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This study aims to understand how the dynamic model of PA signals and the optomechanical properties of hydrogels vary with two molecular weights (MWs) and different concentrations of PVA. The study analyzed the peak-to-peak amplitude, arrival time, and speed of sound of the PA signals. The physical parameters and state-space models (SSM) were estimated from these signals to determine the damping ratio (absorption capacity), natural frequency (energy transfer through the sample), density, and modulus of elasticity. The optical absorption $\left({\mu }_a\right)$ and scattering $\left({\mu }_s\right)$ coefficients were determined using an integrating sphere. Apart from the SSM, the density and the modulus of elasticity were calculated using physical variables and a tensile device. Finally, the ImageJ program and MATLAB were used to calculate the porosity distribution of the microscopic images.

According to the PA theory, the propagation of acoustic pressure $p(\overrightarrow{r},t)$ at position $\overrightarrow{r}$ and time t is governed by the wave equation:

where $c$ is the speed of sound, $\beta$ is the coefficient of thermal expansion, $C_p$ is the specific heat capacity at constant pressure, and $H(\overrightarrow{r},t)$ is the thermal energy distributed within the heated region during radiation per unit time. By considering that thermal and stress confinement conditions were met [24] and following a point-source model [25], the generated PA wave at the surface ($\overrightarrow{r}=0)$ follows that:

where α is a propagation constant, $\rho$ is the density, $\xi$ is the bulk viscosity, $\eta$ is the shear viscosity, and $\Gamma$ is the Gruneisen parameter.

This research aims to provide researchers and clinicians with the tools for constructing hydrogels with varying densities and porosities by adjusting the concentration and molecular weight (MW) to mimic the macro and microenvironment of tissues and enable imaging development. A PVA-based photoacoustic hydrogel was developed using light absorption and speed of sound, optimizing concentration and molecular weight to obtain characteristic optical properties of tissue. The hydrogels were characterized by microscopic imaging methods, mechanical properties, and optical parameters to validate versatility in contrast with a dynamic model.

When the PVA concentration increases, the hydrogel diameter expands, and there is a noticeable change in the time-of-arrival sensitivity of PA signals, as seen in Figure 1 and Table 1. Moreover, this time is related to the speed of sound; therefore, it also experiences a slight variation when changing the concentration of PVA, see Table 1. The amplitude of the PA signal is directly related to absorption. Therefore, an increase in PVA concentration results in a higher amplitude of the PA signal, as observed in Table 1.

Our results are supported by studies showing consistent values for the speed of sound [7,28] and PA amplitude [29]. However, it is important to note that while other researchers changed freezing cycles to determine the speed of sound and PA amplitude, we varied the concentration of PVA.

The N4SID algorithm was applied to the measured signals to provide a dynamic model of the samples under study. According to the algorithm presented in Section 4.3, the model order is determined by the SVD of the block Hankel matrix. The N4SID algorithm retrieved a significant magnitude for the measured PA signals for two singular values [30]. The PA signals are fully described by two states of the SSM in Equations (4) and (5), which leads to a model-order $\widehat{n}=2$, for both MW. This makes sense since, according to the subspace identification, the magnitude of the singular values of the data monotonically decreases to zero after the second component. Moreover, this behavior only allows one to capture helpful information about the system and neglects the small singular value related to measurement artifacts and noise.

In order to evaluate the performance of the estimated SSM, the normalized root mean square error (NRMSE) was computed as follows:

where $y_{\mathrm{P}\mathrm{A}}\left(t\right)$ is the measured signal, $y\left(t\right)$ is the output of the SSM, and ${\overline{y}}_{\mathrm{P}\mathrm{A}}\left(t\right)$ its mean value. Figure 2 depicts the bar plots of the mean NRMSE for the two MWs to assess the similarity between the measured signal and the output of the SSM. In all cases, the estimation accuracy is higher than 95% (horizontal dashed line), thus indicating that a second-order SSM accurately approximates the PA signal for the PVA samples.

In the following, we will discuss the influence of PVA concentrations on the PA signals and their relationship with the SSM parameters.

The eigenvalues of the system matrix give the damping ratio. Physically, the damping ratio is dimensionless, defined as how much energy is consumed or how fast the pressure wave dissipates. Therefore, the damping represents the ability to absorb energy after the excited PVA sample. Interestingly, with the proposed SSM estimation, we can directly retrieve this mechanical feature for the measured samples using the PA effect. Figure 3a,c shows the relation between the damping ratio and PVA concentration for the MW1 and MW2, respectively. From there, one can see that $\zeta$ exhibits a linear relationship with the concentration of PVA, described by a linear model with above 97% accuracy. This trend could be explained by the fact that the damping ratio reflects the ability of the PVA sample to absorb energy from the pressure produced by the PA effect. Therefore, the PVA concentration could also be related to the elastic behavior of the samples.

Hydrogels’ optical stability depends on their absorption and scattering capacity of near-infrared light, which varies with concentration. Increasing PVA concentration improved absorption capacity for both MWs. MW1 has a lower ${\mu }_a$ and higher ${\mu }_s$ coefficient for 7% and 9% concentrations, while higher concentrations of 12%, 15%, and 20% have higher ${\mu }_a$ and lower ${\mu }_s$ coefficients. MW2 has similar ${\mu }_a$ and ${\mu }_s$ coefficients for all concentrations except for 7% and 9%, with higher ${\mu }_a$ and lower ${\mu }_s$ coefficients than MW1. These hydrogels mimic tissue optical properties and aid in diagnosis and treatment using optical techniques such as PA and integrating sphere.

This section presents the results of the porosity distribution (number of pores versus diameter) for the MW1 and MW2 samples. The diameter range of the pores identified in the samples for different concentrations of PVA was from 10 to 80 µm. The increase in the concentration of PVA leads to a decrease in the diameter of the pores and, consequently, the number of pores. High MW samples have greater cross-linking due to more polymer chains; therefore, the pore diameter decreases. The porosity distribution for the samples of the two MWs fits a LogNormal curve with a value R² = 0.998, as shown in Figure 4a,b. Each measurement is the average of four samples.

The histograms in Figure 4b,c display the distribution of porosity, categorized into two groups: small diameters ranging from 0–10 µm, 10–20 µm, and 20–30 µm, and larger diameters ranging from 30–40 µm, 40–50 µm, 50–60 µm, 60–70 µm, and 70–80 µm. The number of pores is higher for smaller diameters, notably within the 0–10 um range. As the concentration of PVA increases, the number and diameter of pores decrease. This trend is also observed in the MW2 samples, where increasing PVA concentration leads to fewer pores, as shown in Figure 4e,f. The differences in the measurements may be attributed to the composition of the various sample cuts.

When the samples have high concentrations and MW, their physical-mechanical properties improve and become more viscous. However, this also leads to a higher reduction in permeability. Our findings can assist in selecting the optimal parameters for PVA hydrogels in biopolymer applications.

Figure 5a shows that higher concentrations increase density due to decreased porosity, as previously discussed. The densities of MW2 are slightly higher than those of MW1. In both MWs, the densities of the 9% samples are slightly higher than those of 7%. However, the increase in density is more significant for the 12% and 15% samples compared to the 7% and 9% samples. The increase in density between 15% and 20% of MW1 is very smooth. The standard deviation of the density values is due to the porosity distribution of the four layers cut from the hydrogel wall for each concentration.

The concentration of PVA directly impacts the modulus of elasticity and density. As the concentration of PVA increases, the modulus of elasticity also increases. The modulus of elasticity increases significantly for higher concentrations of PVA, as seen in Figure 5b. It is important to note that measurement variations may occur due to cutting different sections from the hydrogel wall.

For comparison purposes, we investigated the relationship of the elasticity modulus and density with the damping ratio and natural frequency, respectively, as shown in Figure 6. From Figure 6a, one can see that the damping ratio exhibits a quasi-linear inverse relationship with the elasticity modulus as the concentration grows for both MWs. This situation indicates that the concentration of PVA is related to the samples’ elasticity; therefore, the damping ratio could be an effective parameter to characterize it using the PA effect. On the other hand, from Figure 6b, the natural frequency is inversely related to the sample density for different concentrations. In particular, using the natural frequency of the PVA samples, it is possible to infer the density of the samples, as suggested by the PA model in Equation (2).

This study found that the concentration and MW changes affected the polymeric network’s cross-linking and the hydrogels’ dimensions. For that reason, the arrival time shifted toward longer times with increasing PVA concentration, and the amplitude of the PA signal increased, whereas the latter was due to the superior PVA absorption. A similar behavior was observed for the MW2 samples but with an amplitude slightly more pronounced. Moreover, as the speed of sound is related to arrival time, it displayed variations with increasing PVA concentration. Additionally, a subspace identification (SI) algorithm allowed for the recovery of state-space models from the temporal and frequency profile of the PA signals containing the sample’s biophysical information. There was a similarity between the measured signal and the SSM signal using the NRMSE, which allowed for capturing valuable details about the system and depreciating small noise-related singular values. Based on the damping ratio, it was observed that an increased concentration of PVA resulted in higher energy absorption capacity of the hydrogels under the PA effect for both MWs. Concerning the natural frequency, it was observed that higher PVA concentration led to increased energy transfer in the hydrogels for both MW. Intriguingly, the damping ratio and natural frequency correlated with the modulus of elasticity and density. These properties were observed to increase with higher PVA concentration.

On the other hand, as the concentration of PVA increased, the ${\mu }_a$ coefficients also improved while the ${\mu }_s$ coefficient decreased slightly. Nevertheless, the MW2 samples exhibited minimal changes in their ${\mu }_a$ coefficients and optical ${\mu }_s$, remaining almost steady. These results provide further evidence for the conclusions drawn from the PA method. From the distribution results of porosity, density, and modulus of elasticity, it can be concluded that the physical-mechanical properties of the hydrogels were improved as the concentration of PVA and MW increased. This study aimed to enhance the characterization of hydrogels with varying PVA concentrations and MW through diverse methods. The goal is to create hydrogels as phantoms that can be applied explicitly in tissue simulation and imaging.

To obtain the hydrogels with different concentrations, two steps were followed:

Figure 7c shows a microscopic image (Nikon optical microscope, 40X objective, 10X eyepiece) of a 10 µm thick film that was obtained from the PVA hydrogel wall, and it was dehydrated for 24 h. Figure 1d shows the same film but hydrated with one drop of pure water, where it is observed how the pores are filled with this solution. The PA signals obtained in this study were from the empty hydrogel and with propagation medium, 12 mL of SS or sodium chloride (NaCl), 0.9% (Baxter, Jiutepec, Morelos, Mexico).

As PVA hydrogels have high water absorption capacity, the coupling with immersion transducers is excellent since these are used to being in contact with water.

The PA signal amplitude is related to the PVA concentration and arrival time with the hydrogel’s porosity distribution (number and diameter of pores) by varying the concentration.

The problem here is, given a set of PA signal measurements, to estimate the parameters of the SSM using subspaces of the matrices presented above. For this purpose, the SSM can be rewritten in matrix notation as the following system of linear equations:

where Equations (13a) and (13b) define past and future outputs as a linear combination of previous states and the extended observability matrix. On the other hand, Equation (13c) relates the future and past states given the system matrix $A$.

Now, the algorithm aims to estimate the subspaces of the matrix $\Gamma$ and the state sequence $X_f$, which can be obtained from data projections. Let ${\mathcal{O}}_i$ be an orthogonal projection defined by ${\mathcal{O}}_i={\Gamma }_i{X}_f,$ which can be computed by the projection of the future output onto the subspace of past data in the direction of future input as:

For this setup, we utilized an integrating sphere (ISP-50-8-R-GT Ocean Optics) with an internal diameter of 13.46 cm and a wall reflection coefficient of m = 0.98. This sphere played a vital role in determining the power that the detector received under different illumination conditions. Using mathematical equations based on integrating sphere theory, we estimated this power for each beam and then determined the total reflectance and transmittance of the sample.

To determine the directional-diffuse reflectance, we compared the power reflected in a directional-diffuse illumination setup (beam 2) to the power detected in a diffuse-diffuse illumination setup (beam 1). This reflectance value is represented as:

Table 4 presents the optical properties of the hydrogel samples as a function of the PVA concentration, as described by Equations (19)–(21). The optical properties under consideration include diffuse-directional reflectance (R_cd), diffuse-directional transmittance (T_cd), and coherence transmittance (T_c)—each sample with a width of 100 µm.

In the case of MW1 samples, it was noticed that the R_cd decreased as the concentration of PVA increased. This decrease was particularly significant at higher PVA concentrations, with the lowest value of 0.14 ± 0.01 observed at 15% PVA. However, at 20% PVA, the R_cd increased again to a value of 0.34 ± 0.01. These findings suggest that the concentration of PVA plays a crucial role in determining the reflectance behavior of the samples. Similarly, the T_cd showed a similar pattern, with a general decrease observed as the concentration of PVA increased. At 7% PVA, the T_cd was 0.47 ± 0.01, which gradually decreased to 0.41 ± 0.01 at 15% PVA. For the highest concentration of 20% PVA, the T_cd rose again to 0.44 ± 0.01. These results indicate that the PVA concentration affects the samples’ transmission properties, with a decrease in transmission observed at higher concentrations. Furthermore, the T_c component of these samples is negligible.

The data analysis of MW2 samples shows similar trends to those observed in MW1. When the concentration of PVA increases from 7% to 15%, the R_cd reaches its lowest point at 12% and then slightly increases at 15%. This implies that there is a non-linear relationship between Rcd and PVA concentration. The T_cd remains stable at 0.34 ± 0.09 as the PVA concentration rises from 7% to 12%, indicating that changes in PVA concentration do not significantly impact this range. Nevertheless, at 15%, T_cd decreases to 0.29 ± 0.16, suggesting that higher PVA concentrations could significantly impact this parameter. Finally, Tc generally increases as PVA concentration increases, with a slight decrease at 12% followed by a rise at 15%.

The inverse Monte Carlo method [35] was then employed to calculate the hydrogels’ $\left({\mu }_a\right)$ and $\left({\mu }_s\right)$ coefficients. This method compared the measured values of reflection (R_cd) and transmission (T_cd and T_c) coefficients with the ones generated by proposing a set of optical properties of the medium. The optimization was guided by the downhill simplex algorithm, emphasizing the importance of suitable initial parameter values for convergence.

This study analyzes the distribution of porosity in nine hydrogels (MW1: 7%, 9%, 12%, 15%, 20%, and MW2: 7%, 9%, 12%, 15%). The results section displays histograms for all hydrogels analyzed.

Four samples measuring 1 cm × 1 cm × 0.2 cm were cut from one of the PVA hydrogel walls to determine the density and elasticity modulus (Young’s modulus). The density was measured using the equation $\rho =m/V$ according to Archimede’s principle and the elasticity modulus of Young’s modulus $(E=\sigma /\epsilon )$ was determined using a traction device developed by the Research and Technological Development Unit of the General Hospital of Mexico “Dr. Eduardo Liceaga”.

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