelium. The length of shrinkage along the surface is 2.8 mm. Note that shrinkage is in the direction of the stromal fibers (left-right in photograph), creating a bulge perpendicular to fiber direction. In Fig. 8 all parameters are the same, except the cooling flow rate has been increased to 400 mL/min (estimated convection coefficient is 14 700 W/m2 . K). The epithelium is intact. The indentation in the epithelium left of center was caused by the spacing-string of the heating apparatus. The two dark vertical lines in the epithelium near each edge of the photograph are folds created in the slicing process. Shrinkage extends to 0.4 mm depth, excluding the epithelium; the length of shrinkage along the surface is 1.3 mm. (In this sample, edema has in-
7 ' --
(b)
Fig >. Theoretical iso-temperature lines in the steady-state for absorbes microwave power of 0.376 W, cooling fluid at 17.5"C, and an applicatorcornea gap of 0.67 mm. Isotherms are 2°C apart. (a) Convection coefficient equals 7 x IO3 W / m Z . K. Maximum temperature is 64°C. (b) Convection coefficient equals 1.4 x lo4 W / m 2 . K . Maximum temperature is 60°C.
creased the thickness of the epithelium above its normal value of 0.18 mm.) This heating is the most favorable one of about 50 made with different parameter values: for cases in which the epithelium is intact, the shrinkage region extends deepest. The termination impedance of the applicator was measured as 16 j 22 Q with a saline gap of 0.67 mm; the corresponding power reflection coefficient with a 50 Q feedline is 0.33. Fig. 9 shows theoretical isotherms in the steady-state for a gap of 0.67 mm, a saline temperature of 17.5"C, and absorbed microwave power of 0.376 W. In (a), the surface convection coefficient is 7 x lo3 W/m2 . K (estimated corresponding flowrate is 100 mL/min); in (b) the convection coefficient is
+
1
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. I . JANUARY 1991
TABLE I Summary of Comparison of Theory to Experiment
Method Power' Cooling' Depth' Width4 Max. Temp. Surface Temp.
Experiment 25 w 100 mL/min 0.78 mm 2.8 m m -
Experiment 25 w 400 mL/min 0.59 mm 1.3 mm -
Theory 0.316 W 7 x IO' W/m2 . K 0.84 mm 1.1 mm 64°C 29°C
Theory 0.376 W 1.4 x lo4 W/mZ . K 0.72 mm 0.74 mm 60°C 23°C
'Microwave source power for experiment, cornea absorbed power for theory. 'Flow rate values in experiment yield corresponding convection coefficient values in theory through flat plate Nusselt number correlation. 'Depth of shrinkage effect in experiment, depth of temperature 256°C in theory; measured from top surface of a normal epithelium of thickness 0.18 mm. 4Width of shrinkage effect in experiment, width of temperature 256°C in theory.
1.5 X lo4 W/m2 . K (estimated corresponding flowrate is 400 mL/min). Absorbed microwave power is the power absorbed in the cornea itself; it was adjusted to give a 60°C temperature maximum in case b). The theoretical case of Fig. 9(a) corresponds to the experimental case of Fig. 7; likewise, Fig. 9(b) corresponds to Fig. 8. The correspondence is approximate because in the experiment we know only that the cornea has reached shrinkage temperature, not the exact value attained. The smaller convection coefficient in Fig. 9(a) gives a local maximum 0.4 mm below the cornea surface (including epithelium). The region reaching a temperature 2 5 6 ° C extends to 0.84 mm depth, including the epithelium. The surface temperature is 29°C at the applicator centerline. The width of the region reaching ? 56°C is 1.1 mm. The maximum temperature is 64°C. The larger convection coefficient in Fig. 9(b) reduces region reaching 256°C; the width is 0.74 mm, and the depth is 0.72 mm (including the epithelium). The surface temperature is reduced to 23°C on the centerline, but the local maximum remains at 0.4 mm depth. The results shown in Figs. 7-9 are summarized in Table I. A theoretical calculation of the transient response showed that a temperature within 10% of final value is achieved 10 sec after a step input of microwave power for the cases above.
DISCUSSION This heating system would not deposit significant power in the ocular lens, since virtually all power is absorbed within a tissue radius equal to the microwave applicator radius [ 131. Furthermore, the theoretical calculations show there is no significant temperature rise in the deeper structures due to thermal conduction. The heat flux outward at the cornea surface is roughly seven times the heat flux inward toward deeper structures. Therefore, we would expect little possibility of cataract formation. The required value of microwave source power depends partly on the particular cables we used, since some power was absorbed in the feedlines. In general, source power is much larger than cornea absorbed power, due to feedline losses, reflection at the applicator-saline interface, and absorption in saline. The shrinkage effect is extremely sensitive to applicator-cornea gap. Although the gap can be fixed with a set screw, we believe the device is sensitive to how it is held against the eye. For example, if it were not perpendicular to the eye, then the
effective gap would be smaller on one side, and the shrinkage pattern would be skewed. Placement technique may also affect sparing of the epithelium. Whereas sparing was correlated only weakly with saline flow rate, it may be that mechanical forces were significant in removing the epithelium. Shrinkage in the central stroma is necessary to produce permanent changes in the shape of the cornea [12]. The experimental results in Figs. 7 and 8 indicate that this apparatus produces shrinkage to a depth of 0.6-0.8 mm from the outside of the epithelium, depending on the value of flowrate. This does not span the cornea of the steer, which is more than 1 mm thick; however, this is greater than the thickness of the human cornea, about 0.5 mm, depending on location. Thus, the endothelium (inner layer) of the human cornea could be heated significantly. Since this is undesirable, a smaller diameter applicator scaled to the human cornea could be used to reduce temperature at depth. Swicord and Davis showed that the radius in which a fixed fraction of all radiated power is absorbed is directly related to applicator diameter [ 131. The agreement between theory and experiment is only approximate. The width of shrinkage is roughly twice as great in experiment as it is in theory. The depth of shrinkage in experiment is about 85% of that in theory. Of course, in experiment, the cornea geometry changes when shrinkage occurs. Dimensions parallel to the surface decrease, and perpendicular dimensions increase. Thus, if the experimental temperature distribution were imagined without shrinkage having occurred, it would be even wider parallel to the surface and even shallower perpendicular to the surface. This makes agreement between theory and experiment even poorer. An explanation may be that cornea thermal conductivity is anisotropic, being larger in the direction of fibers (parallel to surface) than across fibers. Also relevant is the possibility that thermal conductivity increases after shrinkage in the direction parallel to the fibers, increasing the diameter of the shrinkage region. The theoretical model predicts a reduction in surface temperature of 6°C when cooling flow rate increases from 7 x lo3 W/m2 K to 1.5 x lo4 W/m2 K. However, the higher flow rate does not necessarily protect the epithelium. In our experiments, there was only a weak correlation between a high flow rate and an intact epithelium. The experimental and theoretical transient response results agree, in that little change in shrinkage was noted between 10 and 20 s heatings in experiment, and in theory the steady state was reached in about 10 s.
~
I1
TREMBLY AND KEATES: MICROWAVE HEATING AND COOLING OF CORNEA
CONCLUSIONS T h e 3.6 m m OD coaxial m i c r o w a v e applicator driven a t 2450 MHz shrinks t h e stroma o f a n excised steer cornea a t least 0.6 m m below the surface w h e n surface cooling b y r o o m temperature saline is enforced; t h e epithelium m a y b e spared. T h e reg i o n of shrinkage is a d i s k about 1 m m i n diameter. Agreement between theoretical predictions a n d experiment is only approximate, possibly d u e to anisotropic thermal conductivity i n t h e
cornea. ACKNOWLEDGMENT
The authors wish to a c k n o w l e d g e t h e assistance of W . L. Withers, B. M. Levin, a n d E. D r a k o u , w h o performed t h e experiments a n d calculations f o r this research a n d created s o m e of t h e figures; L. J . H a m m , w h o coordinated t h e histology a n d preparation of t h e manuscript; Dr. J . D. Fratkin, w h o advised on t h e interpretation of t h e histology; a n d Dr. F. B o w m a n a n d his co-workers, w h o made his equipment available f o r measurement o f thermal properties o f t h e cornea a n d assisted in t h e procedure.
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1121 J. D. Doss and J . I. Albillar, “A technique for the selective heating of corneal stroma,” Contact Lens, vol. 6 , no. 1, pp. 13-17, Jan.-Mar. 1980. 1131 M. L. Swicord and C. C. Davis, “Energy absorption from small radiating coaxial probes in lossy media,” ZEEE Trans. Microwave Theory Tech., vol. 29, no. 11, pp. 1202-1209, Nov. 1981. [14] C. Grabriel, R. J. Sheppard, and E. H. Grant, “Dielectric properties of ocular tissues at 37 degrees C , ” Phys. Med. Biol., vol. 28, no. 1, pp. 43-49, 1983. [I51 R. P. Glovsky, “Instruction manual for THERMAP,” M.S. Thesis, Thayer School of Engineering, Dartmouth College, June 1982. [16] J . R. Welty, C. E. Wicks, and R. E. Wilson, Fundamentals of Momentum, Heat, and Mass Transfer. New York: Wiley, 1976, pp. 169-179, 330-335. [I71 T . A. Balsubramaniam and H . F. Bowman, “Thermal conductivity and thermal diffusivity of biomaterials: a simultaneous measurement technique,” J . Biomech. Eng., pp. 148-154, Aug. 1977. [18] H. F. Bowman, E. G . Cravalho, and M. Woods, “Theory, measurement, and application of thermal properties of biomaterials,” Ann. Rev. Biophy. Bioeng., vol. 4 , pp. 43-80, 1975. [19] F. P. Incropera and D. P. Dewitt, Fundamentals of Heat and Mass Transfer. New York: Wiley, 1985, p. 393.
REFERENCES r11 S. N. Fyodorov and V. V . Durnev, “Operation of dosaged dissection of corneal circular ligament in cases of myopia of mild degree,” Ann. Ophthalmol., pp. 1885-1890, Dec. 1979. [21 J. W. Cowden and L. D. Bores, “ A clinical investigation of the surgical correction of myopia by the method of Fyodorov,” Ophthalmol., vol. 88, no. 8, pp. 737-41, Aug. 1981. [31 K. J . Hoffer and J. J. Darin, “Three years experience with radial keratotomy: the UCLA study,” Ophthalmol., vol. 90, no. 6 , pp. 627-641, June 1983. 141 K. J . Hoffer and J. J . Darin, “UCLA clinical trial of radial keratotomy,” Ophthalmol., vol. 88, no. 8, pp. 729-736, Aug. 1981. 151 M. R. Dietz and D. R. Sanders, “Radial keratotomy-an overview of the Kansas City Study,” Ophrhalmol., vol. 91, no. 5 , pp. 467-478, May 1984. 161 R. E. Smith, “Status of radial keratotomy,” J . Refractive Surg., vol. 3, no. 5, p. 187, Sept.-Oct., 1987. 171 J. J . Salz, “How safe is radial keratotomy,” J . Refractive Surg., vol. 3, no. 5, p. 188-189, Sept.-Oct., 1987. 181 P. A. Asbell, N. Justin, and S. Obstbaum, “Peripheral corneal evaluation post radial keratotomy,” Ophthalmol., vol. 91 (suppl.), p. 122, 1984. [91 S. M. MacRae, M. Matsuda, and L. F. Rich, “The effect of radial keratotomy on the corneal endothelium,” Amer. J . Ophthalmol., vol. 10, pp. 538-542, 1985. 1101 H. Stringer and J. Parr, “Shrinkage temperature of eye collagen,” Nature, vol. 204, no. 4965, p. 1307, Dec. 26, 1964. 1111 J. V . Aquavella, “Thermokeratoplasty,” Ophthalmic Surg., vol. 5, no. 1, pp. 39-47, Spring 1974.
B. Stuart Trembly (M’83) received the B.S. degree from Yale University, New Haven, CT, in 1975, and the Ph.D. degree from Dartmouth College, Hanover, NH, in 1982. He joined the faculty at the Thayer School of Engineering at Dartmouth College in 1982, where he is an Associate Professor. His research interest is the application of electrical engineering to biomedical problems. Dr. Trembly was the recipient of a Presidential Young Investigator Award in 1984.
Richard H. Keates received the B.A. degree from the University of Pennsylvania, Philadelphia, in 1953 and the M.D. degree from Jefferson Medical Center, Philadelphia, PA, in 1957. He holds the Irving H. Leopold Chair of Ophthalmology and is Professor and Chairman of the Department of Ophthalmology at the University of California, Imine.
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38. NO. I . JANUARY 1991
Combined Microwave Heating and Surface Cooling of the Cornea B. Stuart Trembly,
Member, IEEE,
Abstract-We investigated a nonsurgical means of reshaping the cornea to correct hyperopia, keratoconus, or myopia. The object was to heat the central stroma of the cornea to the shrinkage temperature of collagen, 55-58°C. The heating device was an open-ended, coaxial, near-field applicator driven at 2450 MHz; it incorporates cooling of the cornea surface by flow of saline. We investigated the system theoretically by computing the 2-D, axisymmetric temperature distribution with the finite element method. We investigated the system experimentally by heating excised steer corneas. Histology showed the system could shrink the stroma to a depth of 0.6 mm while sparing the epithelium in 75%of cases; the diameter of shrinkage was 1.3 mm. Theory predicted a significantly deeper and narrower region of shrinkage than was observed.
INTRODUCTION
I?
can be useful to change the shape of the cornea to correct disorders such as myopia, keratoconus, and hyperopia. In myopia, the dioptric power of the eye is too great; this can be corrected by flattening the cornea. In keratoconus, the cornea has an abnormal cone-shaped projection at its center; it can also be corrected by flattening. Hyperopia can be corrected by increasing the curvature of, the cornea to increase the dioptric power of the eye. Radial keratotomy is a surgical procedure that can correct myopia by flattening the center portion of the cornea [ 11. Up to 16 nonpenetrating inscisions are made in the cornea, from near the center outward toward the rim. The internal pressure of the eye causes the cornea to bulge where cuts have been made; the uncut center of the cornea flattens as a result. The optical correction can be unpredictable; 10% of procedures yield a correction varying two diopters or more from that intended [6]. Complications include perforation of the cornea, glare, fluctuating vision, infectious keratitis, and loss of endothelial cells [2]-[6]. However, Salz [7] finds that proper technique can reduce incidence of infectious keratitis to 0.2%, and disputes the loss of endothelial cells, based on other studies [8], [9]. Thermal techniques for reshaping the cornea are based on the fact that the cornea stroma shrinks permanently when raised to 55-58OC [lo]. The stroma is the central, thickest layer, consisting of collagen fibers. If the pattern of shrinkage in the stroma were chosen properly, the resulting stresses could be used to reshape the cornea. The cornea has been flattened to correct keratoconus by applying a heated rod to it [ 111. Since the rod heats by conduction, however, the maximum temperaManuscript received December 18, 1987; revised March 29, 1990. This work was supported in part by Frigitronics Corp., Shelton, CT, and NSF Grant ES-8352580. B. S . Trembly is with the Thayer School of Engineering, Dartmouth College, Hanover, NH 03755. R. H . Keates is with the University Hospitals Clinic, The Ohio State University, Columbus, OH 43210. IEEE Log Number 9040627.
and Richard H. Keates
ture must occur at the cornea surface; consequently, the shrinkage effect may not obtain at depth, and the epithelium (the outer layer) may be destroyed. The corneal stroma of excised eyes has been heated by radio-frequency techniques combined with surface cooling [ 121. This produces a local maximum of temperature below the epithelium. However, the system requires two conductors between which the RF current flows. Since one conductor is a groundplane behind the eyeball, the technique may be difficult to apply in the clinic. At microwave frequencies a single applicator of small dimensions can deposit power directly at depth. In this paper, we describe a microwave heating system combined with surface cooling, and report its performance. Our intention was to cause shrinkage in a small dot-shaped region with the tip of a small cylindrical applicator; if the concept were feasible, then other radiators would be investigated to produce extended regions of shrinkage. For example, radial or circumferential lines of shrinkage could be created by an applicator with the diameter of the cornea, in which aperatures with the appropriate pattern are cut.
METHODS We investigated experimentally and theoretically the temperature distribution in the cornea produced by an open-ended, coaxial, near-field applicator in combination with cooling by forced convection of fluid. The microwave applicator and cooling channel are shown in Fig. 1. The microwave applicator is made of a 7.5 cm length of semi-rigid, copper, coaxial cable with an inner conductor OD of 0.91 mm, outer conductor ID of 2.98 mm, and outer conductor OD of 3.6 mm (Micro-Coax Components UT-141A). Its dielectric is PTFE (Teflon). An important feature is the gap between the tip of the microwave applicator and the cornea, through which flowing saline is guided by a channel. The apparatus holds the microwave applicator tip away from the cornea surface by a distance adjustable with a set screw. The cornea tends to bulge upwards when the applicator is applied firmly enough to the surface to form a fluid seal; to prevent this, three nylon threads cross the applicator centerline at the level of the cornea surface, as shown in Fig. 1. The microwave components were a Hewlett Packard 8620C sweep oscillator with 86222A RF Plug-In, Hughes 1277H traveling-wave-tube amplifier, circulator and load, dual directional coupler, Hewlett Packard 435B power meter, double stub tuner, and cables. The cooling system consisted of a reservoir, a positive-displacement pump, a flow integrator to smooth flow pulses, a basin, and tubing. We chose saline as the cooling fluid since it would be innocuous to a living eye. We heated the corneas of excised eyes of steer with this system. This was done within 8 h of excision; in the interval they
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Stand-OHS
SMA
stained with hematoxylin and eosin, sliced with 10 pm thickness through the center of the treated spot, mounted in a slide, and photographed. We noted the treated spot was difficult to locate for histological sectioning in cases where little shrinkage was obtained. The results below show the effect on the region of shrinkage of varying gap size, microwave power, and saline flow rate. Swicord and Davis have derived a theoretical expression for the near field of an open-ended coaxial antenna using a magnetic current sheet as the source [ 131. We programed a form of their equations (22) and (23) on a Vax 11/785, and verified the program by computing the near electric field for a case they computed. In our work, the driving frequency was 2450 MHz; for cornea tissue, relative permittivity E ‘ is then 50.4, and the loss tangent, approximately equal to E ” / E ’ , is 0.343 ( E “ is the relative loss factor) [ 141. The electric field distribution is very nonuniform axially and radially (a typical distribution is given in [13, Fig. 51). The square of the electric field was 1/10 the value at the cornea surface at a depth 1.5 mm in the cornea. In our apparatus, a layer of cooling fluid is present between the microwave applicator and the comea surface. We did not model this separate layer electrically, but treated the medium loading the applicator as all cornea. Although saline relative permittivity is greater than that of cornea (80 versus 50), we assumed the discontinuity would not affect the fields greatly since the largest component of electric field near the end of the coaxial cable is parallel to the interface. The loss tangent of saline is close to that of tissue. The square of the magnitude of the electric field multiplied by ( U W E ~ E )” 2p gives specific absorption rate (SAR, W/kg). The quantity U is electrical conductivity, w is radian driving frequency, is permittivity of a vacuum, and p is mass density. This varying function of position forms a distributed heat source in the thermal analysis of the problem. We used a finite-element-method heat-transfer analysis program [ 151 to compute steady-state and transient temperatures on a 2-D axisymmetric grid representing the cornea and aqueous humour [Fig. 2(a)]. We used a domain of 3 mm depth (z-axis) and 6 mm width ( p axis) made up of 600 rectangular elements with a maximum aspect ratio of 6. The boundary conditions were: a constant temperature at the two edges farthest from the applicator (z = 3 mm and p = 6 mm); no heat flux crossing the domain axis ( p = 0); and a known convection coefficient and cooling fluid temperature at z = 0 [Fig. 2(b)]. The constant temperature boundary condition and cooling fluid temperature were set to 17.5”C, so that results could be compared with experiments conducted at room temperature. The convection coefficient is computed from the empirical correlation for the Nusselt number for laminar flow over a flat plate [ 161. The local Nusselt number is
+
Holding Screw ...
Side View Assembled
(C) Fig. 1. (a) Close-up of coaxial applicator with standoffs to maintain gap between applicator tip and comea. (b) Exploded view of applicator and saline flow channel. (c) Side view of assembly on cornea
were kept in a sealed, humidified container at room temperature. To heat, we fit the curved applicator to the eyeball to make a good seal with the channel, turned on the saline pump, adjusted its flowrate, then applied microwave power for 10 s. We made more than 50 heatings in total. As a practical matter, we found that leakage between the cornea and the sides of the applicator became significant at about 800 mL/min. We found in preliminary studies that a 5 s heating produced little shrinkage effect, and that 20 s was little different from 10 s. After heating, the cornea was excised from the eye, preserved in formalin,
Nu, = h , ( x / k ) = 0.332 Rei/2 Pr’’3, where h, = film coefficient of convective heat transfer at point x (W/m2 . K ) . x = distance from the beginning of plate to the point of interest (m). (The beginning of the plate is the edge of the coaxial applicator, and the point of interest is the center of the applicator).
k = thermal conductivity of cooling fluid (W/m K ) , Pr = Prantl number of fluid at given temperature (dimensionless), Re, = V x / v = Reynolds number at point x (dimensionless),
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TREMBLY AND KEATES: MICROWAVE HEATING AND COOLING OF CORNEA
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and 17 800 W/m2 . K for the same two flow rates. We used the lower values from the flat plate correlation in our calculations. We assumed the temperature of the cooling fluid remains constant along the flow path, since a conservative calculation showed the temperature rise is only 0.6"C. We did not model any variations in thermal properties of the layers of the comea, because we obtained only averaged values for the cornea as a whole. We measured the thermal conductivity as 0.556 W/m . K (five measurements, s.d. = 0.0016 W/m . K) and thermal diffusivity as 1.45 X lo-' m2/s (five meam2/s) using the apparatus of surements, SD = 0.45 x Dr. F. Bowman at M.I.T. described in 1171. Assuming mass density is 1000 kg/m3, this yields 3830 J/kg . K for specific heat. In aqueous humour, thermal conductivity is 0.578 W/m . K and specific heat is 4180 J/kg * K (error not reported in source) [ 181. For simplicity, we used property values for cornea throughout the domain. In the steady-state, only conductivity values affect the temperature distribution, and these values differ by only 4%. We checked for numerical artifacts in the FEM model by making the grid size coarser and by increasing the depth at which the temperature boundary condition was enforced. A grid with half as many elements yielded a temperature profile with maximum temperature only 0.1 mm deeper. Moving the boundary condition T = 17.5"C from z = 3 mm to z = 6 mm changed the location of the temperature maximum by only 0.1 mm. Therefore, we concluded the grid in Fig. 2 was adequate. We used this theoretical model to predict the effect of varying surface cooling, and compared these results to experiment.
....a:i ......
2
3.0 mm'
RESULTS
(b) Fig. 2 . (a) The cornea and aqueous humour modeled as a two-dimensional axisymmetric domain. (b) The boundary conditions for calculation of temperature.
V = average velocity of fluid (m/s), and U = kinematic viscosity of fluid at given temperature (m2/s). With a cross-sectional area for saline flow of 2.4 mm2 for a 0.67 mm applicator-cornea gap, and fluid properties of water computed at 17.5"C, we obtain this relationship between vol; ume flow rate of the cooling fluid and heat transfer coefficient at the cornea surface below the applicator center: h, = [216W/m2
. K][(6.92
X
lo8 s/m')
. q]'/2
or h,
=
[216 W/m2
*
K][(11.6 min/mL)
. q]'I2.
For the two flow rates of interest, 100 mL/min and 400 mL/min, this expression yields h, equal to 7360 W/m2 . K and 14,700 W/m2 . K. These large values are due to the very small flow area. We also computed the convection coefficient using an average Nusselt number correlation for flow in a duct with combined thermal and hydrodynamic entrance effects 119, equation (8.55)]. This correlation yielded 11 200 W/m2 . K
Figs. 3-8 show experimental results. Fig. 3 shows the effect of the microwave source power on the diameter of the region of shrinkage on the cornea. The applicator-cornea gap is 0.67 mm, and the saline flow rate is 200 mL/min. The shrinkage diameters are classified as small ( < 1 mm), medium (1-2.5 mm), or large ( > 2.5 mm). Three heatings were made at each power level, and the number of samples in each size category is plotted. Based on these results, we chose an intermediate power level of 25 W was chosen for following heatings. Note that only a small fraction of the source power was absorbed in the comea since most is absorbed in the feedline before or after reflection at the interface between applicator and saline, or absorbed in the saline. Fig. 4 shows the effect of applicator-cornea gap on shrinkage diameter. Microwave source power is 25 W, and saline flow rate varies between 100 and 300 mL/min to keep the estimated convection coefficient equal to about lo4 W/m2 K. Three heatings were made at each power level. We found the shrinkage diameter is extremely sensitive to gap size. We chose an intermediate gap size of 0.67 mm for following heatings. Fig. 5 shows the effect of saline flow rate on shrinkage diameter. Microwave source power is 25 W, and gap size is 0.67 mm. Estimated convection coefficient varies from 7360 to 20 810 W/m2 . K for these flow rates. Four heatings were made at each flow rate, except the highest value, for which two were made. Shrinkage diameter varies more slowly with flow rate than with microwave source power or gap size. We present the cases of 100 and 400 mL/min flow rate in more detail below. Fig. 6 shows the effect of saline flow rate on the sparing of the corneal epithelium. For each flow rate of Fig. 5, the fraction of samples with an intact epithelium is plotted. We found that
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Size categories.
0 Nothing Visible
0 Small (cl mm d i d Medium (1-2.5 mm dial
W Large b2.5"
15
20 30 Source Power (W)
dia)
35
Fig. 3. Effect of microwave source power on diameter of shrinkage region. Applicator-comea gap is 0.67 mm, and saline flow rate is 200 mL/min. Three measurements made at each power level
Size categories:
IJ
Nothing Visible
0 Small (4 mm dia) Medium (1-2 5mm dia) Large (>2 5 mm d i d
0.25
0.5
0.8
1
Gap size (mm)
Fig. 4. Effect of applicator-cornea gap on diameter of shrinkage region. Microwave source power is 25 W, and estimated convection coefficient is lo4 W/m2 . K (saline flow rate between 100 and 300 mL/min). Three measurements made at each gap size.
4 .-c
Size categories.
n
2 >, 0.L
2 in 'a
0 Nothing Visible
3 0
% E
t !.
S I
Small (clmm d i d
Y
W Medium (1-2 5mm dial
2
'A
W Large (>2 Smm dial
E L 1
100
200
300
400
600
800
Flow Rate, ml/min
Fig. 5 . Effect of saline flow rate on diameter of shrinkage region. Microwave source power is 25 W, and applicator-comea gap size is 0.67 mm. Four measurements made at each flow rate, except the highest, with two.
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TREMBLY AND KEATES: MICROWAVE HEATING AND COOLING OF CORNEA
0
20 0
200
400
600
800
1000
Flow Rate, ml/min Fig. 6. Effect of saline flow rate on sparing of epithelium. Microwave source power is 25 W, and applicator-cornea gap size is 0.67 mm. Four data points at each flow rate, except the highest, with two. Linear regression line shown: r = 0.51.
Fig. 8. Cross-section of a cornea heated with 25 W source power, 400 mL/min saline flow at 17.5"C, and a antenna-cornea gap of 0.67 mm. Scale indicated.
T = 17 5 deg C
\
\
I
\
5 deg C
- P, mm
2
4
6
(cornea surface)
Fig. 7. Cross-section of a cornea heated with 25 W source power, 100 mL/min saline flow at 17.5"C, and a antenna-cornea gap of 0.67 mm. Scale indicated.
a high flow rate tends to spare the epithelium, but note the large scatter from the linear regression line (correlation coefficient r equals 0.51). Fig. 7 shows the cross-section of a steer cornea heated with 25 W of microwave source power, flow rate equal to 100 mL/min, (estimated convection coefficient is 7360 W/m2 * K), and a microwave applicator-cornea gap of 0.67 mm. The epithelium is missing. Shrinkage extends to 0.6 mm depth, excluding the missing epithelium. The length of shrinkage along the surface is 2.8 mm. Note that shrinkage is in the direction of the stromal fibers (left-right in photograph), creating a bulge perpendicular to fiber direction. In Fig. 8 all parameters are the same, except the cooling flow rate has been increased to 400 mL/min (estimated convection coefficient is 14 700 W/m2 . K). The epithelium is intact. The indentation in the epithelium left of center was caused by the spacing-string of the heating apparatus. The two dark vertical lines in the epithelium near each edge of the photograph are folds created in the slicing process. Shrinkage extends to 0.4 mm depth, excluding the epithelium; the length of shrinkage along the surface is 1.3 mm. (In this sample, edema has in-
7 ' --
(b)
Fig >. Theoretical iso-temperature lines in the steady-state for absorbes microwave power of 0.376 W, cooling fluid at 17.5"C, and an applicatorcornea gap of 0.67 mm. Isotherms are 2°C apart. (a) Convection coefficient equals 7 x IO3 W / m Z . K. Maximum temperature is 64°C. (b) Convection coefficient equals 1.4 x lo4 W / m 2 . K . Maximum temperature is 60°C.
creased the thickness of the epithelium above its normal value of 0.18 mm.) This heating is the most favorable one of about 50 made with different parameter values: for cases in which the epithelium is intact, the shrinkage region extends deepest. The termination impedance of the applicator was measured as 16 j 22 Q with a saline gap of 0.67 mm; the corresponding power reflection coefficient with a 50 Q feedline is 0.33. Fig. 9 shows theoretical isotherms in the steady-state for a gap of 0.67 mm, a saline temperature of 17.5"C, and absorbed microwave power of 0.376 W. In (a), the surface convection coefficient is 7 x lo3 W/m2 . K (estimated corresponding flowrate is 100 mL/min); in (b) the convection coefficient is
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. I . JANUARY 1991
TABLE I Summary of Comparison of Theory to Experiment
Method Power' Cooling' Depth' Width4 Max. Temp. Surface Temp.
Experiment 25 w 100 mL/min 0.78 mm 2.8 m m -
Experiment 25 w 400 mL/min 0.59 mm 1.3 mm -
Theory 0.316 W 7 x IO' W/m2 . K 0.84 mm 1.1 mm 64°C 29°C
Theory 0.376 W 1.4 x lo4 W/mZ . K 0.72 mm 0.74 mm 60°C 23°C
'Microwave source power for experiment, cornea absorbed power for theory. 'Flow rate values in experiment yield corresponding convection coefficient values in theory through flat plate Nusselt number correlation. 'Depth of shrinkage effect in experiment, depth of temperature 256°C in theory; measured from top surface of a normal epithelium of thickness 0.18 mm. 4Width of shrinkage effect in experiment, width of temperature 256°C in theory.
1.5 X lo4 W/m2 . K (estimated corresponding flowrate is 400 mL/min). Absorbed microwave power is the power absorbed in the cornea itself; it was adjusted to give a 60°C temperature maximum in case b). The theoretical case of Fig. 9(a) corresponds to the experimental case of Fig. 7; likewise, Fig. 9(b) corresponds to Fig. 8. The correspondence is approximate because in the experiment we know only that the cornea has reached shrinkage temperature, not the exact value attained. The smaller convection coefficient in Fig. 9(a) gives a local maximum 0.4 mm below the cornea surface (including epithelium). The region reaching a temperature 2 5 6 ° C extends to 0.84 mm depth, including the epithelium. The surface temperature is 29°C at the applicator centerline. The width of the region reaching ? 56°C is 1.1 mm. The maximum temperature is 64°C. The larger convection coefficient in Fig. 9(b) reduces region reaching 256°C; the width is 0.74 mm, and the depth is 0.72 mm (including the epithelium). The surface temperature is reduced to 23°C on the centerline, but the local maximum remains at 0.4 mm depth. The results shown in Figs. 7-9 are summarized in Table I. A theoretical calculation of the transient response showed that a temperature within 10% of final value is achieved 10 sec after a step input of microwave power for the cases above.
DISCUSSION This heating system would not deposit significant power in the ocular lens, since virtually all power is absorbed within a tissue radius equal to the microwave applicator radius [ 131. Furthermore, the theoretical calculations show there is no significant temperature rise in the deeper structures due to thermal conduction. The heat flux outward at the cornea surface is roughly seven times the heat flux inward toward deeper structures. Therefore, we would expect little possibility of cataract formation. The required value of microwave source power depends partly on the particular cables we used, since some power was absorbed in the feedlines. In general, source power is much larger than cornea absorbed power, due to feedline losses, reflection at the applicator-saline interface, and absorption in saline. The shrinkage effect is extremely sensitive to applicator-cornea gap. Although the gap can be fixed with a set screw, we believe the device is sensitive to how it is held against the eye. For example, if it were not perpendicular to the eye, then the
effective gap would be smaller on one side, and the shrinkage pattern would be skewed. Placement technique may also affect sparing of the epithelium. Whereas sparing was correlated only weakly with saline flow rate, it may be that mechanical forces were significant in removing the epithelium. Shrinkage in the central stroma is necessary to produce permanent changes in the shape of the cornea [12]. The experimental results in Figs. 7 and 8 indicate that this apparatus produces shrinkage to a depth of 0.6-0.8 mm from the outside of the epithelium, depending on the value of flowrate. This does not span the cornea of the steer, which is more than 1 mm thick; however, this is greater than the thickness of the human cornea, about 0.5 mm, depending on location. Thus, the endothelium (inner layer) of the human cornea could be heated significantly. Since this is undesirable, a smaller diameter applicator scaled to the human cornea could be used to reduce temperature at depth. Swicord and Davis showed that the radius in which a fixed fraction of all radiated power is absorbed is directly related to applicator diameter [ 131. The agreement between theory and experiment is only approximate. The width of shrinkage is roughly twice as great in experiment as it is in theory. The depth of shrinkage in experiment is about 85% of that in theory. Of course, in experiment, the cornea geometry changes when shrinkage occurs. Dimensions parallel to the surface decrease, and perpendicular dimensions increase. Thus, if the experimental temperature distribution were imagined without shrinkage having occurred, it would be even wider parallel to the surface and even shallower perpendicular to the surface. This makes agreement between theory and experiment even poorer. An explanation may be that cornea thermal conductivity is anisotropic, being larger in the direction of fibers (parallel to surface) than across fibers. Also relevant is the possibility that thermal conductivity increases after shrinkage in the direction parallel to the fibers, increasing the diameter of the shrinkage region. The theoretical model predicts a reduction in surface temperature of 6°C when cooling flow rate increases from 7 x lo3 W/m2 K to 1.5 x lo4 W/m2 K. However, the higher flow rate does not necessarily protect the epithelium. In our experiments, there was only a weak correlation between a high flow rate and an intact epithelium. The experimental and theoretical transient response results agree, in that little change in shrinkage was noted between 10 and 20 s heatings in experiment, and in theory the steady state was reached in about 10 s.
~
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TREMBLY AND KEATES: MICROWAVE HEATING AND COOLING OF CORNEA
CONCLUSIONS T h e 3.6 m m OD coaxial m i c r o w a v e applicator driven a t 2450 MHz shrinks t h e stroma o f a n excised steer cornea a t least 0.6 m m below the surface w h e n surface cooling b y r o o m temperature saline is enforced; t h e epithelium m a y b e spared. T h e reg i o n of shrinkage is a d i s k about 1 m m i n diameter. Agreement between theoretical predictions a n d experiment is only approximate, possibly d u e to anisotropic thermal conductivity i n t h e
cornea. ACKNOWLEDGMENT
The authors wish to a c k n o w l e d g e t h e assistance of W . L. Withers, B. M. Levin, a n d E. D r a k o u , w h o performed t h e experiments a n d calculations f o r this research a n d created s o m e of t h e figures; L. J . H a m m , w h o coordinated t h e histology a n d preparation of t h e manuscript; Dr. J . D. Fratkin, w h o advised on t h e interpretation of t h e histology; a n d Dr. F. B o w m a n a n d his co-workers, w h o made his equipment available f o r measurement o f thermal properties o f t h e cornea a n d assisted in t h e procedure.
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1121 J. D. Doss and J . I. Albillar, “A technique for the selective heating of corneal stroma,” Contact Lens, vol. 6 , no. 1, pp. 13-17, Jan.-Mar. 1980. 1131 M. L. Swicord and C. C. Davis, “Energy absorption from small radiating coaxial probes in lossy media,” ZEEE Trans. Microwave Theory Tech., vol. 29, no. 11, pp. 1202-1209, Nov. 1981. [14] C. Grabriel, R. J. Sheppard, and E. H. Grant, “Dielectric properties of ocular tissues at 37 degrees C , ” Phys. Med. Biol., vol. 28, no. 1, pp. 43-49, 1983. [I51 R. P. Glovsky, “Instruction manual for THERMAP,” M.S. Thesis, Thayer School of Engineering, Dartmouth College, June 1982. [16] J . R. Welty, C. E. Wicks, and R. E. Wilson, Fundamentals of Momentum, Heat, and Mass Transfer. New York: Wiley, 1976, pp. 169-179, 330-335. [I71 T . A. Balsubramaniam and H . F. Bowman, “Thermal conductivity and thermal diffusivity of biomaterials: a simultaneous measurement technique,” J . Biomech. Eng., pp. 148-154, Aug. 1977. [18] H. F. Bowman, E. G . Cravalho, and M. Woods, “Theory, measurement, and application of thermal properties of biomaterials,” Ann. Rev. Biophy. Bioeng., vol. 4 , pp. 43-80, 1975. [19] F. P. Incropera and D. P. Dewitt, Fundamentals of Heat and Mass Transfer. New York: Wiley, 1985, p. 393.
REFERENCES r11 S. N. Fyodorov and V. V . Durnev, “Operation of dosaged dissection of corneal circular ligament in cases of myopia of mild degree,” Ann. Ophthalmol., pp. 1885-1890, Dec. 1979. [21 J. W. Cowden and L. D. Bores, “ A clinical investigation of the surgical correction of myopia by the method of Fyodorov,” Ophthalmol., vol. 88, no. 8, pp. 737-41, Aug. 1981. [31 K. J . Hoffer and J. J. Darin, “Three years experience with radial keratotomy: the UCLA study,” Ophthalmol., vol. 90, no. 6 , pp. 627-641, June 1983. 141 K. J . Hoffer and J. J . Darin, “UCLA clinical trial of radial keratotomy,” Ophthalmol., vol. 88, no. 8, pp. 729-736, Aug. 1981. 151 M. R. Dietz and D. R. Sanders, “Radial keratotomy-an overview of the Kansas City Study,” Ophrhalmol., vol. 91, no. 5 , pp. 467-478, May 1984. 161 R. E. Smith, “Status of radial keratotomy,” J . Refractive Surg., vol. 3, no. 5, p. 187, Sept.-Oct., 1987. 171 J. J . Salz, “How safe is radial keratotomy,” J . Refractive Surg., vol. 3, no. 5, p. 188-189, Sept.-Oct., 1987. 181 P. A. Asbell, N. Justin, and S. Obstbaum, “Peripheral corneal evaluation post radial keratotomy,” Ophthalmol., vol. 91 (suppl.), p. 122, 1984. [91 S. M. MacRae, M. Matsuda, and L. F. Rich, “The effect of radial keratotomy on the corneal endothelium,” Amer. J . Ophthalmol., vol. 10, pp. 538-542, 1985. 1101 H. Stringer and J. Parr, “Shrinkage temperature of eye collagen,” Nature, vol. 204, no. 4965, p. 1307, Dec. 26, 1964. 1111 J. V . Aquavella, “Thermokeratoplasty,” Ophthalmic Surg., vol. 5, no. 1, pp. 39-47, Spring 1974.
B. Stuart Trembly (M’83) received the B.S. degree from Yale University, New Haven, CT, in 1975, and the Ph.D. degree from Dartmouth College, Hanover, NH, in 1982. He joined the faculty at the Thayer School of Engineering at Dartmouth College in 1982, where he is an Associate Professor. His research interest is the application of electrical engineering to biomedical problems. Dr. Trembly was the recipient of a Presidential Young Investigator Award in 1984.
Richard H. Keates received the B.A. degree from the University of Pennsylvania, Philadelphia, in 1953 and the M.D. degree from Jefferson Medical Center, Philadelphia, PA, in 1957. He holds the Irving H. Leopold Chair of Ophthalmology and is Professor and Chairman of the Department of Ophthalmology at the University of California, Imine.
