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. I. Direct problem on image characteristics
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. Measurement of surface temperature with an IR sensor
5. Conclusions

3. 1. Introduction
Thermography is the measurement of an object temperature with further goal to make some conclusions on its conditions. The temperature is usually determined by an IR sensor or thermal vision device. Thermographic methods on the base of IR images are now actively introduced into medical practice. Their essence is the detection of IR radiation from an open biotissue surface and making the conclusion on an inflammatory pathology of an internal region with enhanced temperature or on a dystrophic pathology for lowered temperature. The temperature distribution over volume of a living biological object is well known to depend on the depth of an observation point in the medium. For example, skin surface is usually colder than tissue interior. Meanwhile, an optical thermal imaging device measures the temperature on the base of infrared radiance of light exiting the tissue surface. To what tissue portion does this temperature correspond? The problem becomes more intricate, if there is an internal localized heat source in tissue, which leads to an even more non-uniformity of the temperature.

4. The answer to the posed question depends on the working wavelength or spectral range of the device, the source parameters such as its depth and thermal power, heat exchange conditions at the surface, etc. We estimated temperature distributions over tissue volume created by heat sources of several simple shapes, namely by spherical, cylindrical and plane ones. The sources simulate a pathological region inside a living organism and can have a higher or lower temperature than that of its surrounding tissue. The temperature fields are calculated on the base of analytical solutions to the corresponding heat transfer equations. Then infrared luminosity of the skin surface created by the whole tissue volume is evaluated with accounting for the spectral absorptivity (due to water, mainly) of the medium. So, the thermal images of tissue surface are investigated for various spectral observation conditions to elaborate some approaches for solving the inverse problem on the retrieval of heat source parameters.

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):
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.
The similar equation can be written for a line cylindrical source (Awbery J.H., 1929, Phil. Mag., 7, 1143).

6. There are known investigations of temperatures inside soft tissue created by the said ideal sources (heated point or line). No studies have been made for sources with finite dimensions simulating heated internal organs or blood vessels. Meanwhile, the corresponding distributions can be obtained by integration of the above temperatures over respective spatial coordinates. This is especially important for studying IR radiative fluxes recorded by a thermal vision device.
Surface temperature excess Tp as a function of radial coordinate r at h = 0.05 см-1 (dotted curves) and 0.8 (solid), Та = 293 К and Тs = 306 К for a point source Q=1 mW located at different depths а , cm: 1 – 0.5, 2 – 1, 3 – 1.5, 4 – 2, 5 – 3
For real Q values, Tp can be several K.

7. Our calculations for finite dimension 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 (see below).

8. Temperature excess inside spherical heat source, Q=1 mW with diameter d=0.1 cm (left) and 1 cm (right) located at depth a=0.05 cm (solid curves) and 0.5 cm (dotted), h=0.05 (curves 1) and 0.8 cm-1 (2). Here are shown temperatures along diameter normal to tissue surface

9. 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 od radiance at specific l and polychrome one by InSb (l_max=5.3 mm) and by CdxHg1-xTe (l_max=9 mm) detectors.

10. IR thermal images (radiance excess В as a function of r, cm) of tissue surface. Internal spherical 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
Therefore, the optical images bear information on source sizes as opposite to the temperatures. Besides, source depth can be retrieved.

11. 4. Measurement of surface temperature with an IR sensor
Tissue temperature is non-uniform over its volume. Usually it is lower at the surface and higher in deep layers. An optical receiver records integral light fluxes emitted by all tissue layers of its volume. So, if temperature inside tissue is higher than at the surface, an IR sensor will give a larger signal as compared with the case of uniform temperature over the volume. Therefore, the surface temperature value Ts* reconstructed from the data of the sensor will be higher than real surface temperature Ts. Let difference Ts*- Ts=dT be a temperature correction that should be accounted for to determine Ts. The correction is obviously to depend on the tissue absorption coefficient, heat exchange conditions at the surface, as well as on the power of localized heat source, its depth, sizes, etc.

12. Temperature correction dT at r=0 as a function of absorption coefficient k (wavelength) for small (a) and large (b) k values, a=0.05 (curves 1), 0.06 (2), 0.07 (3), 0.1 (4), 0.5 (5), 3 cm (6), without localized source (7), Q=11 mW, h = 0.05 сm-1, d = 0.1 сm
Curves 1 and 2 correspond to a spherical source located very closely to the surface. Source temperature at the surface is higher than that of internal layers. So the correction is negative. As the source becomes deeper in tissue, the temperature of internal layers is higher than at the surface. The correction is positive.
For large absorption, the luminance of tissue volume makes smaller contributions to IR light exiting the surface. The correction is small.

13. Temperature correction at r=0 as a function of d for k , сm-1: curve 1 – 13.2 (l=2.2 mm), 2 – 21.5 (2.1 mm), 3 – 40 (2.4 mm), 4 – 55.3 (2 mm), а = 0.1 (solid curves), 0.3 (dashed), 0.6 сm (dotted), average source temperature 5 K, h = 0.05 сm-1

14. 5. Conclusions
1. Temperature fields inside soft biological tissue with localized heat source of finite dimensions are simulated. It is shown that source sizes cannot be retrieved by measuring temperature distribution over the tissue surface.
2. Spectral radiance values at wavelengths 2 – 10 mm exiting tissue surface are simulated. . It is shown that these measurements provide an opportunity to retrieve source sizes.
3. Measurements of the temperature of tissue surface by the data of an IR vision device should be corrected. The temperature correction can be up to several degrees. It can be positive or negative. This factor should be accounted for, while calibrating an IR sensor by using a standard tissue as an etalon.

15. Thank you for the attention!