Continuum modeling perspectives of non-Fourier heat conduction in biological systems
CCONTINUUM MODELING PERSPECTIVES OF NON-FOURIER HEAT CONDUCTION INBIOLOGICAL SYSTEMS ´A. SUD ´AR , , G. FUTAKI , R. KOV ´ACS , , Abstract.
The thermal modeling of biological systems has increasing importance in developing more advanced,more precise techniques such as ultrasound surgery. One of the primary barriers is the complexity of biologicalmaterials: the geometrical, structural, and material properties vary in a wide range, and they depend on manyfactors. Despite these difficulties, there is a tremendous effort to develop a reliable and implementable thermalmodel. In the present paper, we focus on the continuum modeling of heterogeneous materials with biological origin.There are numerous examples in the literature for non-Fourier thermal models. However, as we realized, they areassociated with a few common misconceptions. Therefore, we first aim to clarify the basic concepts of non-Fourierthermal models. These concepts are demonstrated by revisiting two experiments from the literature in which theCattaneo-Vernotte and the dual phase lag models are utilized. Our investigation revealed that using these non-Fourier models is based on misinterpretations of the measured data, and the seeming deviation from Fourier’s laworiginates in the source terms and boundary conditions.
Keywords : non-equilibrium thermodynamics, heat conduction, biological systems1.
Introduction
Thermodynamics is one of the most crucial fundamental blocks of interdisciplinary researches. It is proved tobe useful in numerous coupled problems such as thermo-electricity, thermo-diffusion, and thermo-acoustics [1–3],thanks to the universal definition of the thermodynamic laws. Regarding the topic in this paper, we consider onlythe I. and the II. laws of thermodynamics for heat conduction in a continuum medium. Moreover, we also assumethe material to be rigid, although it is not characteristic of biological systems. We want to keep the discussion assimple as possible to clearly present the essential aspects of non-Fourier heat conduction for such complex systems.Otherwise, mechanics must be included together with its consequences: including source and coupling terms in thebalance and constitutive equations, respectively.These equations serve the basis for continuum models, i.e., the I. law of thermodynamics describes the balanceof internal energy: ρ∂ t e + ∇ · q = 0 , (1)with q being the heat flux, ρ is the mass density, ∇ is the usual nabla operator, e = cT is the internal energydensity with c standing for the specific heat, and T is the temperature. Also, the zero on the right-hand side in(1) expresses the lack of source terms. This equation requires a ‘closure,’ an expression between the temperature T and the heat flux q . Such expression is called constitutive equation, describing a material’s behavior, and strictlyrestricted by the II. law of thermodynamics [4, 5]. In the simplest model, one can use Fourier’s law: q = − λ ∇ T, (2)in which λ is the thermal conductivity. However, as observed in many cases [6–8], this relation is restricted on acertain time, and spatial scale [9], and can lose its validity. Therefore, one must find a suitable, thermodynamicallycompatible generalization.One often referred example in papers on bioheat transfer is the dual phase lag (DPL) model, q ( x, t + τ q ) = − λ ∇ T ( x, t + τ T ) , (3)introduced by Tzou [10]. Here, τ q and τ T are the corresponding relaxation times (time lags), vaguely described inmany papers. According to the original argument of Tzou, τ q can be either smaller or higher than τ T , depending onthe phenomenon. Sadly, numerous works concluded that these relaxation times are not independent of each otherand cannot be arbitrary. Furthermore, the DPL model can violate the basic physical principles [11–15], dependingon τ q and τ T . This problem originates in the Taylor series expansion of equation (3), considered up to arbitraryorders, which is not a thermodynamically compatible way to generalize and derive constitutive relations. a r X i v : . [ phy s i c s . c l a ss - ph ] J a n ´A. SUD´AR , , G. FUTAKI , R. KOV´ACS , , Instead of the DPL model, we propose a similar one with stronger physical background, called Guyer-Krumhansl(GK) equation. In what follows, we first shortly gather the thermodynamic background of the GK model. After,we briefly summarize the known experiences on non-Fourier heat conduction experiments, restricting ourselves onheterogeneous macro-scale materials and room temperature conditions. Then, we revisit two experiments in whichboth Fourier and non-Fourier models are utilized, showing common misconceptions about the possible non-Fouriereffects. With revisiting these experiments, we aim to highlight the less-known properties of non-Fourier models andpresent our perspectives of modeling heterogeneous materials.2.
Thermodynamics of constitutive relations
As we previously mentioned, the II. law of thermodynamics restricts the constitutive equations. In fact, it offersa constructive methodology to find a ‘closure’ and derive the possible set of constitutive equations. The frameworkof Classical Irreversible Thermodynamics [1, 16] formulate the entropy density s as a potential function of thestate variables. For heat conduction, it reads s = s ( e ), and the Gibbs relation d e = T d s expresses the functionalrelationship between e and s . The II. law can be formulated similarly to the I. law: ρ∂ t s + ∇ · J s = σ s ≥ , (4)where J s stands for the entropy flux, and σ s is the entropy production. Eventually, that inequality leads to the setof constitutive equations. ‘Set’ is intentional: there are infinitely many solutions of such inequality. For instance,in case of s = s ( e ) with J s = q /T , it yields σ s = q ∇ T ≥ , (5)for which the simplest linear solution is the Fourier’s law (2). Although many other solutions exist in the nonlinearregime (i.e., nonlinear functions respect to ∇ (1 /T )), but we keep our focus on the linear ones. Such linear solution of(4) still allows to have a state variable-dependent transport coefficient in the constitutive equation such as λ = λ ( T ).In the non-Fourier case, either the set of state variables can be extended, e.g., with the heat flux q , or the entropyflux J s can be more general [17, 18]. Among the various approaches, the non-equilibrium thermodynamics withinternal variables (NET-IV) is applied here due to its useful generality [19–23]. In other words, while the kinetictheory-based Rational Extended Thermodynamics (RET) [24, 25] builds its heat conduction models on phononinteractions, the continuum formulation of NET-IV leaves the particular transport mechanism aside. Consequently,in the RET model, the transport coefficients can be calculated, while in the framework of NET-IV, these coefficientscan be determined through a fitting procedure. Therefore, while it is easier to determine the transport coefficients inRET, it is restricted to systems with high Knudsen number (Kn ≥ . s = s ( e, q ) and J s = Bq , in which B is called current or Ny´ırimultiplier [17], helping to introduce the so-called nonlocal terms (e.g., the ∂ xx q ). For the detailed derivation,explanation, and background, we refer to [26]. The GK equation for a one-dimensional situation in Cartesiancoordinates is τ q ∂ t q + q = − λ∂ x T + κ ∂ xx q, (6)where κ is a sort of ‘dissipation parameter’, usually connected to the mean free path in the kinetic theory. Eq. (6)has two special cases. The first one is the Cattaneo-Vernotte (CV) equation (with κ = 0), the other is a less-knownone, called Ny´ıri equation (with τ q = 0): q = − λ∂ x T + κ ∂ xx q, (7)which is analogous with the Brinkman equation (the nonlocal generalization of Darcy’s law). Although the CVequation is well-applicable for dissipative wave propagation (‘second sound’), it is not observed for macro-scaleheterogeneous materials on room temperature. At that point, we note that many other procedures are possible,depending on the purpose and on the thermodynamic framework in which the problem is formulated [27–32].2.1. Why the Guyer-Krumhansl model?
In our recent heat pulse experiments, we investigated the transientthermal behavior of several heterogeneous materials such as metal foams, rocks, and capacitor specimen [33, 34].Among them, the capacitor sample has the most regular structure in which good conductor and insulator layersare arranged periodically, two materials with different characteristic thermal time scales. The presence of multipletime scales attributes the heterogeneous materials, in general. In the T -representation of the GK model, τ q ∂ tt T + ∂ t T = α∂ xx T + κ ∂ xxt T, (8)the thermal diffusivity α = λ/ ( ρc ) is comparable with the ratio of κ /τ q . When κ /τ q = α , it is called Fourierresonance, and it recovers the Fourier behavior exactly. Otherwise, non-Fourier solutions appear, and these GK ONTINUUM MODELING PERSPECTIVES OF NON-FOURIER HEAT CONDUCTION IN BIOLOGICAL SYSTEMS 3 parameters represent the existence of parallel heat conduction time scales. On the contrary to the usual beliefthat merely wave propagation is possible when non-Fourier effects come into the picture, we found the so-calledover-diffusion for which κ /τ q > α , and Figure 1 shows its characteristics. Remarkably, it seems similar to theclassical case, but the Fourier equation cannot describe the data with acceptable accuracy. Moreover, both thethermal diffusivity and the non-Fourier effects can be size-dependent [35]. Overall, based on our experimentalfindings, the GK equation stands as the next candidate beyond the Fourier equation. Neither of its special casesprovided acceptable predictions. Figure 1.
The signs of non-Fourier heat conduction for heterogeneous materials at room temperature.3.
Bioheat models
Roughly speaking, the related heat conduction models belong to two groups, based on the considered constitutiveequation, but all aim to predict the tissue temperature. In the first group, Fourier’s law is utilized with variousbiologically motivated extensions, mostly being source terms in the balance of internal energy. Particular examplesare the internal heat transfer by blood perfusion, which appeared first in the Pennes’ model [36]. Later, more andmore physiological details are tried to be implemented. A few well-known examples are related to the coupledtwo-temperature Chen-Holmes [37] and the Weinbaum-Jiji-Lemons models [38]. However, the methodology ofimplementing the detailed structure (e.g., the artery-vein pairs, various tissue structures) does not seem to beadvantageous due to the more complicated equations, the unknown variables, and limited validity region. Also, insome cases, the extensions are physically contradictory: in Wulff’s model [39], the mean blood velocity u b explicitlyappears: q = − λ t ∇ T t + ρ b f b u b (9)with f b being the blood enthalpy, and the indicies t and b are referring to the tissue and blood properties. Suchcoupling between q and u is not possible, unfortunately, due to objectivity reasons [40–42]. Similar shortcomingsappear in Klinger’s approach [43, 44].The second group of biologically motivated heat conduction models consists of the DPL and the CV equationsin the vast majority of problems, with keeping specific source terms. Sadly, the DPL model suffers from physicaland mathematical shortcomings, and the CV equation predicts waves, which are not observed so far.Beyond these models, there are more specific ones in the literature. For instance, finding the thermal responseof the cornea is a long-lasting question. In this respect, we mention the work of Taflove and Brodwin [45] due totheir remarkable idea: it is more advantageous to use ‘effective’ models. Effective in the sense that the complexheterogeneous structure is substituted with a homogeneous one; consequently, the model parameters are also effec-tive, i.e., ‘averaging’ the complete thermal behavior. This methodology agrees with the GK model’s perspective:it introduces two new, physically strongly motivated parameters, which characterize the entire medium, despiteits particular heterogeneities. From a purely thermodynamic point of view, rocks do not differ significantly frombiological materials: they both porous, having various heterogeneities and irregularities, thus parallel time scalesare present. If such material shows non-Fourier thermal behavior, then it should be observable when all biologicalspecific effects are excluded, and no heat source is present. In other words, the difficulties with complicated heatsources and boundary conditions must be separated from the constitutive model. It stands as the motivation torevisit two preceding experiments in which these aspects appear and could lead to the misinterpretation of theobserved data. ´A. SUD´AR , , G. FUTAKI , R. KOV´ACS , , Experiment I.: processed meat samples
The first experiment we re-evaluate is performed by Tang et al. [46] on processed meat samples, including fatas inclusions. They prepared three samples with 2, 3 and 4 mm thickness, and all are having the same 10 mmdiameter. They applied a heat pulse on the front side uniformly, and it lasts 1 s while they registered the rear sidetemperature history for 90 s. Figure 2 shows their results partially; for their complete investigation, we refer to [46].According to their interpretation, convection cannot be responsible for the measured deviation from Fourier’s law.They investigated two approaches to explain this phenomenon. On one hand, they proposed to use the DPL modelwith adiabatic (zero flux) boundaries, hence neglecting the possible heat transfer to the environment. On the otherhand, they introduced the size of the fat inclusion as a variable parameter and performed a detailed numericalanalysis using the Fourier equation.In the end, they concluded that the Fourier equation seems to be a better choice together with these newparameters than the DPL model. These models differ mostly at the end of the temperature history. Despite thevaluable insight they provided by these experiments, there are a few misinterpretations, which we found to beimportant presenting here.
Figure 2.
The observed temperature history for the thinnest sample (2 mm). A) presents thesolutions for Fourier equation with various heat transfer coefficients. B) shows the outcomes ofother approaches. The complete figures and for further information, we refer to [46].We re-evaluate that data using a novel algorithm developed explicitly for the GK equation using its analyticalsolution [47]. Shortly, this evaluation procedure first finds the heat transfer coefficient and the thermal diffusivity.If needed, it continues to find the GK parameters ( τ q and κ ). We found that omitting the heat transfer to theenvironment yields significant errors in the modeling, i.e., Fourier’s law and convection boundary condition provide asuitable model to adequately describe the observed data. In other words, no generalized heat conduction equation isneeded, merely a proper optimization on the effective thermal diffusivity. Consequently, despite that they found therelaxation times in the mathematically valid domain ( τ q > τ T ), it remains misleading to interpret the temperaturehistory as proof for the wave nature of heat conduction under such conditions. ONTINUUM MODELING PERSPECTIVES OF NON-FOURIER HEAT CONDUCTION IN BIOLOGICAL SYSTEMS 5
Figure 3.
Comparing the Fourier model to the sample with 4 mm thickness. The measured datais recovered using a plot digitizer.
Figure 4.
Comparing the Fourier model to the sample with 2 mm thickness. The measured datais recovered using a plot digitizer.Nevertheless, the size-dependent behavior is indeed present within this spatial scale (2 − − [m /s]2 mm 0 .
663 mm 0 .
734 mm 1 . Table 1.
The found thermal diffusivities with the Fourier equation.5.
Experiment II.: skin tissue samples
In this section, we revisit the experiment of Jaunich et al. [48], in which they prepared various cylindrical samplesmade from the mixture of araldite, DDSA resin, and DMP-30 epoxies in order to mimic the behavior of real tissue,moreover, they added titanium dioxide to this mixture as a ‘scatterer’ for the laser light. They assumed that thismixture also thermally substitutes the real skin tissue, therefore utilizing the same optical and thermal parameters,detailed in [48]. A skin tissue has a promising structure in regard to non-Fourier heat conduction due to the paralleltime scales (the presence of various layers and heat transfer mechanisms). Nevertheless, its observation might ´A. SUD´AR , , G. FUTAKI , R. KOV´ACS , , also require the heat flux to be parallel with these layers [33]. Contrary to the previous experiment, the thermalexcitation now lasts 10 s instead of 1 s, and registering the temperature history for 10 s.Regarding the aspects of non-Fourier heat conduction, all these are extremely important. The material structurehas no sharp interfaces inside due to the relatively homogenous mixture of epoxy components, therefore it is differentfrom real heterogeneous porous materials. Furthermore, the excitation is unusual in the sense that it is not ‘pulse-like’ but much longer, significantly absorbed under the surface, and the temperature history is measured only inthe period of laser heating, and there is no available data after. Consequently, the entire experiment is heat sourcedominated, which hides heat conduction and, eventually, almost any other thermal transport effects.According to their findings, Fourier’s law is seemingly inadequate for this problem since it predicts the temper-ature field quite far from the measured one. Figure 5 presents the measured and fitted results partially, only thedata, which we also tried to reproduce. The assumption that other constitutive equations should be used insteadof Fourier’s law requires accurate knowledge of any other influencing property, most importantly, on the internalheat generation caused by the laser light. In their study, the CV (hyperbolic) equation is tested and found to bebetter than the Fourier equation, providing more accurate predictions. Unfortunately, even the CV equation failsto offer the proper values, and at some points, it is still the same as the Fourier equation. Figure 5.
The partial results of Jaunich et al., for their detailed description and for the completefigures, we refer to [48]. Triangles denote the measured temperature increase. A) presents thetemperature history at 2 mm depth on the axis of the cylinder. B) and C) show the radial andaxial temperature distribution at specific points.We also tried to model this experiment by solving the Fourier, CV, and GK equations in cylindrical coordinatesbased on the scheme published in [9]. We found that the problem indeed originates in the heat source since neither
ONTINUUM MODELING PERSPECTIVES OF NON-FOURIER HEAT CONDUCTION IN BIOLOGICAL SYSTEMS 7 the CV nor the GK model could improve the fitting, as it was supposed from the beginning. Therefore we triedvarious volumetric heat generation models from which the following one provides the closest values with the Fourierequation: Q v = (cid:40) q m (cid:0) (cid:0) πr e /r p (cid:1)(cid:1) r e < r p , (10)in which r e = (cid:112) r + ( z − z c ) (11)is the effective radius where heat generation Q v occurs, r p stands for the pulse radius; z c appears in the case whenthe center of heat generation is under the surface, in our case, it is found to be z c = 0; q m is the maximum heatgeneration. The value of q m is assumed to be the only adjustable parameter due to the unknown amount of absorbedheat, and it is found to be q m = 1 . · W/m , which means that around the half of the irradiated energy is turnedinto heat generation. Furthermore, z and r are the axial and radial coordinates, respectively. Although this is anempirical model, it is motivated by statistical physics: photons propagate with equal probability in every directionafter enough number of scattering, resulting in a spherically symmetric absorption. Figures 6 and 7 present thebest outcome we managed to achieve. Figure 6.
Comparing the Fourier model to the measured values in both axial and radial directions.The measured data is recovered using a plot digitizer.
Figure 7.
Comparing the Fourier model to the temperature history recorded at z = 2 mm. Themeasured data is recovered using a plot digitizer.As it is apparent from Fig. 6, we also applied a convection boundary on the surface, which was irradiated bythe laser pulse, the heat transfer coefficient is assumed to be 10 W/(m K). Unfortunately, the measured transientcharacteristic at z = 2 mm is still uncovered, and non-Fourier models cannot offer a better description, in agreementwith [48]. ´A. SUD´AR , , G. FUTAKI , R. KOV´ACS , , Discussion
The thermal modeling of biological materials is a challenging task and has increasing attention due to its numerousapplications. In the present paper, we aimed to offer a brief overview of the essential aspects of non-Fourier equationsin continuum thermodynamics, focusing on their observability in room temperature experiments.First, we want to emphasize that one must pursue to separate the heat sources from the heat conduction effectsas much as possible. Although volumetric heat generation is inevitable in many practical situations and also has asignificant influence in medical diagnostics [49–51], it could dominate the time evolution of the temperature field.Therefore, the experiments, which aim to measure biological material’s thermal properties, should be designed tolet the heat conduction be the dominant heat transport mechanism instead of including complex heat source andheat generation mechanisms. In such a preferable situation, it is easier to decide whether we observed a non-Fourier phenomenon or not; consequently, the role of a constitutive equation becomes apparent in the thermalmodel. Moreover, the non-Fourier thermal parameters are more comfortable to determine. Otherwise, one caneasily misinterpret the measurements and concluding in a different outcome, and it is recommended to always testall the possibilities in a model before implementing a generalized constitutive equation.One must be careful since the observation of non-Fourier propagation depends both on the boundary conditionsand material properties. In other words, it requires the coexistence of suitable spatial and time scales, probablyhaving a size-dependent behavior as well. This field is still under development, and there are countless questions,which wait to be solved in the future. Also, as our re-evaluation attempts are standing here, the Fourier equationcan explain most of the observations, and most of the seeming deviation disappeared.Overall, we aim to collect and call attention to these modeling aspects. The research field of non-Fourier heatconduction is vast, continuously growing, and increasingly challenging to keep up with the up-to-date results, mainlybecause these constitutive equations require a deeper understanding of boundary conditions as well. The classicaldefinitions established for the Fourier equation do not work, and even for a thermodynamically strongly motivatedmodel (such as the GK equation) can lead to unphysical, false solutions with inadequate boundary conditions.Therefore, despite the attractive results of non-equilibrium thermodynamics, it is suggested to be careful withgeneralized models.
Acknowledgement
We dedicate our paper to the memory of our respected colleague, Prof. Jos´e Casas-V´azquez, who passed awayrecently.The research reported in this paper and carried out at BME has been supported by the grants National Research,Development and Innovation Office-NKFIH FK 134277, and by the NRDI Fund (TKP2020 NC, Grant No. BME-NC) based on the charter of bolster issued by the NRDI Office under the auspices of the Ministry for Innovationand Technology. This paper was supported by the J´anos Bolyai Research Scholarship of the Hungarian Academyof Sciences.
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Journal of Physics: Conference Series , 1224(1):012001, 2019. , , G. FUTAKI , R. KOV´ACS , , Department of Energy Engineering, Faculty of Mechanical Engineering, BME, Budapest, Hungary, Departmentof Theoretical Physics, Wigner Research Centre for Physics, Institute for Particle and Nuclear Physics, Budapest,Hungary,3