Reconstruction of thermal field in target tissue during the therapy of high intensity focused ultrasound

International Communications in Heat and Mass Transfer 108 (2019) 104325

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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt

Reconstruction of thermal field in target tissue during the therapy of high intensity focused ultrasound Yanhao Li, Deping Zeng, Jianwen Tan

T

⁎

State Key Laboratory of Ultrasound Engineering in Medicine Co-Founded by Chongqing and the Ministry of Science and Technology, College of Biomedical Engineering, Chongqing Medical University, Chongqing 400016, PR China

ARTICLE INFO

ABSTRACT

Keywords: High intensity focused ultrasound Temperature reconstruction Heat transfer inverse Model prediction

High intensity focused ultrasound (HIFU) is a minimally invasive medical procedure, obtaining the transient temperature field of the target tissue is a key to study HIFU ablation. In this work, an inverse heat transfer scheme is proposed to investigate thermal effect of HIFU and further reconstruct the transient temperature field of the target tissue. In the inverse scheme, a prediction model is established between the measured temperature and heat source in the target tissue. Then, based on the prediction model, a model predictive inverse scheme is established to estimate heat source induced by HIFU in the target tissue. Finally, the inversed heat source is used to solve biological heat transfer equation to reconstruct temperature distribution in the target tissue of HIFU ablation. Numerical experiments are performed to study the effects of the form of heat source, the number of future time steps and measured error on the inverse results. The results have shown that the inverse scheme can significantly reduce the sensitivity of the inverse results on measurement noise, and the temperature distribution in the target tissue can be obtained by the inverse scheme, which provides theoretical guidance for treatment planning and efficacy evaluation of HIFU.

1. Introduction As a minimally invasive medical procedure (Fig. 1), high intensity focused ultrasound (HIFU) has shown considerable potential for a variety of therapeutic applications, which include thermal ablation of tumors and uterine fibroids, and gene activation [1–4]. As shown in Fig. 1, the principle of HIFU technology is based on the ultrasonic beam emitted by the external transducer, which is focused in the target tissue after passing through the human tissue, and the tissue temperature is raised to above 60 °C in a few seconds by the thermal effect of the ultrasonic wave, thereby causing protein coagulative necrosis of the tissue. During the therapy of HIFU, some lesion tissues are ablated by ultrasonic irradiation. The generated heat during ablation process could cause thermal damage to surrounding normal tissues [5–7]. Therefore, obtaining the transient temperature field of the target tissue is a key to study HIFU ablation [8–10]. At present, the most important means of temperature measurement of the target tissue is magnetic resonance image (MRI)-guided HIFU in HIFU ablation [11,12]. However, the MRI-guided HIFU technique requires ultrasonic equipment to be adapted to work in strong magnetic fields, along with the need for maintenance of MRI system, which makes the MRIguided HIFU arrangement bulky and relatively expensive [13,14]. In addition to MRI, through establishing two different models of acoustic propagation and heat transfer of the tissue, numerical simulation is used to ⁎

Corresponding author. E-mail address: [email protected] (J. Tan).

https://doi.org/10.1016/j.icheatmasstransfer.2019.104325

0735-1933/ © 2019 Published by Elsevier Ltd.

calculate thermal field of the target tissue. Zhang et al. [15] used biological heat transfer equation to approximate acoustic pressure of HIFU and temperature field for brain therapy. Li et al. [16] obtained the distribution of heat source induced by HIFU within the tissue in vitro, and then calculate transient temperature field of the tissue. However, the ultrasonic propagation path, ultrasonic parameters and the control strategy of the transducer are very complicated [17], it is very difficult to establish an effective ultrasonic propagation model. Therefore, it is necessary to find other ways to determine the temperature field of the target tissue in HIFU ablation. In this work, the temperature field distribution of the tissue is determined by processing an inverse problem. Due to the measurement method and the limitation of measuring environment, the boundary conditions, heat source, thermal physical parameters cannot be obtained directly, the inverse heat transfer problem (IHTP) could inverse the parameters based on the partial temperature information [18]. In the past few decades, many methods for solving inverse heat transfer problems have been developed, which include conjugate gradient method (CGM) [19,20], sequential function specification method (SFSM) [21,22], Tikhonov regularization method (TRM) [23,24], artificial neural networks [25,26], stochastic optimization methods [27,28]. A few researchers have applied the inverse method specifically to HIFU studies. Hariharan et al. [29] employed an inverse algorithm to back calculate the acoustic intensity on the basis of the HIFU-induced

International Communications in Heat and Mass Transfer 108 (2019) 104325

Y. Li, et al.

Fig. 1. Schematic diagram of HIFU therapy.

Fig. 2. Schematic geometry of the individual tissues structure.

streaming velocities measured in a liquid medium. More recently, an inverse heat transfer method was developed to predict the temperature rise in a tissue phantom [14]. However, these methods have to establish an acoustic propagation model. In the following, not need to establish an acoustic propagation model, the heat source induced by HIFU in target tissue is inversed by using inverse heat transfer method, and the thermal field is reconstructed, which can simplify many complicated problems. In the therapy of HIFU, the problem of solving the temperature distribution in target tissue is essentially a kind of typical heat transfer problem of distribution parameter system [5,9], the following challenges are encountered when solving this problem using existing inverse methods: (1) In the actual HIFU ablation, considering obvious radiation and convective thermal effects between the blood, bones and muscles of the human body and complex acoustic environment, the temperature changes in the tissue caused by focused acoustic energy can also cause the changes of acoustic parameters [15,17]. It can be seen that the inverse problem of temperature distribution in target tissue is essentially a kind of inverse problem of time-varying systems. (2) The temperature of the target region in HIFU therapy have showed time and space distribution [12,13]. The model needs to be calculated repeatedly in the inverse process, which may have higher computational cost and lower computational accuracy. Therefore, in order to reconstruct the transient temperature distribution in the target tissue, it is necessary to find other schedule with good adaptability and better accuracy. This paper is aimed at the temporal and spatial distribution characteristics of the temperature of the target tissue in the HIFU therapy. Based on the characteristics of the inverse problem and the inadequacies of the existing inverse methods, the model prediction schedule is proposed. The overall concept of the schedule is as follows: firstly, for the characteristics of HIFU heat transfer, a prediction model is established between the measured temperature and heat source in the target tissue. Then, based on the prediction model, a model predictive inverse scheme is established to estimate heat source induced by HIFU in the target tissue. Finally, the inversed heat source is used to solve biological heat transfer equation to reconstruct temperature distribution in the target tissue of HIFU ablation. Numerical experiments are performed to study the effects of the form of heat source, the number of future time steps and measured error on the inverse results.

T (x , y, 0) = T0 (x , y ),

T (x , y , ) = 0, n

=

2 T (x , y,

) + wb cb [Tb (x , y, )

(x , y

1;

> 0)

(x , y

;

> 0)

(3)

where τ is time. Q(x, y, τ)indicates heat source induced by HIFU in target tissue. T0(x, y)is the initial temperature distribution, χ, ρand C are the thermal conductivity coefficient, the density and the heat capacity of target tissue, respectively. wb is the blood perfusion rate in the heated region. cb is the specific heat capacity of blood, and Tb(x, y, τ) is the temperature of the arterial blood, Ω1 is the region of target tissue, Γ is the boundary of target tissue. n represents the outer normal direction to the corresponding boundary. In the absence of perfusion and any metabolic process, the second term on the right hand side of Eq.(1) can be neglected [30]. The heat source, Q(x, y, τ)，is equivalent to the absorbed HIFU power in the tissue, which can be considered as a Gaussian distributed heat source and it is defined as shown in Eq.(4).

( ) exp

Q (x , y , ) = 0,

y2 x2 + 2 a2 b

,

(x , y

(x , y

2;

> 0)

2;

> 0) (4)

where Θ(τ)denotes the maximum thermal energy at the focus region of HIFU, Θ(τ) is related with the sound speed, the attenuation coefficient, and the acoustic pressure distribution. Ω2 is the region of heat source; a and b represent the 1/2 long axis and the 1/2 short axis of the focal region, which can be determined as follows:

b = F /4d

(5a)

a = 3.6Fb/ d

(5b)

where F and d are the focal length and the aperture radius of HIFU transducer, respectively; λis the acoustic wavelength. 3. Inverse problem In the direct problem, the thermo-physical properties, the sound speed, the attenuation coefficient, and the acoustic pressure distribution are known to determine the transient temperature field of the tissue. Differently, in the inverse problem, the sound speed, the attenuation coefficient, and the acoustic pressure distribution are unknown, in other words, the heat source Θ(τ) is unknown and need to be recovered. The additional information required is the temperature field, which is the known measured data or simulated by the solution of the thermal model. This paper has established model predictive algorithm to solve the inverse problem.

The heat transfer system of target tissue is studied during thermal therapy of HIFU, which is as shown in Fig. 2. The transient temperature field of target tissue during thermal therapy of HIFU has often been computed using a bioheat transfer equation [30–32]: T (x , y, )

(2)

1)

Boundary condition:

2. Thermal model of tissue

C

(x , y

4. Model predictive inverse algorithm

T (x , y, )] + Q (x , y, ),

(1)

The objective function is adopted to construct the above inverse problem:

Initial condition: 2

International Communications in Heat and Mass Transfer 108 (2019) 104325

Y. Li, et al.

min J ( ) = (T mea

T pre)T (T mea

T pre) +

(6)

T

6

T pre = T

pre

lateral direction/mm

In Eq.(6), Θ = (Θk, Θk+1, ⋯, Θk+r−1)Tis the vector of the inversed parameters, r is the number of future time steps, Θk+i(i = 0, ⋯, r − 1)is the estimated heat source at (k + i)th moment, Tmea = (Tkmea, Tk+1mea, ⋯, Tk+r−1mea) and Tpre = (Tkpre, Tk+1pre, ⋯, Tk+r−1pre) are the vectors of measured and predictive temperatures, respectively. αis regularization parameter. In the present work, the model predictive inverse algorithm in Ref. [33, 34] is adopted for minimizing the rolling objective function in Eq. (6). However, Tpre can be obtain by establishing the predictive model between inversed heat source and measured temperature by Eq.(7). pre pre (Tk , Tk + 1,

pre

0 -2 -4

0

k

T k+1 k 1

T k+r

k

380

0

k+1 1

T k+r

k+1

1

) 1H T (T mea

T

pre

0

(

= 9000 kW

5000 kW ( ) rec = 16000 kW 5000 kW

0

50

60

Set the heat source intensity Q(x, y, τ) = f{Θ(τ)constant, x, y}, the temperature field of the target tissue can be calculated as simulation values of measured temperature. Fig. 3(a) and (b) are the heat source intensity distribution of focus region and the simulated measurement temperature. Fig. 4 are the thermal fields of tissue at τ = 15 s, τ = 30 s, τ = 45 s and τ = 60 s, respectively. From Fig. 4, with the increase of the heat time, the temperature get higher in the focus region. However, the temperature outside the focal region has barely increased, which is because the focused energy is concentrated in the focal region in the HIFU ablation.

In the numerical experiments, according to Ref. [32], χ=0.59W/ (m ⋅ K), ρ=1079kg/m3and C=3540J/(kg ⋅ K); The initial temperature is 37 °C; The parameters of HIFU transducer are, respectively, F=170 mm, and d=50 mm, and λ=1.54 mm as shown in Ref. [7]. The thermal energy Θ(τ) is, respectively, as shown in Eq.(10a), Eq.(10b) and Eq.(10c).

)con

40

5.2. The calculation of temperature field

5.1. The condition of numerical experiments

2000

30

2.576], and σ denotes the standard deviation of the measured temperature error.

In this paper, the accuracy and robustness of the inverse algorithm for estimating heat source of the tissue are evaluated through numerical experiments. The following simulation examples are given here. The simulation is worked on a standard PC (3.4 GHz, 4GB RAM).

= 90000 sin

20

Fig 3. (a) Heat source by simulation calculation. (b) Measured temperature by simulation calculation.

(9)

)

10

Time/s

5. Results and discussion

(

340

(8)

Where Ψ is unit matrix.

)sin

Measured temperature 360

320

k+r 1

where the solution of the step response coefficient matrix has been shown previously in Ref. [33] in the present study, and hence not shown explicitly again in this paper. Order dJ(Θ)/dΘ = 0 by taking Eq. (7) into Eq. (6), the optimal Θ can be obtained by Eq.(9).

= (H T H +

0

(b)

0

T k+1

-10

Axial direction/mm

pre , Tk + r 1)

Temperature/K

Tk

T k+r

2

-6 -20

In Eq.(7), T = is the vector of initial predictive values, Ηstands for step response coefficient matrix as shown in Eq.(8).

=

4

(7)

+

Heat source/(W/m2) 9x106 8x106 8x106 7x106 6x106 5x106 5x106 4x106 3x106 10 20

(a)

+ 60000 kW

0

60

5.3. The effect of the form of heat source In this case, we mainly discuss the effect of the form of heat source for inverse results. In order to evaluate the inverse results, the root mean square error and the average relative error of the inverse heat energy are introduced and defined by Eq.(12a) and Eq.(12b), respectively.

(10a) (10b)

60

0 < 10 10 20 20 < 60

1 K

k=1

PT ( ) =

1 K

k=1

( )

(10c)

In the inverse analysis, the measured point is illustrated in Fig. 2, which is located at 0.01 m position along the axial direction, and 0.01 m position along the lateral direction. The simulation values of the measured temperature Tkmea are produced by Eq. (11):

K

K

[(

k )exa

(Tkmea

k ]2

Tkcal )2 /

/

1 K 1 K

K k=1 K k=1

[(

k ) exa]2

(Tkmea ) 2

(12a) (12b)

Take σ = 0，r = 5, the effect of the form of heat source on the inverse results is investigated with Eq.(10), respectively. Fig. 5(a) and Fig. 5(b) respectively represent the inverse results of heat energy Θ(τ) and measured temperature under different the form of heat source. For different heat energy, Θ(τ)sin, Θ(τ)con and Θ(τ)rec, the PΘ(τ) are 6.41%, 6.49% and 13.55%, respectively, PT(τ) are 0.2%, 0.14% and 0.13%, respectively.

(11)

Tkmea = Tkexa +

=

P

exa

where the “exact temperature” Tk at measured point is obtained by solving the direct heat conduction problem. ω is the random number obeying the standard normal distribution within the interval [−2.576, 3

International Communications in Heat and Mass Transfer 108 (2019) 104325

Y. Li, et al.

Fig. 4. (a) Temperature distribution of tissue at τ = 15 s. (b) Temperature distribution of tissue at τ = 30 s. (c) Temperature distribution of tissue at τ = 45 s. (d) Temperature distribution of tissue atτ = 60 s.

From Fig. 5(a), for different the form of heat source, the inverse method can get better inverse results of heat source. The inverse method has significantly decreased the sensitivity of inverse results to the form of heat source. From Fig. 5(b), the measured results and the inverse results of the temperature at measured point are consistent with different the form of heat source.

Thermal energy/(W/m2)

(a) 2.4x107 2.0x107

Exactsin Inversesin

Exactcon Inversecon

5.4. The effect of the number of future time steps Take σ = 0, the specific form of boundary heat flux is taken asΘ(τ)rec. The effect of the number of future time steps r on the inverse results is investigated with r = 2, r = 5 and r = 10 respectively. Fig. 6(a) and Fig. 6(b) respectively represent the inverse results of heat energy Θ(τ)and measured temperature under different number of future time steps. For different the number of future time steps, the PΘ(τ) are 14.75%, 13.55% and 15.92%, respectively, PT(τ) are 0.12%, 0.13% and 0.18%, respectively. From Fig. 6(a), for smaller r, due to the delay and damping of heat transfer of HIFU, the tracking ability of the inverse results of heat source is insufficient. With the increase of r, the tracking ability of the inverse results is insufficient. However, the accuracy and stability of the inverse results are maintained using r = 5 [Fig. 6(b)].

Exactrec Inverserec

1.6x107 1.2x107 8.0x106

5.5. The effect of measured error

4.0x106 0

10

20

30

40

50

60

Take r = 5, the specific form of heat source is taken asΘ(τ)rec. The effect of measurement error σ on the inverse results is investigated withσ = 0.05, σ = 0.10 and σ = 0.15 respectively. Fig. 7(a) and Fig. 7(b) respectively represent the inverse results of heat energy Θ(τ) and measured temperature obtained under different measurement error. For different measured error, the PΘ(τ)are 16.43%, 16.44% and 17.32%, respectively, PT(τ) are 0.13%, 0.14% and 0.19%, respectively. As can be seen from Fig. 7(a), the inverse method can obtain satisfactory heat source with smaller measurement error. With the increase of measurement error, the inverse results still have a good stability with larger measurement error, which has indicated that the inverse method significantly reduces the sensitivity of the inverse results on measurement noise [Fig. 7(b)].

Time/s

Temperature/K

400

(b) Exactsin Inversesin

380

Exactcon Exactrec con Inverse Inverserec

360 340 320 0

10

20

30

40

50

60

6. Conclusion and future work

Time/s Fig. 5. (a) Inverse results of heat energy Θ(τ) with different form of heat source. (b) Inverse results of measured temperature with different form of heat source.

In this study, based on the heat transfer system of HIFU, the model predictive inverse scheme is established to obtain the transient 4

International Communications in Heat and Mass Transfer 108 (2019) 104325

Y. Li, et al.

Thermal energy/(W/m2)

Thermal energy/(W/m2)

(a) 1.6x107 7

1.2x10

optimization. Finally, the temperature distribution is reconstructed in the target tissue of HIFU ablation. Numerical experiment is performed to discuss the effects of different form of heat source, future time steps r and measurement error on the inverse results of heat source. The results have shown that the inverse results improve the anti-interference ability of the inverse results on measurement error. At present, this work only considers the two-dimensional tissue heat transfer system. In fact, for practical HIFU ablation, the tissue is threedimensional structure, which is the follow-up research direction of this paper. In addition, the intensity estimation of non-Gaussian heat source is another research direction of the paper.

2x107 2x107 2x107 8

12 16 20 Time/s

Exact value r=2 r=5 r=10

6

8.0x10

4.0x106 0

10

20

30

40

50

60

Time/s

Declaration of Competing Interest

(b)

360

The authors declare that there are no conflicts of interest.

Temperature/K

350

Acknowledgements

340

Measured value r=2 r=5 r=10

330 320 310 0

10

20

30

40

50

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Time/s Fig. 6. (a) Inverse results of heat energy Θ(τ) with different r. (b) Inverse results of measured temperature with different r.

Thermal energy/(W/m2)

(a) Exact value

1.6x107

σ=0.05 σ=0.10 σ=0.15

7

1.2x10

8.0x106 4.0x106 0

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Time/s

Temperature/K

360

(b)

350 340 Measured value σ=0.05 σ=0.10 σ=0.15

330 320 310 0

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Time/s Fig. 7. (a) Inverse results of heat energy Θ(τ) with different σ. (b) Inverse results of measured temperature with different σ.

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International Communications in Heat and Mass Transfer 108 (2019) 104325

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