Ultrasound-modulated optical tomography (UOT), ^{[1– 3]} also known as acousto-optical tomography (AOT), is an acousto-optic interaction imaging modality. With high spatial resolution of ultrasonic location and high sensitivity of optical detection, UOT is a promising noninvasive biological tissue imaging technology. With the improvement of experimental apparatuses and technical routes over the last 30 years, the imaging depth of UOT has reached 3– 5 cm, ^{[4, 5]} with its spatial resolution determined by ultrasound (up to the maximum of dozens of micrometers)^[6] and its contrast ratio provided by optics. In contrast with photoacoustic imaging technology (PAT), ^{[7, 8]} it can provide not only the optical absorption coefficient^[9] but also the scattering coefficient^[10] of tissue, and thus it is able to gain much imaging information of tissue. However, its theoretical foundation, or what is known as the “ acousto-optic interaction mechanism” in the scattering medium, still remains a mystery. The previous ultrasonic modulation model cannot explain all experimental phenomena. Therefore, further exploration of its mechanism is still needed.

Biological tissue is a multiple random scattering medium. The acousto-optic interaction mechanism in tissues is not only a fundamental problem of UOT that has to be solved, but it is also a principal problem vital to the final image processing and reconstruction. Currently, there are three possible mechanisms. (i) The optical properties of the scattering medium (including absorption coefficient, scattering coefficient, and refractive index) change after being sensed by the mechanical wave in the ultrasonic field, resulting in the modulation of scattering light. This mechanism is an intensity modulation mechanism.^{[11– 13]} Both coherent and incoherent light can be used for this mechanism. However, this mechanism has not been reported by experiments yet. (ii) Due to the forced vibration of scatters in the scattering medium with an ultrasonic field, the optical distance and phase of coherent scattering light change correspondingly, which is finally manifested by the intensity variation of coherent light speckles with the ultrasonic wave. This mechanism has been studied by using dynamic light scattering (DLS).^[14] (iii) The refractive index variation of a medium caused by the ultrasonic field also influences the optical distance change of scattering light, which is manifested by modulation of the scattering light. The latter two mechanisms are phase modulation mechanisms, requiring a light source with good coherence.^[15] These two mechanisms were verified by using Monte Carlo simulation.^[16] The interaction mechanism between scattering light and impulse ultrasonic field^{[17, 18]} in the scattering medium with anisotropy^[19] and uneven space^{[20, 21]} were given successively.

Based on an intensity modulation mechanism, an analytic equation interpreting the intensity of scattering light subjected to ultrasonic modulation is derived in this paper based on the diffusion theory. Furthermore, ultrasonic modulation of incoherent light in the scattering medium is developed, which supplements the existing research on the ultrasonic modulation mechanisms of scattering light.

Considering the strong light scattering of tissues, an understanding of light traveling in biological tissue does not need to involve the interaction among optical fields and a diffraction effect, but attention has to be paid to the transmission of light intensity (light power) in the tissue.

The time-dependent transmission equation without a light source can be obtained under approximate diffusion^[22]

where Δ is the Laplace operator, φ (r, t), v, μ _a, and μ _s are the light energy flow rate, the light velocity, the absorption coefficient, and the scattering coefficient, and D = 1/3[μ _a+(1− g)μ _s] is the diffusion coefficient.

Generally, biological tissues always scatter light more than absorb it, that is, satisfying 3Dμ _a ≪ 1. Under the non-ultrashort pulsed lights, the distribution of the light energy flow rate can be regarded simply as a postponed time variable. Therefore, equation (1) can be simplified into

Ultrasonic waves are a kind of mechanical stressed elastic longitudinal wave. When an ultrasonic wave travels in a continuous elastic medium, elastic strain or stress is produced in the medium as time and space vary, thus changing the medium density. If the ultrasonic wave travels as a plane wave forward along the y axis, the medium density can be expressed as^[23]

where ρ ₀ and ρ _m are the static medium density and the amplitude of density variation, ω _a and k_a are the angular frequency and the wave vector of ultrasound. The relationship between ρ _m and p_m (amplitude of acoustic pressure) is^[23]

where u is the sound velocity.

The variation of medium density will surely change the densities of absorption and scattering grains in the medium in the same law. Take the simplest one-dimensional space for example. Such a one-dimensional space is equal to the modulated light intensity detected on the z axis (optical axis) when a wide beam of continuous light is vertically irradiated on a semi-infinite scattering medium. Since the common focused ultrasound has a focal length of about 2 cm and very small width (submillimeter or millimeter), the optical parameters of a medium on the z axis can be approximately considered as only changing with time under the effect of an ultrasonic field. Therefore, the absorption coefficient and scattering coefficient of the medium in the focal area can be expressed as

where μ _a0 and μ _s0 are the initial absorption and scattering coefficient of the medium, μ _am and μ _sm are the variation of absorption and the scattering coefficient of the medium, and

It can be known from Eq. (7) that M′ is determined by the acoustic property of the medium, which is proportional to the amplitude of acoustic pressure but inversely proportional to the square of sound velocity. Suppose u = 1500 m/s (u is the travel speed of ultrasound in biological tissues), p_m = 2× 10⁵ Pa, and ρ ₀ ∼ 1000 kg/m³, then it can be estimated that M′ ∼ 10^{− 4}. This demonstrates the little change of the absorption coefficient and the scattering coefficient of the medium caused by the ultrasonic field in biological tissues.

When the incident light source is a wide beam of continuous light, the light traveling in the tissue with an ultrasonic field satisfies the diffusion equation

where

Solve Eq. (8) by using the separation of variables. Substitute φ (t, z) = Z(z)T(t) into Eq. (8), and set

where λ is a constant.

Then, equation (10) can be separated into Eqs. (11) and (12)

Compose the eigenvalue problem from boundary conditions Z(z)| _{z→ ∞} = 0 and Z(z)| _z=0 = Z₀, then

where the eigenvalue λ > 0.

Compare Eq. (13) with the most common and simplest steady diffusion solution [], then the eigenvalue (λ ) can be gained

Substitute Eq. (14) into Eq. (11), then we obtain

Considering the practical estimate M′ ∼ 10^{− 4}, equation (15) can be approximately expressed as

Combining Eqs. (13), (14), and (16), the analytical solution of Eq. (8) can be given by

where the initial energy flow rate is φ ₀ = T₀Z₀.

Given a fixed light source size and observation direction, there is a linear relationship between the light intensity and the energy flow rate in the same place of the medium. Hence, the scattering light intensity in the medium modulated by an ultrasonic field is

where I₀ is the initial light intensity, and M′ ∝ P_m/u².

Equation (18) is applicable to both coherent and incoherent sources. Supposing that μ _ao = 100m^{− 1}, v ∼ 3× 10⁸ m/s, and M′ ∼ 10^{− 4}, equation (18) can be approximately expressed as

In accord with Eq. (19), figure 1(a) shows the rule of the scattering light intensity with time when the ultrasonic frequency is 1 MHz or 2 MHz. Figure 1(b) shows the modulation depth of modulated light intensity versus the frequency of ultrasound. The modulation depth is defined as the peak-to-peak value divided by the average value of the ultrasound-modulated light intensity.

Fig. 1.

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	Fig. 1. (a) Rule of the scattering light intensity with time. (b) Modulation depth of modulated light intensity versus the frequency of ultrasound.

As shown in Eq. (18) and Fig. 1, the diffuse light intensity decays forward along the z axis. Under the effect of an ultrasonic field, the scattering light intensity presents a periodic variation as the ultrasonic frequency changes, and its modulation frequency is the same as the ultrasonic frequency (Fig. 1(a)). This is in good accordance with general experimental findings.^{[14, 24]} The modulation depth of modulated light intensity, known as modulation efficiency, is determined by the acoustic properties and optical properties of the medium, and it increases with the increase of the amplitude of acoustic pressure (p_m) and the optical absorption coefficient of medium (μ _a0), but it declines with increasing acoustic angular frequency (Fig. 1(b)). In another paper, ^[25] we compared the imaging sensitivity and signal-to-noise with 1, 2.25, 5, and 10 MHz center frequency of impulse ultrasound in UOT. Experimental results validate Eq. (18), namely, a higher modulation efficiency of scattering light can be gained by the use of lower ultrasound frequencies and increasing ultrasonic energy. The conclusion that modulation efficiency is closely correlated with the absorption coefficient, but not with the scattering coefficient, is right, assuming a wide beam of continuous light vertically irradiated on a semi-infinite medium.

Considering the high scattering characteristic of light in biological tissues, this paper derives an analytical equation interpreting the intensity of ultrasound-modulated scattering light, based on the diffusion theory. This analytical equation is a time-space function. When a wide beam of continuous light is vertically irradiated on a semi-infinite scattering medium, the intensity of diffused light not only decays with its travel distance (z) in the medium but also changes periodically as the ultrasonic frequency changes. The modulation efficiency of scattering light is determined by the acoustic properties and optical properties of the medium, which is proportional to the amplitude of acoustic pressure (p_m) and the optical absorption coefficient of the medium (μ _a0), but it is inversely proportional to the square of sound velocity and acoustic angular frequency. Finally, the ultrasonic modulation of incoherent light in the scattering medium is developed through the experiment, which confirms the validity of the intensity modulation mechanism of the acoustic-optic interaction in the scattering medium and supplements existing research on the ultrasonic modulation mechanism of scattering light. Compared with a coherent source, an incoherent source has lower efficient ultrasonic modulation and its modulation depth is less sensitive to the absorption coefficient and scattering coefficient of the medium. Therefore, an incoherent source is not the ideal light source for UOT.