Effect of the ultrasound wave propagation regime in the heat source term of Penne s bio-heat transfer equation

Transcription

1 Effect of the ultrasound wave propagation regime in the heat source term of Penne s bio-heat transfer equation Guillermo Cortela, Carlos Negreira, and Wagner C. A. Pereira Citation: Proc. Mtgs. Acoust. 28, (2016); View online: View Table of Contents: Published by the Acoustical Society of America Articles you may be interested in Micro-Perforated materials for the reduction of flow-induced noise Proceedings of Meetings on Acoustics 28, (2017); /

2 Volume nd International Congress on Acoustics Acoustics for the 21 st Century Buenos Aires, Argentina September 2016 Biomedical Acoustics: Paper ICA Effect of the ultrasound wave propagation regime in the heat source term of Penne s bio-heat transfer equation Guillermo Cortela and Carlos Negreira Laboratorio de Acústica Ultrasónica, University of the Republic, Montevideo, Uruguay; gcortela@fisica.edu.uy; carlosn@fisica.edu.uy Wagner C. A. Pereira Biomedical Engineering Program, COPPE/Federal University of Rio de Janeiro, Brazil,wagner@peb.ufrj.br This work analyses the influence of the ultrasound wave propagation regime on the temperature increase on biological media. Simulated and experimental temperature values from bovine ex-vivo tissue in two different depths, 5 and 40 mm, respectively smaller and bigger than the mean free propagation path (l S= 8.52 mm), are obtained. Temperature curves measured during US therapeutic application (High Intensity Therapeutic Ultrasound, HITU, 1 MHz, Wcm -2 ) for 10 min, are compared to simulated ones using the Penne s bio-heat transfer equation (BHTE) at the same frequency and intensities and using the standard source term (ultrasonic absorption coefficient). At 5 mm, estimated and measured temperature diverge no more than 1%, while at 40 mm, when the scattering starts being not negligible, the simulated temperature is 20% smaller than the measured one. The results indicate that absorption increases when the wave propagation regime changes for depths greater than the l S value. Published by the Acoustical Society of America 2017 Acoustical Society of America [DOI: / ] Proceedings of Meetings on Acoustics, Vol. 28, (2017) Page 1

3 1. INTRODUCTION Therapeutic ultrasound has been used extensively since 1955 for a variety of conditions, such as treatment of musculoskeletal pain [1], soft tissue injury, and joint dysfunction [2]. Additional applications such as accelerated tissue repair and wound healing, edema reduction and treatment of scar tissue have been reported in the literature [3]. In general, the effects of ultrasound may be divided into thermal and nonthermal effects. Typically, thermal effects are performed for the pain treatment, reduction of subacute and chronic inflammation and muscle spasm, and stretching of collagenous tissue in joint and connective tissue contracture. Low dose nonthermal ultrasound is used for stimulation of tissue repair [4] and reduction of edema [3]. Classically, tissue temperature evolution over time is obtained by solving Pennes bioheat transfer equation (BHTE) [5]. Specific temperature increases are required to achieve beneficial effects in tissue. Based on previous studies [6], where the baseline muscle temperature was C, an increase of 1 C (mild heating) accelerates metabolic rate in tissue. An increase of 2-3 C (moderate heating) reduces muscle spasm, pain, and chronic inflammation, and increases blood flow. Strong heating ( 4 C) decreases viscoelastic properties of collagen. ter Haar [7] has attempted to plot the heating rate of ultrasound via theoretical models. According to her postulations to increase temperature, specific energy requirements exist based on the tissue absorption coefficients, size of the area being treated, and the frequency of the ultrasound beam. She has suggested that tissue temperature in an area equal to the radiating area of the ultrasound applicator will increase at a rate of 0.86 C per minute when delivered with a stationary applicator at 1 MHz and at an intensity of 1.0 Wcm -2. With respect to tissue absorption coefficients, it is responsible to convert ultrasonic into thermal energy. Several studies based on the rate of heating method, measure the slope of tissue temperature rise due to US insonation. By embedding invasive thermocouples into a block of tissue, the slope of the recorded temperature rise is related to the US pressure absorption coefficient [8]. The models used so far to determine tissue temperature elevation based on changes in the absorption of ultrasound (energy increase per unit of tissue volume) do not distinguish the energy deposited by ultrasound from that originated by other processes. In particular, the increase in temperature due to ultrasonic irradiation has been only associated with the absorption without taking into consideration the phenomenon of scattering of the ultrasonic waves by tissues. Recent studies show that for sample thicknesses bigger than the mean free path, scattering regime increases and starts having an important contribution to ultrasonic energy absorption [9]. In this study, we have investigated the effect of scattering on the thermal source in ex-vivo biological tissue. Specifically, the purposes are: show that for physiotherapy treatment depths, the ultrasonic propagation regime may differ from single scattering; show how this change may influence the thermal source of BHTE model; and verify its impact on the thermal dose (TD), calculated from the temperature curves according to the method of Sapareto and Dewey [10]. 2. THEORETICAL CONCEPTS In this section, the basic concepts involved in this work are exposed A. THERMAL DOSE The thermal dose (TD) is a suitable index to quantify the relationship between treatment efficacy and the desired temperature as a function of time. The calculations of TD presented in this report are based on extensive experimental results, the following TD relationship has been proposed [10]. Proceedings of Meetings on Acoustics, Vol. 28, (2017) Page 2

4 TD( r, t) = R 0 t [43 T( r,ζ)] dζ (1) where t is the time, r a spatial position, T the temperature, and R (empirical value) is the thermal normalization constant, with R = 0.5 if T 43 C and R = 0.25 otherwise. B. BIOHEAT THERMAL MODEL Several Bioheat Transfer equations (BHTE) have been proposed. The BHTE model relates temperature field to several parameters of tissue and describes the tissue temperature evolution over time, and according to Pennes, it can be expressed as [5]: ρc P T t = (k t T) + ω b ρ b C b (T b T) + Q + q m (2) where T is the temperature, ρ is the density, CP is the specific heat, k is the thermal conductivity, ρb is the density of blood, Cb is the specific heat of blood, b is the blood perfusion rate, Tb is the temperature of the blood, Q is the external heat source and qm is the metabolic heat source. The Q, the absorbed ultrasound energy by the tissue, it can be expressed as: Q = 2α A (r, t)i (3) where A is the absorption coefficient, which depends on the position and temperature and I is the acoustic intensity. It is assumed that the acoustic wave propagation is linear and also that the amplitude of shear waves in the tissue are much smaller than that of the pressure waves. Nonlinear effects and shear waves are therefore neglected [11]. If we consider that the ultrasonic wave propagation occurs in ex-vivo tissue, b=0, q m = 0 or Q q m, considering the expression (3), Eq. (2) becomes: ρc P T t = (k t T) + 2α A ( r, t)i (4) C. ULTRASONIC SCATTERING The scattering mean free path (ls) can be inferred from the amplitude transmission coefficient of the coherent wave, that relates to ls and sample thickness as [12]: T C = e z 2l S (5) where z is the depth of the sample [13]. TC is defined as the spectral ratio of the incident and dispersed pulses in the bandwidth of interest. Proceedings of Meetings on Acoustics, Vol. 28, (2017) Page 3

5 3. MATERIALS AND METHODS A. MEAT SAMPLE To determine the acoustic properties, calculate the scattering mean free path and measure temperature, several samples of fresh bovine skeletal muscle were used (from the same piece of meat). The meat sample (MS) used is the square cut (Uruguayan cut). For the temperature measurements the samples used had approximately parallelepiped shape with dimensions 10(w) 10(h) 20(l) cm. For the acoustical measurement two smaller cuts of the same samples were taken. The two samples (2-mm and 40-mm thick) were used to estimate the US velocity and global attenuation. With the 2-mm thickness sample, we can consider that the attenuation was mainly due to US absorption [14]. The other one was used to measure the total ultrasonic attenuation (absorption and scattering). For the scattering mean free path calculation, fifteen samples were cut into cylindrical shapes (by using a tubular cutter). The thickness of the tissue sample (parallel faces) was varied by cutting successive slices of about 5 mm each. The muscle fibers were positioned perpendicularly to the cylinder axis. The samples were degassed for 40 minutes, using a vacuum pump at very low pressure. The sample were finally immersed in another degassed saline solution (0.9% NaCl) where they rested for approximately 2 hours before any measurement at ambient temperature (25 C). B. ACOUSTICAL PROPERTIES Ultrasound velocity and attenuation coefficient were measured by a transmission/reception technique [15] and the substitution method was chosen by comparing the sample and water transmission [16]. In the current study, the attenuation coefficient ( ) was calculated from the spectral amplitudes ratio (Eq. 6), obtained from the RF signals acquired with and without the sample, AM and AW, respectively. α = 1 L ln A M A W (6) where L is the distance separating the source and receiver (sample thickness). The mean speed of ultrasound (vm) of the sample was obtained comparing the time of flight between the source and the receiver (Eq.7), through the MS (tm) and through pure water (tw): v M 1 = t M t w L + 1 v w (7) The vw is the velocity in water was estimated by literature [17]. The acoustical properties are determined in the temperature range 22 C to 43 C, the rate of increase per minute is 0.2 /min. C. ULTRASOUND i. Ultrasonic equipment The HITU (High Intensity Therapeutic Ultrasound) commercial physiotherapeutic equipment is an Ultrasound therapy Electromedicarin (M 212K model). The circular transducer is unfocused, 1.0-MHz central frequency, 14.5-mm radius and W.cm -2 nominal intensity. Proceedings of Meetings on Acoustics, Vol. 28, (2017) Page 4

6 ii. Ultrasonic scattering The experimental method used is described in the literature [9]. The emitter transducer (Olympus, 1MHz) is excited in broad band (US-Key, Lecoeur Electronique). The US-Key generates an electric pulse with 90V, 2 s duration to excite the ultrasonic transducers. A needle hydrophone (Precision acoustics LTD, 0.5mm) receives the RF signal. The digitization of the signals were performed using a digital oscilloscope (Tektronix TDS2024B, USA) with a maximum sampling frequency of 200 MSa/s. iii. Temperature The temperature was measured by a K-type thermocouple, using a high speed multiplexer NI 9213 (National Instruments; Austin, Texas, USA). Signals were saved using USB interface with a program developed in Matlab (The MathWorks Inc., Natick, MA, USA) and further processed. The thermocouples were used to measure temperature inside the sample at 5-mm and 40-mm depths. 4. RESULTS AND DISCUSSION The scattering mean free path is calculated from the logarithm of the TC (Eq. 5), and was plotted as a function of sample thickness (Figure 1). The measurements were carried out at 37 C. The thickness of the tissue sample (parallel faces) was varied by cutting successive slices of about 5 mm each. The linear fit applied shows that the single scattering regime is up to ls=8.52 mm ln(t C ) -2-4 l s = 8.52mm Depth (mm) Figure 1. Evolution of the transmitted amplitude versus sample thickness for determination of the scattering mean free path, l S=8.52mm. Meat sample is irradiated at a frequency of 1.0 MHz at 37 C. Triangle: ballistic amplitude averaged on disorder. Full circle ballistic amplitude received on hydrophone. Proceedings of Meetings on Acoustics, Vol. 28, (2017) Page 5

7 1.56 Ultrasonic velocity (mm. s -1 ) Temperature ( C) Figure 2. Measured ultrasonic velocity as a function of temperature in a 40-mm thickness meat sample; linear fit: v = T (in mm. s -1 ). The simulated temperature curves were estimated from Eq. (4) using the finite difference method; being the knowledge of the temperature dependence of ultrasonic velocity essential in this simulation. This dependence shows a positive linear trend for the meat sample (Figure 2). A MS of 40-mm thickness was used. Velocity values are matched to the ones from literature [14]. To estimate the global attenuation, a sample of 5-mm thickness and another sample of 40-mm thickness were used. With the one of smaller thickness, we can consider that the attenuation was mainly due to the US absorption. The sample of bigger thickness includes scattering effects. So for the sample of 40 mm the attenuation was 40= 8.12±0.09 Np.m -1 while for the lower thickness, 5= 8.00±0.11 Np.m -1, measurements made at 37 C. The US field was generated by a physiotherapeutic transducer operating in CW. The frequency of 1.0 MHz was applied for 10 min. The nominal intensity varied between 1.5 and 2.0 Wcm -2. The typical physiotherapy transducer has a metal matching layer. The numerical (uses the concept of spatial impulse responses [18,19] and is developed with MatLab ) ultrasonic intensity field sample tissue is shown in Figure 3. Figure 3. Simulated ultrasonic intensity field for physiotherapy transducer (1.0 MHz). Proceedings of Meetings on Acoustics, Vol. 28, (2017) Page 6

8 When applying 1.5 W.cm -2 or 2.0 W.cm -2 US intensity, in the meat sample at 40 mm depth, the measured and simulated temperatures curves present not negligible differences (Figure 4 upper panel). For the case at a 5 mm depth (Figure 4 lower panel), there is almost no difference. All the simulated temperatures curves only consider the ultrasonic absorption. 41 Temperature ( C) Time (min) 46 Temperature ( C) Time (min) Figure 4. Temperature curves at 40-mm (upper panel) and 5-mm (lower panel) depths of the meat sample. Ultrasonic Intensity applied: 2.0 Wcm -2 (numerical: continue line; measured: full circle) and 1.5 Wcm -2 (numerical: dash line; measured: triangle). The TD for both depths was calculated by using the numerically simulated and experimental curves (Figure 4) and their values are presented in the Table 1. A percentage error (E%TD) can be calculated for the TD from the experimental curves, taking the TD from the simulated curve as reference. Proceedings of Meetings on Acoustics, Vol. 28, (2017) Page 7

9 Table 1. Thermal dose (equivalent minutes) values calculated from temperature curves. Depth [mm] Intensity [W.cm -2 ] Numerical TD Measured TD E%TD Considering the results of the intensity 2 Wcm -2. It is observed that a TD error is 1.3% for small depth when only absorption is present (scattering component is negligible), but increased to 21% for greater depth (scattering component becomes important to the heating). The E%TD values for the other ultrasonic intensity (1.5 Wcm -2 ) present a similar behavior. 5. CONCLUSION It is important to note that the Born approximation is usually assumed for US propagation in biological tissues. This experimental work has demonstrated that, besides contributing to the global attenuation, scattering may play a non-negligible role in the absorption of US energy once it exceeds the scattering mean free path for the US wave. Regarding the dependence on the propagation regime, for small thicknesses (typically less than a mean free propagation path) the contribution from scattering is negligible (single scattering regime), while for sample thicknesses bigger than the mean free path, the scattering regime increases and starts making an important contribution to absorption. So it is reasonable to suppose that there will be some level of implication on the source term. REFERENCES 1 R. Williams, Production and transmission of ultrasound, Physiotherapy. 73, (1987). 2 K. Hattori, K. Ikeuchi, Y. Morita, Y. Takakura, Quantitative ultrasonic assessment for detecting microscopic cartilage damage in osteoarthritis, Arthritis Res Ther. 7, 1-9 (2004). 3 T. Watson, Ultrasound in contemporary physiotherapy practice, Ultrasonics. 48, (2008). 4 B.A. Scheven, R.M. Shelton, P.R. Cooper, A.D. Walmsley, A.J. Smith, Therapeutic ultrasound for dental tissue repair, Med. Hypotheses. 73, (2009). 5 H.H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forearm, J. Appl. Physiol. 1, (1948). 6 J.H. Demmink, P.J. Helders, H. Hobæk, C. Enwemeka, The variation of heating depth with therapeutic ultrasound frequency in physiotherapy, Ultrasound Med. Biol. 29, (2003). 7 G. ter Haar, Basic physics of therapeutic ultrasound, Physiotherapy. 64, (1978). 8 C.R. Hill, Front Matter, in: Ellis Horwood Limited (Ed.), Phys. Princ. Med. Ultrason., John Wiley & Sons, Ltd, Chichester, UK, (2005). 9 G.A. Cortela, M.A. von Krüger, C.A. Negreira, W.C.A. Pereira, Influence of ultrasonic scattering in the calculation of thermal dose in ex-vivo bovine muscular tissues, Ultrasonics. 65, (2016). 10 S.A. Sapareto, W.C. Dewey, Thermal dose determination in cancer therapy, Int. J. Radiat. Oncol. 10, (1984). 11 E.A. Filonenko, V.A. Khokhlova, Effect of acoustic nonlinearity on heating of biological tissue by high-intensity focused ultrasound, Acoust. Phys. 47, (2001). Proceedings of Meetings on Acoustics, Vol. 28, (2017) Page 8

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