Influence of Blood Vessels on Temperature during High-Intensity Focused Ultrasound Hyperthermia Based on the Thermal Wave Model of Bioheat Transfer

The coupled effects of blood vessels and thermal relaxation time on temperature and thermal lesion region in biological tissue during high-intensity focused ultrasound (HIFU) hyperthermia are numerically investigated. Considering the non-Fourier behavior of heat conduction in biological tissue, the traditional Pennes bioheat equationwasmodified to thermalwavemodel of bioheat transfer (TWMBT). Consequently, a joint physical model, which combines TWMBT for tissue and energy transport equation for blood vessel, is presented to predict the evolution of temperature and the thermal lesion region. In this study, pulsatile blood flow is first introduced into numerical study of HIFU hyperthermia, and thermal relaxation time, ultrasonic focus location, blood vessel radius, and blood flow velocity are all taken into account. The results show that the thermal relaxation time plays a key role in the temperature and the thermal lesion region. Larger thermal relaxation time results in lower temperature and smaller thermal lesion region, which indicates that TWMBT leads to lower temperature and smaller thermal lesion region compared to Pennes bioheat transfer model. In addition, we found that the ultrasonic focus location and blood vessel radius significantly affected the temperature and thermal lesion region, while the heartbeat frequency and amplitude factor of pulsating blood flow as well as the average velocity of blood flow had only a slight effect.


Introduction
High-intensity focused ultrasound (HIFU) is a promising noninvasive technology, which can rapidly produce a local high temperature of more than 70 ∘ C in target tissues for the purpose of thermal ablation [1,2].In 1942, Lynn [3] designed a focused ultrasound generator to produce focal heating to destruct the focal area deep in the fresh liver tissue without damage to the intervening tissue.Thereafter, HIFU technology had been paid more and more attention by scientists and doctors, especially since the rapid development of ultrasonic imaging technology in 1990s [4].HIFU hyperthermia had been used to ablate solid tumors, including soft tissue sarcomas and cancers of the prostate, liver, kidney, breast, and pancreas [5].The accurate thermal dose at the lesion location plays a decisive role in the clinical success of HIFU hyperthermia.Accordingly, it is necessary to study the temperature and the thermal dose of lesion region [6,7].
In general, the temperature of biological tissue was predicted by Pennes bioheat transfer model because of its simplicity and practicability.It is well known that the model was built on the classical Fourier's law, implying an infinite thermal propagation velocity and an instantaneous thermal effect [8,9].That is to say, any heat perturbation in the biological tissue can be reached anywhere at the same time, which had aroused controversy among many scientists.To overcome this physically unreasonable drawback, Cattaneo and Vernotte independently proposed a generalized non-Fourier law heat conduction equation by introducing a lagging time called "relaxation time" [10,11].In addition, the non-Fourier behavior of heat conduction in non-homogenous medium requiring a relaxation time had been experimentally verified by several researchers [12][13][14].The reasonable relaxation time was in the range of 0.464-6.825saccording to the convective heat transfer coefficient and the available properties of blood and tissue in Zhang's research [15].In addition, TWMBT had many applications.For example, Dai studied skin burn injury subjected to radiation heating [16].Jaunich analyzed the temperature distributions in the skin tissue medium during short pulse laser irradiation [17].However, to our knowledge, few studies have been done on HIFU hyperthermia employing TWMBT until now, especially considering biological tissue with blood vessel.
Recently, Jiang employed HIFU to ablate tumors near significant blood vessels clinically [18].In addition, several numerical studies of the effects of blood vessels on temperature and thermal lesion region in ultrasound hyperthermia had attracted the interests of many researchers.Pennes treated the blood vessel and bone mathematically exactly as the soft tissue and presumed that the blood and surrounding tissue were completely thermal equilibration [19].This approach is valid for tissue with capillaries.Nevertheless, several researches implied that the thermal equilibration between the large blood vessels (diameters larger than 0.2 mm) and surrounding tissues was broken [20][21][22], and the large vessels in biological tissue should be considered.For instance, Kolios [20] examined the effects of blood flow on the thermal lesion dimensions and temperature distribution during focused ultrasound surgery.The blood vessel was coaxial with acoustic axis, and the ultrasonic focus was located in the center of the blood vessel.Hariharan [21] presented a three-dimensional physical model to investigate the efficacy of high-intensity focused ultrasound procedures targeted near large blood vessel, which was located outside the 6 dB width of the beam.Solovchuk [22] put forward an acoustic-thermal-fluid coupling model to study the influence of blood vessel on temperature, taking the effect of acoustic streaming into account.However, the temperature field computation was based on Pennes bioheat transfer model in most previous studies, neglecting the non-Fourier effects on thermal transfer, and the quantitative effects of the blood vessel on temperature and thermal lesion region in the heated tissue are still ambiguous.In our work, the effects of blood vessels on temperature and thermal lesion region based on TWMBT during HIFU hyperthermia will be comprehensively investigated, including various factors associated with blood vessels.In addition, pulsatile blood flow generated by the periodic pumping of heart contraction will be taken into consideration, which is firstly introduced into numerical study of HIFU hyperthermia.We believe that this study is significant for HIFU hyperthermia.

Theory
The HIFU transducer is a spherical cap with an aperture radius  of 35 , a focal length  of 62.64 , and a center frequency  of 1 , and the transducer and biology tissue are placed in the water.The geometric configuration of physical model is shown in Figure 1.

Acoustic Model for Ultrasound
Wave Propagation.To model the ultrasound wave propagation in thermoviscous medium incorporating the effects of absorption, diffraction, and nonlinearity, a widely used Westervelt equation was employed, which can be written as follows [23]: where ∇ 2 , ,  0 ,  are Laplace operator, acoustic pressure, ultrasonic velocity, and time, respectively;  = 1 + (/2) is the nonlinearity coefficient; and  = 2 3 0 / 2 is the acoustic diffusivity accounting for thermoviscous effect in the fluid, where  is the acoustic angular frequency and  is the acoustic absorption coefficient.The values of acoustic parameters used in this study are listed in Table 1 [24].

Thermal Energy Model for Tissue
Heating.The heat conduction based on the classic Fourier is as follows: where  denotes heat flux; ,  →  , and ∇ the thermal conductivity, position vector, and temperature gradient, respectively; minus denotes that the direction of heat transfer is opposite to the temperature gradient.Generally, the bioheat transfer equation can be shown below: Combining formula ( 2) with (3), a famous Pennes bioheat transfer equation can be obtained [19]: where   and   are the specific heat and density of tissue, respectively;   ,   , and   are the specific heat, perfusion rate, and initial temperature of blood, respectively; and all the values of thermal parameters in this study are listed in Table 2 [24].  is the ultrasound heat deposition source term which can be calculated by employing time-averaged over one acoustic period by numerical integration [25]: It is well known that the heat conduction in the Pennes bioheat transfer equation is based on Fourier law.To incorporate the non-Fourier behavior, Cattaneo and Vernott proposed a modified heat conduction equation as follows [10,11]: where  is thermal relaxation time, which denotes a time lag between heat flux and temperature gradient, leading to significant non-Fourier thermal behavior.Based on ( 3) and ( 6), TWMBT can be expressed as follows [26]: In this paper, the physical model discussed in the next is the perfused tissue containing a large blood vessel.To compute the temperature field, the physical model should be split into two regions, one is the tissue region with perfusion [20], and the other is the blood region with a large blood vessel.
In the region without large blood vessel, TWMBT is used to compute the temperature field in the perfused tissue region.In the region with a large blood vessel resulting in the local cooling, an advective term −    ()(/) is added in the heat diffusion equation.The energy transport equation is as follows [20]: In this study, the pulsatile blood flow in the blood vessel is considered, with the hypothesis that the blood vessel is rigid and the blood flow is laminar, incompressible, and Newtonian fluid.The pulsatile blood flow resulting from the periodic pumping of heart contraction is divided into a steady part and an oscillatory one [27]: where (, ) is the velocity of pulsatile blood flow.  () is steady parabolic velocity of blood flow, which is relation to the corresponding Poiseuille flow velocity in steady blood flow;   (, ) represents the oscillatory velocity of blood flow in the rigid blood vessel;  V is the average velocity of blood flow;  is dynamic viscosity of blood;  =  0 /√/    is the Womersley number;  characterizes the relative intensity of the pulsatile flow;   is the angular frequency of heartbeat;   =   /2 denotes the heartbeat frequency varied from 1 to 3Hz [28]; and  0 is zero-order Bessel function of the first kind.
To evaluate the performance of the HIFU treatment, thermal dose is usually used to estimate the tissue damage.The thermal dose depends on the final time   and temperature level , which is developed by Sapareto and Dewey [29]: where  of an isothermal dose value of 240 min at 43 ∘ C was usually selected to predict the size of the thermal lesion region [30].
The initial condition is where   ,   are temperature of tissue and blood flow, respectively.At the interface Γ between the tissue and blood vessel, the continuity condition of temperature is imposed.

Thermal Relaxation Time.
Here, the influences of thermal relaxation time  on hyperthermia treatment are investigated.To simplify the physical problem, we neglected the boiling cavitations.The ultrasonic transducer is excited by sinusoidal wave and the amplitude of acoustic pressure  0 at the surface of the transducer is 1.5 × 10 5 , the ultrasound heating time  ℎ is 1 , and the thermal relaxation time  is set to 0, 0.464, 1.756, 6.825, 10  [15].The heartbeat frequency   is set to 1Hz [28], and amplitude factor  is 0.5.When  = 0, the thermal wave model of bioheat transfer becomes Pennes bioheat transfer model.
Figure 2 shows the time variation of the maximum temperature under different thermal relaxation time.The  peak temperatures in space ,  ∈ Ω are 87.258∘ C, 82.432 ∘ C, 73.209 ∘ C, 61.88 ∘ C, 58.872 ∘ C at time 1, 1.565 , 2.939 , 5.87 , 7.147  for  = 0, 0.464, 1.756, 6.825, 10 , respectively.The greater the thermal relaxation time, the lower the peak temperature in biological tissue, and the greater the delay time reaching to the peak temperature.Besides, the peak temperature decreases immediately when the ultrasound power source is turned off at time  = 1  for  = 0, but continues to increase for  ̸ = 0 (e.g.,  = 0.464, 1.756, 6.825, 10 ).This phenomenon is mainly due to infinite thermal propagation speed in biological tissue when  = 0 and finite thermal propagation speed when  ̸ = 0.The finite thermal propagation speed means that the thermal energy needs a certain amount of time to spread within the biological tissue, which is the physical significance of the thermal relaxation time.Meanwhile, a larger thermal relaxation time results in a larger delay time because of smaller thermal propagation speed in biological tissue.
In Figure 3, we present the thermal lesion region in space ,  ∈ Ω with different thermal relaxation time.The thermal lesion is an elliptical shape with the size 0.82  × 0.16 , 0.8  × 0.16 , 0.72  × 0.15 , 0.52  × 0.11 , 0.41  × 0.08  for  = 0, 0.464, 1.756, 6.825, 10 , respectively.There is only tiny difference to lesion size between  = 0 and  = 0.464 , but the lesion size decreases from 0.82  × 0.16  to 0.41  × 0.08  when thermal relaxation time varies from 0 to 10 , which is almost reduced 75%.This can be easily understood that the peak temperature is 87.258 ∘ C for  = 0 and 58.872 ∘ C for  = 10 from Figure 2. Consequently, it can be concluded that TWMBT results in lower temperature and smaller thermal lesion region compared to the classical Pennes bioheat transfer model.

Pulsatile Blood Flow.
Figure 4 shows effect of heartbeat frequency   on maximum temperature-time change and thermal lesion region.Figure 5 shows the effects of amplitude factor  and different blood flow velocity forms (steady parabolic velocity   () and pulsatile velocity (, )) on maximum temperature-time change, respectively.There is almost no difference in maximum temperature evolution and thermal lesion region for different heartbeat frequencies in Figure 4. Subgraphs of maximum temperature variations in the time range from 2.9 s to 3.0 s are shown in Figures 4(a) and 5, respectively.Noteworthily, the largest difference in peak temperature is only about 0.02 ∘ C, indicating that heartbeat frequency   and amplitude factor  almost have no difference on maximum temperature evolution and thermal lesion region.Meanwhile, although there is oscillatory velocity   (, ) in pulsatile blood flow, the steady parabolic velocity   () can still be instead of time-dependent pulsatile velocity (, ) for simplicity.

The Distance between Ultrasonic Focus and Central
Axis of Blood Vessel.In Figure 6, the simulated maximum temperature versus time is presented with different distance between ultrasonic focus and central axis of blood vessel.The peak temperature is 47.85 ∘ C when the distance  is 0.5 mm (the focus is at the midpoint between blood vessel center and blood vessel wall); 62.92 ∘ C when the distance  is 1 mm (the focus is just right at the blood vessel wall); 74.65 ∘ C when the distance  is 2.0 mm; and the difference of peak temperature is very small when  is 2.0 mm and 2.5 mm.As the distance  increases ( ≤ 2.0 ), the peak temperature increases, which can be easily explained by the fact that the smaller distance  leads to the larger effect of blood flow cooling.When the distance  is greater than 2.0 mm, there is little effect of blood flow cooling on peak temperature.
Figure 7 demonstrates the thermal lesion region in tissue with different distance between different ultrasonic focus and central axis of blood vessel.When the ultrasonic focus is at the midpoint between the blood vessel center and blood vessel wall (i.e.,  = 0.5 ), there is no thermal lesion region; when the ultrasonic focus is just right at blood vessel wall (i.e.,  = 1 ), the thermal lesion region is an elliptical shape with the size 0.48  × 0.07  excluding the region in the blood vessel; when  = 1.5 , the thermal lesion region is an elliptical shape with the size 0.71  × 0.14  excluding the region in the blood vessel; and the thermal lesion region is an elliptical shape with the size 0.75  × 0.21  and 0.75  × 0.24  for  = 2.0  and  = 2.5 , respectively.The greater the distance , the lower the cooling effect of blood flow, and the larger the thermal lesion region, which also has clinical significance.When the tumor is adjacent to a significant blood vessel, the doctor should choose the suitable location of ultrasonic focus, not too close to the vessel wall, especially not in the blood vessel.Otherwise, there is a high probability that the tumor will not be thermal ablated completely.

Blood Vessel Radius.
When the ultrasonic focus is right at the center of the blood vessel, the smaller radius gives rise to the greater peak temperature, as shown in Figure 8.
When the blood vessel diameter is less than 0.2 mm, there is thermal equilibrium between blood vessel and surrounding tissues, and the effects of the blood vessel on temperature and thermal lesion region can be ignored.In Figure 9, it can be seen that thermal lesion region has only a slight difference between the vascular radius of 0.1 mm and without blood vessel and covers the whole blood vessel.When  0 = 0.2 , thermal equilibrium between blood vessel and surrounding tissues is broken.Due to the cooling effect of the blood flow, the thermal doses in some areas of the biological tissue are less than 240 min equivalent time at 43 ∘ C, resulting in deficit of thermal lesion region.In addition, the part of the thermal lesion region is shaped like tail as shown in the dotted box of Figure 9(a), which may be caused by the comprehensive influence of heat conduction, convective blood cooling, and heat source.It also gives us a hint that HIFU hyperthermia most probably hurts the normal tissue because of the existence of tail-like thermal lesion region.When  0 = 0.3 , it has a greater deficit of thermal region and smaller tail-like thermal lesion region compared with  0 = 0.2 .When the radius of the vessel varies from 0.4  to 0.6 , the thermal lesion region split into two parts.Accordingly, the thermal lesion region with blood vessel radii of 0.2 mm and 0.3 mm can be considered as a transition stage in the heated tissue with large vessels and without blood vessel.As shown in Figure 9, the smaller blood vessel radius results in the larger thermal region, and the thermal lesion region is very sensitive to the blood vessel radius.Even if the radius of the blood vessel just changes 0.1 mm, it also causes very different thermal lesion region.

Blood Flow Velocity.
When the ultrasonic focus is right at the center of the blood vessel, Figure 10 shows the maximum

Conclusions
In this paper, TWMBT, improved from the traditional Pennes bioheat transfer model, is employed to study the effects of blood vessel and thermal relaxation time on temperature and thermal lesion region in biological tissue during the HIFU hyperthermia.The heartbeat frequency   and amplitude factor  almost have no effect on temperature and thermal lesion region, and there is almost the same thermal lesion size between steady parabolic velocity and pulsatile velocity.The greater thermal relaxation time leads to smaller thermal lesion region.This phenomenon indicates that TWMBT results in lower temperature and smaller thermal lesion region compared to the classical Pennes bioheat transfer model in the HIFU hyperthermia.The distance between the ultrasonic focus and the central axis of blood vessel also has an important influence on the HIFU hyperthermia treatment.The larger the distance , the larger the thermal lesion region.The blood vessel radius is very sensitive to the thermal lesion region.When the blood vessel radius  0 is between 0.2  and 0.3 , it has part of thermal lesion region like a tail, which may hurt the normal tissue.The thermal lesion region is insensitive to blood velocity during the HIFU hyperthermia.All the numerical simulation results are meaningful to guide the doctors to perform HIFU thermal ablation of tumor.

Figure 1 :
Figure 1: Geometric configuration of physical model.The tissue containing a large blood vessel, a cylinder with radius of 35 mm and length of 50 mm, is placed at  1 = 40 .The blood vessel is at the center of the tissue,  0 is the radius of the blood vessel, and  is the distance between ultrasonic focus and central axis of blood vessel.The computational domain Ω is −35  ≤  ≤ 35  and 0 ≤  ≤ 90 , abbreviated as ,  ∈ Ω.
The tissue containing a large blood vessel, a cylinder with radius of 35 mm and length of 50 mm, is placed at  1 = 40 .The blood vessel is at the center of the tissue,  0 is the radius of the blood vessel, and  is the distance between ultrasonic focus and central axis of blood vessel.The computational domain Ω is −35  ≤  ≤ 35  and 0 ≤  ≤ 90 , abbreviated as ,  ∈ Ω.

Table 1 :
Values of acoustic parameters in this study.

Table 2 :
Values of thermal parameters in this study.