1. Belarus National Academy of Sciences
Institute of Physics
Belarus National Academy of Sciences
Thermographic imaging of a localized heat source hidden in biological tissue. II. Retrieval of the source parameters
V.V. Barun and A.P. Ivanov

2. Content
1. Introduction
2. Temperature fields in biological tissue with an internal heat source
3. IR radiance exiting tissue surface
4. Effective radiative depth
5. Retrieval of source and tissue parameters
6. Conclusions

3. 1. Introduction
Thermographic images of tissue surface are currently used in biology and medicine for diagnostics of various pathological conditions of a human organism. However, the applicability range of the employed infrared optical methods is not so wide, because they give rather qualitative, not quantitative information. The reason is that a thermal imaging camera measures infrared fluxes emerging from the surface, but the source of interest is essentially shadowed by infrared light emerging from tissue layers between the region and the surface. So the known thermographic methods can provide only indirect data on the source parameters. What is required for obtaining quantitative information on the source is the “clearing” of the thermal images of the surface from the noisy background created by the said intermediate tissue layers.

4. To simulate the infrared imaging, we believe that a local heat source with a higher of lower temperature than that of its surroundings is in biotissue. This source represents a pathological state of the organism owing to various abnormal metabolic processes and some other reasons. The source can be an internal organ, blood vessel, etc. Several experimental procedures and schemes are mentally proposed here for doing so. They are based on the solutions to the heat and radiation transfer equations for a heat source of some simple shapes. It is theoretically shown that, in principle, one can retrieve the source depth in tissue, its thermal power or temperature from infrared images of tissue surface. The limiting capabilities of the proposed procedures are estimated for a number of thermal imaging cameras operating over different spectral ranges. .

5. 2. Temperature fields in biotissue with an internal heat source
Spatial temperature distribution inside tissue can be represented as
T(r,z) = T0(z) + DT(r,z),
where r is the radial coordinate, z – depth, T0(z) = Ts + hz(Ts – Ta) – temperature without source, h – heat exchange parameter at the surface, Ts and Ta – temperature values of the surface and environment, DT – temperature increment due to the source action. We will use the known analytical solutions for temperature increment DTp from a point spherical source (J. Draper, J.W. Boag, Phys. Med. Biol., 1971, 16, 201) located at z = a, r = 0 :
Here Q is the heat power, W of the source, a – its depth , k – heat conduction of tissue, J0 – Bessel function of zero-th order.

6. The similar equation can be written for a line cylindrical source with power Q*, W/cm per unit length (Awbery J.H., 1929, Phil. Mag., 7, 1143) located at z = a, x = 0.
By integrating over source volume, one can get spatial temperature distributions from spherical and cylindrical heat sources with finite dimensions (diameter d of the sphere or of the cylinder base).
It follows from the above two equations that one can pose an inverse problem on the retrieval of source parameters Q/k or Q*/k, d and a, as well as of heat exchange parameter h at the tissue surface.

7. Our calculations for finite dimension d heat sources have showed that temperatures outside the source are the same as for ideal point or line ones. The differences can be observed at spatial coordinates inside the source only. The important conclusion is followed from this fact. IT IS IMPOSSIBLE TO ESTIMATE SOURCE DIMENSIONS BY MEASURING TEMPERATURE EXCESS OF TISSUE SURFACE.
However, an IR sensor provides optical data on the thermal luminance of tissue surface. This opens some opportunities for retrieving source sizes.
For the simplicity, only spherical source will be considered below.

8. 3. IR radiance exiting tissue surface
Above non-uniform temperature distributions enable one to calculate IR radiance exiting surface with using the Plank formula and Kirchhoff law
r0 is the surface reflectance, k – tissue absorption coefficient (water), f(T) = Mexp(-N/T)=f(Ts)(1 + NDT/Ts2) – Plank formula at DT<<Ts.
Radiance excess В, W/(cm2sr mm K) due to internal heat source as a function of wavelength ,mm for h = 0.05 (curve 1) and 0.8 (2) сm-1
We will consider below monochrome recording of radiance at specific l and polychrome one by InSb (l_max=5.3 mm) and by CdxHg1-xTe (l_max=9 mm) detectors.

9. 4. Effective radiative depth
Let effective radiative depth (ERD) be a tissue thickness, from which emitting 99 % of radiation power exiting the surface. The calculations have shown that it is independent of the source shape, its depth, power and very weakly depends on parameter h.
The only factor making dominant contribution to this quantity is the absorption coefficient (see Fig).
For the above polychrome detectors the ERD values are 0.23 and 0.11 cm, respectively
Note that ERD values determine the limiting source dimensions, which can be retrieved in the IR spectral range 2 – 10 mm.

10. 5. Retrieval of source and tissue parameters
Consider first the common retrieval of the source depth a and heat exchange parameter h. An example is shown below for the case, where IR radiance exiting the surface is independent of source size d.
Let one measures radiance excess DB(r) =B(r) – B0, where B(r) is the radiance created by a spherical heat source at radial distance r along the surface, B0 is the radiance without a localized heat source(or at large distance from it), at three points r = 0, 1 and 3 cm. Construct ratios DB(r)/B(0). Calculate them for various a and h values. The corresponding nomograms are shown below.
Nomograms for common retrieval of a and h values. Bold (near “horizontal”) and thin (near “vertical”) lines correspond to constant depth a and parameter h. Abscissa is for r=1 cm, ordinate for r=3 cm.
These and similar data enable one to commonly retrieve depth a and parameter h.

11. Then let source size d be retrieved under known a and h values.
Measure the radial pattern of radiance excess DB(r).
THESE AND SIMILAR DATA ENABLE ONE TO RETRIEVE SOURCE DEPTH
Radiance excess В as a function of r, cm of tissue surface. Internal heat source with d =1 cm (dotted curves, upper abscissa axis) and 0.1 cm (solid, lower abscissa axis) for a, сm: curve 1 – 0.5, 2 – 1, 3 – 2, 4 – 3, 5 – 0.05, - 6 – 0.1, 7 – 0.25, 8 – 0.5

12. At the final step, source heat power Q/k or average temperature of the source can be retrieved under known a, h and d values. Let the temperature T or temperature excess DT be measures at the surface immediately above the source (at r=0). Then
Q/k = DTΨ(h,a,d),
where Ψ(a,h,d) is the known function. In particular, for point source
These function can be easily calculated.

13. 5. Conclusions
Here is proposed a complete algorithm for retrieving heat source parameters by measuring IR images of tissue surface. The parameters include source depth a, source diameter d, and heat power Q. The errors of the retrieval scheme and its practical implementation are beyond the scope of the presentation. The maximal source depth, at which the scheme will probably work successfully, depends on many factors. At this stage, we can estimate it as 2 – 3 cm from the tissue surface.

14. Thank you for the attention!