Temperature Distribution Inside the Human Eye with Tumor Growth
Let us assume that Eq. (7.6) is valid for all types of tumors. Since the doubling time is a unique property of tumors, the only variable in Eq. (7.6) that may change to account for different types of tumors is the constant . Instead of using the range of metabolic heat generation of breast tumor, it may be more appropriate to allow the value of to vary within a certain range. In this study, we shall assume to have a value that ranges from 1 × 106 to 5 × 106 W · day · m−3. For an average eye tumor doubling time of 63 days,43 the corresponding range of volumetric metabolic heat calculated using Eq. (7.6) is found to be approximately 15,000–80,000 Wm−3.
By establishing the range of thermal conductivity, blood perfusion rate, and metabolic heat generation of various tumors of the human body, we may now assume that the thermal properties of an eye tumor lie within the corresponding range of values given in Table 7.2. The values given in Table 7.2 have been obtained for different types of tumors that grow on various parts of the human body, and this assumption is likely to remain valid, as we do not expect the thermal properties of the tumors to be significantly different from one another.
7.5. Numerical Scheme
7.5.1. Integro-Differential Equations
The boundary element method begins with the derivation of the integral representations of the governing equations. Following the steps outlined by Ang,44 the integral representations of Eqs. (7.1) and (7.2) are respectively given as
λ(ξi, ηi, ζi)T(ξi, ηi, ζi) = |
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− i 3D(x, y, z; ξi, ηi, ζi) ∂n |
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for (ξi, ηi, ζi) Ri i and |
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for i = 1, 2, 3, 4, 5 and 6, |
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Ooi, E.H. and Ng, E.Y.K. |
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Ti(x, y, z) ∂n [ 3D(x, y, z; ξi, ηi, ζi)] |
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× dA(x, y, z) − i |
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(x, y, z; ξi, ηi, ζi) ∂n |
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× [Ti(x, y, z)]dA(x, y, z)
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3D(x, y, z; ξi, ηi, ζi) |
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× [αTi(x, y, z) − β]dV(x, y, z) |
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for (ξi, ηi, ζi) Ri i and |
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for i = 7, |
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where α = (ρblcbl bl)/κi, β = (ρblcbl blTbl + Qm)/κi, i denotes the curve boundary of region Ri, dA(x, y, z) denotes the area of an infinitesimal portion of surface i, dV (x, y, z) denotes the volume of an infinitesimal portion of region Ri, λ(ξi, ηi, ζi) = 1, if point (ξi, ηi, ζi) lies at the interior of region Ri, λ(ξi, ηi, ζi) = 1, if point (ξi, ηi, ζi) lies on the smooth part of surface i, and if 3D(x, y, z; ξi, ηi, ζi) and ∂ 3D(x, y, z; ξi, ηi, ζi)/∂n are the fundamental solution of the 3D Laplace equation and its normal derivative, respectively, which are given as:
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4π [(x − ξi)2 + (y − ηi)2 + (z − ζi)2] 23 |
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where nx, ny, and nz are the components of the outward unit normal vector on i in the direction of x, y, and z, respectively.
The boundary element method is implemented by dividing the boundaryi into small triangular elements. The domain integral appearing in Eq. (7.8) is converted into an equivalent boundary integral using the dual reciprocity method.21 For a detailed derivation, one may refer to Ooi et al.45
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