周　浩,鄭音飛

(浙江大學(xué) 生物醫(yī)學(xué)工程教育部重點(diǎn)實(shí)驗(yàn)室,浙江 杭州 310027)

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非均勻組織醫(yī)學(xué)超聲非線性傳播仿真

周浩,鄭音飛

(浙江大學(xué) 生物醫(yī)學(xué)工程教育部重點(diǎn)實(shí)驗(yàn)室,浙江 杭州 310027)

摘要:為了實(shí)現(xiàn)仿真醫(yī)學(xué)超聲波在非均勻組織中的傳播過程,建立超聲非線性傳播計(jì)算模型.由軟組織中一階非線性波動(dòng)方程推導(dǎo)得出“聲壓-質(zhì)點(diǎn)振動(dòng)速度”耦合超聲非線性波動(dòng)方程以降低求解復(fù)雜度.采用k空間方法對(duì)非線性波動(dòng)方程組求解,在保證數(shù)值計(jì)算精度的同時(shí)降低計(jì)算的內(nèi)存占用量和計(jì)算時(shí)間.通過與一維問題的解析解和二維問題的時(shí)域有限差分(FDTD)求解結(jié)果對(duì)比,驗(yàn)證所述模型的精度.在空間采樣率為聲波波長(zhǎng)的1/9、Courant-Friedrichs-Lewy(CFL)數(shù)為0.3的情況下,所述模型的平方誤差為0.012 5%,而時(shí)域有限差分方法(FDTD)的平方誤差為42.5%.利用體腹壁組織數(shù)字模型,進(jìn)行醫(yī)學(xué)超聲諧波成像仿真,驗(yàn)證諧波成像較基波成像能夠提高深部組織區(qū)域的圖像質(zhì)量.

關(guān)鍵詞：非線性聲學(xué)；波動(dòng)方程；超聲成像；k空間方法；諧波成像

臨床醫(yī)學(xué)中應(yīng)用的醫(yī)學(xué)超聲通常處于較高頻帶范圍(1～10 MHz),這導(dǎo)致超聲波在生物組織的傳播過程中伴隨有顯著的非線性現(xiàn)象.醫(yī)學(xué)超聲的非線性效應(yīng)已被應(yīng)用于醫(yī)學(xué)超聲成像、高強(qiáng)度聚焦超聲治療和低強(qiáng)度超聲理療等方面[1-4].生物組織中超聲非線性傳播仿真可應(yīng)用于醫(yī)學(xué)超聲換能器設(shè)計(jì)、新型成像算法開發(fā)、聲束畸變校正研究和醫(yī)師培訓(xùn)等領(lǐng)域[5-9].相較于傳統(tǒng)線性聲場(chǎng)的積分求解方法[10],生物醫(yī)學(xué)超聲非線性傳播仿真的關(guān)鍵在于求解超聲傳播的非線性微分方程.利用算子分裂求解非線性Khokhlov-Zabolotskaya-Kuznetsov(KZK)方程是超聲非線性傳播仿真的常用方法[11-13],但KZK方程為拋物線型方程,僅能在較小的發(fā)射孔徑內(nèi)保證仿真精度,難以獲得后向散射回波[14].利用時(shí)域有限差分方法求解非線性全波方程(full-wave equation),在仿真超聲前向傳播的同時(shí)可以獲得后向散射回波,但由于超聲波速與頻率的限制,需要極高的時(shí)間-空間采樣率以保證仿真精度[14-16].與時(shí)域有限差分(finite-difference time-domain, FDTD)方法不同,k空間方法采用傅里葉變換計(jì)算波動(dòng)方程中的空間導(dǎo)數(shù)項(xiàng),利用修正的差分算子計(jì)算時(shí)間導(dǎo)數(shù)項(xiàng),從而可在較低的計(jì)算開銷下保證仿真精度[17-18]；然而,傳統(tǒng)的k空間方法難以獲得超聲非線性傳播時(shí)介質(zhì)內(nèi)質(zhì)點(diǎn)振動(dòng)速度分布情況,也難以直接應(yīng)用完全匹配層(perfect matched layer, PML)方法抑制聲波在仿真計(jì)算區(qū)域邊界處的周期性入射.

本文在推導(dǎo)出一階耦合超聲非線性波動(dòng)方程組的基礎(chǔ)上,采用k空間方法模擬超聲在組織中的非線性傳播,同步獲得生物組織中聲壓與質(zhì)點(diǎn)速度的分布特性.此外,采用聲壓-速度耦合的波動(dòng)方程組,可以直接將PML應(yīng)用于k空間方法.通過超聲非線性傳播的解析解和FDTD解驗(yàn)證所提出方法的準(zhǔn)確性.最后將該方法用于醫(yī)學(xué)超聲諧波成像仿真.

1理論

生物軟組織中一階非線性波動(dòng)方程[19]為

(1)

式中：ρ′為超聲傳播導(dǎo)致的介質(zhì)密度變化量,u為質(zhì)點(diǎn)速度,p為聲壓,B/A為非線性參量,c0和ρ0分別為介質(zhì)的平衡聲速和密度,μ與μB分別為切變和膨脹黏滯系數(shù),κ為熱傳導(dǎo)系數(shù),cv與cp分別為定容和定壓比熱，t為時(shí)間.式(1)的數(shù)值求解較為復(fù)雜,為降低計(jì)算復(fù)雜度,式(1)可進(jìn)一步合并：

(2)

(3)

式中:β=1+Β/(2Α)為介質(zhì)非線性參數(shù),

δ1=κ(1/cv-1/cp)/ρ0,δ2=(μB+4μ/3)/ρ0,

兩者分別為由熱傳導(dǎo)和組織黏滯導(dǎo)致的超聲衰減參數(shù)；δ1+δ2=δ,δ為聲衰減參數(shù),由于在生物組織中熱傳遞導(dǎo)致的聲衰減占總衰減比例甚少[19],在本研究中經(jīng)驗(yàn)性地認(rèn)為δ2=0.01δ.為了對(duì)式(3)以k空間方法求解,需要對(duì)等號(hào)右邊的非線性項(xiàng)進(jìn)行改寫.式(3)中的非線項(xiàng)為二階小量,而聲壓與質(zhì)點(diǎn)振動(dòng)速度線性關(guān)系在一階精度下成立：

(4)

可將式(4)代入式(3)中的非線性項(xiàng)而不影響等式推導(dǎo)的精度[13]，則式(3)可寫為

(5)

利用時(shí)間交錯(cuò)網(wǎng)格,設(shè)Δt為時(shí)間步進(jìn)長(zhǎng)度,采用時(shí)域顯格式在時(shí)域采樣點(diǎn)nΔt (n=1, 2, …)上對(duì)聲壓p進(jìn)行離散、在時(shí)域采樣點(diǎn)(n+1/2)Δt上對(duì)質(zhì)點(diǎn)運(yùn)動(dòng)速度u進(jìn)行離散,則離散后的一階非線性波動(dòng)方程組(2)與(5)可寫為

ξ=x, y,z.

(6)

(7)

式中：n為時(shí)間節(jié)點(diǎn),ξ為質(zhì)點(diǎn)速度矢量方向標(biāo)志.為了抑制時(shí)間導(dǎo)數(shù)離散引起的數(shù)值色散,利用k空間方法計(jì)算式(6)與(7)中的空間導(dǎo)數(shù)：

(8)

(9)

由于離散傅里葉變換具有周期性,從計(jì)算邊界一側(cè)入射的聲波會(huì)從對(duì)側(cè)邊界出射,本研究中直接利用一階完全匹配層方法使聲波在計(jì)算邊界處發(fā)生衰減[18,20].

2仿真實(shí)驗(yàn)

2.1均勻介質(zhì)中的非線性傳播仿真

為了驗(yàn)證所述方法對(duì)聲波非線性傳播模擬的精度,對(duì)單頻平面波在衰減介質(zhì)中的傳播進(jìn)行仿真.在單頻連續(xù)平面聲波非線性傳播的情形下,式(1)轉(zhuǎn)化為Burgers方程,其解析解由文獻(xiàn)[21]給出.在本研究中,設(shè)定源點(diǎn)聲壓為

p=p0sin(2πf0t).

圖1　單頻平面波在衰減介質(zhì)中非線性傳播仿真結(jié)果Fig.1　Plane wave nonlinear propagation simulation in homogeneous lossy medium

2.2非均勻介質(zhì)中的非線性傳播仿真

圖2　非均勻介質(zhì)超聲傳播仿真Fig.2　Acoustic wave nonlinear propagation simulation in a heterogeneous medium

由于人體組織聲學(xué)參數(shù)的非均勻性,醫(yī)學(xué)超聲波在人體組織中傳播時(shí)將發(fā)生折射、散射等現(xiàn)象.為了驗(yàn)證所述方法對(duì)非均勻介質(zhì)聲波非線性傳播的準(zhǔn)確性,令聲波在如圖2(a)所示的二維非均勻介質(zhì)中傳播.此非均勻介質(zhì)由圓形異質(zhì)物和均勻背景組成,均勻背景為50mm×50mm矩形,圓形異質(zhì)物半徑為4mm,位于均勻背景中央,距圓形異質(zhì)物中央20mm處設(shè)置聲波接收點(diǎn).為了與人體各類型組織間聲學(xué)參數(shù)相似,均勻背景介質(zhì)聲速設(shè)為1 500m/s,密度為1 000kg/m3,非線性參數(shù)為3.5,并設(shè)其為無衰減介質(zhì)；圓形異質(zhì)物的聲速設(shè)為1 575m/s,密度為1 050kg/m3,非線性參數(shù)為4.0,超聲衰減參數(shù)為1×10-4m2/s.令超聲發(fā)射孔徑位于均勻背景介質(zhì)的底端,以平面波方式向介質(zhì)內(nèi)發(fā)射聲波.聲波發(fā)射孔徑為20mm,發(fā)射波形為加Hanning窗的2周期正弦波,正弦波頻率為1MHz,發(fā)射波形最高幅度為0.25MPa.在使用k空間方法時(shí),設(shè)定空間采樣間隔為非均勻介質(zhì)中聲波最小波長(zhǎng)的1/9,CFL=0.3.采用同樣的仿真參數(shù),利用FDTD方法進(jìn)行仿真以對(duì)比結(jié)果,分析在較低空間分辨率下兩種方法的精度.同時(shí),以空間采樣間隔為非均勻介質(zhì)中聲波最小波長(zhǎng)的1/27,CFL=0.1為仿真參數(shù),進(jìn)行高時(shí)間-空間分辨率下FDTD仿真以對(duì)非均勻介質(zhì)中聲波傳播仿真提供基準(zhǔn).圖2(b)為3種仿真方法下,聲波接收點(diǎn)接收到的聲壓波形,可見k空間方法所得結(jié)果與基準(zhǔn)結(jié)果高度符合,體現(xiàn)出k空間方法在低空間分辨率情況下的優(yōu)勢(shì).圖2(c)為聲波接收點(diǎn)接收波形的頻譜A(f).k空間方法所得波形頻譜與基準(zhǔn)波形在基頻與一次諧波附近吻合較好；由于k空間方法采用的空間采樣率較低(9倍基頻波波長(zhǎng)),對(duì)4MHz以上的頻率成分不能有效采樣.由圖2(c)還可見,在低空間分辨率下FDTD方法幾乎不能有效包含聲波的非線性頻率成分.

2.3超聲諧波成像仿真

在驗(yàn)證了所述方法對(duì)超聲波在非均勻介質(zhì)中傳播仿真準(zhǔn)確性的基礎(chǔ)上,對(duì)醫(yī)學(xué)超聲線陣探頭所發(fā)聚焦聲波在非均勻人體組織中的傳播進(jìn)行模擬,并采集后向散射回波,進(jìn)行超聲諧波成像仿真.

圖3　超聲諧波成像仿真Fig. 3　Simulation of tissue harmonic ultrasound imaging

仿真中設(shè)定空間采樣間隔為33.87μm,CFL=0.3,模擬的換能器型號(hào)為UltrasonixL9-4/34高頻線陣換能器,陣元數(shù)N=128,陣元中心間距為0.304 8mm,換能器中心頻率fc=5.0MHz.模型采用文獻(xiàn)[22]提供的數(shù)字化二維人體腹壁組織圖為仿真介質(zhì).該腹壁組織結(jié)構(gòu)如圖3(a)所示,包含水(W)、結(jié)締組織(C)、肌肉(M)及脂肪(F).各類介質(zhì)所對(duì)應(yīng)的聲學(xué)參數(shù)已由實(shí)驗(yàn)測(cè)得(見表1)[17].在此基礎(chǔ)上,對(duì)聲速和密度分別加以幅度為相應(yīng)物理量平均值3%的隨機(jī)波動(dòng),以在超聲傳播過程中產(chǎn)生后向散射回波,并在22mm深度處設(shè)置充滿水的均勻圓形區(qū)域作為囊性病變.相對(duì)k空間方法仿真的空間采樣率要求,原始組織切片圖像分辨率相對(duì)較低(300dpi),仿真前須對(duì)圖像進(jìn)行最鄰近點(diǎn)插值以提高空間采樣率.仿真時(shí),設(shè)定區(qū)域的上邊界為換能器發(fā)射表面,通過水與腹壁組織進(jìn)行聲學(xué)耦合；換能器發(fā)射信號(hào)賦值于y軸方向質(zhì)點(diǎn)速度采樣點(diǎn),以仿真換能器縱向振動(dòng)模式；換能器振動(dòng)信號(hào)為加Hanning窗的2周期4MHz正弦信號(hào),換能器振動(dòng)的最高速度為0.5m/s(可以在焦點(diǎn)處等效產(chǎn)生0.5MPa的空間峰值聲壓).

表1　腹壁組織聲學(xué)參數(shù)

設(shè)定換能器發(fā)射焦點(diǎn)距離F=22mm,發(fā)射陣元數(shù)為32(發(fā)射孔徑數(shù)F/#=2.3),發(fā)射線數(shù)為32.在仿真聚焦聲波傳播過程中,同步記錄計(jì)算區(qū)域上邊界的聲壓值作為陣列換能器接收到的回波信號(hào).對(duì)回波信號(hào)進(jìn)行波束合成、解調(diào)、抽取、對(duì)數(shù)壓縮、雙線性插值,獲得基頻B型超聲圖像[23],如圖3(b)所示.利用Butterworth帶通濾波器濾取波束合成信號(hào)的二次諧波成分,可獲得諧波圖像[1],如圖3(c)所示.對(duì)比圖3(b)與(c)可見,諧波成像所獲得的囊性暗區(qū)邊界比基頻成像所獲囊性暗區(qū)邊界清晰,驗(yàn)證了諧波成像的優(yōu)勢(shì).由圖3(c)可見,諧波成像模式不能對(duì)5mm以內(nèi)的淺層組織良好顯像,這是由于超聲諧波須在組織中傳輸一定距離后才能生成,較淺深度范圍內(nèi)回波中的諧波能量仍較弱.

3討論

本文所述方法使用基于傅里葉變換的k空間算子計(jì)算超聲非線性方程組中的空間導(dǎo)數(shù)(式(8)與(9)),計(jì)算精度要高于FDTD中的空間、時(shí)間差分運(yùn)算,因而允許仿真時(shí)采用較FDTD方法更低的時(shí)間-空間采樣率.本文方法還可精確獲取超聲傳播時(shí)介質(zhì)內(nèi)質(zhì)點(diǎn)的振動(dòng)速度,從而可用于超聲傳播時(shí)組織內(nèi)聲強(qiáng)矢量分布和組織升溫的仿真計(jì)算[24].

(10)

(11)

除與離散傅里葉變換相關(guān)運(yùn)算過程外,式(6)與(7)所描述的迭代步進(jìn)計(jì)算可以在空間各采樣點(diǎn)上并行完成.為了展示并行運(yùn)算對(duì)仿真的加速效果,改變2.2節(jié)中所述問題的采樣點(diǎn)數(shù)目,在計(jì)算機(jī)操作系統(tǒng)中將CPU分別設(shè)置為單核與雙核運(yùn)行條件下,測(cè)量k空間方法在時(shí)間域上單步步進(jìn)所需的計(jì)算時(shí)間.測(cè)量結(jié)果如圖4所示,可見隨著網(wǎng)格數(shù)量(Gsize)的上升,雙核對(duì)仿真計(jì)算的加速效果逐漸明顯.尤其是可利用在通用圖形處理器(generalpurposegraphicprocessingunit,GPGPU)中運(yùn)行的并行快速傅里葉變換程序[25],將本文所述方法移植到GPGPU中進(jìn)一步提高運(yùn)算速度.

圖4　不同矩陣規(guī)模下單、雙線程程序單步計(jì)算時(shí)間對(duì)比Fig.4　Comparison of compute time between single and double-thread programs under different computation grid sizes

4結(jié)語

本文在推導(dǎo)出一階超聲非線性波動(dòng)方程的基礎(chǔ)上,提出了利用k空間方法仿真超聲在非均勻組織中傳播的模型,同步獲得介質(zhì)內(nèi)聲壓與質(zhì)點(diǎn)振動(dòng)速度分布情況.與FDTD方法相比,k空間方法能夠顯著降低仿真時(shí)的內(nèi)存占用量和計(jì)算用時(shí).通過仿真聲波在一維衰減組織與二維非均勻組織中的傳播,驗(yàn)證了k空間方法的準(zhǔn)確性,并在此基礎(chǔ)上進(jìn)行了超聲諧波成像的仿真.超聲諧波成像的仿真結(jié)果表明,諧波成像較基波成像能夠明顯提高深部組織區(qū)域的圖像質(zhì)量.

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DOI:10.3785/j.issn.1008-973X.2016.03.023

收稿日期：2015-01-04.

基金項(xiàng)目：中央高?；究蒲袠I(yè)務(wù)費(fèi)專項(xiàng)資金資助項(xiàng)目(2015FZA5019,2016FZA5015).

作者簡(jiǎn)介：周浩(1984-)，男，博士生，從事醫(yī)學(xué)超聲成像技術(shù)研究. ORCID：0000-0001-6894-1139. E-mail:bmezhou@zju.edu.cn 通信聯(lián)系人：鄭音飛，男，副教授. ORCID：0000-0001-6837-2634. E-mail:zyfnjupt@126.com

中圖分類號(hào)：TN 98

文獻(xiàn)標(biāo)志碼：A

文章編號(hào)：1008-973X(2016)03-06-0574

Simulation of nonlinear ultrasound propagation in heterogeneous tissue

ZHOU Hao, ZHENG Yin-fei

(KeyLaboratoryforBiomedicalEngineeringofMinistryofEducation,ZhejiangUniversity,Hangzhou310027,China)

Abstract:A numerical model was proposed for the simulation of the nonlinear ultrasound propagation in heterogeneous tissue. First, the coupled nonlinear wave equations for pressure and velocity were obtained based on 1-st order nonlinear wave equations in soft tissue to reduce the complexity of numerical computation. Then, k-space method was used to solve the derived nonlinear wave equations to reduce the memory usage and the computation time of the simulation, while preserving the computation accuracy. Compared with the analytic solution of a 1-dimensinal problem and the finite-different time-domain (FDTD) results of a 2-dimensinal problem, and the accuracy of the proposed model was validated. With grid size of 1/9 of the wavelength and Courant-Friedrichs-Lewy (CFL) of 0.3, the square errors of the proposed model and the FDTD method are 0.0125% and 42.5%, respectively. Medical harmonic ultrasound imaging was simulated using the proposed method based on a digital human abdominal map. The results show that image quality can be improved in the deeper tissue by using the harmonic signal．

Key words:nonlinear acoustics; wave equation; medical ultrasound imaging; k-space method; harmonic medical ultrasound imaging