Development of a new theoretical model for blood-CNTs effective thermal conductivity pertaining to hyperthermia therapy of glioblastoma multiform

Computer Methods and Programs in Biomedicine 172 (2019) 79–85

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Computer Methods and Programs in Biomedicine journal homepage: www.elsevier.com/locate/cmpb

Development of a new theoretical model for blood-CNTs effective thermal conductivity pertaining to hyperthermia therapy of glioblastoma multiform L. Benos a, L.A. Spyrou a, I.E. Sarris b,∗ a b

Biomechanics Group, Institute for Bio-Economy and Agri-Technology, Centre for Research and Technology Hellas (CERTH), 38333 Volos, Greece Department of Mechanical Engineering, University of West Attica, 12210 Athens, Greece

a r t i c l e

i n f o

Article history: Received 7 January 2019 Accepted 12 February 2019

Keywords: Hyperthermia Glioblastoma Thermal conductivity Blood cells CNTs

a b s t r a c t Background and objective: The present study deals with the hyperthermia therapy, which is the type of treatment in which tissues are exposed to high temperatures in order to destroy cancer cells with minimal injury to healthy tissues. In particular, it focuses on glioblastoma multiform, which is the most aggressive cancer that begins within the brain. Conventional treatments display limitations that can be overcome by using nanoparticles for targeted heating. Out of the proposed nanoparticles, this investigation focuses on a new field that utilizes carbon nanotubes (CNTs) which are able to selectively heat the cancer cells since they can convert near infrared light into heat. In the absence of any experiment or theoretical model for the estimation of an effective thermal conductivity of blood and CNTs, a first principle model is developed in this study which takes into account the blood micro-structure. Besides, a number of factors are included, namely the shape and the size of the nanoparticles, the interfacial layer formed around them and their volume fraction. Methods: Firstly, assuming that the blood consists of blood cells and plasma, the thermal conductivity of the former is estimated. Then, the effective thermal conductivity of plasma/CNTs is calculated for various parameters. Finally, the resulting “bio-nanofluid” consisting of plasma/CNTs and blood cells is formed. Results: It is ascertained that thin and elongated CNTs with relatively large nanolayer thickness as well as large concentrations of CNTs contribute to the increase of the thermal conductivity and, thus, in the enhancement of the heat transfer. Conclusions: Investigating of how design parameters pertaining to CNTs, such as their size and shape, affect the effective thermal conductivity of blood-CNTs, possible regulating ways are suggested regarding the hyperthermic treatment. Finally, the present simple estimation of the effective thermal conductivity can be used as an effective property of the nanofluid when it comes to numerical investigations regarding heat transfer occurring during hyperthermia or other potential clinical uses (for example targeted heat of living tissues). © 2019 Elsevier B.V. All rights reserved.

1. Introduction Given the global burden of cancer worldwide, which is estimated to be 9.6 million cancer deaths in 2018 [1], the need for novel methods for cancer treatment has been increased more than ever. Out of the several kinds of cancer, glioblastoma multiforme (GBM) is considered to be the most malignant cancer with median survival being less than 15 months [2]. The treatment of GBM is extremely difficult owing to the limited capacity of the brain to

∗

Corresponding author. E-mail address: [email protected] (I.E. Sarris).

https://doi.org/10.1016/j.cmpb.2019.02.008 0169-2607/© 2019 Elsevier B.V. All rights reserved.

repair itself and GBM nature of being both invasive and resistant to therapies [3]. The conventional treatments include surgery as well as chemotherapy and radiation, which use drugs and high doses of radiation, respectively, to kill cancer cells and shrink tumors. Surgery tries to improve the quality of patient‘s life and prolong its survival while the objectives contain the confirmation of the diagnosis or the elevation of symptoms. Radiotherapy, which is the most common treatment, is based on electrons and free radicals via ionizing radiation to destroy DNA [4]. Finally, chemotherapy uses mainly temozolomide as a drug, which displays cytotoxic effects via methylation of specific DNA sites, with bevacizumab being an adjuvant therapy pertaining to GBM via acting as angiogenesis

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inhibitor [5,6]. At the moment, chemotherapy in conjunction with radiotherapy is considered as the most effective cancer therapy [7]. In brief, surgery treatment is limited by its invasive nature while radiotherapy suffers from the risk of necrosis and resistance of certain types of tumors. Finally, chemotherapy severe side effects include nausea, hair loss, nerve damage and infertility to mention but a few [7]. The present study focuses on the use of above normal physiological temperatures (41 °C–46 °C) in order to cause the death of cancer cells with the injury of normal tissues being as minimum as possible [8]. This treatment, known as “hyperthermia”, even being aware by Hippocrates [9] is considered a relatively new and promising approach. Generally, cells die when they are exposed to high dose-time combinations by necrosis, while for relatively mild exposures undergo apoptosis [10]. When the heat is not able to cause either necrosis or apoptosis, it can sensitize cancer cells to radiation [11] and many chemotherapeutic drugs [12]. Hyperthermia can be utilized by using external devices to transfer thermal energy to tumors via irradiation with light or electromagnetic waves. Conventional treatments include tubes with hot water, infrared irradiation, microwaves and ultrasound to mention than a few [7]. Nevertheless, the aforementioned techniques display restrictions such as low penetration of heat in the tumor, excessive heating of healthy tissue and dissipation of heat via the blood, which is a drawback in tumors which are wellvascularized [13]. For the purpose of overcoming the above limitations advances in chemistry and physics have provided some nanoparticles in order to be used for targeted heating of GBM, namely magnetic nanoparticles, gold nanorods and carbon nanotubes (CNTs). In brief, the hyperthermic therapy using magnetic nanoparticles involves injection of iron oxide nanoparticles (Fe3 O4 , α -Fe2 O3 , β Fe2 O3 and γ -Fe2 O3 ) inside the cancer cell and exposure of the patient in an alternating magnetic field. This, leads to an increased tumor temperature, thermal ablation of cancer cells and, as a consequence, tumor shrinkage [7,14]. Concerning gold nanorods, they are intravenously injected and then irradiated with near infrared laser light in order to cause thermal death of tumor via apoptosis or necrosis. Finally, hyperthermia using CNTs is a new field in comparison to the aforementioned ones. CNTs are able to selectively heat the tumor because of they are impressively able to convert near infrared light into heat [15]. Since the present investigation deals with the usage of CNTs as hyperthermic working nanoparticles, only some demonstrative experiments pertaining to CNTs are going to be referred next while the applications of magnetic nanoparticles as well as gold nanorodes can be found in recent review papers such as [7]. Generally, carbon nanotubes are classified into carbon nanotubes having a single wall (SWCNTs) or multiple walls (MWCNTs). The former are individual cylinders consisting of a single rolled graphene sheet whereas MWCNTs are identified as being similar to a Russian “matryoshka doll”. Thus, they constitute of several concentric grapheme cylinders around a central hollow core, with weak Van der Waals forces binding the tubes together [16]. CNTs, owning to their extremely small aspect ratio (diameter over tube length) enhance absorption of electromagnetic energy and cause rapid heating of the tube [17,18]. Markovic et al. [19] utilized CNTs and nanoparticles made from graphene for the purpose of killing human GBM via hyperthermia in vitro. Their investigation indicated that CNTs triggered the death of cancer cells via mitochondrial membrane depolarization. Furthermore, Wang et al. [20,21] using antibody-conjugated CNTs caused targeted ablation of glioblastoma and neuroblastoma cells while the viability of control cells did not affected. CNTs have also been used for thermal ablation [22,23]. However, according to Burlaka et al. [24] the multi-walled CNTs (MWCNTs) perform broad absorption spectra in

comparison with the resonance absorptions of single-walled CNTs (SWCNTs), rendering them suitable for stimulation via a range of near infrared energy sources. Moreover, MWCNTs can absorb significantly more near infrared irradiation compared with SWCNTs [24]. Given the growing interest in the field of targeting hyperthermia, to our knowledge, there is no theoretical model for the estimation of an effective thermal conductivity of blood containing MWCNTs for the purpose of investigating the very important problem of heat transfer during the hyperthermic treatment. Due to the absence of an experimental study, the present theoretical model can provide useful insight into the thermal conductivities of such bio-nanofluids. The present model takes into account the composition of the blood, namely the blood cells and plasma. Also, a first attempt is made in order to potentially regulate this effective thermal conductivity via altering nanoparticles’ concentration, shape, size and interfacial nanolayer which is formed between MWCNTs and blood. 2. Brief theory concerning blood composition The blood is roughly considered to be a mixture which consists of plasma and blood cells. Plasma, that comprises 55% of blood, is mainly water (92% water, 8% plasma proteins and trace amounts of other material) while the blood cells constitute the remaining 45% of blood. Blood cells (BC) are principally red blood cells (RBC), white blood cells (WBC) and platelets with their volume percentage being equal to 97%, 2% and 1%, respectively [25]. Fig. 1a gives a qualitative illustration of the blood [26] while Fig. 1b depicts a test tube containing blood [27]. As it can be noted, the blood separates into three layers, namely RBC, WBC and plasma, as the denser constituents locate to the bottom of the tube and plasma remains uppermost. 3. Estimation of thermal conductivities Up to date, the direct estimation of an effective thermal conductivity that takes into account the micro-structure of blood and the suspended MWCNTs has never been the subject of extensive experimental or theoretical study. Thus, in this investigation the methodology of Ghassemi et al. [28] is mainly followed in which, in contrast with the present one, an Al2 O3 -blood nanofluid is considered without investigation of the crucial effect of the shape of nanoparticles as well as the variation of the interfacial nanolayer which exists around the nanoparticle. Ghassemi et al. [28] calculated the thermal conductivity of blood cells since there are no known values concerning WBC and platelets. Thus, utilizing the parallel mixture rule [29]:

kblood = ϕ plasma k plasma + ϕBC kBC

(1)

estimated the only unknown parameter, kBC , where the subscripts blood, plasma and BC indicate where either the thermal conductivity, k, or the blood volume fraction, ϕ , refer to. Thus, they calculated kBC = 0.4 W/mK via considering kplasma = 0.57 W/mK, kblood = 0.492 W/mK, φ plasma = 0.55, and φ BC = 0.45 according to the relative literature [30]. The aforementioned value pertaining to the thermal conductivity of BC, i.e. kBC = 0.4 W/mK, is also incorporated in the present analysis while its validity is going to be addressed in Section 3.1. Subsequently, by ignoring the BC, the effective thermal conductivity of plasma-MWCNTs, kBC − CNT is calculated in Section 3.1 considering a renovated Hamilton-Crosser model. The latter was recently developed by Jiang et al. [31] and was proved specifically valid for CNT-water nanofluids after successfully comparison against experimental data. Finally, the effective thermal conductivity of the resulting bio-nanofluid, kBio , is evaluated in

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Fig. 1. (a) A roughly depiction of the blood [26] and (b) Separation of blood into three layers in a test tube [27].

Section 3.2 using the Maxwell’s well-known relationship [32] considering the plasma-MWCNTs as the base fluid and BC as the solid particles. 3.1. Effective thermal conductivity of plasma-MWCNTs As far as the theoretical description of the nanoparticle-base fluid effective thermal conductivity is concerned, it is still in elementary level mainly due to a plethora of factors associated with it. In particular, these factors include the shape, the size and the volume fraction of the nanoparticles as well as the thermal conductivities of the nanoparticles and the base fluid. Briefly, some indicative models for thermal conductivity of suspensions are the Maxwell model [32] (considering spherical nanoparticles) and Hamilton-Crosser model [33] (using an empirical shape factor). In addition, Brownian motion-based models have been presented by researchers such as Koo and Kleinstreuer [34] and Benos and Sarris [35]. However, real nanofluids acquire much greater thermal conductivities than most of the theoretical predictions. The role of the nanolayer, formed around the nanoparticle because of the adhesion of liquid molecules to it, was found to be quite significant. In fact, this ultrathin layer seems to behave much like a solid [36] and play a key role regarding heat transfer from the solid nanoparticle to the adjacent liquid [37-40]. While the effect of interfacial layer is negligible for micrometer particles, intriguingly this effect becomes very crucial when solid nanoparticles are dispersed in the base fluid. The effective thermal conductivity model, which was incorporated in the present analytical approach, is that of Jiang et al. [31]. Jiang et al. developed a renovated Hamilton–Crosser model for the purpose of introducing not only the effect of the nanoparticle shape but that of the interfacial nanolayer as well. As a matter of fact, the theoretical predictions of the thermal conductivity of a CNT based nanofluid were found to be in quite good agreement with the experiments [31]. Thus, mono-sized ellipsoidal particles of semi-axes a, b, and c (a ≥ b ≥ c) are considered having an interfacial nanolayer, which is formed around them. In particular, the case of a cylindrical particle is illustrated in Fig. 2 with a ≥ b = c, which is more relevant to the present analysis, as it is going to be elaborated below. The thickness of the interfacial layer, t, its microstructure and its physicochemical properties depend on the nanoparticle, the fluid and the interaction between them. The nanolayer is expected to have an intermediate thermal conductivity which is between that of the base liquid and that of the nanoparticle (kplasma and kCNT , respectively in the present investigation). This occurs because the layered molecules are in an intermediate physical state between a liquid

Fig. 2. Schematic for the case of a cylindrical nanoparticle with an interfacial nanolayer.

and a solid. As a consequence, revising the Hamilton and Crosser model [33], Jiang et al. [31] derived the following form regarding the effective thermal conductivity (kplasma − CNT in this study):



k plasma−CNT = k plasma

k pe + k plasma (n − 1 ) + (n − 1 ) k pe − k plasma



k pe + k plasma (n − 1 ) − k pe − k plasma





ϕe

ϕe (2)

where kpe stands for the thermal conductivity of the equivalent nanoparticle, φ e is the effective nanoparticle volume fraction and n is an empirical shape factor. Concerning kpe , φ e and n, they are estimated via the following expressions [31]:

k pe =

2kCNT + 2kl +





1+

1+



t 2 b



t 2 b



− 1 (kCNT + kl )



− 1 (kCNT + kl )

kl

(3)

  t 2 ϕe = 1 + ϕ

(4)

n = ψ −g

(5)

b

In the above relationships, kl represents the average thermal conductivity of the interfacial nanolayer which, based on linear

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variation within it, was simplified in [31] for the case of a cylindrical particle (a ≥ b = c), Fig. 2, as follows:

kl = k plasma

1+

t b

a·

 

− a ln 1 + t b

ln

  t t b

(6)

1+ b a

According to Yu and Choi [38] the sphericity, ψ , can be expressed for a cylindrical particle as:



ψ=



k(W/mK)

Volume fraction

0.492 [41] 0.57 [41] 0.4 (calculated via Eq. (1)) 30 0 0 [42]

1 0.55 0.45 0 ≤ ϕ ≤ 0.04

(7)

e(δ ) 1 − e2 (δ ) + arcsin e(δ )

4. Results and discussion

where e stands for the eccentricity of the nanoparticle and δ being between 0 and t [31]. For elongated particles, the dependence of n on ψ is stronger than for spherical ones with g = 1.55 being a representative value for CNTs [31]. In a nutshell, according to the present incorporated theoretical model, the effective thermal conductivity of plasma-CNT mixture depends on the nanoparticle size and shape, the nanolayer thickness, the thermal conductivities of the nanoparticle, the base fluid (i.e. plasma) and the nanolayer as well as the nanoparticle volume fraction. 3.2. Effective thermal conductivity of bio-nanofluid (plasma/MWCNTs-Blood cells) After estimating the thermal conductivity of plasma containing MWCNTs, kplasma − CNT , via Eq. (2) for a range of CNT volume fractions, thicknesses of the interfacial layer, shape factors as well as CNT diameters and lengths, the resulting thermal conductivity of the bio-nanofluid, kBio , is calculated by utilizing the following Maxwell equation [32]:

 

 ϕBC  (8) kBC + 2k plasma−CNT − kBC − k plasma−CNT ϕBC

kBC + 2k plasma−CNT + 2 kBC − k plasma−CNT

considering the plasma-CNT mixture as the base fluid and the BC as the solid particles. In this fashion, it should be emphasized that using the Maxwell equation, the validity of the estimation of kBC can be verified, since it is one of the major assumptions made in this study similarly to [28]. To this end, assuming plasma as being the base fluid and BC as the solid particles:

kblood = k plasma

Blood Plasma Blood Cells (BC) MWCNTs

1 / 6

2e ( δ ) 1 − e2 ( δ )

kBio = k plasma−CNT

Table 1 Thermal conductivity and volume fraction values incorporated in the present study.

 

 ϕBC   0.489 W/mK kBC + 2k plasma − kBC − k plasma ϕBC

kBC + 2k plasma + 2 kBC − k plasma

(9) This calculated value is only 0.59% lower than the experimentally measured value of 0.492 W/mK reported by Kreith et al. [41]. In addition, taking into account that approximately 92% of plasma is water in conjunction with no available experimental data pertaining to plasma-CNT effective thermal conductivity, the theoretical model of Jiang et al. [31] can be regarded, to a first extent, a rational approximation for the present first principle investigation. Concerning Maxwell equation, it gives reliable results when the size of the particles is in the micrometer range [28]. Indeed, the size of RBC, WBC and plackets is approximately 8, 15 and 3 μm, respectively [24]. As a consequence, the usage of Eqs. (8) and (9) also stands to reason. Finally, the values which were incorporated in the present study regarding the thermal conductivity and the volume fraction of blood, plasma, BC and MWCNTs can be shown in Table 1.

Firstly, the effect of the diameter of the CNT, d ≡ 2b, as well as the thickness of the interfacial layer, t, is investigated on the thermal conductivity of the interfacial nanolayer formed around the nanoparticle, kl . The accurate calculation of the average kl according to Eq. (6) is very important since it determines, to a great extent, not only the effective thermal conductivity of plasmaCNT mixture, kplasma − CNT , but the resulting thermal conductivity of the bio-nanofluid as well. Thus, in contrast with Ghassemi et al. [28] who took only a simple linear dependence of kl with kplasma , in the present approach also the dependence of kl on the size of the nanoparticle (i.e. diameter and length) and the thickness of the interfacial nanolayer is considered. Finally, all the values of the parameters used next are in the parameter range used in the experimental study of Burlaka et al. [24] while the range pertaining to the thickness of the interfacial nanolayer, t, along with the shape factor, g, were derived by Jiang et al. [31]. Fig. 3 depicts the thermal conductivity of the interfacial nanolayer, kl , for a range of values pertaining to d and t that was presented in the relative literature. Concerning the reference values used in Fig. 3, the length of the CNT was equal to 2 μm, the CNT volume fraction was 0.02 and g = 1.55 (the most suitable shape factor for CNT [31]). Considering a specific diameter of the CNT, namely d = 12 nm, the nanolayer thermal conductivity, kl , seems to increase linearly when its interfacial layer thickness increases, Fig. 3a. Taking a constant nanolayer thickness, namely t = 1.5 nm, kl decreases as the CNT diameter increases, Fig. 3b. Furthermore, Fig. 3b implies that when the diameter of the CNT is relatively small the impact of the nanolayer on the resulting effective thermal conductivity is anticipated to be more obvious, as it was highlighted in studies such as [31]. The large values of thermal conductivity of the interfacial nanolayer, kl , is commensurate with smaller values of thermal resistance. This thermal resistance, which is known as “Kapitza resistance”, appeared to be very crucial concerning the resulting heat transfer [43]. The effective thermal conductivities pertaining to plasma-CNT, kplasma − CNT , and bio-nanofluid (plasma/CNT-BC), kBio , corresponding to the cases illustrated in Fig. 3a and b, are simultaneously depicted in Fig. 4a and b, respectively. In particular, as it can be gleaned from Fig. 4a, both kplasma − CNT and kBio follow the increase of the thermal conductivity of the nanolayer, kl , mentioned above with the increase of nanolayer thickness. Moreover, as the CNT diameter increases both kplasma − CNT and kBio decrease following again the lowering of kl noted in Fig. 3b. Interestingly, the anticipation pertaining to the impact of thinner carbon nanotubes on the resulting thermal conductivity not only of plasma-CNT but that of the bio-nanofluid is verified. It appears that smaller ratios of the nanolayer thickness over CNT diameter, t/d, contribute to larger thermal conductivities, thus, enhancing the heat transfer. This finding definitely can be taken into account in the “regulation” of hyperthermia via properly manufacturing the solid nanoparticles. Consequently, surface chemistry could be considered for the purpose of decreasing or increasing the nanolayer thickness depending on the desired result. Finally, in all cases presented in Fig. 4a

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Fig. 3. Effect of (a) the nanoparticle thickness, t, and (b) the nanolayer diameter, d, on the thermal conductivity of the interfacial nanolayer, kl .

Fig. 4. Effect of (a) the nanoparticle thickness, t, and (b) the nanolayer diameter, d, on the thermal conductivity of the plasma-CNT mixture, kplasma-CNT , and bio-nanofluid, kBio .

Fig. 5. Effect of (a) the nanoparticle length, L, and (b) the shape factor, g, on the thermal conductivity of the plasma-CNT mixture, kplasma-CNT , and bio-nanofluid, kBio .

and b the thermal conductivity of the plasma-CNT mixture is consistently larger than the corresponding bio-nanofluid one. Subsequently, the reference values which are used in Fig. 5a and b are the CNT volume fraction (ϕ = 0.02), the nanoparticle diameter (d = 12 nm) and the nanolayer thickness (t = 1.5 nm). Keeping a specific shape factor, namely g = 1.55, in Fig. 5a both kplasma − CNT and kBio appear to increase as the length of the CNT increases, thus, indicating the characteristic property of CNTs that makes them

so outstanding when it comes to hyperthermic applications. Taking a constant CNT length, namely L = 2 μm, and altering g it can be clearly seen that the shape of the CNT plays also a key role. More specifically, as the nanoparticle is elongated (g increases) an increase is observed in both thermal conductivities, Fig. 5b. This fact, in conjunction with the extremely large aspect ratios (L/d) mentioned above, reinforces the usage of CNTs as nanoparticle within base fluids when extreme values of thermal conductivities

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These factors can be selectively used as design parameters before or during the hyperthermic treatment of glioblastoma or other kinds of cancer for the purpose of regulating the thermal conductivity of the blood-CNT nanofluid, which is denoted as kBio . Besides, it can be used by other researchers when it comes to theoretical or numerical investigations regarding heat transfer occurring during hyperthermia or other potential clinical uses (for example targeted heat of living tissues). Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.cmpb.2019.02.008. References Fig. 6. Effect of the CNT volume fraction, ϕ , on the thermal conductivity of the plasma-CNT mixture, kplasma-CNT , and bio-nanofluid, kBio .

are required. These factors are also crucial and can be used as designed parameters in hyperthermic applications. Finally, as the CNT volume fraction, ϕ , increases, the effective thermal conductivities increase, as can be illustrated in Fig. 6, in accordance with all theoretical models pertaining to nanofluids. When there are no CNTs within the blood kplasma − CNT results in the value describing plasma, i.e. 0.57 W/mK, while the kBio results in the value of 0.489 W/mK characterizing blood (Eq. (9)). The reference values which are used in Fig. 6 are again d = 12 nm, L = 2 μm, t = 1.5 nm, g = 1.55 while the volume fraction of the CNTs are between 0 and 0.04. 5. Conclusions A first principle theoretical model is presented in this study pertaining to the estimation of the effective thermal conductivity of the blood-CNT nanofluid. It is the first time, at least to our knowledge, that so many factors are included in a bio oriented model, namely the shape and the size of the nanoparticles, the interfacial layer formed around them, their volume fraction as well as the microstructure of blood. The theoretical model is based mainly on three steps. The first one, in the absence of experimental data on BC (only the RBC thermal conductivity is known), is the estimation of the thermal conductivity of BC, kBC via the rule of mixtures. Afterwards, considering the plasma containing CNTs, the effective thermal conductivity of its mixture, kplasma − CNT , is calculated based on a model that takes into account a number of factors, namely the shape and the size of the nanoparticles, the interfacial layer and their volume fraction. Finally, considering plasma-CNT as the base fluid the resulting kBio is calculated via the Maxwell equation [32] which is valid for micrometer-sized solid particles such as those considered here, namely BC [24]. Both kplasma − CNT and kBio are estimated for a range of CNT sizes and shapes, volume fractions and thicknesses of the interfacial layer. The latter is proved to be particularly crucial as it was stressed in the results section. In a nutshell, thinner, bigger and elongated CNTs in conjunction with relatively large amount of their concentrations seem to enhance the thermal conductivity. In all figures, which were presented and analyzed above, the behavior of the curves resembles that of water-based nanofluid. As it was anticipated, the thermal conductivity of kplasma is very close to that of water (kwater is approximately equal to 0.6 W/mK) which lies on the fact that almost 92% of plasma is water. The quite simple estimation of kBio according to the current theoretical model takes into account the microstructure of blood, namely plasma and blood cells as well as a plethora of factors.

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