# Global Thermal Modeling of Lung Heat Transfer with Blood Perfusion

Jean-François Duhé, Stéphane Victor, Pierre Melchior, Youssef Abdelmounen, François Roubertie

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Jean-François Duhé, Stéphane Victor, Pierre Melchior, Youssef Abdelmounen, François Roubertie. Global Thermal Modeling of Lung Heat Transfer with Blood Perfusion. Proceedings of the International Conference on Fractional Differentiation and its Applications (ICFDA'21), Springer International Publishing, pp.231-238, 2022, 978-3-031-04383-3. 10.1007/978-3-031-04383-3_26. hal-03717297

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# Global thermal modeling of lung heat transfer with blood perfusion

Jean-François Duhé<fnref n="1" />, Stéphane Victor<fnref n="1" />, Pierre Melchior<fnref n="1" />, Youssef Abdelmounen<fnref n="2" />,<fnref n="3" />, and François Roubertie<fnref n="2" />,<fnref n="3" />

**Abstract.** Thermal models often consider low-frequency approximations and are largely based upon RC circuits. If temperature fluctuations are non-negligible, a more accurate model is needed and is proposed through thermal two-port network which is direct extension of the heat equation into matrix formalism. As blood flow plays a major role for heat transfers in biological tissue, a second model is obtained from the bio-heat equation. These two type of models are combined into a global thermal modeling of human lung by considering blood perfusion.

**Keywords:** Heat equation, bio-heat equation, blood perfusion, fractional calculus, fractional order system, human lung

## 1 Introduction

There is a wide range of applications in which temperature is critical, such as chip cooling, power systems, building comfort or biomedical systems. Very accurate models can be obtained via finite element methods, but they are computationally demanding and not well-suited for real-time applications.

RC circuits are simple and well-known models. They can be used for power electronics [1], human heat losses [2] or bio-impedance models [3]. Most of these applications assume low-frequency behavior. The thermal two-port network is introduced [4] in order to obtain a circuit model directly derived from the heat equation. Such a model greatly improves the accuracy of the time response and widens the frequency validity range.

On the other hand, this paper is focused on heat transfers in human lungs. During cardiac surgery, lungs will be under extra corporeal circulation (ECC). Organs are therefore exposed to harsh conditions and can get irreversible damages [6]. The therapeutic effect of a controlled hypothermia to protect the lungs has been studied [7] and usually real-life scenarios involve the use of thermal pads over the lung exposed surface. However, these techniques are empirical and

<footnote n="1">¹ Univ. Bordeaux, CNRS, IMS-UMR 5218, Bordeaux INP/enseirb-matmeca, 351 cours de la Libération, 33405 Talence Cedex France (e-mails: name.lastname@ims-bordeaux.fr)</footnote>

<footnote n="2">² IHU Liryc, Electrophysiology and Heart Modeling Institute, Fondation Bordeaux Université 33000 Bordeaux Cedex France</footnote>

<footnote n="3">³ INSERM, Centre de recherche Cardio-Thoracique de Bordeaux, 33000 Bordeaux Cedex France (e-mails: name.lastname@u-bordeaux.fr)</footnote>
a better understanding of thermal transient will allow a better thermal control to minimize injuries.

Lungs exhibit complex geometry and its fractal nature was originally proposed by Mandelbrot [8]. Moreover, mechanical impedance models were explored in [9], where fractional operators were used to characterize lung fractality. Blood flow prevents tissue injuries as well as temperature regulation. Perfusion is taken into account thanks to Pennes' bio-heat equation, which is described as the following:

$$ \rho c \frac{\partial T}{\partial t} = k \nabla^2 T + \rho_b c_b \omega_b (T_a - T) + q_m + q_{ext}, \quad (1) $$

where T and $T_a$ respectively stand for tissue and blood temperature. Equation (1) introduces blood perfusion coefficient $\omega_b$ as well as a metabolic term $q_m$ to take into account cell breathing and $q_{ext}$ to include external heat sources (expressed as power per volume unit).

The paper is organized as follows: section 2 presents thermal two-port networks, the global lung model is presented in section 3, simulations and results are presented in section 4 and finally conclusions are given in section 5.

## 2 Two-port network models

Consider a thermal wall with 1D heat conduction as shown in figure 1, with section $S_w$ and length $L$. Its properties are given by conductivity $k$, density $\rho$ and specific heat capacity $c$. The blood thermal properties are given by $\rho_b$, $c_b$ and the perfusion coefficient is $\omega_b$.

<figure><img src="image_2.png" /><figcaption>Fig. 1: 1D thermal system</figcaption></figure>

### 2.1 Classic two-port network

The thermal two-port network [4] allows to relate inputs and outputs of a thermal system, namely the heat equation, into a matrix form. The considered variables are the temperature $T$ and the heat flow $\dot{Q}$. Put under matrix form, the two-port network yields (see [5]):

$$ \begin{bmatrix} T_{in}(s) \\ \dot{Q}_{in}(s) \end{bmatrix} = M \begin{bmatrix} T_{out}(s) \\ \dot{Q}_{out}(s) \end{bmatrix}, \quad \text{with} \quad M = \begin{bmatrix} \cosh(\delta_c L) & \frac{1}{kS_w \delta_c} \sinh(\delta_c L) \\ kS_w \delta_c \sinh(\delta_c L) & \cosh(\delta_c L) \end{bmatrix} \quad (2) $$
where $\delta_c = \sqrt{\frac{s}{a}}$ and $a = \frac{k}{\rho c}$.

## 2.2 Bio-heat two-port network

Bio-heat equation is used to take into account the blood influence. Metabolic heat rate and external sources are considered negligible. Variable change $\dot{T} = T - T_b$ is introduced to analyze deviations around the body core temperature $T_b$. The blood temperature $T_a$ is supposed constant and equal to the body core one. This assumption lead to the same two-port network as in equation (2), with the following difference:

$$ \delta_{bio} = \sqrt{\frac{s}{a} + h_b} \qquad (3) $$

where $h_b = \frac{\rho_b c_b \omega_b}{k}$.

Note that $\delta$ involves the natural presence of fractional order derivatives.

## 2.3 T circuit

Matrix $M$ obtained for both the classic and bio-heat two-port network leads to a T circuit representation of the system (see figure 2).

<figure><img src="image_3.png" /><figcaption>Fig. 2: Thermal two-port network</figcaption></figure>

Impedances are given by:

$$ Z_1(s) = Z_2(s) = \frac{1}{kS_w\delta}[\coth(\delta L) - \cosh(\delta L)] \qquad (4) $$

and

$$ Z_3(s) = \frac{1}{kS_w\delta}\cosh(\delta L) \qquad (5) $$

where $\coth$ and $\cosh$ respectively are the hyperbolic cotangent and cosecant functions. The variable $\delta = \delta_c = \sqrt{\frac{s}{a}}$ will be used for systems modelled from the classic heat equation and as $\delta = \delta_{bio} = \sqrt{\frac{s}{a} + h_b}$ if the model is based upon the bio-heat equation.
## 3 Lung global model

Figure 3 shows a lung scheme. Each branch $n$ has a length $L_n$ as well as its cross-section $S_n$. A combination of Murray’s law with a mechanical impedance model [10] allows to establish a relationship between two consecutive cross-sections:

$$ \frac{S_{n+1}}{S_n} = 2^{-\frac{2}{3}}, \qquad (6) $$

as well as two consecutive lengths:

$$ \frac{L_{n+1}}{L_n} = 2^{-\frac{1}{3}}. \qquad (7) $$

<figure><img src="image_4.png" /><figcaption>Fig. 3: Lung scheme</figcaption></figure>

A symmetric lung model may be represented as the circuit shown in figure 4. On a same branching level $n$, the equivalent impedance is the parallel combination of $Z_1$ and $Z_3$, each lung level having $2^n$ branches which give:

$$ Z_{i-eq,n}(s) = \frac{Z_{i,n}(s)}{2^n} \quad \text{for } i = 1, 2, 3 \text{ and } n = 1, 2, \dots, 23. \qquad (8) $$

## 4 Simulations and results

Lung blood circulation is usually non-uniform. Normally the first branches are considered as a dead space used only for passive air transport. The gas exchange zone begins around the 15-th branch. Therefore, three different models are simulated for comparison (see table 1). Simulation parameters are shown in table 2, where the subscript $t$ stands for tissue, $b$ for blood and $a$ for air. Figure 5 shows
<figure><img src="image_5.png" /><figcaption>Fig. 4: Global circuit model for a lung</figcaption></figure>

<table><caption>Table 1: Global model propositions</caption><thead><tr><th>Model</th><th>Dead space (n &lt; 15)</th><th>Exchange space (n ≥ 15)</th></tr></thead><tbody><tr><td><i>Air</i></td><td>Classic</td><td>Classic</td></tr><tr><td><i>Mix</i></td><td>Classic</td><td>Bio-heat</td></tr><tr><td><i>Bio</i></td><td>Bio-heat</td><td>Bio-heat</td></tr></tbody></table>

the equivalent impedance for the three models from the trachea and from n = 13 (interior of the lungs).

As observed, air model behaves as a capacitance for low-frequencies, which would allow slow enough thermal disturbances to fully propagate through the lungs. Mixed and biological models have a constant static gain and similar behavior, though the higher perfusion in the biological model led to a lower static gain. High-frequency behavior coincides for air and mixed model, which means dead space acts as a bypass at such frequencies. The equivalent impedance for the 13-th level is shown in figure 5 to illustrate the differences in the interior of the lungs. As one approaches the exchange zone, capacitance behavior is more present in the air model, which implies more heat accumulation. All proposed models show a slope of -10 dB/dec for the high frequency limit, which means the three equivalent impedances have a fractional-order component. By observing figure 5, one can derive the general structure of impedance for both models
<table><caption>Table 2: Simulation parameters for the lung impedances</caption><tr><td>a<sub>a</sub></td><td>2.2e-5 [m<sup>2</sup>][s<sup>-1</sup>]</td><td>k<sub>a</sub></td><td>0.025 [W][m<sup>-1</sup>][K<sup>-1</sup>]</td></tr><tr><td>ρ<sub>t</sub></td><td>550 [kg][m<sup>-3</sup>]</td><td>ρ<sub>b</sub></td><td>1060 [kg][m<sup>-3</sup>]</td></tr><tr><td>c<sub>t</sub></td><td>3718 [J][kg<sup>-1</sup>][K<sup>-1</sup>]</td><td>c<sub>b</sub></td><td>3900 [J][kg<sup>-1</sup>][K<sup>-1</sup>]</td></tr><tr><td>k<sub>t</sub></td><td>0.28 [W][m<sup>-1</sup>][K<sup>-1</sup>]</td><td>ω<sub>b</sub></td><td>0.0043 s<sup>-1</sup></td></tr><tr><td>L<sub>0</sub></td><td>0.1 m</td><td>d<sub>0</sub></td><td>1.4 cm</td></tr></table>

(mix and bio models exhibit the same mathematical structure):

$$Z_{eq-air}(s) = \frac{K(1 + as)}{s(1 + \tau\sqrt{s})} \qquad (9)$$

and

$$Z_{eq-mix-bio}(s) = \frac{K}{1 + \tau\sqrt{s}}. \qquad (10)$$

<figure><img src="image_6.png" /><figcaption>Fig. 5: Input equivalent impedance for lung system at n = 1 (left) and n = 13 (right)</figcaption></figure>

## 5 Conclusions

A circuit approach for a global model of heat transfer in human lung was presented and three different models were obtained. The equivalent impedance of the models were compared. The air model implies that as one approaches gas
exchange zone, capacitance behavior becomes dominant. This neglects dissipa-
tion and implies strong energy storage on alveolar level. On the other hand, slow
temperature fluctuations may propagate through the whole lung which is not
realistic. Mix and bio models have a static gain which is inversely proportional
to blood perfusion. This means that higher perfusion would require a larger
heat input in order to significantly increase temperature in the lung, which is
coherent with blood thermal regulation role. Research perspectives may include
analyzing more complex scenarios (as respiratory obstructions or illness) to re-
late impedance parameter variations to different medical conditions. This could
lead to a medical application of equivalent thermal impedance identification.

References

1. R. Kunzi: Thermal design of power electronic circuits, Proceedings of the CAS-CERN Accelerator School: Power Converters, Baden, Switzerland. (2015)

2. N. Ismail, N. Ghaddar, and K. Ghali: Electric circuit analogy of heat losses of clothed walking human body in windy environment, International Journal of Thermal Sciences, vol. 127, pp. 105 – 116, (2018) doi/10.1016/j.ijthermalsci.2018.01.025

3. K. Chinen, I. Kinjo, A. Zamami, K. Irei, and K. Nagayama: New equivalent-electrical circuit model and a practical measurement method for human body impedance, Bio-Medical Materials and Engineering, vol. 26, pp. S779-S786, (2015)

4. D. Maillet, S. Andre, J. Batsale, A. Degiovanni, and C. Moyne: Thermal Quadrupoles: Solving the Heat Equation through Integral Transforms. John Wiley & Sons (2000)

5. J.-F. Duhé, S. Victor, P. Melchior, Y. Abdelmoumen, F. Roubertie: Thermal modeling using two-port network impedance fractional-order approximations, In: 17th ASME/IEEE MESA International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE 2021 (2021)

6. Badenes, R., Lozano, A., and Belda, F: Postoperative pulmonary dysfunction and mechanical ventilation in cardiac surgery. Critical Care Research and Practice, 420513. (2015) doi:10.1155/2015/420513

7. Giesinger, R. and McNamara, P. The Impact of Therapeutic Hypothermia on Pulmonary Hemodynamics of Meconium Aspiration Syndrome. American Journal of Respiratory and Critical Care Medicine, 198(2). (2018) doi:10.1164/rccm. 201802-0207LE.

8. Weibel, E. R.: Mandelbrot's Fractals and the Geometry of Life : A Tribute to Benoît Mandelbrot on his 80th Birthday. Fourth International Symposium on Fractals in Biology and Medicine (2014)

9. Ionescu, C.M., Copot, D., and De Keyser, R. A fractional order impedance model to capture the structural changes in lungs. IFAC Proceedings Volumes, 47(3), 53635368. (2014) doi:10.3182/20140824-6-ZA-1003.01124.

10. Kuwahara, F., Sano, Y., Liu, J., and Nakayama, A. A Porous Media Approach for Bifurcating Flow and Mass Transfer in a Human Lung. Journal of Heat Transfer, 131(10). (2009) doi:10.1115/1.3180699.