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predictions obtained from the present study are in agreement with the experimental results obtained by Bourjat and Gautherie,47 Bogdasarov et al.48 and Wittig et al.49

Along with the experimental results found by other researchers,47−49 the numerical results obtained from the present study suggest that the growth of a tumor inside the human eye may be detected using IR thermography based on the thermal abnormality displayed on the corneal surface. From the results presented in Secs. 7.6.2 and 7.6.3, the average increase in the corneal surface temperature for category T1, T2, and T3 eye tumors are found, for Case 1, to be 0.066, 0.198, and 0.273◦C, respectively and, for Case 2, to be 0.028, 0.082, and 0.112◦C, respectively. Other than the category T1 eye tumor in Case 2, the increases in the corneal surface temperature are within the range of thermal sensitivity of a highly sensitive IR camera today.50 The thermal asymmetry on the corneal surface may also indicate the location of tumor growth. For instance, a higher temperature at the superior of the corneal surface may imply that the tumor is growing at the positive y region inside the human eye (see Case 1).

It is worth noting that the corneal surface temperature is sensitive to factors such as the environmental condition (ambient temperature and ambient convection coefficient) and physiological variations among individuals (age, blood convection coefficient, and tear evaporation rate). According to Scott36 and Ng and Ooi,32 these factors have been found to produce changes in the corneal surface temperature that are of the same order of magnitude as those caused by the eye tumor. Therefore, to accurately measure the effects of the eye tumor on the corneal surface, it is important to ensure that changes in the corneal surface temperature recorded using an IR camera are solely caused by the eye tumor and are free from the effects of other factors that are classified as “noise.” A parametric optimization based on ANOVA and the Taguchi method may be carried out to explain the precedence and the importance of the various factors that may affect the corneal surface temperature. These methods are presented next.

7.7. Parametric Optimization

The effects of eye tumor on the ocular temperature distribution have been demonstrated in the previous section. A tumor inside the human eye has

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been shown to cause an increase in the temperature of surrounding ocular tissue. Furthermore, an increase in temperature and a small degree of thermal asymmetry on the corneal surface have been predicted. We have also mentioned that factors pertaining to the environment and the physiological condition of the human eye can produce changes in the corneal surface temperature that are of the same order of magnitude as those caused by the eye tumor. In this section, a statistical analysis based on ANOVA is carried out to identify the precedence and importance of various factors that may affect the corneal surface temperature of an eye that is infected with a tumor. An optimization procedure based on the Taguchi method is then performed to determine the optimal setting of factors that maximizes the signal from the eye tumor, while isolating the effects of noise factors. The statistical study carried out in this section assumes that the use of IR thermography for detecting eye tumors is a well-established technique.

7.7.1. Analysis of Variance

ANOVA is a statistical tool that is often applied in the design of experiments and is generally used for deciding whether and which factor(s) or interaction(s) has a significant effect on the response of a system. In the context of the numerical investigation carried out in this chapter, ANOVA is used to determine the factors or interaction of factors that have dominant effects on the corneal surface temperature.

From the results obtained in Sec. 7.6, it is found that the metabolic heat generation, size, and location are among the factors of the eye tumor that determine the magnitude of temperature increase on the corneal surface. Effects of the tumor-blood perfusion rate and thermal conductivity are found to be small. From the results of the sensitivity analysis carried out by Ng and Ooi, the factors such as ambient temperature, ambient convection coefficient, blood convection coefficient, body temperature, and tear evaporation rate have been shown to affect the temperature on the corneal surface significantly.32 The precedence and importance of each of these factors are investigated here.

We consider five factors: the ambient convection coefficient, the blood convection coefficient, the tear evaporation rate, the tumor size, and tumor metabolic heat generation, which we denote by A, B, C, D, and E,

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respectively. Effects of ambient and body temperature are not considered. It is assumed that the ambient temperature remains constant when the measurements of the corneal surface temperature are carried out. Likewise, the growth of an eye tumor is assumed to have no significant effect on the temperature of the human body. Although the location of the tumor growth has been found to play a major role in the thermal abnormality on the corneal surface, it is not considered in the present analysis due to the irregular shape of the choroidal surface, which makes a consistent establishment of distance between the eye tumor and the corneal surface difficult. In the present study, ANOVA is carried out only for the location of the eye tumor in Case 1.

A 2n factorial design is used, where n represents the number of factors considered, i.e. five. Each of the five factors has two levels defined by low and high, which leads to 32 (25) runs. Selections of the low and high values of each factor are described as follows. According to Ng and Ooi,32 a value of 10 Wm−2K−1 has been selected to represent the value of the ambient convection coefficient at the control level, i.e. the condition depicting normal environmental condition. It is assumed that the value of the ambient convection coefficient may fluctuate as much as ±20% from the control as measurements is carried out. Values of 80 and 120% of the control, which are given by 8 and 12 Wm−2K−1, respectively, are, thus, considered the low and high values of the ambient convection coefficient. The control value of the blood convection coefficient is given as 65 Wm−2K−1, as suggested by Lagendijk38 in 1982. More recently, Flyckt et al.51 discovered that values between 250 and 300 Wm−2K−1 better represent the blood convection coefficient inside the human eye. Based on this information, the values of 65 Wm−2K−1 and 300 Wm−2K−1 are selected to be the low and high values of the blood convection coefficient, respectively. The heat loss due to tear evaporation in a normal human eye is represented as 40 Wm−2 (see Scott36). According to Mishima and Maurice,52 the tear evaporation rate of a normal human eye may range from 2.8 to 14.7 mg·hr−1cm−2, which corresponds to a heat loss of 20–100 Wm−2. The low and high values of the heat loss due to the tear evaporation rate are, thus, chosen to be 20 and 100 Wm−2, respectively.

Categories T1 and T3 eye tumors are chosen to represent the sizes of eye tumors at levels low and at high, respectively. The diameters of these tumors are given as 10 mm and 17 mm. In the analysis carried out in Sec. 7.6, the

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metabolic heat generation of the eye tumor has been assumed to have a value of between 15,000 and 80,000 Wm−3. The range of metabolic heat generation is obtained using Eq. (7.6), where values of between 1 × 106 and 5 × 106 W·day·m−3, and an eye tumor doubling time, DT of 63 days have been considered. According to Eskelin et al.43 the doubling time of a typical eye tumor ranges from 34 to 220 days. Assuming the value of is 3 × 106 W·day·m−3, the range of metabolic heat generation corresponding to the range of eye tumor doubling time (34–220 days) is calculated as approximately 13,000–90,000 Wm−3. Thus, the lower and upper values of this range are taken to be the low and high values of the eye tumor metabolic heat generation, respectively.

Table 7.5 summarizes the low and high values of each of the five factors considered. The upper case letters (A, B, . . ., E) denote the factors while the lower case letters (a, b, . . ., e) denote the runs. The following 32 runs are carried out: run 1, a, b, c, . . ., ab, abc, . . ., bcde, abcde. Run 1 corresponds to the time at which all factors are at low levels. For instance, the lower case letter appearing in a run would indicate that the letter’s corresponding factor(s), represented in upper case letters, is at a high level. Run “ab” implies that the simulation is carried out for factors A and B at a high level, while factors C, D, and E are at a low level. Similarly, run “de” represents the simulation that is executed when factors A, B, and C are low, while factors D and E are high. In the 32 runs we carried out, the values of the other variables, such as thermal conductivity, ambient temperature, and body temperature are taken at the control level, such as stated by Ng and Ooi.32 The thermal conductivity and the blood perfusion rate of the eye tumor are taken to be 0.5 Wm−1K−1 and

Table 7.5. Low and high values of the five factors considered in the ANOVA.

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Table 7.6. Results obtained from the ANOVA.

Total DOF = 31(32 − 1)

aSS: sum of squares; bDOF: degree of freedom; cMS: mean square; and dMSE: mean square of error.

0.0063 m3s−1m−3, respectively. Observations are made at the superior of the corneal surface (see Fig. 7.2) where temperature rise is expected to be the largest.

Results obtained from ANOVA are summarized in Table 7.6. The interactions of factors beyond the 10th rank are collectively treated as errors due to their small effects when clubbed together. From the results presented in Table 7.6, the tear evaporation rate, (C), appears to be the most dominant factor. This factor is followed by blood convection coefficient, (B), the metabolic heat generation of eye tumor, (E), and the ambient convection coefficient, (A). The combination of factors D and E, (DE), ranks fifth, while the size of the eye tumor, (D), ranks sixth. The effects of both the tear evaporation rate and the ambient convection coefficient are negative; implying that the larger values of A and C yield a cooler corneal surface. The fact that DE ranks higher than D is not surprising, since the size and metabolic heat generation of the eye tumor are mathematically related, whereby a larger tumor produces a greater amount of metabolic heat for the same amount of volumetric metabolic heat generation.

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