Ultrasound, Ultrasound therapy, Mathematical model, Temperature, Ultrasound probe


Background. An important place among the methods of treatment in physiotherapy is the treatment of ultrasound. The process of ultrasound therapy consists of contacting the interaction of ultrasound from an ultrasonic emitter with superficial skin layers. Therefore, patient safety in this procedure is a dominant factor. During ultrasonic action on the biological tissue, the ultrasound emitter is heated. The range of heating of the ultrasonic emitter is determined by measuring and controlling the temperature values. When performing the procedure of ultrasound therapy, it is problematic to measure the temperature of the probe directly before the piezoelectric element on the external contact surface, which, in turn, has the highest temperature values across the entire working surface.

Objective. Development of a mathematical model that allows for the temperature values from the peripheral parts of the working surface of the ultrasonic probe to get the temperature value of the central part of its working surface.

Methods. In the study used, the method of randomization of the experiment, the theory of processing of experimental data, the regression analysis, which allowed to develop a mathematical model and evaluate its adequacy and accuracy, was applied. In the course of the study, the main values of temperature at the points on the working surface of the ultrasonic probe were determined, which were applied in the analysis of experimental data and determination of regression coefficients of the mathematical model.

Results. An analysis of experimental data and a regression analysis of the received values of the temperature of the working surface of the ultrasound probe showed that there is a significant mathematical dependence between the temperature in the central part of the probe and its peripheral parts, and this mathematical dependence has been determined, among other basic dependencies, and its verification has been made and errors have been calculated.

Conclusions. The mathematical model and estimated values of temperature on the working surface of an ultrasonic emitter are developed, which makes it possible to successfully use them in the development of new ultrasonic treatment heads for ultrasound therapy.

Author Biographies

Anatolii Yu. Kravchenko, Igor Sikorsky Kyiv Polytechnic Institute

Анатолій Юрійович Кравченко

Mykola F. Tereshchenko, Igor Sikorsky Kyiv Polytechnic Institute

Микола Федорович Терещенко

Sergiy P. Vysloukh, Igor Sikorsky Kyiv Polytechnic Institute

Сергій Петрович Вислоух

Gregory S. Tymchik, Igor Sikorsky Kyiv Polytechnic Institute

Григорій Семенович Тимчик


M. Tereshchenko et al., Ultrasound Physiotherapy Devices and Machines. Kyiv, Ukraine: Politekhnika, 2018, 184 p.

L. Orlova, Medical Encyclopaedia. Minsk, Belarus: Harvest, 2019, 896 p.

T. Watson, “Ultrasound in contemporary physiotherapy practice”, Ultrasonics, vol. 48, no. 4, pp. 321–329, 2008. doi: 10.1016/j.ultras.2008.02.004

T. Leighton, “What is ultrasound?”, Progress Biophys. Molec. Biol., vol. 93, no. 1-3, pp. 3–83, 2007. doi: 10.1016/j.pbiomolbio.2006.07.026

M. Tereshchenko et al., “Effect of ultrasound of therapeutic intensities on the cluster structure of distilled water”, Bulletin of Kyiv Polytechnic Institute. Ser. Instrument Making, vol. 1, no. 51, pp. 126–131, 2016.

V. Tsapenko et al., “Complex emitter of ultratonotherapy”, in Proc. Instrument Making – 2015, Minsk, Belarus, Nov. 25–27, 2015, pp. 158–159.

W. Obrien, “Ultrasound-biophysics mechanisms”, Progress in Biophysics and Molecular Biology, vol. 93, no. 1-3, pp. 212–255, 2007. doi: 10.1016/j.pbiomolbio.2006.07.010

A. Donskoy et al., Ultrasonic Electrical Engineering Devices. Leningrad, SU: Energoizdat, 1982, 208 p.

G. Tymchik et al., “Investigation thermal conductivity of biological materials by direct heating hermistor method”, in Proc. IEEE 38th Int. Conf. ELNANO, 2018, pp. 429–434.

C. Doody et al., “Prediction of the temperature rise at the surface of clinical ultrasound transducers”, BMUS Bulletin, vol. 11, no. 3, pp. 26–28, 2003. doi: 10.1177/1742271x0301100307

F. Duck et al., “Surface heating of diagnostic ultrasound transducers”, British J. Radiol., vol. 62, no. 743, pp. 1005–1013, 1989. doi: 10.1259/0007-1285-62-743-1005

M. Tereshchenko and A. Kirilova, “Investigation of parameters of influence of ultrasonic signal on biological structures”, Bulletin of Kyiv Polytechnic Institute. Ser. Instrument Making, no. 41, pp. 152–161, 2011.

L. Crum et al., “Therapeutic ultrasound: Recent trends and future perspectives”, Physics Procedia, vol. 3, no. 1, pp. 25–34, 2010. doi: 10.1016/j.phpro.2010.01.005

F. Ahmadi et al., “Bio-effects and safety of low-intensity, low-frequency ultrasonic exposure”, Progress in Biophysics and Molecular Biology, vol. 108, no. 3, pp. 119–138, 2012. doi: 10.1016/j.pbiomolbio.2012.01.004

J. Demmink et al., “The variation of heating depth with therapeutic ultrasound frequency in physiotherapy”, Ultrasound in Medicine & Biology, vol. 29, no. 1, pp. 113–118, 2003. doi: 10.1016/s0301-5629(02)00691-9

L. Spicci and G. Vigna, “Heat drain device for ultrasound imaging probes”, in Proc. COMSOL Conference in Cambridge, 2014, pp. 200–207.

SPSS Statistics – Overview (2019). [Online]. Available: