Ultrasound-modulated optical tomography

Compared to traditional optical technology, UOT provides sufficient penetration depth. UOT relies on all the photons within deep tissue. No matter how many times photons are scattered, as long as they are not absorbed, those photons are still useful for UOT. By applying this method, it is possible to generate images at quasi-diffusive or diffusive layer for biological tissue.

UOT basically built upon the modulation of light due to ultrasonic wave effect happened within testing medium. The target within the testing medium will be irradiated by a laser beam and a focused ultrasonic wave. The reemitted light propagating through ultrasonic wave will then carry information of local optical and acoustic properties. These properties will be used to regenerate images showing the inside view of the medium.

In analytic model, two approximations are made. (1) the optical wavelength is much shorter than the mean free path (weak-scattering approximation) and (2) the ultrasound-induced change in the optical path length is much less than the optical wavelength (weak-modulation approximation). [3]

With the first approximation, we can assume the ensemble-averaged correlations between electric fields are the same. Since the differences between correlation results from different paths are negligible. With this assumption, the correlations for electric fields G1(ꚍ) can be written as the following:

${\displaystyle G1(\tau )=\int p(s)\langle E_{s}(t)E_{s}^{*}(t+\tau )\rangle ds}$[3]

In this equation the parenthesis represents the ensemble and time averaging. Es demsontrates the unit-amplitude electric field of the scattered light of a path of length s, and p(s) denotes the probability density function of s.[3] In analytic UOT model, we treat the light source as an optical plane wave. We assume the plane wave light normally hit a slab of thickness d. The transmitted light will be captured by a point detector. After applying diffusion theory, the original G1 equation can be further modified as:

${\displaystyle G1(\tau )=\left({\frac {\left({\frac {d}{l_{t}'}}\right)(\sinh({\varepsilon [1-cos(\omega _{a}\tau )]})^{1/2})}{\sinh(d/l_{t}')({\varepsilon [1-cos(\omega _{a}\tau )]})^{1/2}}}\right)}$[3]

Where ${\displaystyle \varepsilon =6(\delta _{n}+\delta _{d})(n_{0}k_{0}A)^{1/2}}$ ${\displaystyle \delta _{n}=(\alpha _{n1}+\alpha _{n2})\eta ^{2}}$ ${\displaystyle \alpha _{n1}=1/2k_{a}l_{t}'\arctan(k_{a}l_{t}')}$ ${\displaystyle \alpha _{n2}={\frac {\alpha _{n1}}{k_{a}l_{t}'/\arctan(k_{a}l_{t}')-1}}}$ ${\displaystyle \delta _{d}={\frac {1}{6}}}$

In these equations, ωa represents the acoustic angular frequency, and the n0 is the background index of refraction; k0 is the magnitude of the optical wave vector in vacuo; A is the acoustic amplitude, which is proportional to the acoustic pressure; ka is the magnitude of the acoustic wavevector; l_t' is the optical transport mean free path; η is the elasto-optical coefficient; ρ is the mass density;[4] ${\displaystyle \delta _{n}\delta _{d}}$ represents how ultrasound averagely change the light per free path via index of refraction and displacementare respectively.

After deriving the autocorrelation equation, Wiener-Khinchin theorem is applied. With this theorem, we can further connect G1 with the spectral density of the modulated speckle. Their relationship in frequency space is shown as the following Fourier transformation equation.

${\displaystyle S(\omega )=\int _{-{\infty }}^{\infty }G1(\tau )\exp(i\omega \tau )d\tau }$[4]

For simplicity, the Fourier transformed term exp(-iω₀t) is dropped, and the ω here represents relative angular frequency of unmodulated light. For example, if ω=0 this equation is calculating the spectral density with absolute angular frequency ω₀. Since G1 is an even autocorrelation function, the spectral intensity at ω_a can be written as:

${\displaystyle I_{n}={\frac {1}{T_{a}}}\int \limits _{0}^{T_{a}}\cos(n\omega _{a}\tau )G1(\tau )d\tau }$[4]

Here, the n and Ta represent the acoustic period. Since the frequency spectrum is symmetric about ω₀, the one side modulation depth is defined as:

${\displaystyle M_{1}={\frac {I_{1}}{I_{0}}}}$[5]

Then we can consider the condition under the second approximation (weak-modulation approximation). In this situation, the term ${\displaystyle (d/l_{t}')\varepsilon ^{1/2}}$ is much smaller than 1. By applying the feature of sinh function, which is the main component of G1 function, the original autocorrelation G1 can be further simplified as:

${\displaystyle G1(\tau )=1-{\frac {1}{6}}({\frac {d}{l_{t}'}})^{2}\varepsilon [1-cos(\omega _{a}\tau )]}$[5]

Therefore, the one side modulation depth can be simplified as ${\displaystyle M_{1}={\frac {1}{12}}({\frac {d}{l_{t}'}})^{2}\varepsilon }$. From previous equation, we can see ${\displaystyle \varepsilon =6(\delta _{n}+\delta _{d})(n_{0}k_{0}A)^{1/2}}$. Therefore, in conclusion, A and M1 has quadratic relationship. Such quadratic modulation can be captured by a Fabry-Perot interferometer. Or, we can calculate the ratio between observed AC signal and he observed DC signal (also named as apparent modulation depth) which can carry enough information to represent such modulation as well.

In conclusion, in UOT analytic model, with the help of weak-scattering approximation, weak-modulation approximation, diffusion theory, and Wiener-Khinchin theorem, the relationship between acoustic amplitude and modulated light can be successfully observed.

For single-frequency UOT, the axial resolution along ultrasonic axis is always limited by the elongated ultrasonic focal zone. To improve the axial resolution, Ultrasonic frequency-swept UOT model is designed. In this system, the object is placed in a tank full of UOT scattering medium. There will also be an ultrasound absorber at the bottom of the tank to avoid rebound of ultrasound. Basically, a function generator will produce a frequency signal relating to time. After passing through a power amplifier and a transformer, such frequency command will be sent to the ultrasonic transducer to generate ultrasonic beam with different frequencies. After a brief calculation, a focused ultrasonic beam will be sent to the medium and the target. Meanwhile, a laser beam which is perpendicular to the ultrasonic beam will also illuminate the scattering medium. Then, on the other side of the light source, the PMT, modulated by frequency signal sent through the first function generator will detect transmitted light signal within the tank and transfer the optical signal to electrical signal. The electrical signal will then pass through an amplifier, an Oscilloscope and be stored in the data base.

With such data base, spectral intensity vs frequency plots at multiple points can be generated (The first spectrum is generated as a reference, produced by optical signal far from the object.). Each of the spectrum can then be further converted to a 1D image showing the interior of the medium in the direction perpendicualr to the tank (z direction). In the end, all the 1D image will be pieced together to generate a full view inside the medium.

In summary, a frequency-swept (chirped) ultrasonic wave can encode laser light traversing the acoustic axis with various frequencies. Decoding the transmitted light provides resolution along the acoustic axis. This scheme is analogous to MRI.[6]

Was first proposed as a method for virus detection in 2013.[7]