- 學位論文
短脈衝波之頻寬對微氣泡的低頻成分響應與次諧振響應—使用兩種頻率的近似
The bandwidth of short-pulse to low frequency response and subharmonic response of micro-bubbles—using two-frequency approximation
摘要
微氣泡的非線性行為次諧振響應，可以提供一個較好的區分在微氣泡和人體組織上，特別在遠場的應用中。然而，現在還沒有一個適當的分析對於在次諧振訊號於短脈衝波的入射。在這篇論文裡，我們延續Newhouse et al.利用兩個頻率的近似理論分析解[2]，來推倒微氣泡在有限頻寬入射下的響應。利用傅立葉理論(Fourier theory)，一個有限頻寬的訊號將可以用多個不同頻率的弦波合成。兩個頻率的近似有限頻寬的訊號是最簡單的例子。在這篇論文裡，給定一個微氣泡，他的共振頻率為 ，傳送訊號是兩個訊號(f1,f2,f1>2f0>f2)的合成，假設傳送訊號有頻譜上的連續。因此，傳送訊號頻寬將可以近似為這兩個頻率的距離。依照Eller次諧振的理論解[1]，我們發現如果 f1，f2 遠離 2f0(也就是說傳送訊號頻率增加)，次諧振的振幅將迅速減小。我們的理論分析說明了次諧振訊號的減小跟傳送訊號頻寬的關係。此外，在應用聲壓為0.1Mpa，當分數頻寬(fraction bandwidth)增加到8%，次諧振振幅趨近於零。也就是說，這個推倒理論上證明了只有窄頻寬(時域上長脈波)的訊號才能激發微氣泡產生次諧振訊號。 以微氣泡的非線性為基礎，除了產生次諧振訊號，我們也兩種頻率的近似方法來推導不同的響應。這種不同的響應叫做微氣泡低頻響應，頻段在於fL=f1-f2 。如果這兩個訊號非常靠近，這個響應相當靠近DC。而且，低頻響應的振幅也可以被推知，藉由增加傳送訊號的頻寬，這裡不同於上列所示的次諧振訊號。它可以顯現次諧振的訊號減小，當傳送訊號分數頻寬重1%拉到40%，發射頻率5Mhz，聲0.1Mpa。剛好相反地，當傳送訊號的頻寬增加時，低頻響應增加。必然地，分佈解析度(range resolution)和對組織的對比，可以互相藉由低頻響應跟次諧振響應來改善。
關鍵字
次諧振並列摘要
The subharmonic response due to the nonlinear behavior of microbubble can be used to provide good discrimination between microbubble and surrounding tissue, especially in deep region. However, there is no proper analysis about the subharmonic response under short-pulse insonification. In this work, we extend the two-frequency approximated analytic solution of Newhouse et al. to derive the subharmonic response of microbubble under band-limited insonification. [2]. Based on Fourier theory, a band-limited signal can be synthesized by multiple sinusoids, with a two-frequency approximation being the simplest case. In this work, for a given bubble whose resonance frequency denoted as , the transmission is modeled as the composition of two-frequency (f1,f2,f1>2f0>f2 ) under the constraint condition at which the transmission has spectral continuity. Therefore, the bandwidth of the transmission can be approximated as the width of the spectral region spanned by theses two frequencies. Based on Eller’s analytic solution of the amplitude of the subharmonics, we found that if and are away from (i.e. the bandwidth of the transmission is increased), the amplitude of the subharmonics will decrease rapidly. Our theoretical analysis illustrates that the amplitude of the subharmonics decrease with the transmitted fractional bandwidth. Moreover, under an applied pressure of 0.1 MPa, it approaches zero when the fractional bandwidth is increased to 8 %. In other words, this proves theoretically that only narrowband transmission can excite the microbubble to generate the subharmonics. Based on the nonlinear behavior of microbubble, in addition to the generation of subharmonics, we also derived that the resulting echo also contains difference response under two-frequency insonification. It is named as low-frequency response at the frequency, fL= f1- f2, in this work. It will reside nearby DC in spectrum if these two frequencies are close. Furthermore, the amplitude of low-frequency response can be derived to increase with the fractional bandwidth, which is different from that of subharmonics. It can be shown that the amplitude of the subharmonics decrease with the transmitted fractional bandwidth ranging from 1 % to 40 % when the emitted frequency is 5 MHz and the acoustic pressure is 0.1 MPa. On the contrary, the low-frequency response increases with the transmitted bandwidth. Consequently, both of range resolution and contrast to surrounding tissues can be achieved by using low-frequency response and subharmonics alternatively in imaging.