Feasibility study of the employment of laser Doppler Vibrometry for photoacoustic imaging

Photoacoustic imaging (PAI) is a biomedical imaging method which seems very promising for the detection of tumors in breast or in brain. Currently the detection of these signals is performed with piezoelectric transducers, which are very sensitive but have several limitations. Laser Doppler Vibrometry (LDV) represents a good alternative because it is a noncontact technique. The purpose of this paper is to investigate the potentiality of LDV for PAI in comparison with current transducers, and understand, in which measuring condition it is reasonable to adopt it. First, we make an introduction about the theory of PAI, we present a model of PAI for LDV and then, we investigate the theoretical limits for PAI with LDV. We derive a minimal detectable tumour radius of 390 μm for a commercial LDV with He-Ne laser.


Introduction
Photoacoustic imaging (PAI) is an imaging technique based on the photoacoustic effect. The main applications of PAI are in imaging of molecules, microvasculature, tumors, brain, and small animals [1][2][3]. This technique provides the use of a short-pulse light source for the irradiation of the tissue, this leads to the generation of a broadband photoacoustic (PA) wave. The PA wave propagates through the tissue and it is detected with ultrasound (US) transducers, and an image is computed.
The ultrasound transducers used for PAI present a high sensitivity but they have limitations due to the narrow frequency bandwidth, the finite aperture size, the contact nature of the device and the need of a coupling medium [4]. Optical detection of ultrasound could represent a valid alternative, thus several research groups investigated different interferometric techniques for PA applications [4][5][6][7]. In particular reference [4] presents a study of an all-optical non-contact PAI by using a commercial laser Doppler vibrometer (LDV) [8]. In reference [4] the authors present experiments on a pig brain with an artificial tumor as absorbing object, showing that LDV offers a promising method to achieve a noncontact PAI.
In this work, we investigated for which conditions it is reasonable to use LDV for PAI. We investigated which are the limits of detection for LDV depending on the metrological characteristics of the device and on the geometry of the investigated object. We introduce at first the typical signals generated from a spherical absorbing object and we examined the suitability of the LDV in comparison with ultrasonic transducers.

Photoacoustic imaging theory
Photoacoustic imaging is based on the photoacoustic effect. A short laser pulse irradiates the tissue, the light is locally absorbed, and converted into heat. The temperature rise of the irradiated object induces a pressure rise through thermoelastic expansion. This process is widely described in [1][2][3][9][10][11][12][13][14]. In particular the fractional volume expansion of the heated region is d , N is the isothermal compressibility, E is the coefficient of volume expansion, and T is the change in temperature. Neglecting volume expansions, d 0 V V during the fast heating, i.e. assuming that the laser pulse is shorter than the thermal and stress relaxation times, [1][2][3] the initial pressure can be calculated as The temperature increase T can be expressed as the specific optical energy deposition e A divided by the density of the material U and the heat capacity at constant volume V C . The originated pressure propagates as an ultrasound wave in the tissue according with the wave equation (4) where s v is the sound velocity in the material and ( , ) p t r is the pressure at the location r and time instant t . Equation (4) can be solved with the Green's function approach, as the response to a temporal and spatial impulse [1][2][3] leading to (5) where G is the delta function and r' is the position of the source location. This equation is expressed in spherical coordinates and it is valid to calculate the photoacoustic pressure generated by an arbitrary heterogeneous optically absorbing object heated by a short laser pulse in stress and thermal confinement conditions. A detailed description of the mathematics can be found in [1][2][3].
2.1. Typical photoacoustic signal generated from a spherical absorbing object Equation (6) describes the pressure propagation of a homogeneously heated sphere which could represent an early stage tumor in breast tissue [9]. For the case of a small sphere, the pressure p at the time instant t and at the distance r from the center of the sphere ( Figure 1) [1], can be derived from where U is the Heaviside function and S R is the radius of the small sphere [1][2][3]. Therefore, the signal generated by a small sphere with higher absorption has a bipolar shape. The amplitude is inversely proportional to the distance r from the sphere/tumor and directly proportional to the initial pressure 0 .
p In particular, the duration W of the bipolar signal depends on S R s 2 . Figure 1 shows the set-up for measurements performed in transmission, i.e. the transducer is positioned in the opposite side of the excitation at the distance d r from the sphere. Figure 2 shows the signal generated by a small heated and expanded sphere with 0.5 mm S R acquired for different detector distances ( ) d r r according with equation (6). Acoustic attenuation was not considered for this simulation. The duration of the signal W remains the same since S R and s v are fixed, but the amplitude is damped by a factor 1 2 .
r The amplitude of the positive peak of the bipolar signal when the photoacoustic wave reaches the detector position d r is [9][10][11] The acoustic attenuation of the wave travelling through a medium plays an important role in ultrasound imaging. Soft tissue has a typical acoustic attenuation coefficient [2,10] ac MHz (11) includes also the acoustic attenuation through the tissue by multiplying equation (8) with the damping. The frequency spectrum of the bipolar signal presents a characteristic sequence of oscillations which decrease in amplitude. In particular the central frequency u f depends on the sound velocity and the dimension of the object [10][11][12] The major portion of acoustic energy resides in the half-power bandwidth (-3dB), i.e. between

Photoacoustic signal detection with LDV
The LDV is a noncontact interferometric technique that employs a heterodyne detection scheme [8]. It allow the detection of real time velocity and displacement signals with a resolution down to picometer range. An LDV sensor measures the Doppler shift of the back-scattered measurement beam which impinges the moving object. Reference [8] provides a detailed description of the measurement principle. LDV sensors are widely used for noncontact surface vibration measurements in mechanical, civil and industrial engineering. Due to its metrological properties and its non-contact nature, this technique finds also several application in the biomedical field like the identification of the cardiovascular parameter, studies of the ear, and dentistry. Recently LDV was employed for PA detection [4].

Measurement condition for the detection of PAI signals with LDV
Photoacoustic detectors are usually broadband ultrasound sensor, here we present the measurement condition for the alternative solution with a LDV proposed in reference [4]. While ultrasound sensors measure the pressure, LDV sensors measure the velocity or the displacement. In order to be able to detect the PA signal without degradation, two main conditions need to be fulfilled: a) The maximal detectable frequency of the sensor has to be greater than the characteristic frequencies of the photoacoustic signals, which are strictly related to the minimal dimension of the absorbing object. According with [2], to acquire PA signals properly a bandwidth of 150% in respect to the central frequency u f is necessary (13) This represent a good compromise between the detection of high frequency components and the noise level. The typical bandwidth of LDV has a lower frequency L =0 Hz f and an upper frequency of H =2 MHz f [8]. The minimal detectable absorbing object is calculated by combining equation (12) and (13). b) The pressure, the velocity or the displacement at the boundary needs to be larger than the resolution of the sensor. In the case of the LDV the minimal detectable velocity or the minimal detectable displacement has to be greater than the velocity/displacement at the boundary. The minimal detectable displacement of a heterodyne interferometer is related to the signal to noise ratio referred to 1 Hz bandwidth ' SNR . In particular the noise equivalent mean square displacement min ' d after demodulation [8] is O is the wavelength of the laser source. In theoretic calculations min ' d can reach -1/2 4 fm Hz [8]. However, commercially available class II LDVs are in combination with digital decoding that yields to a resolution of about -1/2 min ' 50 fm Hz d [8]. Thus we made the assumption that the displacement resolution of LDV at a given bandwidth B , min@ B d , can be expressed in terms of power spectral density PSD of a white noise 2 1 (15) where 1 Q and 2 Q are respectively the lower and the upper frequencies of the bandwidth. As mentioned before, the typical bandwidth of commercial LDV is 0-2 MHz, thus

Resolution comparison between US sensors and LDV
An important aspect is to understand when it is reasonable to apply the LDV for PA detection in comparison with US sensors. While the typical resolution for commercial LDV is where Z is the acoustic impedance of the tissue. The factor 2 depends on the impedance mismatch between skin and air which results in an asymmetric displacement in direction of the skin-air interface [14].
In figure 3 the resolution of the different sensors are compared. In particular the resolution of a broadband ultrasound sensor ( -1/2 0.6 mPa Hz ) and of a commercial class II He-Ne LDV are reported. In addition, since the newest LDVs have an infrared (IR) laser(1550 nm) and this technology seem very promising, we show in figure 4 also the theoretical resolution limit for a 1550 nm heterodyne interferometer =0.88 fm Hz d [4]. Figure 3 reveals that for frequencies lower than ~1 kHz the commercial LDV has better resolution than broadband US transducers. This limit can go theoretically up to almost 100 kHz with IR technology. Since typically the ultrasonic frequency spectrum of PA signals has components between 100 kHz-10 MHz [12], US transducers have generally better performances than LDV for the detection of PA signals. However the detection of PA signals with LDV is possible according to the conditions described in paragraph 3.1.

Limits of LDV for the detection of PA signals
According to the condition a) in paragraph 3.1, for a proper reconstruction of a PA signal with LDV, its central frequency has to be lower as u,max f The velocity signal is directly proportional to the pressure and thus, except for an amplitude factor, it has the same characteristics in time and in frequency domain of the pressure signal. The displacement signal is presented in figure 4. Its maximum amplitude is max d  (11) and (22) [9][10][11]14]. We chose a pulsed laser with 1064 nm wavelength as laser excitation. The maximal allowed laser fluence for application in human body according with the laser safety norm [15] is  c) The theoretical resolution limit of a 1550 nm heterodyne interferometer with the bandwidth varying with the radius of the sphere ( )

S B R
The curves of case a), b) and c) are graphically, compared with the curves of max d over the radius

Conclusion
In this paper we present a model of PAI for LDV and the potential of LDV for PAI in comparison with current transducers. We explore the measuring conditions for a reasonable adoption of the LDV. In addition, we derived the theoretical limits for PAI with LDV. The analysis indicates that the commercial LDV has better resolution than broadband US transducer for frequencies lower then 1 kHz. This limit can go theoretically up to almost 100 kHz with IR technology. Since typically the ultrasonic frequency spectrum of PA signals are about between 100 kHz -10 MHz [2], US broadband transducers have generally better resolution than LDV for the detection of PA signals. However, since LDV presents advantages due to the non-contact nature of the technology and it has sufficient resolution to detect PA signals, it is still reasonable to use LDV for PAI acquisition. LDV allows the detection of absorbing spherical objects with a radius down to 0.39 mm, depending on its depth in the tissue, and the absorption coefficient at the excitation wavelength.
Future works are oriented on experimental tests to verify the theoretical limits described in this work. We would like also to include in our model the frequency dependent attenuation and evaluate its effects on the limits of LDV for the detection of PA signals.