Jet impingement modeling of cryogen spray cooling: analysis of 2D cryogen temperature distribution

The main goal of this study was to demonstrate the feasibility of using backside infrared imaging to estimate the spatial cryogen temperature distribution during a cryogen spurt. Calculations from numerical models showed that the frontside temperature distribution could be identified at the back side of a thin aluminum sheet. Infrared images were obtained at various timepoints during a cryogen spurt from the back side of an 800-micrometers aluminum sheet and the temperature distribution estimated. The temperature distribution was approximately gaussian in shape. A secondary goal was to calculate the temperature distribution in skin for two cases: 1) uniform cryogen temperature distribution, essentially representative of a 1D geometry assumption; and 2) nonuniform distribution. At the end of a 100-ms spurt, calculations showed that, for the two cases, large discrepancies in temperatures at the surface and at a 60-micrometers depth were found at radii greater than 2.5 mm. These results suggest that it is necessary to consider spatial cryogen temperature gradients during cryogen spray cooling of tissue.


INTRODUCTION
Cryogen spray cooling (CSC) is currently used in conjunction with pulsed laser radiation for treatment of Port Wine Stains (PWS) [1] and wrinldes [2]. Laser heating of target chromophores such as hemoglobin or water in the papillary dermis runs the risk of significant thermal injury to the overlying epidermis due to absorption of incident light by melanin and water. Rapid cooling of the epidermis with a cryogen spurt prior to laser irradiation can result in a marked reduction of the epidermal temperature rise with minimal alteration of the temperature rise in the dermis.
Most previous studies on CSC assumed a 1-D tissue geometry because the sprayed area is much larger than the expected depth over which significant cooling occurs [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. A recent study by Verkruysse et al. [16] investigated the possibility of radial thermal gradients during CSC. Thermocouples were embedded at a 90-pm depth at the central axis of a cryogen spurt and at a lateral distance of 3 mm away from the central axis. The temperature-time history at these two positions differed considerably from one another, suggesting that CSC dynamics were different at these two points.
Knowledge of the surface temperature of skin during CSC is essential for proper treatment of cutaneous abnormalities. However, both contact and non-contact methods of estimating surface temperatures are limited in the amount of knowledge they provide. Thermocouples offer only a point temperature measurement, and their relatively large size (e.g. 30-90 .tm bead diameters [3,[14][15][16]) compared to the effective depth of cooling ( 200-400 m [1 1) may lead to errors in temperature measurements, especially in regions of sharp temperature gradients [1 8]. Infrared temperature measurements are limited due to cryogen film formation on the surface; tI-us film attenuates infrared emission from superficial layers of skin. Since the film thickness as a function of time and radius is unknown, it is not possible to obtain an accurate 2-D surface temperature map from surface infrared images. An alternate technique for estimating the spatial cryogen temperature distribution is to apply a cryogen spurt to the front side of a sample and take infrared images from the back side. If the thermal conductivity of the sample is high and the initial spatial temperature gradients oriented such that axial heat conduction is the dominant heat transfer mechanism, then back-side infrared images can provide a reasonable estimate of the initial temperature distribution.
The goal of this study was to test the predictive capability of 1-D CSC models. We 1) estimated the surface temperature distribution induced by a cryogen spurt using back-side infrared imaging and 2) predicted the difference in transient internal surface temperatures between assuming a uniform (e.g. 1 -D) and non-uniform cryogen temperature.

FEASIBILITY STUDY OF BACK-SIDE INFRARED IMAGING
To determine the feasibility of back-side infrared imaging, an explicit finite-difference model [19] was used to predict the temperature distribution within a thin aluminum sheet subjected to a cryogen spurt of varying duration. The heat conduction equation was solved using cylindrical coordinates and assuming a convective surface boundary condition. Relevant spatial and thermal parameters incorporated into the model are listed in Table 1 . Thermalproperties of the aluminum sheet were assumed to be comparable to that ofAlloy-2024 [19]. In the model, the cryogen spurt covered an area that was 3 cm in diameter. For regions directly under the spurt, the cryogen heat transfer coefficient h was applied; for regions under air, the air heat transfer coefficient was used. In the literature, values of h range between 0.24 and 9 W/cm2/K and cryogen temperature T between -7 and -49°C [4,10,14,16]. For our initial model, we used h 0.24 W/cm2/K and T -44°C, which were estimated from temperature measurements using thermocouples embedded in an epoxy tissue phantom [14].
Three cryogen temperature distributions were assumed: 1) uniform, 2) linear, and 3) gaussian (see Figure 1). Simulations were run using MATLAB 5. The calculated back-side temperature distribution at the end of a 100-ms spurt is shown in Figure 2. Each plot in Figure 2 represents a convolution of a cryogen temperature distribution ( Figure 1) with a thermal blurring function due to radial heat conduction. Blurring is more pronounced with the uniform cryogen temperature distribution then for the linear or gaussian distributions due to the sharp boundary demarcating cooled versus non-cooled regions of the surface. A 3-D surface plot of the calculated temperature distribution inside the aluminum sheet after a 100-ms spurt is shown in Figure 3. In this case, the cryogen temperature distribution was assumed to be uniform. Note that for a given radial position, the temperature values as a function of axial position are relatively constant. Infrared temperature measurements of an object are accurate only if no superficial temperature gradients exist [18,20,21]. This model run suggests that temperatures estimated from back-side infrared images may be fairly accurate representations of temperatures at the back surface.
Results of this feasibility study indicate that back-side infrared imaging is an adequate technique for estimating the cryogen temperature distribution during CSC. The measured temperature distribution may be a slightlyblurred representation of the actual cryogen temperature distribution. Since the difference in shape between the blurred temperature distribution ( Figure 2) and initial cryogen temperature distribution ( Figure 1) is small, the measured temperature distribution seves as a good first approximation.

INFRARED IMAGING DURING CSC OF ALUMINUM SHEET Experimental Setup
The experimental setup is shown in  Infrared image sequences were obtained for five spurt durations: 20, 40, 60, 80 and 100 ms The images were acquired directly into PC memory and were saved to hard disk for subsequent analysis. A total of 75 image sequences were acquired and analyzed.
Blackbody calibration was performed using a thermoelectric cooler (lEG) (Model 2 SC 040 050-127-63, Melcor, Trenton, NJ). The same black paint used on the aluminum sheet was applied to the TE surface. The surface temperature of the TE was monitored with a thermocouple-based digital thermometer system (HH2O2A, Omega Engineering Inc., Stamford, CT'). Blackbody images were acquired at temperatures ranging between -.s°c and 24.2°c.

Image Processing
All image processing was done using LabVIEW 6i software (National Instruments, Austin, TX). The average gray level value of each blackbody image was calculated and a third-order polynomial equation relaling gray level and blackbody temperature derived.
Gray levels of images acquired during csc were converted to temperature values using the polynomial fit. A median filter was applied to each image to remove salt-and-pepper noise caused primarily by bad pixel elements on the focal plane array. Surface plots of average back-side temperature distributions 1 0, 40, and 1 00 ms after the onset of a 100-ms cryogen spurt are shown in Figure 5. Note that the temperature distribution is approximately gaussian in shape.
Temperature line profiles from the approximate centers of the cold regions were obtained. Average line profiles for varying spurt durations are shown in Figure 6. At 100 ms, the temperature distribution is approximately uniform over a 4-mm central region.

VALUES
Results of the thermal model predicted a minimum temperature of 14-1 6°C (Figure 2). From the infrared imaging experiment, the minimum temperature was approximately -6°C, or about 20°C lower ( Figure 6). One possible explanation for this difference is that the value of h we used in the model is too low. Since the radial temperature gradient over a central 2-mm radius is approximately zero (Figure 6), we used a 1-D analytical approximation to calculate the value of h that is necessary for the minimum temperature at the end of a 100ms spurt to be -6°C. For convective cooling of an infinitely-wide plane wall, the temperature at depth is [19]: A plot of back-side temperature as a function of radial position during CSC of an aluminum sheet is shown in Figure 7. A uniform cryogen temperature distribution (Figure 1) at -44°C and heat transfer coefficient of 1.04 W/cm2/K was assumed. Note that the minimum temperature of -5.3°C in Figure 7 is comparable to the niinimum value of -6°C obtained from the infrared images ( Figure 6). Since this value of h approximated the measured minimum temperature better than 0.24 W/cm2/K, we used h 1 .04 W/cm2/K in the model described in the next section.

PREDICTION OF TEMPERATURE DISTRIBUTION WITHIN SKIN DURING CRYOGEN SPURT
Models were run to compare the predicted differences in internal temperature distribution of skin during a 100ms cryogen spurt. The cryogen temperature distribution was assumed to be either uniform or nonuniform. For the uniform case, the cryogen temperature was set to -44°C. For the nonuniform case, the minimum cryogen temperature (e.g. at the center) was set to -44°C. The shape of the modeled cryogen temperature  Table 2. The internal temperature distributions within a 5-mm radius at the end of a 100-ms spurt are shown in Figure 9 for the two cases. Figure 1 0 shows a difference plot (Tof the temperature distributions. Cooling effects are similar for both cryogen temperature distributions within a 2-mm radius. At larger radii, the difference in internal temperature distributions is as great as 40°C. Figure 1 1 shows radial line profiles at different depths; the depth positions were selected based on those chosen by Verkruysse et al. [17]. A depth of 60 .tm corresponds to the basal layer; 150 and 400 tm are two potential PWS vessel target depths. The temperatures at the PWS vessel depths are largely unaffected by the nonuniform cryogen temperature distribution. At the basal layer, however, the uniform temperature assumption predicts basal layer temperatures that are 16 and 28°C cooler at radii of 3.5 mm and 5 mm than for the nonuniform case.  Table 2. Parameters used in finite-difference model of CSC of skin. Figure 9. Skin temperature distribution at end of 100-ms cryogen spurt with the following surface boundary condition: a) uniform cryogen temperature and b) nonuniform cryogen temperature.

DISCUSSION
Previous studies on CSC have been conducted based on the assumption that a 1-D geometry is valid. Recent work from our lab has shown that the 1 -D geometry assumption for laser irradiation studies is reasonable only for relatively wide, un,form temperature changes [22}. If the induced temperature change contains radial gradients over a lateral distance that is not sufficiently large relative to the effective depth of temperature change, then this assumption is no longer valid.
It is very difficult to measure spatial temperature distributions during cryogen spray cooling by using surface temperature measurements. Results of preliminary models simulating CSC of an aluminum sheet demonstrated that spatial temperature gradients induced at the front surface of a thin aluminum sheet can be identified by obtaining infrared images at the back side of the sheet (Figure 2). The approximate shape of the temperature distribution is gaussian (Figures 5 and 6) during a cryogen spurt. At the end of 100 ms, the temperature is relatively constant over a 4-mm-diameter spot ( Figure 6). Recent clinical PWS studies [23,24] have involved the use of 7-mm-diameter spot size. As light propagates through a highly-scattering medium, the effective spot size is larger than the nominal value at the surface. Thus, it is important to consider spatial cryogen temperature gradients during CSC of skin.
By assuming a uniform cryogen temperature distribution, the amount of cooling at the basal layer at radial positions larger than approximately 2.5 mm is considerably overestimated (Figure 11). In order to avoid epidermal damage, this discrepancy should be taken into consideration when devising treatment plans. If the incident laser spot is gaussian-shaped, then the effect of radial cryogen temperature gradients may be mitigated due to the lower incident energy at the edges of the gaussian beam. However, for a fiattop beam profile, radial gradients may play a large role in determining the overall thermal effects during treatment.
In this study, h was assumed to be constant. In reality, the surface heat transfer coefficient may vary as a function of radial position [17,25]. From temperature measurements during CSC of an epoxy block, Verkruysse et al. [16] calculated h 1 W/cm2/K, which is approximately the value estimated in this study. They used a range of h values centered around 1 W/cm2/K as input into a 1 -D model to determine the effect of different he.,, values on the predicted temperature distribution in the block. They found that the calculated temperature values varied by less than 0.5°C when h was varied between 0.59 and 3 W/cm2/K, indicating that relatively large changes in h have a negligible effect on the overall temperature distribution. Thus, as a firstorder approximation, it is reasonable to assume that h is constant. Future studies need to be conducted to assess the relationship between h and radial position.

CONCLUSIONS
Back-side infrared imaging can be used to estimate the cryogen temperature distribution at the front surface during a cryogen spurt. Results of this study indicate that spatial cryogen temperature gradients need to be considered during modeling of CSC. Experimental and detailed modeling studies need to be conducted to determine the ultimate effect of these gradients 9. ACKNOWLEDGMENTS