# Bone Chemical Composition Assessment with Multi-Wavelength Photoacoustic Analysis
## 2 Materials and Methods
### Theory  
The amplitude of the PA signal generated by the bone after ignoring the reflected acoustic wave from the boundary can be expressed as [29; 30]:  
\[p(r,\lambda)=k\cdot\Gamma\cdot\eta_{th}\cdot\mu_{a}(\lambda)\cdot F_{0}\cdot e^{- \mu_{eff}(\lambda)\cdot\tau}e^{-\alpha\cdot(z_{0}-\tau)} \tag{1}\]  
where \(\Gamma\) is the Gruneisen parameter, which can be expressed as \(\Gamma=\frac{\beta\cdot c^{2}}{C_{p}}\), where \(\beta\) is the thermal coefficient of volume expansion, \(c\) is the SOS of tissue, and \(C_{p}\) is the heat capacity at constant pressure; \(k\) is a constant accounting for the impulse response of the detection system; \(\eta_{th}\) is the heat conversion efficiency; \(\mu_{a}\) is optical absorption of bone tissue; \(F_{0}\) is the light fluence generated by the laser; \(\mu_{eff}\) is the effective optical attenuation coefficient in the bone; \(e^{-\mu_{eff}(\lambda)\cdot\tau}\) is the light attenuation in the bone; \(\alpha\) is the ultrasound attenuation coefficient in the bone; \(e^{-\alpha\cdot(z_{0}-\tau)}\) is the ultrasound attenuation in the bone. Since both the ultrasound and optical attenuation are large in bone tissue [4; 31; 32], the PA signal received by the transducer is mostly generated from the bone surface (\(r=0\)). Then, the amplitude of the PA signal received by the transducer could be simplified as  
\[p(\lambda)=k\cdot\Gamma\cdot\eta_{th}\cdot\mu_{a}(\lambda)\cdot F_{0}\cdot e^{ -\alpha\cdot z_{0}}. \tag{2}\]  
In this equation,\(F_{0}\) can be calibrated by the recorded laser energy at different wavelengths, while \(k\), \(\Gamma\), \(\eta_{th}\), and \(\alpha\) are independent of the wavelength. Therefore, \(k\), \(\Gamma\), \(\eta_{th}\), and \(\alpha\) can be removed by dividing \(\mu_{a}(\lambda_{0})\) at the reference wavelength of \(\lambda_{0}\):  
\[\frac{p(\lambda)}{p(\lambda_{0})}=\frac{k\cdot\Gamma\cdot\eta_{th}\cdot\mu_{a} (\lambda)\cdot F_{0}\cdot e^{-\alpha\cdot z_{0}}}{k\cdot\Gamma\cdot\eta_{th} \cdot\mu_{a}(\lambda_{0})\cdot F_{0}\cdot e^{-\alpha\cdot z_{0}}} \tag{3}\]  
From this equation, we can get the \(\mu_{a}(\lambda)\) by using  
\[\frac{\mu_{a}(\lambda)}{\mu_{a}(\lambda_{0})}=\frac{p(\lambda)}{p(\lambda_{0})} \tag{4}\]  
Equation (4) indicates that, by performing multi-wavelength PA measurement of bone samples, we can measure the relative optical absorption spectrum in the trabecular bone. With the relative optical absorption spectrum of a bone measured, spectral unmixing was then be performed to derive the relative contribution of each chemical component to the optical absorption spectrum [33]. A least-squares method was used, which tries to fit the PA measured optical absorption spectrum of bone via a linear  
Figure 2: X-ray imaging results for three bone groups. (**a**) The X-ray image of an untreated bone from Group 1. (**b**) The X-ray image of a bone from Group 2 treated with EDTA for 13 hrs. (**c**) The X-ray image of a bone in Group 3 treated with EDTA for 26 h. (**d**) The grayscale value for the three bone groups; * stands for \(p<0.05\).  
combination of the optical spectra of all the relevant chemical components leading to the minimum sum of the squared residuals of the measurement points from the fitting line.