Non-Fourier Bio-heat Transfer

About this note

A plain-language summary of the 2 publications listed under this topic on Academic Publications (Topic-wise). Full citations are under References.

Why this matters

Fourier’s law, the basis of almost every heat conduction calculation, says that heat flux responds instantly to a temperature gradient, so a thermal disturbance travels infinitely fast. For metals the error is negligible, as the delay is of the order of picoseconds. Biological tissue, a heterogeneous mix of cells, fibres and fluid, is different: its thermal relaxation times are estimated at roughly 1 to 100 s (Narasimhan and Sridhar, 2013). When a laser deposits heat within milliseconds, as in retinal surgery or photo-thermal cancer therapy, Fourier’s law may mislead.

The Cattaneo-Vernotte (hyperbolic) model adds a time lag on the heat flux, making conduction wave-like. The dual phase lag (DPL) model adds a second lag, on the temperature gradient. With lags ${\tau }_q$ (heat flux) and ${\tau }_T$ (temperature gradient), transport is wave-like when ${\tau }_T/{\tau }_q$ is much less than 1, Fourier-like when it equals 1, and “over-diffusive” (faster spreading than Fourier) when it exceeds 1 (Narasimhan and Sridhar, 2013). Reliable values of these lags for human tissue are not known.

What we did

We combined the DPL relation with the Pennes bio-heat equation, which adds the cooling effect of blood perfusion (blood flowing through the tissue’s fine vessels) to heat conduction.

With Sridhar Sadasivam, we modelled a 0.2 W retinal laser shot in a one-dimensional, seven-layer eye (Narasimhan and Sridhar, 2013). Most of the energy is absorbed in the retinal pigmented epithelium (RPE), a layer about 10 microns thick. The code was validated against a semi-analytical DPL solution and, with both lags zero, against our earlier Fourier results. Thermal damage was computed with the Arrhenius damage integral, which converts a temperature history into the fraction of tissue destroyed. We held ${\tau }_q$ at 2 s and set ${\tau }_T$ to 0.1, 2 and 5 s to cover all three regimes.

With Nilamani Sahoo, Soham Ghosh and Sarit K. Das, we took the question to the laboratory (Nilamani et al., 2013). Collagen tissue mimics, plain or loaded with gold nanostructures, were heated with a 1064 nm laser, and measured surface temperatures were compared with Fourier and DPL predictions (using first and higher order expansions of the DPL relation) and a new laser source model based on multiple scattering of light.

What we found

In the retina model, the lags changed the required heating time dramatically: full damage of the RPE took about 4 ms when ${\tau }_T$ = 0.1 s but about 48 ms when ${\tau }_T$ = 5 s (Narasimhan and Sridhar, 2013). A small ${\tau }_T$ traps heat in the RPE, with a tiny heat-affected zone but uneven damage; a large ${\tau }_T$ spreads it, with uniform damage that extends about 10 µm into the neighbouring choroid and vitreous. Put simply, ${\tau }_q$ governs localisation of laser energy and ${\tau }_T$ its diffusion away. Damage became significant only above about 55 °C, and blood perfusion had no meaningful effect, since laser heating was about three orders of magnitude larger than perfusion cooling. Fourier and DPL results converged once heating outlasted the lags, so non-Fourier effects matter only for processes faster than the relaxation time.

The experiments agreed (Nilamani et al., 2013). The DPL model with our source term matched measured surface temperatures well, and its small departure from Fourier predictions appeared during early heating, after which diffusion dominated. The data suggested lags of about 2 to 8 s for heat flux and 0.045 s for temperature gradient, lower than some earlier estimates. Both studies argue that tissue conducts heat in a non-Fourier way at short times, and that measuring these lags properly remains an open task.

Related

• Bio-Heat Transfer in Cancerous Tumour Ablation
• Bio-Heat Transfer in Human Eye

References

1. Nilamani, S., Ghosh, S., Narasimhan, A. and Das, S. K., (2013), “Investigation of non-Fourier effects in bio-tissues during laser assisted photothermal therapy,”   Int. J. Thermal Sci., 76, 208-220. [DOI: 10.1016/j.ijthermalsci.2013.08.014 | Abstract]
2. Narasimhan, A. and Sridhar, S., (2013), “Non-Fourier bio heat transfer modelling of thermal damage during retinal laser irradiation,”   Int. J. Heat and Mass Transfer, 60, 591-597. [DOI: 10.1016/j.ijheatmasstransfer.2013.01.010 | Abstract]