Modeling of the spatiotemporal distribution of temperature fields in skin and subcutaneous adipose tissue after exposure to ultrasound waves of different frequencies

Temperature fields produced in the skin and adjacent subcutaneous white adipose tissue (sWAT) during and after exposure to ultrasound (US) waves are significantly dependent on the US frequency. In this study, we present theoretical descriptions of temperature fields appearing in composite skin/sWAT after exposure to US at frequencies of 3 MHz, 10 MHz, and 19 MHz. While the temperature increased by approximately 1.5°C in skin during US exposure at intensities up to 10.0 W/cm2 and a frequency of 3 MHz, this increase reached 9.0°C and 16.0°C at US frequencies of 10 MHz and 19 MHz, respectively. Because of the large difference in heat capacitances and US attenuation coefficients in the skin and adjacent sWAT, the interface between these two layers was subjected to a temperature gradient that increased with US frequency. This gradient was low after applications of US at 3 MHz but was as high as 7.5°C/mm at 10 MHz and 14.0°C/mm at 19 MHz for US intensities of 10.0 W/cm2. High temperature gradients produced by U...


I. INTRODUCTION
Absorption of ultrasound (US) waves during their propagation in a homogeneous biological tissue causes temperature increase in the tissue, which exponentially decays with US penetration depth. Depending on the absolute value of the temperature increase, such heating can cause physiological reactions or morphological changes in the tissue. However, in inhomogeneous tissue, the attenuation of US is significantly dependent on the US frequency and on the microscopic tissue structure, especially at boundaries between single layers with different acoustic impedances.
Skin and adjacent subcutaneous white adipose tissue (sWAT) are two mutually confined layers with very different mechanical, thermal and acoustic properties and with distinct US attenuation coefficients. Mechanically, the bilayer skin/sWAT can be considered a composite with some adherence between the layers. 1 Corresponding adherence strength is different in distinct facial fat compartments: adherence is weak in the medial and lateral midface, in some parts of the periorbital area, and in the temple, forehead, and neck. However, adherence is much stronger in the perioral and nasal facial compartments. 2 The skin/sWAT interface appears as a boundary between the dermis and sWAT with high and several times smaller US attenuation coefficients at the same frequency, respectively. Consequently, under suitable sonication conditions, cells located near this interface should be exposed to significant temperature gradients in a thin superficial layer of sWAT adjacent to the reticular dermis. This superficial sWAT layer is known as dermal (or interfacial) WAT. 3,4 This layer contains adipocytes with transient phenotypes, which more readily undergo differentiation, hyperplasia, and hypertrophy than normal adipocytes from deeper layers of sWAT. These processes can be stimulated, for example, by local heating and mechanical stress, [5][6][7] which are present during US wave propagation.
Very high frequency US (VHF-US) with frequencies of 10 MHz and above has been proposed for treating different skin efflorescences (e.g., acne, eczema, perioral dermatitis, and chronic wounds) and for anti-aging therapy. [8][9][10][11][12] The appearance of skin efflorescences has recently been connected with adipocyte activity in the interfacial WAT layer. 3,4,13,14 At least for a few efflorescences, therapeutic effects can be provided by US-induced mild hyperthermia. For example, mild hyperthermia has been shown to improve human psoriatic lesions. 15 To treat chronic psoriatic plaques, 5.3 MHz US produced dermal heating up to 42-45 • C was successfully applied. 16 Similarly, application of 8 MHz US produced temperatures of 42-44 • C at a depth of 2-3 mm beneath the skin surface, thereby demonstrating significant improvements in palmoplantar psoriasis. 17 The utilization of 10 MHz VHF-US successfully treated psoriatic lesions by concentrating heating in the dermis and superficial sWAT. 12 Very recently, sonoelastographic measurements of adipose tissue adjacent psoriatic plaques after US treatment revealed significant differences in the elastic properties of interfacial WAT layers in treated and non-treated psoriatic plaques. 18 While the attenuation of US waves generally produces thermal effects in tissue, the optimal heating of the skin/sWAT interface can be achieved only with US waves within a frequency range. Too low frequency US will pass the skin without sufficient attenuation, whereas extremely high frequency US will primarily be absorbed in the dermis. Here, we presented a model describing the temperature fields produced in composite skin/sWAT at different US frequencies. This model can be used for identifying optimal treatment conditions at the skin/sWAT interface.

A. Bioheat transfer
Let x be the depth under skin, with x = 0 corresponding to the skin surface. A US wave at frequency f having energy density I 0 on the skin surface enters the skin perpendicular to its surface and penetrates the sWAT through the skin/sWAT interface, which is oriented parallel to the skin surface. The US beam is assumed to have a homogenously distributed intensity over the radiating surface of the applicator. This simplification reduces the description of the temperature distribution and temperature gradients in the composite skin/sWAT to a one-dimensional problem.
US energy absorbed in the tissue is assumed immediately transformed into thermal energy causing a local temperature increase in the tissue. To describe the temperature evolution in the composite skin/sWAT, the following heat transfer equation is solved: 19 where index i = s refers to the skin and i = a to the sWAT; T i (x,t) is the temperature at depth x; ρ i is the mass density; c pi is the specific heat capacity at constant pressure; k i is the thermal conductivity; Q i (x,t) is the distributed heat source due to US energy absorption; and Q Pi (x,t) is the distributed heat sink from blood perfusion. The distributed heat source Q i (x,t) is due to US energy absorption in the tissue. For exponential US absorption, the distributed heat source is represented as where α i = α i ( f ) is the frequency-dependent attenuation coefficient of US; τ is the duration of the US pulse; h d is the dermis thickness; θ(τ t) = 1 for 0 ≤ t ≤ τ, and θ(τ t) = 0 for other τ. Generally, skin demonstrates different attenuation coefficients in the epidermis and upper and lower dermis; similarly, sWAT has superficial and deep layers known to have distinct mechanical properties. 4,20 However, we first assume the skin and the sWAT as homogeneous layers having spatially constant thermal and mechanical characteristics. The distributed heat sink Q Pi (x,t) can be described by Pennes' equation: where index b refers to blood; and W(T) is the blood perfusion rate, which is strongly temperaturedependent. Furthermore, the following approximation for W(T) is used: 21 where W s0 = 0.45 kg m 3 s and W a0 = 0.36 kg m 3 s are normal blood perfusion rates in skin and sWAT, respectively.
Because the single tissue layers in the composite skin/sWAT have different acoustic impedances, US wave reflection on the skin/sWAT interface can reduce the energy penetrating the sWAT and thus, diminish the heating of the superficial sWAT layer. To estimate this effect, we consider the acoustic impedances, Z i , of the skin and sWAT, which are the products of their mass densities with corresponding US velocities. If a US wave of intensity I 0 falls perpendicular to the skin/sWAT interface, one part (I tr ) will be transmitted into the sWAT and another part (I ref ) will be reflected back into the dermis: Acoustic impedances of the skin and sWAT were reported to be approximately Z s = 1.99x10 6 kg/sec/m 2 and Z a = 1.38x10 6 kg/sec/m 2 , respectively. 22 From (6), 96.7% of US energy will be transmitted into the sWAT and only 3.3% will be reflected on the interface skin/sWAT. Thus, at first approximation, we can neglect the reflection of US waves on this interface. We will also neglect reflections on sWAT/muscle and muscle/bone interfaces. This approach is a good approximation for sufficiently thick sWAT. We are primarily interested in the temperature increases in the skin and the skin/sWAT interface. Temperature field calculations have been provided for the mass densities, heat capacities of the skin and sWAT 23 and for the thermal conductivities of these tissues which are summarized in Table I.
To solve equation (1) and find the temperature distributions and gradients in the skin and sWAT for US waves of different frequencies, skin thickness and frequency-dependent US attenuation coefficients in both layers are required.

B. Variations of the facial dermis thickness
The absolute value of the dermis thickness h d is dependent on the measurement procedure and is different in vitro and in vivo. This parameter has high inter-areal and inter-subject variations. 6 Facial h d shows strong spatial variations; dermal thickness is larger in the cheek and chin areas and smaller in the neck area. 24 For example, h d in the neck area has been reported to vary in the range 0.25-0.80 mm, in malar eminence area in the range 0.57-1.62 mm, and in the cheek area in the range 1.04-1.20 mm. In the non-wrinkled areas of human skin, h d was 0.95±0.28 mm with lower and upper limits of 0.35 mm and 1.65 mm, respectively. 25 Due to this variability, h d = 0.5 mm, 1.00 mm, and 1.5 mm were used to model the influence of the skin thickness on the temperature distribution at the skin/sWAT interface.

C. Attenuation coefficients of VHF-US in the skin and sWAT layers
While attenuation coefficients have demonstrated some temperature dependence, 26 we will first consider the attenuation coefficients constant for all temperatures in this study.

Skin
The dependence of the attenuation coefficient α on 1-5 MHz frequency in human skin is described by the equation 27 The US attenuation coefficient at 3 MHz in skin can be assessed as α s (3) 29 While the attenuation coefficient has been shown to be age and gender-dependent, 30 we will also neglect these effects.

sWAT
The dependence of the attenuation coefficient α on 3-7 MHz frequency in human sWAT is described by the equation α a ( f ) = 0.07f 1. Attenuation coefficients of the US in the skin and sWAT at three US frequencies considered in this study are summarized in Table II. These values were used for numerical solution of Eq. (1).

A. Method
Finite element analysis was applied to solve Eqs. (1)-(5). All simulations were performed with inhouse code. This code was developed to simulate non-stationary problems of heat transfer in systems with non-linear, spatially distributed heat sources and spatially varying tissue properties. The code uses a fully implicit numerical scheme 32 with a linear solver based on the TDM algorithm. 33 The implemented algorithm can be applied to non-uniform meshes with arbitrary space discretization. In the present study, collocated meshes were used, i.e., the interface between different materials was assumed to be located between calculation nodes. The code was tested and verified against analytical solutions for uniform media for stationary and non-stationary cases.
In this study, the skin surface was assumed to be located at spatial coordinates equivalent to 0.0 mm. The calculated cell size in the skin layer was set constant at 5.0·10 -5 m, whereas an exponentially increasing cell size was used for the mesh in the sWAT. Temporally and spatially dependent solutions for temperature and temperature gradients were obtained within the first 20 sec after US application with 0.01-sec steps for a 0-3 mm distance at 0.05-mm steps, respectively. Temperature fields were calculated as temperature differences from baseline, which was set for 36.6 • C. Temperature gradients were calculated and are presented in • C/mm.
Calculations were performed at three US frequencies of 3 MHz, 10 MHz, and 19 MHz for three skin thicknesses of 0.5 mm, 1.0 mm, and 1. influence of variable energy densities on the temperature fields, additional simulations were provided for all frequency-thickness combinations at 1.0 W/cm 2 and application times of 7.5 sec and 15.0 sec. All simulations were performed with and without blood perfusion. To confirm the stability of the simulation results with varying skin thermal conductivity, two additional calculations with thermal conductivities 25% lower and higher than the average value (see Table I) were performed on skin with thickness 1 mm at 19 MHz, 2.0 W/cm 2 , and 5 sec.
The results of each simulation are presented as video clips demonstrating the temporal evolution of temperature and the temperature gradients over 20 sec and up to a 3 mm depth under the skin surface (up to 1,000 frames per clip). We determined the maximum temperature and temperature gradients for each given set of treatment parameters and calculated the corresponding two-dimensional spatial-temporal fields. significantly varied with sonication time. At the skin/sWAT interface, these parameters can reach 5.5 • C and 5.0 • C/mm (Fig. 2 a,b) and 7.0 • C and 8.5 • C/mm (Fig. 2 c,d) at 10 MHz and 19 MHz, respectively. The temperature fields and temperature gradients in the composite skin/sWAT after US application at 2.0 W/cm 2 and 5 sec and 10 MHz and 19 MHz for different skin thicknesses of 0.5 mm, 1.0 mm, and 1.5 mm are shown in Fig. 3. While variations in skin thickness for facial skin typically range from 0.5-1.5 mm and thereby affect temperatures in the dermis, the temperature gradient at the skin/sWAT interface is nearly the same for all thicknesses but strongly increase with increasing US frequency. Fig. 4 shows the influence of the total US energy density on the temperature field. As the energy density increased from 10.0 J/cm 2 to 15.0 J/cm 2 at 10 MHz and 19 MHz and at 1 W/cm 2 , the temperature in the dermis significantly increased. However, only moderate temperature gradient changes were observed at skin/sWAT interface.

C. Influence of thermal conductivities on the temperature gradient at the interface skin/sWAT
Based on the calculations thus far, the thermal conductivities of the skin and sWAT were observed to be 0.4 W/m/K and 0.2 W/m/K, respectively (Table I), which corresponded with those reported elsewhere. 34,35 The thermal conductivities showed significant variations dependent on temperature 19 and water and collagen contents in tissue. For example, to describe heat transfer in the composite skin/sWAT, the following values were used by different authors for skin and sWAT, respectively: 0.445 and 0.185 W/m/K, 36,37 0.34 and 0.22 W/m/K, 21 and 0.53 and 0.16 W/m/K. 38 Because variations in the skin and sWAT thermal conductivity can influence the development of temperature gradients at the interface between these layers, we calculated the temperature distribution and gradients at three skin thermal conductivities k s = 0.3 W/m/K, 0.4 W/m/K, and 0.5 W/m/K and at constant sWAT thermal conductivity k a = 0.2 W/m/K. The results are presented in Fig. 5. Variations in the skin thermal conductivity only slightly influenced the composite skin/sWAT temperature fields.

D. Spatiotemporal distributions of temperature and temperature gradients in the composite skin/sWAT
The spatiotemporal distributions of temperature and temperature gradients in the composite skin/sWAT with a skin thickness of 1.0 mm after application of VHF-US at 10 MHz and 19 MHz are presented in Fig. 6. The maximum temperature gradient was localized near the surface skin/sWAT and temporally corresponded with the US exposure time. This gradient quickly reduced with increasing distance from the interface.
A spatiotemporal analysis of the temperature demonstrated a relatively slow decay of the dermis temperature with significant heating of the skin at least during the time corresponding to two times that of US exposure.

IV. DISCUSSION
The dermal-sWAT interface plays an important role in affecting the mechanical properties of skin. Cells located in this area demonstrate strong reactions to different physical factors, such as mechanical stress and heat, and can undergo trans-differentiation, which can significantly modulate the local mechanical parameters of composite skin/sWAT. [1][2][3][4] Moreover, the effective mechanical properties of the composite are significantly dependent on adherence between the two layers. Stronger adherence more effectively increases the influence of sWAT in determining the mechanical properties of the skin by decreasing the effective Young's modulus of the composite and thus, increasing the susceptibility of the skin to the onset of structural instabilities typical for aging skin. 1 The ability to control adherence should be considered as an important tool in anti-aging strategies, and local heating of the interfacial area can sufficiently modify this property of composite skin/sWAT.
Modification of the structural and mechanical properties of the dermis-sWAT interface can be achieved by sufficient heating. The incorporated energy should be primarily absorbed within the vicinity of the interface. We recently showed that sufficient energy levels can be achieved by applying radio-frequency currents using optimally placed electrodes on the skin surface. 5 In the present study, we investigated whether similar effects could be achieved using very high frequency US waves. Temperature fields produced in the composite skin/sWAT under US exposure were significantly dependent on the US frequency. Whereas the application of 3 MHz US did not produce any substantial thermal effect in single layers or at the skin/sWAT interface, 10  Because of large differences in thermal conductivities and US attenuation coefficients in the skin and sWAT, the interface between these two layers experiences temperature gradients. At 19 MHz, this temperature gradient was approximately 5.0 • C/mm and 14.0 • C/mm at US intensities of 1.0 W/cm 2 and 10.0 W/cm 2 , respectively. Corresponding temperature gradients were significantly lower at 10 MHz, reaching 2.5 • C/mm and 7.0 • C/mm, respectively. Temperature gradients after US exposure reached maximum values at the skin/sWAT interface and quickly reduced with increasing distance from the skin/sWAT interface.
The application of 19 MHz VHF-US to heat the dermis and produce high temperature gradients at the skin/sWAT interface could be utilized in the treatment of different dermatological conditions and aesthetic issues of normal skin thicknesses. The application of 10 MHz US could be beneficial for treating the skin/sWAT interface for very thick skins with thicknesses of more than 2.0 mm. These differences should be taken into account when developing dermatological treatments based on the application of very high frequency US waves.

V. CONCLUSIONS
Temperature fields produced in composite skin/sWAT during and after exposure to VHF-US were strongly dependent on the US frequency. At a US intensity of 10.0 W/cm 2 , maximum temperatures in the dermis reached approximately 46 • C and 53 • C at US frequencies of 10 MHz and 19 MHz, respectively. These temperature fields were spatially inhomogeneous and characterized by high temperature gradients near the skin/sWAT interface, reaching 7.5 • C/mm at 10 MHz and 14.0 • C/mm at 19 MHz. These temperature gradients can significantly affect adherence between the skin and sWAT and thus modulate the effective mechanical properties of skin.

ACKNOWLEDGMENTS
I thank Dr. Alexei Kotchourko for his generous help with numerical calculations and thoughtful discussions.
ILK is the managing partner of Wellcomet GmbH. Wellcomet GmbH provided support in the form of salaries for ILK, but did not have any additional role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript. The commercial affiliation of ILK with Wellcomet GmbH does not alter the adherence to all AIP Advances policies on sharing data and materials.