CN107049252A - A kind of biological magneto-optic sound joint endoscopic imaging method - Google Patents

Abstract

A biomagnetoacoustic combined endoscopic imaging method, the method is based on the establishment of a multi-layer cavity tissue cross-sectional model, the process of ultrasonic echo imaging, photoacoustic imaging and inductive magnetoacoustic imaging in the cavity Through numerical simulation, the ultrasonic echo signal reflected by the tissue on the cavity cross section and the photoacoustic and magnetoacoustic signals generated by the tissue are obtained, and then the three ultrasonic signals are optimally weighted and summed to obtain the fused joint imaging signal. The present invention fuses the ultrasonic echo signals reflected/scattered by the cavity tissue received by the ultrasonic transducer in time-division and the photoacoustic signal and magnetoacoustic signal generated by the tissue at the signal layer. Compared with the traditional method, the combined Imaging signals can retain more morphological structure and composition information of tissues, and the reconstructed combined images have extremely high spatial resolution, contrast, sensitivity and contrast resolution, and can accurately display the position, Morphology and its functional components.

Description

A Biomagneto-Photoacoustic Combined Endoscopic Imaging Method

technical field

The invention relates to a method for performing magneto-optoacoustic combined endoscopic imaging on biological cavity tissue, and belongs to the technical field of medical imaging.

Background technique

Biophotoacoustic (PA) imaging is based on the photoacoustic effect of biological tissue as the physical basis, that is, the tissue absorbs short-pulse laser light and then heats up and expands to generate ultrasonic waves (ie, photoacoustic signals). Inductive magnetoacoustic tomography (magnetoacoustictomography with magnetic induction, MAT-MI) imaging is based on the magnetoacoustic effect of biological tissues, that is, the target is placed in a static magnetic field, and magnetic pulse excitation is applied in the same direction as the static magnetic field. The action of the eddy current induced in the medium and the static magnetic field produces the Lorentz force, and the charged particles vibrate under the action of the Lorentz force, thereby emitting ultrasonic waves, which are then imaged after being received by the ultrasonic probe. A single imaging technique cannot describe the structural and functional information of biological tissues in a comprehensive and detailed manner. Ultrasonic (US), PA and MAT-MI imaging are all acoustic imaging techniques based on ultrasound, and have complementary characteristics. Combine them for magneto-photo-acoustic (MPA) joint imaging. MPA imaging combines the higher resolution of ultrasonic detection in the photoacoustic signal transmission stage and the higher resolution and sensitivity of magnetoacoustic signal detection, which can accurately locate and image functional components of early lesion tissues.

At present, MPA combined imaging is in vitro imaging, that is, the ultrasonic transducer is placed around the organism to receive ultrasonic signals outside the body. Compared with endoscopic imaging, this method cannot observe and diagnose biological cavity tissues (such as digestive tract, intestinal tract and blood vessels, etc.) in a timely and effective manner. Therefore, it is necessary to study an endoscopic magneto-photo-acoustic (EMPA) imaging method.

Contents of the invention

The purpose of the present invention is to address the drawbacks of the prior art, to provide a biomagneto-optoacoustic combined endoscopic imaging method to obtain high-resolution and sensitive images of biological cavity tissue, and to help medical staff timely and effectively detect biological cavity tissue Observation and diagnosis.

Problem described in the present invention is realized with following technical scheme:

A biomagnetoacoustic combined endoscopic imaging method, the method is based on the establishment of a multi-layer cavity tissue cross-sectional model, the process of ultrasonic echo imaging, photoacoustic imaging and inductive magnetoacoustic imaging in the cavity Through numerical simulation, the ultrasonic echo signal reflected by the tissue on the cavity cross section and the photoacoustic and magnetoacoustic signals generated by the tissue are obtained, and then the three ultrasonic signals are optimally weighted and summed to obtain the fused joint imaging signal.

The above biomagnetism photoacoustic combined endoscopic imaging method, the method comprises the following steps:

a. Establish a cross-sectional model of multi-layer cavity tissue

The imaging catheter is located at the center of the multi-layer cavity tissue cross-section model, and the ultrasonic transducer is located at the top of the imaging catheter. The coordinate system of the model is the θ-l polar coordinate system, where the origin of the coordinates is the center of the top of the imaging catheter, and θ is the polar angle. l is the polar diameter, the horizontal direction to the right is the positive direction of the l-axis, ignoring the aperture effect of the ultrasonic transducer, the ultrasonic transducer is regarded as an ideal point detector, and its scanning trajectory is a circular trajectory parallel to the imaging plane , taking the center of the model as the starting point, divide the model into N parts at equal angles, and each part is approximately a multi-layer cavity tissue parallel to each other, and the imaging angle at the catheter is

θ _i =360(i-1)/N

Where i=1,2,...,N, the angular range of the imaging area corresponding to θ _i is [θ _ia ,θ _ib ], where θ _ia =θ _i -180/N, θ _ib =θ _i +180/N;

b. Numerical simulation of the ultrasonic echo signal reflected by the imaging tissue and the photoacoustic signal and magnetoacoustic signal generated by the tissue;

①Ultrasonic echo signal:

Ultrasonic pulses are emitted radially from the center of the cavity, and the ultrasonic echo signals reflected/scattered by the tissue are simulated according to the difference in acoustic impedance of different components of the tissue and the impulse response of the ultrasonic probe;

② Photoacoustic signal:

Laser pulses are emitted radially from the center of the cavity to the surrounding tissue, and the tissue generates photoacoustic signals due to the photoacoustic effect. According to the light absorption coefficient and scattering coefficient of tissues with different components, combined with Monte Carlo simulation and photoacoustic wave equation, the simulation results of tissue generation photoacoustic signal;

③Magnetoacoustic signal:

Static magnetic field and pulsed magnetic excitation are applied along the axial direction of the cavity, and the tissue generates magnetoacoustic signals due to the magnetoacoustic effect. According to the conductivity of different components of the tissue, combined with the magnetoacoustic wave equation, the magnetoacoustic signal generated by the tissue is simulated;

c. Fusion of ultrasonic echo signal, photoacoustic signal and magnetoacoustic signal

For the angle θ _i (i=1,2,…,N) in the imaging tissue, the ultrasonic echo signal at position r photoacoustic signal and magnetoacoustic signal Fusion is performed using the following formula:

in is the fusion signal of ultrasonic echo, photoacoustic and magnetoacoustic signals, that is, the joint imaging signal, W _1i , W _2i , and W _3i are the weighting factors of ultrasonic echo, photoacoustic and magnetoacoustic signals, respectively, and W _1i +W _2i +W _3i =1.

The biomagnetoacoustic combined endoscopic imaging method, in order to make the fusion signal of ultrasonic echo, photoacoustic and magnetoacoustic signal The mean square error is the smallest, and the weighting factors W _1i , W _2i , and W _3i of ultrasonic echo, photoacoustic and magnetoacoustic signals should take the optimal value The formula for calculating the optimal value is:

where k is the number of measurements of ultrasonic echo, photoacoustic and magnetoacoustic signals, is the optimal weighting factor for ultrasonic echo, photoacoustic and magnetoacoustic signals when the total mean square error is minimized, for Variance, for Variance, for The variance of R _UU is The autocorrelation coefficient of R _UP is with The cross-correlation coefficient of R _PP is The autocorrelation coefficient of _RPM is with The cross-correlation coefficient of R _MM is The autocorrelation coefficient of R _MU is with The correlation coefficient of .

The present invention fuses the ultrasonic echo signals reflected/scattered by the cavity tissue received by the ultrasonic transducer in time-division and the photoacoustic signal and magnetoacoustic signal generated by the tissue at the signal layer. Compared with the traditional method, the combined Imaging signals can retain more morphological structure and composition information of tissues, and the reconstructed combined images have extremely high spatial resolution, contrast, sensitivity and contrast resolution, and can accurately display the position, Morphology and its functional components.

Description of drawings

The present invention will be further described below in conjunction with accompanying drawing.

Figure 1 is an example of a cross-sectional model of a blood vessel containing calcified plaque;

Fig. 2 is a schematic diagram of receiving ultrasonic echo signals, photoacoustic signals and magnetoacoustic signals when the EMPA imaging catheter is at an angle _θi ;

FIG. 3 is a schematic diagram of approximating the imaging region corresponding to the angle θ _i in FIG. 2 as multi-layer blood vessel wall tissue.

The symbols in this paper are: θ and l are the polar angle and polar diameter of the θ-l plane polar coordinate system, where the imaging catheter is located at the coordinate origin (the center of the cavity), and the horizontal direction to the right is the positive direction of the l-axis; N is the cavity The number of equiangular divisions of the cross-sectional model; θ _i is the i-th imaging angle; θ _ia and θ _ib are the lower limit and upper limit of the angle range of the imaging area corresponding to θ _i , where i=1,2,...,N; F(r, θ) is the ultrasonic echo signal collected by the ultrasonic probe at position r and angle θ; T(r, θ) is the acoustic impedance difference function of the imaging tissue at position r and angle θ; h(r, θ) is the point spread function of the ultrasonic detector, that is, the impulse response; σ _r is the pulse width of the emitted ultrasonic wave; σ _θ is the width of the emitted ultrasonic beam; k ₀ is the wave number; f ₀ is the center frequency of the emitted ultrasonic wave; G( r, θ) is Gaussian white noise with a mean of 0 and a standard deviation of 1; E(r, θ) is the difference in acoustic impedance of tissues at position r and angle θ; is the ultrasonic echo signal collected at angle θ _i and position r, where i=1,2,...,N; is the Hamiltonian; p(r,t) is the sound pressure at time t and position r; c is the sound velocity in biological tissue; A(r) is the light energy deposition distribution function; β is the volume expansion temperature coefficient of the tissue ; C _P is the specific heat capacity of the tissue; I(t) is the intensity of the incident laser pulse in photoacoustic imaging; (j,k) is the coordinate of the point r on the tissue in the polar coordinate system θ-l; Δθ, Δl are θ Δt is the discrete time interval; n is the discrete time; p ⁿ (j,k) is the sound pressure at time n and position (j,k); v _θ ⁿ (j,k) and v _l ⁿ (j,k) are the vibration velocities of the particle at the time n and the position (j,k) in the θ direction and the l direction respectively; ρ is the tissue density; c(j,k) is the position (j,k) ) at the sound velocity; A(j,k) is the light energy deposition value at the position (j,k); I ⁿ is the laser pulse intensity at time n; c _max is the maximum sound velocity in the tissue; λ _min is the ultrasonic minimum wavelength; is the photoacoustic signal at angle θ _i and position r, where i=1,2,…,N; J is the induced current density; B ₀ is the static magnetic field strength; J(j,k) is the position (j,k) The induced current density at the position; B ₀ (j,k) is the static magnetic field intensity at the position (j,k); is the magnetoacoustic signal at angle θ _i and position r, where i=1,2,…,N; k is the number of measurements of ultrasonic echo, photoacoustic and magnetoacoustic signals; R _UU is The autocorrelation coefficient of ; R _PP is The autocorrelation coefficient of ; R _MM is The autocorrelation coefficient of ; R _UP is with The cross-correlation coefficient; R _PM is with The cross-correlation coefficient of R _MU is with The correlation coefficient of for Variance; for Variance; for Variance of ; W _1i , W _2i , W _3i are weighting factors of ultrasonic echo, photoacoustic and magnetoacoustic signals; is the fusion signal of ultrasonic echo, photoacoustic and magnetoacoustic signals; is the total mean square error of the fusion signal; is the minimum value of the total mean square error; is the optimal weighting factor for ultrasonic echo, photoacoustic and magnetoacoustic signals when the total mean square error is minimized.

detailed description

Endoscopic magneto-photo-acoustic (EMPA) imaging uses the same imaging system to simultaneously perform ultrasound, photoacoustic and inductive magnetoacoustic imaging inside the biological cavity. The ultrasonic transducer directly collects the ultrasonic signal reflected, scattered or generated by the tissue in the cavity, and then the combined image is obtained after fusion by the computer.

1. Establish a cross-sectional model of multi-layer cavity tissue:

As shown in Figure 1, taking the cross-sectional model of a blood vessel as an example, the imaging catheter is located in the center of the model, and the ultrasonic transducer is located at the top of the imaging catheter. Surrounding the catheter from the inside to the outside along the radial direction are the lumen of the blood vessel and the atherosclerotic plaque. (calcified, lipid, fibrous or mixed plaque), vessel wall intima/media and adventitia. The coordinate system where the model is located is the θ-l polar coordinate system, where the origin of the coordinates is the center of the imaging catheter, θ is the polar angle, l is the polar diameter, and the horizontal direction to the right is the positive direction of the l-axis. The method of the invention ignores the aperture effect of the ultrasonic transducer, and regards the ultrasonic detector (that is, the ultrasonic transducer) as an ideal point detector, and its scanning trajectory is a circular trajectory parallel to the imaging plane.

As shown in Figure 2, the center of the blood vessel model is taken as the starting point, and the model is divided into N parts at equal angles, and each part is approximately a multi-layered blood vessel wall tissue parallel to each other (as shown in Figure 3) . Ultrasonic pulses, laser pulses, and excitation magnetic fields are respectively applied to the model (as shown in Figure 2, including static magnetic fields and pulsed magnetic fields, where the direction of the static magnetic field coincides with the axial direction of the cavity). The imaging angle at the catheter is

θ _i =360(i-1)/N (1)

where i=1,2,...,N. The angle range of the imaging area corresponding to θ _i is [θ _ia , θ _ib ], where θ _ia =θ _i −180/N, θ _ib =θ _i +180/N.

2. Ultrasonic echo signal of simulated cavity tissue:

Ultrasonic endoscopic imaging uses ultrasonic beams to scan in the circumferential direction of the cavity, and detects diseased tissues by ultrasonic pulse reflection. Biological soft tissue has layers, and different layers have different tissue components, which manifest as differences in acoustic characteristic parameters. When ultrasonic waves propagate in tissues, when encountering the interface between different tissues, reflected echoes will be generated, which include Location and structure information of different tissues.

The ultrasonic probe emits ultrasonic signals radially from the center of the cavity cross-sectional model. To simplify the problem, the absorption attenuation and higher-order echoes of the tissue are not considered, then:

F(r,θ)=h(r,θ)*T(r,θ) (2)

Among them, F(r, θ) is the ultrasonic echo signal collected by the ultrasonic detector at the position r and angle θ; h(r, θ) is the point spread function of the ultrasonic detector, that is, the impulse response, which is the time Invariant and separable, expressed as

In the formula, σ _r is the pulse width of the emitted ultrasonic wave, σ _θ is the width of the emitted ultrasonic beam, the wave number k ₀ =2πf ₀ /c, c is the sound velocity in the tissue, and f ₀ is the center frequency of the emitted ultrasonic wave. T(r, θ) in formula (2) is the acoustic impedance difference function of the imaging tissue at position r and angle θ:

T(r,θ)=G(r,θ)*E(r,θ) (4)

Among them, G(r, θ) is Gaussian white noise with mean value of 0 and standard deviation of 1; E(r, θ) is the difference value of acoustic impedance of tissue at position r and angle θ. If the E value of tissue is larger, It shows that the difference in acoustic impedance is large, and the echo signal generated is also strong.

The ultrasonic echo signals collected by the ultrasonic probe at N angles can be obtained from formula (2), and the ultrasonic echo signals collected at the angle θ _i and position r are recorded as (i=1,2,...,N).

3. Simulate the photoacoustic signal generated by the short pulse laser acting on the cavity tissue:

The short-pulse laser is emitted radially from the center of the cavity cross-section, and the light energy deposition distribution function is obtained through Monte Carlo simulation of the light in the tissue according to the light absorption coefficient and scattering coefficient of the tissue. The photoacoustic signal (which is essentially ultrasonic wave) produced by the tissue due to the photoacoustic effect satisfies the photoacoustic wave equation

in, is the Hamiltonian; p(r,t) is the sound pressure at time t and position r; c is the sound velocity in biological tissue; A(r) is the light energy deposition distribution at position r; β is the volume of the tissue The temperature coefficient of expansion, C _P is the specific heat capacity of the tissue; I(t) is the intensity of the incident laser pulse.

Combined with the relationship among the three physical quantities of ultrasonic sound pressure, particle vibration velocity and tissue density, the formula (5) is discretized using the time domain finite difference method to obtain

Among them, (j,k) is the coordinate of point r on the tissue in the polar coordinate system θ-l; Δθ and Δl are the unit lengths of θ axis and l axis respectively; Δt is the discrete time interval; n is the discrete time; p ⁿ (j,k) is the sound pressure at time n and position (j,k); with is the vibration velocity of the particle at time n and position (j, k) in the θ direction and l direction respectively; ρ is the tissue density; c(j, k) is the sound velocity at position (j, k); A(j, k) is the light energy deposition value at position (j,k); In is the laser pulse intensity at time ⁿ .

Equation (6) needs to satisfy the Courant stability condition:

Among them, c _max is the maximum sound velocity in the tissue; λ _min is the minimum wavelength of ultrasound.

The photoacoustic signals at N angles can be obtained from formula (6), and the photoacoustic signals at angle θ _i and position r are recorded as (i=1,2,...,N).

4. Simulate the magnetoacoustic signal generated by static magnetic field and pulsed magnetic field acting on cavity tissue:

In MAT-MI imaging, the biological tissue is excited by the external pulsed magnetic field to generate induced eddy current, the induced eddy current interacts with the static magnetic field applied by the outside to generate the Lorentz force, and the charged particles in the tissue are generated under the action of the Lorentz force The mechanical vibration is transmitted outward in the form of ultrasonic waves, that is, magnetoacoustic signals.

For the sake of simplicity, the tissue is approximated as an ideal homogeneous fluid, its viscosity and incompressibility are ignored, its initial state is static, and the heat exchange is ignored, and the acoustic wave propagation process is assumed to be an adiabatic process. Based on the above assumptions, the magnetoacoustic signal generated by the tissue satisfies the magnetoacoustic wave equation

Among them, p(r,t) is the sound pressure at time t and position r; c is the sound velocity in biological tissue; J is the induced current density, and B ₀ is the static magnetic field strength.

The time domain finite difference form of formula (8) is:

Among them, (j,k) is the coordinate of point r on the tissue in the polar coordinate system θ-l; Δθ and Δl are the unit lengths of θ axis and l axis respectively; Δt is the discrete time interval; n is the discrete time; p ⁿ (j,k) is the sound pressure at time n and position (j,k); with is the vibration velocity of the particle at time n and position (j, k) in the θ direction and l direction respectively; ρ is the tissue density; c(j, k) is the sound velocity at position (j, k); J(j, k) is the induced current density at position (j,k); B ₀ (j,k) is the static magnetic field strength at position (j,k).

Formula (9) also needs to satisfy the Courant stability condition in formula (7). The magneto-acoustic signals at N angles can be obtained from formula (9), and the magneto-acoustic signals at angle θ _i and position r are recorded as (i=1,2,...,N).

5. Fusion of ultrasonic echo, photoacoustic signal and magnetoacoustic signal:

Under the condition that the total mean square error is minimum, the method of the present invention is used for the measurement of ultrasonic echo signals, photoacoustic signals and magnetoacoustic signals of imaging tissues at angles θ _i (i=1, 2,..., N) and positions r Value, find the optimal weighting factor corresponding to each signal in an adaptive way, and get the optimal fusion result. Specific steps are as follows:

Assume with The number of measurements is k, and the variances of the measured values are with The specific calculation method is as follows:

for with The autocorrelation coefficient of

with The correlation coefficient of

but

for with The autocorrelation coefficient of

in

with The correlation coefficient of

but

for with The autocorrelation coefficient of

with The correlation coefficient of

but

The weighting factors of ultrasonic echo, photoacoustic and magnetoacoustic signals are W _1i , W _2i and W _3i respectively, and satisfy

W _1i +W _2i +W _3i ＝1 (19)

The fused signal is

The total mean square error is

By finding the extremum of the multivariate quadratic function in formula (21), the total mean square error is obtained minimum value of

When the total mean square error is minimized, the optimal weighting factors of ultrasonic echo, photoacoustic and magnetoacoustic signals are respectively:

Substitute Equation (23) into Equation (20) to get the fused signal. The fusion algorithm does not require any prior knowledge about the signals to be fused, and only needs to estimate the variation of the variance of each signal according to the measured value of the signals to be fused, and adjust the weighting factors of each signal participating in the fusion in time to obtain the fusion signal with the smallest mean square error . The mean square error of the fused signal is not only smaller than the mean square error of a single signal, but also the fusion result is better than the traditional mean estimation algorithm in terms of accuracy and fault tolerance.

Claims (3)

1. A biological magneto-optical-acoustic combined endoscopic imaging method is characterized in that numerical simulation is carried out on ultrasonic echo imaging, photo-acoustic imaging and induction type magneto-acoustic imaging processes in a cavity on the basis of establishing a multilayer cavity tissue cross section model, ultrasonic echo signals reflected by tissues on the cross section of the cavity and photo-acoustic and magneto-acoustic signals generated by the tissues are obtained, and then optimal weighted summation is carried out on the three ultrasonic signals to obtain a combined imaging signal after fusion.

2. The bio-magneto-optical-acoustic combined endoscopic imaging method according to claim 1, comprising the steps of:

a. establishing a multi-layer cavity tissue cross section model

The imaging catheter is positioned at the center of a cross section model of the multilayer cavity tissue, the ultrasonic transducer is positioned at the top end of the imaging catheter, a coordinate system where the model is positioned is a theta-l polar coordinate system, wherein the origin of coordinates is the center of the top end of the imaging catheter, theta is a polar angle, l is a polar diameter, the horizontal rightward direction is the positive direction of an axis l, the aperture effect of the ultrasonic transducer is ignored, the ultrasonic transducer is taken as an ideal point detector, the scanning track of the ultrasonic transducer is a circular track parallel to the imaging plane, the center of the model is taken as a starting point, the equal angles of the model are divided into N parts, each part is approximately a multilayer cavity tissue parallel to layers, and the imaging angle at the catheter is

θ_i＝360(i-1)/N

Where i is 1,2, …, N, theta_iThe corresponding imaging area has an angular range of [ theta ]_ia,θ_ib]Wherein theta_ia＝θ_i-180/N，θ_ib＝θ_i+180/N；

b. Carrying out numerical simulation on ultrasonic echo signals reflected by the imaging tissues and photoacoustic signals and magnetoacoustic signals generated by the tissues;

and fourthly, ultrasonic echo signals:

transmitting ultrasonic pulses from the center of the cavity along the radial direction, and simulating to obtain ultrasonic echo signals reflected/scattered by the tissues according to the acoustic impedance difference values of the tissues with different components and the impulse response of the ultrasonic detector;

photoacoustic signal:

emitting laser pulses to surrounding tissues from the center of the cavity along the radial direction, generating photoacoustic signals by the tissues due to the photoacoustic effect, and simulating the photoacoustic signals generated by the tissues according to the light absorption coefficient and the scattering coefficient of the tissues with different components and by combining Monte Carlo simulation and the photoacoustic wave equation;

magnetic acoustic signal:

applying a static magnetic field and pulse magnetic excitation along the axial direction of the cavity, generating magnetoacoustic signals by the tissues due to the magnetoacoustic effect, and simulating to obtain the magnetoacoustic signals generated by the tissues according to the electric conductivity of the tissues with different components and in combination with a magnetoacoustic kinetic equation;

c. fusing ultrasonic echo signals, photoacoustic signals and magnetoacoustic signals

For angles theta in imaged tissue_i(i ═ 1,2, …, N), ultrasound echo signal P at position r_i ^(U)(r) photoacoustic signal P_i ^(P)(r) and a magnetoacoustic signal P_i ^(M)(r) fusing using the formula:

whereinAs a fusion signal of ultrasonic echo, photoacoustic and magnetoacoustic signals, i.e. joint imaging signal, W_1i、W_2i、W_3iWeighting factors for ultrasonic echo, photoacoustic and magnetoacoustic signals, respectively, and having W_1i+W_2i+W_3i＝1。

3. A bio-magneto-optical-acoustic combined endoscopic imaging method according to claim 2, wherein the ultrasonic echo, the opto-acoustic and the magneto-acoustic signals are fused to form a signalIs minimum, the weighting factor W of the ultrasonic echo, the photoacoustic and the magnetoacoustic signals_1i、W_2i、W_3iShould take the optimum valueThe optimal value is calculated by the formula:

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<mrow> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <msub> <mi>R</mi> <mrow> <mi>P</mi> <mi>P</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>R</mi> <mrow> <mi>P</mi> <mi>M</mi> </mrow> </msub> </mrow>

<mrow> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>3</mn> <mi>i</mi> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <msub> <mi>R</mi> <mrow> <mi>M</mi> <mi>M</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>R</mi> <mrow> <mi>M</mi> <mi>U</mi> </mrow> </msub> </mrow>

<mrow> <msub> <mi>R</mi> <mrow> <mi>U</mi> <mi>U</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>U</mi> <mi>U</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> <msubsup> <mi>P</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mi>P</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>R</mi> <mrow> <mi>U</mi> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>U</mi> <mi>P</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> <msubsup> <mi>P</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mi>P</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>P</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>R</mi> <mrow> <mi>P</mi> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>P</mi> <mi>P</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> <msubsup> <mi>P</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>P</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mi>P</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>P</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>R</mi> <mrow> <mi>P</mi> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>P</mi> <mi>M</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> <msubsup> <mi>P</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>P</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mi>P</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>M</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>R</mi> <mrow> <mi>M</mi> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>M</mi> <mi>M</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> <msubsup> <mi>P</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>M</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mi>P</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>M</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>R</mi> <mrow> <mi>M</mi> <mi>U</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>M</mi> <mi>U</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> <msubsup> <mi>P</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>M</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mi>P</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow>

where k is the number of measurements of the ultrasonic echo, photoacoustic and magnetoacoustic signals,the optimal weighting factors of the ultrasonic echo, the photoacoustic signal and the magnetoacoustic signal when the total mean square error is minimum,is P_i ^(U)(r) the variance of the (r),is P_i ^(P)(r) the variance of the (r),is P_i ^(M)Variance of (R), R_UUIs P_i ^(U)(R) autocorrelation coefficient, R_UPIs P_i ^(U)(r) and P_i ^(P)(R) cross-correlation coefficient, R_PPIs P_i ^(P)(R) autocorrelation coefficient, R_PMIs P_i ^(P)(r) and P_i ^(M)(R) cross-correlation coefficient, R_MMIs P_i ^(M)(R) autocorrelation coefficient, R_MUIs P_i ^(M)(r) and P_i ^(U)The cross-correlation coefficient of (r).