Numerical simulation of the effect of superparamagnetic nanoparticles on microwave rewarming of cryopreserved tissues.

Cryobiology xxx (2014) xxx–xxx

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Cryobiology journal homepage: www.elsevier.com/locate/ycryo

Numerical simulation of the effect of superparamagnetic nanoparticles on microwave rewarming of cryopreserved tissues q Tao Wang a, Gang Zhao a,⇑, Xin M. Liang a,b, Yunpeng Xu a, Yang Li a, Heyu Tang a, Rui Jiang a, Dayong Gao b,⇑ a b

Centre for Biomedical Engineering, Department of Electronic Science & Technology, University of Science and Technology of China, Hefei 230027, China Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA

a r t i c l e

i n f o

Article history: Received 10 October 2013 Accepted 4 February 2014 Available online xxxx Keywords: Finite element method (FEM) Superparamagnetic (SPM) nanoparticles Cryopreserved tissues Microwave rewarming

a b s t r a c t In this study, the microwave rewarming process of cryopreserved samples with embedded superparamagnetic (SPM) nanoparticles was numerically simulated. The Finite Element Method (FEM) was used to calculate the coupling of the electromagnetic field and the temperature field in a microwave rewarming system composed of a cylindrical resonant cavity, an antenna source, and a frozen sample phantom with temperature-dependent properties. The heat generated by the sample and the nanoparticles inside the electromagnetic field of the microwave cavity was calculated. The dielectric properties of the biological tissues were approximated using the Debye model, which is applicable at different temperatures. The numerical results showed that, during the rewarming process of the sample phantom without nanoparticles, the rewarming rate was 29.45 °C/min and the maximum temperature gradient in the sample was 3.58 °C/mm. If nanoparticles were embedded in the sample, and the cavity power was unchanged, the rewarming rate was 47.76 °C/min and the maximum temperature gradient in the sample was 1.64 °C/mm. In the presence of SPM nanoparticles, the rewarming rate and the maximum temperature gradient were able to reach 20.73 °C/min and 0.68 °C/mm at the end of the rewarming under the optimized cavity power setting, respectively. The ability to change these temperature behaviors may prevent devitrification and would greatly diminish thermal stress during the rewarming process. The results indicate that the rewarming rate and the uniformity of temperature distribution are increased by nanoparticles. This could be because nanoparticles generated heat in the sample homogeneously and the time-dependent parameters of the sample improved after nanoparticles were homogeneously embedded within it. We were thus able to estimate the positive effect of SPM nanoparticles on microwave rewarming of cryopreserved samples. Ó 2014 Elsevier Inc. All rights reserved.

Introduction Tissue cryopreservation has been wildly utilized in modern clinical treatments and academic research [33,34]. Typical tissue cryopreservation process consists of the following five steps, cryoprotective agent (CPA) addition, cooling down, low temperature storage, rewarming, and CPA removal [23,53,57]. Successful cryopreservation requires not only the ultrastructures, but also the biological functions of the tissues to be maintained. Over the years, major attention has been focused on studying either CPAs [3,6,7,13,15,18,22,24,35,36,39,49,52] or the cooling down process

q Statement of funding: This work was supported by the National Natural Science Foundation of China (Nos. 51076149, 51276179), and the Fundamental Research Funds for the Central Universities. ⇑ Corresponding authors. E-mail addresses: [email protected] (G. Zhao), [email protected] (D. Gao).

[1,17,26,28,32,44,55,56]. Only a few studies that investigated the thawing process in part because the lack of readily accessible computational or experimental approaches [10,11,19,30]. Previous findings suggest the cryopreserved tissues must be rewarmed rapidly enough so that they reach their melting points before the occurrence of any devitrification and/or recrystallization [8,51]. However, given the thermal conductivities of biological tissues are usually very small, conventional rapid rewarming techniques would cause large thermal gradients [20]. So far, electromagnetic heating is considered as the only effective heating mechanism capable of providing uniformed rewarming for cryopreserved tissues or organs [45,47]. During microwave rewarming, tissues are volumetrically heated [2,21,37], therefore, the temperature uniformity is dramatically improved while the temperature gradient and thermal stress are greatly suppressed [29,40,42]. The fundamental heating mechanism of tissue rewarming in a microwave resonant cavity has been theoretically analyzed using finite difference time domain (FDTD) method [19]. Currently, there

http://dx.doi.org/10.1016/j.cryobiol.2014.02.002 0011-2240/Ó 2014 Elsevier Inc. All rights reserved.

Please cite this article in press as: T. Wang et al., Numerical simulation of the effect of superparamagnetic nanoparticles on microwave rewarming of cryopreserved tissues, Cryobiology (2014), http://dx.doi.org/10.1016/j.cryobiol.2014.02.002

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T. Wang et al. / Cryobiology xxx (2014) xxx–xxx

are two types of electromagnetic heating cavities have been reported that directly heat the samples via microwave. The first was a multimode cavity that utilizes complex multiple electromagnetic fields for heating [2,4,9,12,25,37,41,48,50], which is extremely troublesome to calculate the electromagnetic distribution inside the cavity. Furthermore, it has been reported that multimode cavity, i.e., typical microwave oven, suffers greatly from high temperature gradient within the bulk biomaterials and prolonged heating process [37,48]. The second was a single mode cavity [30,42,45]. The main advantage of such single mode microwave resonant cavity is the exact peak value of the electric field at any given location can be precisely calculated. Moreover, the sample is usually placed at the center of the cavity, which will more likely for the sample to experience optimal heating effect, i.e., high heating rate and uniformed temperature distribution [30]. Despite the attractive features provided by the latter single mode cavity, such microwave rewarming approach also suffers from the inherent uneven heating rate problem when dealing with bulk tissue samples (center vs. surface) [11]. Recent progress in nanotechnology has raised the possibility that we may overcome the nonuniform heating rate problem associated with thawing bulk tissue samples using pre-embedded nanoparticles. Currently, two studies have been reported that investigated the heating effect of using pre-embedded nanoparticles in electromagnetic field [11,31]. Here, we present a numerical simulation study about the effect of nanoparticles on the microwave rewarming process in a microwave resonant cavity. The electromagnetic field distribution in the microwave cavity was determined by Maxwell’s equations and boundary conditions. The heat was generated in the sample and in the nanoparticles via interaction with the microwave field, which obeyed the modified Pennes’ equation and Debye’s law, respectively. The variations in permittivity and conductivity of the sample at different frequencies were considered. The Finite Element Method (FEM) was applied to calculate the coupling of the electromagnetic field and the temperature field. The rewarming rate and the temperature gradient of the sample were simulated at various concentrations of nanoparticles and cavity powers. Methods Electromagnetic field distribution in the microwave cavity To vary the electromagnetic field in a sinusoidal manner, the * vector used for*electric field intensity E ðx; y; z; tÞ and the magnetic field intensity Hðx; y; z; tÞ * were time harmonics. Using separation of * variables, E ðx; y; z; tÞ and Hðx; y; z; tÞ were expressed as follows [43]: *

*

E ðx; y; z; tÞ ¼ E0 ðx; y; zÞejxt

*

ð1Þ

*

Hðx; y; z; tÞ ¼ H0 ðx; y; zÞejxt

ð2Þ

Thus, the Maxwell’s equations for the electromagnetic field for the different dielectric properties in the microwave cavity and sample are expressed as [43]: *

Heat generated in the sample by the interaction of the microwave field Since the sample will be removed from the living body and placed in the microwave cavity, metabolic heat is considered negligible during the whole rewarming process. In addition, perfusion heat is ignored due to its negligible influence on frozen tissue and the rewarming process. Heat transfer during the rewarming process of biological tissue can be accurately described by the simplified Pennes’ equation [8]:

qt C

@T ¼ r ðkrTÞ þ Q wave þ Q nano @t

ð7Þ

where T is the temperature, t the time, qt the density of tissue, C the effective thermal capacity of the tissue, k the effective thermal conductivity, Q wave the energy generated by direct electromagnetic wave in tissue, and Q nano the energy produced by the pre-embedded nanoparticles. Without nanoparticles, the energy generated by direct electromagnetic wave in tissue, Q wave , depends solely on the conductivity * of the tissue, r1, and the amplitude of the electric field, E ðx; y; zÞ. Therefore, time-dependent volumetric heating may be approximately as below [31]:

Q wave ðx; y; z; tÞ ¼

2 * * * ¼ r1 j Ex j2 þ j Ey j2 þ j Ez j2 E ðx; y; zÞ 2 2

r1 *

ð8Þ

Phase-transition temperature of the sample during the rewarming process If the temperature interval of the phase-transition temperature was ½T 1 ; T 2 , where T 1 and T 2 were the lower and upper phase-transition temperatures of the tissue, respectively, then the whole tissue during the rewarming process could be divided into the frozen zone, the mushy zone, and the unfrozen region. To find the corresponding heat and mass transfer equations, the detailed heat and mass transfer model during the rewarming process was constructed assuming equivalent thermal capacity and thermal conductivity. An apparent heat capacity method was used to study the heat and mass transfer model during phase-transition and the rewarming process. Additionally, the effective thermal capacity and the effective thermal conductivity can be expressed as follows [10]:

CðTÞ ¼

8 > < > :

C 01 Q0 ðT 2 T 1 Þ

T < T1 C 01 þC 02 2

þ

T1 6 T 6 T2

C 02

ð9Þ

T > T2

*

r E ¼ jxl0 H *

where x is the angular frequency of the electromagnetic wave, e the dielectric constant of the material, l0 permeability of vacuum (4p 107 N=A2 ), and j the unit imaginary number. The boundary conditions of the microwave cavity were assumed to be perfect electric conductors; therefore, the electromagnetic field distribution in the cavity could be accurately calculated using numerical stimulation methods.

ð3Þ

*

r H ¼ jxe E

ð4Þ

kðTÞ ¼

8 > < k01

k01 þ k02 > 2

:

k02

T < T1 T1 6 T 6 T2 T > T2

ð10Þ

*

rE ¼0

ð5Þ

*

rH ¼0

ð6Þ

where C 01 and C 02 are the thermal capacities of the frozen and unfrozen tissues, respectively, k01 and k02 the thermal conductivity of the frozen and unfrozen tissues, and Q 0 the latent heat of the tissues.


3


1

Heat generated by SPM nanoparticles in the microwave field

r2 Assume the interactions between the nanoparticles at the molecular level are negligible, the magnetization of the ferrofluid in an alternating electromagnetic field can be represented by the 00 complex susceptibility, x ¼ x0 þ ix , where x is the susceptibility 0 00 of nanoparticles, x and x are both frequency dependent. The heat generated by nanoparticles is then governed by [31,46]:

2 * 00 Pnano ¼ l0 pfx Hðx; y; zÞ

ð11Þ

Subsequently, the magnitude of the magnetic field intensity is described as follows [5,31]:

* Hðx; y; zÞ ¼

1 1 þ NðxÞ

* E ðx; y; zÞ

ð12Þ

l0 pfR

where f is the frequency of the electromagnetic wave, R the radius of the magnetic induction loop, N(x), the demagnetizing factor of tissue for a spherical composite. The total heat generated in the electromagnetic field by nanoparticles and the tissue, Qnano and Qwave, can be equivalent to the heat generated by another tissue with the effective conductivity of r3 [31,38,46]:

Q nano þ Q wave

( 2 ) * 1 4 r2 4 3 3 ¼ nVP pr þ V nv pr E ðx; y; zÞ 3 v 3 SPM 2 " 2 # * 3nr 3 x00 r2 4 r3 *2 3 E ðx; y; zÞ ¼ þ E 1 npr ¼ 2 3 2 2 4l0 fR

¼

ð1 gÞ

r1

þ

g r3

ð19Þ

The density, specific heat, and thermal conductivity of tissue in the presence of nanoparticles were calculated as previously described [31,45]. C1 and r1 have been illustrated in Fig. 1(a). According to previously reported data [31,45], the typical properties of tissues and SPM nanoparticles are as follows: q1 = 1000 kg/m3, k1 = 0.47 W/m °C (T P 273 K), k1 = 1.75 W/ m °C (T < 273 K), L = 0.08 m, q3 = 5180 kg/m3, r3 = 25,000 S/m, x0 = 18, r = 10 nm, k3 = 40 W/m °C, C3 = 4000 J/kg °C, R = 0.01 m, f = 2.45 GHz, and N(x) = 1/3. According to Eq. (13), if the heat generated by nanoparticles would be more than the heat generated by the tissue, i.e., Qnano P Qwave,

2 2 * 3nr 3 x00 * P r2 1 4 npr3 E ðx; y; zÞ E ðx; y; zÞ 2 3 2 4l0 fR

ð20Þ

then, r2

n P n0 ¼

2 3r 3 x00 4l0 fR2

þ 23 r2 pr 3

ð21Þ

In order to directly analyze the function of SPM nanoparticles, we selected its concentration, n, to be integer multiples of n0, i.e., n = n0, 2n0, 3n0, and 4n0, which leads to the heat generated by nanoparticles is integer multiples of the heat generated by tissues.

ð13Þ where r is the radius of the nanoparticles, n the concentration of nanoparticles in the tissue, V the control volume in the frozen tissue, r2 the conductivity of the SPM nanoparticles pre-embedded tissue, and r3 the effective conductivity of an unknown tissue. According to Eq. (13), r3 can be expressed as:

"

r3 ¼ 2

3nr 3 x00

r2

þ 2 2 4l0 fR

# 4 1 npr3 3

ð14Þ

Therefore, using Eqs. (8) and (14), the influence of nanoparticles on heat transfer process may be attributed to the change of the conductivity as follows:

Dr ¼ r3 r1

ð15Þ

Due to volume proportions, g ¼ 4npr =3 stands for the volume concentration of nanoparticles inside spherical frozen tissue. After homogeneously embedding the nanoparticles in tissue, the dielectric constant e2 and permeability l2 of tissues were treated as normal tissues because the concentration of the particles was too low to affect the bulk material properties [31]. Also, the assumption of homogeneously embedded SPM nanoparticles was based upon the high diffusivity of SPM nanoparticles in tissue due to their ultra-small size [54]. Hence, in most cases, it is more likely for these nanometer-sized particles to distribute evenly in a 6 mm sized small biomaterial at a reasonable short period of time. Furthermore, C 2 was the average specific heat; q2 the density; and r2 the electrical conductivity of frozen tissues embedded with nanoparticles. These variable quantities could be approximated by a serial arrangement of the two materials [31]: 3

q2 ¼ ð1 gÞq1 þ gq3

ð16Þ

C 2 ¼ ð1 gÞC 1 þ gC 3

ð17Þ

1 ð1 gÞ g ¼ þ k2 k1 k3

ð18Þ

Fig. 1. Temperature and frequency dependent thermal and electrical properties of biomaterials. (a) Temperature dependent apparent specific heat, conductivity, and relative dielectric constant of biological tissues [45]. (b) Frequency dependent dielectric properties of biological tissues [16,53].


4


Variation in permittivity and conductivity of the sample at different frequencies The interaction between the tissue and the frequency of electromagnetic field is described by Debye’s law [16,53], as shown in Eq. (22), since the dielectric constant is frequency dependent,

e ¼ e0 je00 ¼ e1 þ

et e1 1 þ jxs

ð22Þ

where et is the dielectric constant of the tissue in low frequency microwaves, e1 the dielectric constant of the tissue in high frequency microwaves, and s the relaxation time. The relationship between frequency and the dielectric constant of the tissue is shown in Fig. 1(b). Although limited measurements regarding the dielectric properties of muscle and fat at 2.45 GHz were reported [37], the dielectric properties of these tissues at 434 MHz have been measured with high accuracy [45]. Furthermore, the dielectric properties of CPAs have been measured at frequencies ranging from 27 MHz to 3 GHz and at temperatures ranging from 75 to 25 °C [14]. Because the dielectric properties of tissue at different microwave frequencies greatly correlate with the water content of the tissue [14,37,45], in this study, we assumed Debye’s law remained applicable for the whole rewarming process. Since the accurate parameters for the tissue at 2.45 GHz were not accessible, the absolute values of our theoretically stimulated results are not exact. However, investigating the effect of SPM nanoparticles on microwave rewarming is more important than obtaining the absolute values.

Results and discussion The tissue was rewarmed in a cylindrical single mode microwave resonant cavity with a high-powered antenna through a cuboid waveguide. Using Finite Element Method (FEM), the coupling of the electromagnetic field and temperature field was modeled with the working frequency of the microwave set at 2.45 GHz. When the temperature of any point in the sample was above zero, the rewarming process was suspended. The parameters of the tissue used in the simulations are shown in Fig. 1(a). Since these data were obtained from the resonant cavity and perturbation method at 434 MHz, Debye’s law worked as an approximation. The geometric model for the microwave rewarming system is shown in Fig. 2(a). The diameter and height of the cylindrical cavity was 320 and 150 mm, respectively. The cuboid waveguide was 18 39 50 mm. The diameter of the spherical frozen tissue was 12 mm. Due to the symmetrical characteristic of the cylindrical cavity and the cuboid waveguide, the cross section of the system was considered as a rectangle with the sample shown in this plane as circle (Fig. 2(b)). In the manufacture of the microwave rewarming system simulated in this study, a circulator should be installed at the middle of the waveguide. Typically, the insertion loss and isolates degree of the circulator may be 0 and 30 dB, respectively. The microwave can pass through the waveguide without power loss from the source to the cavity. While even if 100% of the input power is reflected, 99.9% of the power will be absorbed by the circulator, and only 0.1% of the power will be finally reflected into the source. With this design, the microwave source could be well protected and the whole system could work smoothly. Most of the ‘‘cavity power’’ is used to generate the electric field *

in the cavity, i.e. E , which is typically approximately 104 V/m. The 2 * power density of the sample absorbing energy is 12 r E 6 3

6000 kW/m (since 0 < r < 0:12 S/m), while the sample is so small

Fig. 2. The CAD model of the single mode microwave resonant cavity used for rewarming frozen tissues at 2.45 GHz. (a) The microwave system is composed of a cylindrical microwave cavity and a cuboid waveguide. The dimension of the cavity is shown in the Cartesian coordinate system. L = 320 mm, h = 150 mm, a = 18 mm, b = 39 mm, and c = 50 mm. (b) The side view of the cross section. The length and width of the rectangle is L = 320 mm and h = 150 mm, respectively. The circle represents the profile of the sample in 2-D. To amplify the outline of the circle, the coordinates of the circle are not based on the proportions of the rectangle. There are three sampling points in the sample, shown as point a, b, and c. Point a is located at the center of the sample. Line segment ac denotes the radius of the circle, which is 6 mm. Point b is located at the midpoint of segment ac. (c) Schematic diagram of the microwave rewarming system at 2.45 GHz. 3

in volume (43 pð6 103 Þ 9:05 107 m3) that the absorbed power of the sample is very small, 5 W. To avoid the interference modes of the low order mode, the single working mode in the cavity is TM411. The adjacent modes are TM220 and TM320, with resonant frequency of 2.093 and 2.511 GHz, respectively. Comparison of tissues with SPM nanoparticles and without nanoparticles during the rewarming process Since the effect of nanoparticles is not evident when n = n0 (n, the concentration of nanoparticles), n = 2n0 was used as a starting value. According to Eq. (21), Qnano = Qwave if n = n0, so Qnano = 2Qwave when n = 2n0. The initial temperature of the tissue was set at 60 °C. Using FEM and the parameters given above, the electric field intensity distribution and the displacement distribution in the cross section during the rewarming process were modeled, shown in Fig. 3(a) and (b), respectively. No change was observed after the nanoparticles were embedded in the sample. The effect of SPM nanoparticles on electric polarization intensity distributions and electromagnetic power loss density distributions in the cross section during the rewarming process are shown in Fig. 3(c)–(f). Electric polarization intensity in the cross section increased with the addition of SPM nanoparticles. The magnitude of electromagnetic power loss density reflected the microwave energy absorbed by the profile of the sample. According to the definition of quality factor (Q-factor) of the microwave cavity [27,35], Q-factor was expressed as follows:

Q ¼ 2p

W0 W1

ð23Þ


5


Fig. 3. The electromagnetic field distributions under a fixed cavity power is 8000 W at the cross section (Fig. 2(b)) during the rewarming process. (a) Field intensity. (b) Displacement. The central ‘dot’ represents the displacement distribution in the cross section of the tissue during rewarming process. The displacement is not equal to zero at the center of the tissue, because the relative dielectric constant of tissue is far more than empty space. (c) Electric polarization intensity of the sample without nanoparticles. (d) Electric polarization intensity of the sample with embedded nanoparticles (2n0). (e) Electromagnetic power loss density of the sample without nanoparticles. (f) Electromagnetic power loss density of the sample with embedded nanoparticles (2n0).

where W0 is the stored electromagnetic energy, and W1 the dissipated electromagnetic energy. Since the cycle time of the microwave, Dt, is very small, Q-factor can be expressed as the ratio of cavity power and the electromagnetic power loss density:

Q ¼ 2p

W0 p Dt p ¼ 2p 0 ¼ 2p 0 p1 Dt W1 p1

ð24Þ

where P0 is the cavity power, which was set to be 8000 W in this rewarming process. The electromagnetic power loss density, P1, is a position- and time-dependent variable. According to Eq. (24), Fig. 3(e) and (f), the interval estimation of the Q-factor during the rewarming process were calculated as follows:

Interval estimation of Q ¼

½20350;213675 without nanoparticles ½10010;20652

with nanoparticles ð25Þ

In order to valid the estimation, the calculated Q-factor interval was checked with a previously reported value that was obtained experimentally. Luo et al. reported the Q-factor to be 136,410 with a microwave resonant cavity working at 434 MHz [30], which is within our estimated range, indicating our calculations were correct. The effect of SPM nanoparticles on temperature distributions in the cross section during the rewarming process are shown in Fig. 4(a)–(d). The maximum temperature gradient in the sample


6


Fig. 4. Temperature distributions under a fixed cavity power is 8000 W at the cross section (Fig. 2(b)) during the rewarming processes. (a) Sample without embedded nanoparticles after 164 s of rewarming. (b) Sample embedded with nanoparticles after 87 s of rewarming. (c) Sample without embedded nanoparticles after 82 s of rewarming. (d) Sample embedded with nanoparticles after 43 s of rewarming.

with nanoparticles was 1.64 °C/mm, while it was 3.59 °C/mm without nanoparticles. The whole rewarming process took 87 s (the midpoint was approximately 43 s) with nanoparticles, while it would take 164 s to complete without the SPM. The temperature of the sample went from 9.71 to 7.57 °C in 43 s, suggesting the temperature gradient within the sample was relatively small. Numerical results of three different sampling points in the rewarming processes with nanoparticles vs. the sample without nanoparticles are shown in Fig. 5. The numerical result of the same tissue during rewarming from a previous study [19] is also included in Fig. 5 for comparison purpose. It is evident that, after embedding with nanoparticles, the warming rate increased and the temperature gradient decreased. Both properties are attractive for fast and uniform rewarming. Comparison of the heat generated by samples without nanoparticles when the cavity power is changed According to Eq. (8), the energy absorbed by the sample can be modified by varying the electric field intensity in the cavity and the electrical conductivity of the sample. The change of the cavity power would also influence the rewarming process due to the change in the electric field intensity in the cavity. Using FEM, the maximum temperature gradient in the sample was 2.05 °C/mm when the cavity power was 4000 W and the initial temperature of the tissue was at 60 °C, while it got reduced to 0.77 °C/mm when the cavity power was 2000 W (Fig. 6(a) and (b)). Electric field intensity distributions in the cross section of the two cavity power conditions are shown in Fig. 6(c) and (d), respectively. No difference in term of the electric field distribution was observed when the sample was embedded with nanoparticles.

Fig. 5. Simulation results of the three sampling points (as shown in Fig. 2(b)) during the two different rewarming processes. The three blue lines represent the three points rewarmed with the presence of nanoparticles (n = 2n0). The three black lines represent the same three points rewarmed without the presence of nanoparticles. A close-up view of the curves is shown at the bottom right of the figure. The red lines are adopted from the previous published numerical study of the center point and the edge point of the same tissue undergoes similar rewarming process. The same tissue was rewarmed in the microwave cavity at 434 MHz with an input electric field intensity of 300 V. The 15 mm in radius spherical sample exhibited a maximum temperature gradient of 0.83 °C/mm [19]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The rewarming process took 310 s with 4000 W, while it lasted 500 s when the power was 2000 W. The numerical result of the



7

Fig. 6. Temperature and electric field intensity distributions at the cross section (Fig. 2(b)) during the rewarming process with various cavity power. (a) Temperature distribution with a 4000 W cavity power. (b) Temperature distribution with a 2000 W cavity power. (c) Electric field intensity distribution with a 4000 W cavity power. (d) Electric field intensity distribution with a 2000 W cavity power.

Although the thermal stress in the sample was decreased, the probability of devitrification was increased, which means the tissue had a higher probability to be injured during the rewarming process. It might be impossible to achieve both high warming rate and uniformity by solely adjusting the cavity power. The effect of heat generated by SPM nanoparticles and the tissue

Fig. 7. Simulation results of the three sampling points (as shown in Fig. 2(b)) during the two different rewarming processes. The three blue lines represent the three points rewarmed with a 4000 W cavity power. The three black lines represent the three points rewarmed with a 2000 W cavity power. A close-up view of the curves is shown at the bottom right of the figure. The red lines are adopted from the previous published numerical study of the center point and the edge point of the same tissue undergoes similar rewarming process [19]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

same tissue in a similar rewarming process from a previous study [19] is included (Fig. 7) for comparison purpose. A decrease of the cavity power has caused a decrease in temperature gradient in the sample, but an increase in rewarming time.

Here we let n equal to 4n0, the heat generated by nanoparticles was anticipated to be quadruple the heat generated by tissue. The initial temperature of the tissue was set to be 60 °C. As shown in Fig. 8(a) and (b), the maximum temperature gradient in the sample changed from 4.4 to 1.18 °C/mm when the cavity power changed from 8000 to 2000 W, respectively. Furthermore, the duration of the rewarming process was also increased from 75 to 300 s, respectively. The numerical result of the same tissue in a similar rewarming process from a previous study [19] is presented in Fig. 9 for comparison. The blue lines in Fig. 9 represent the rewarming process of three points within the sample while the cavity power was at 8000 W. The rewarming rate was improved dramatically. The slowest rewarming process was completed within 75 s. Unfortunately, the temperature gradient in the sample was unacceptable. Such great temperature gradient was because of high energy generated by SPM nanoparticle (four times the concentration of nanoparticles in the sample, 4n0) and limited thermal conductivity of the tissue. The optimal parameters to obtain acceptable rewarming results Based on the findings shown above, in order to obtain the optimal combination of both high rewarming rate and low


8


Fig. 8. Temperature distributions at the cross section (Fig. 2(b)) during the rewarming processes with a fixed nanoparticle concentration (n = 4n0) and a various cavity power. (a) Temperature distribution when cavity power is 8000 W. (b) Temperature distribution when cavity power is 2000 W.

Fig. 10. Temperature distributions at the cross section (Fig. 2(b)) during the rewarming processes with a fixed nanoparticle concentration (n = 3n0) and a various cavity power. (a) Temperature distribution when cavity power is 5000 W. (b) Temperature distribution when cavity power is 3000 W. (c) Temperature distribution when cavity power is 2000 W.

Fig. 9. Numerical results of the three sampling points (as shown in Fig. 2(b)) during the two different rewarming processes. The three blue lines represent the three points rewarmed with the presence of nanoparticles (n = 4n0) when the cavity power is 8000 W. The three black lines represent the three points rewarmed with the presence of nanoparticles (n = 4n0) when the cavity power is 2000 W. A close-up view of the curves is shown at the bottom right of the figure. The red lines are adopted from the previous published numerical study of the center point and the edge point of the same tissue undergoes similar rewarming process [19]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

temperature gradient, the concentration of SPM nanoparticles was selected to be 3n0. Also, in order to make a better comparison with previously reported results [45], the initial temperature of the tissue was changed from 60 to 70 °C. All parameters of the tissue were assumed to be temperature-independent within the temperature range of 70 and 60 °C. The maximum temperature gradient in the sample changed from 1.35 to 0.68 °C/mm and to 0.44 °C/mm when the cavity power changed from 5000 to 3000 W and to 2000 W, respectively (Fig. 10). Furthermore, the duration of the rewarming process was also changed from 130 s to 215 s and 325 s, respectively. Both



9

embedded SPM nanoparticles, the heat generated by nanoparticles in the sample will increase linearly and the rewarming rate of the sample will also increase accordingly. Unfortunately, the maximum temperature gradient is also elevated due to the limited thermal conductivity of the tissue. However, the uniformity of temperature distribution can be improved if the cavity power is lowered; (2) to achieve fast and uniform rewarming, optimized combination of both embedded SPM nanoparticle concentration and cavity power were attempted. Our stimulation suggested that the most favorable choices were when n was set to 3n0 with 3000 W of cavity power. The rewarming rate and the maximum temperature gradient were able to be maintained at 20.73 °C/min and 0.68 °C/mm at the end of the rewarming, respectively. Experimentally obtaining a better understanding of the dielectric coefficient and thermal conductivity of tissues, so that a more precise simulation of the heat generation and distribution of nanoparticles inside the biomaterial can be theoretical calculated, which is the subject matter of another ongoing parallel effort. Fig. 11. Simulation results at the two sampling points (point a and point c in Fig. 2(b)) during the two different rewarming processes. The two blue lines represent the two points rewarmed from 70 °C with the presence of nanoparticles (n = 3n0) and a 5000 W cavity power. The two black lines represent the two points rewarmed from 70 °C with the presence of nanoparticles (n = 3n0) and a 3000 W cavity power. A close-up view of the curves is shown at the bottom right of the figure. The red lines are adopted from the previous published numerical study of the center point and the edge point of the same tissue undergoes similar rewarming process. The same tissue was rewarmed in the microwave cavity at 434 MHz with an input electric field intensity of 300 V. The 15 mm in radius spherical sample exhibited a maximum temperature gradient of 0.83 °C/mm. The green lines represent the experimental results of the center point and the edge point of the tissue during the rewarming process. The sample exhibited a maximum temperature gradient of approximately 1.11 °C/mm [45]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

experimental and FDTD stimulation results of the same tissue in a similar rewarming process at 434 MHz from previous studies were included in Fig. 11 as references [19,45]. Our results indicate that we were able to achieve either a 0.68 °C/mm maximum temperature gradient with a rewarming rate of 20.73 °C/min, or a 1.35 °C/mm maximum temperature gradient with a rewarming rate of 36.32 °C/min. Conclusions In this work, the effect of SPM nanoparticles in microwave rewarming of cryopreserved tissues has been investigated through numerical simulation. Although the accurate electrical and thermal parameters of the tissues at 2.45 GHz are not known and the absolute values of the presented simulation work are not exact, the trend of using single mode microwave resonant cavity coupled with SPM nanoparticles to thaw frozen biomaterials has been precisely analyzed in detail. The impact of nanoparticles was reflected mainly in two aspects. One is the heat generated by nanoparticles directly; and two is the alteration of the biomaterials’ electrical and physical properties. The variation of electric field intensity in single mode microwave resonant cavity cannot solely lead to rapid and uniform rewarming. Our work confirmed that the adjuvant use of nanoparticles may be a promising approach to deliver uniform and rapid rewarming of biomaterials. Our simulations suggested a positive effect that the rewarming rate and the uniformity of temperature distribution both were improved by embedding SPM nanoparticles. Furthermore, using SPM nanoparticles led to a significantly reduced maximum temperature gradient in the sample. From the presented results, the following insightful comments can be stated: (1) with the increase of the concentration of

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