Effects of optical attenuation, heat diffusion, and acoustic coherence in photoacoustic signals produced by nanoparticles

In recent years, pulsed laser-induced ultrasound (US), better known as the photoacoustic (PA) effect, has had a major resurgence because its wide range of applications, mainly in the biological and medical areas,¹ for instance, PA imaging² and as monitoring method in thermo-therapy of cancer.³ Further, the analogies between optical and acoustic phenomena led to advancements in confocal PA microscopy,⁴ creations of new methodologies to detect US,⁵ and generation of new materials to achieve thermal and/or acoustic contrasts.⁶ The PA effect is produced by the absorption of CW modulated pulsed optical radiation by a medium. This absorption raises non-radiative decays that increase the temperature and causes mechanical waves typically in the range of US. The major advantages of PA techniques are their sensibility to distinguish different optical contrast and the US penetration in the tissue.⁷ Nowadays, metallic nanoparticles (NPs) play an important role as enhancers of the PA signal, and the design and application of these materials are subject to the type of applications desired, as well as to the available laser source.^1,8,9 Previous reports have shown that the PA amplitude is not always proportional to materials concentration; for example, the first reports of PA saturation are described in the work of Lin¹⁰ and Wang;¹¹ later, the authors employed a continuous absorption model to explain the saturation in red ink, but such model depends on multiple parameters (without physical meaning) adjustment to correctly fit the experimental data. Notwithstanding this deficiency, this reference has been the explanation for the PA saturation obtained in dyes, pigments, and tissue. On the other hand, despite the different physical properties exhibited by NPs, a similar saturation effect is observed. In several cases, a linear dependence between the amplitude and concentration has been obtained; for example, gold (Au) nano-vesicles,³ Au nano-rods,⁹ Au bio-conjugated nano-spheres,^12,13 silica-coated Au nano-rods,¹⁴ and Au nano-carbon-tubes and Fe₂O₃ nano-spheres;¹⁵ while, in many other examples, a saturated behavior is observed: Au nano-spheres,¹⁶ Au nano-cages,¹⁷ and Au nano-beacons.¹⁸ The origin of this discrepancy has been never analyzed in detail, and there are few reports where the PA response from a discretized system in microscale was obtained.^19–21 Also, the PA signal shape at low concentrations has been observed to be symmetric whereas at high concentrations to be asymmetric; this behavior has been extensively reported in the literature.^{4,13,19,21–28} Therefore, the goal of this study is to explain the causes of these concentration-dependent effects. Herein, it is proposed that the key characteristics of the nonlinear saturated behavior in the peak-to-peak (P-P) PA amplitude and its asymmetry at high concentrations are explained by means of a photo-thermo-acoustic model.²⁹ Sigrist²² previously reported a model that explained the PA signal produced by a continuous media (liquids); in this paper, the model is extended to a discrete case from continuous media to NPs. The NPs are represented by single point that absorbs the incident radiation instantaneously.³⁰ With this model, a simulation code was developed, taking into account the heat source size, the light attenuated by the NPs, and the coherence of a single PA signal source.^29,31,32 The light losses, by absorption and scattering, can be described by the well-known Beer-Lambert (BL) law, which corresponds to an exponential decay in the amount of light. We hypothesize that the asymmetric shape of the PA signals is due to the interference of the individual PA signals generated by the NPs.³² To verify this, it is theoretically estimated that the spatial region is where interference of two PA waves occurs. These results are of great importance in applications where the NPs are used as a PA enhancer, since they allow to identify a dynamic range for PA amplitude generation and an optimal contrast agent concentration at which the maximum PA signal contrast is possible.

For this, PA experiments were conducted with 5, 10, and 100 nm Au NPs (spheres, nanoComposix) in 2 mM sodium citrate dihydrate aqueous solution (detailed information about the samples characteristics can be found in the supplementary material). The suspensions optical density (OD)^33,34 were measured as a function of the NPs concentration for each aliquot by using an UV-VIS spectrometer (Lambda 900 UV/VIS/NIR, Pekin Elmer). The experimental setup for the detection of PA signals is shown in Fig. 1. The second harmonic from a Nd:YAG pulsed laser (Brilliant, Quantel) was employed to provide 532 nm light with a pulse duration of 10 ns and a repetition rate of 10 Hz. This beam was focused into the PA cell using a couple of lenses such that a large Rayleigh waist length was obtained to be approximately constant inside the cell (0.7 mm of diameter). The energy per pulse was set at 1 mJ (±5%, SD). The laser beam was set perpendicular to the transducer 2 mm away. The laser energy was monitored using a thermopile (1917-R, Newport) coupled to a power meter (818P-030–19, Newport). PA signals were detected with a homemade transducer (20 MHz central response and a bandwidth of 20 MHz)³⁵ and displayed by a 200-MHz oscilloscope (TDS5104B, Tektronix, Wilsonville, OR) triggered by a photo-diode (DET10A; Thorlabs, Newton, NJ) with a 1-ns rise time. The signals were amplified with a gain of 25 dB via 500 MHz amplifier (ZFL-500LN-BNC+, Mini-Circuits). NP suspensions were diluted from stock concentration (100%) in steps of 10% using a sodium citrate aqueous solution at 2 μM (6028, Karal).

Figure 2(a) shows the PA signals generated by the NPs suspensions at stock concentrations: 4 × 10¹³ for 5 nm, 5 × 10¹² for 10 nm, and 5 × 10⁹ for 100 nm. The signals generated by the NPs are observed at 1.3 μs. For lower NPs concentrations, the PA signals exhibited similar shape, but smaller amplitudes. Figures 2(b)–2(d) show the respective normalized P-P PA amplitudes as a function of the NPs concentration for each NP size.

Here, it is assumed that the measured pressure p(R, t) in r generated by a single nanoparticle (NP) at $r′$ is described by^22,30,36,37

with $p0=4E0βc2/π3/2Cpd03$ and $R=r−r′$⁠. Here, E₀β, c, and C_p are the energy per pulse, the thermal expansion coefficient, the sound speed medium propagation, and the heat capacity at constant pressure of the fluid sample, respectively. In Sigrist paper, d₀ was defined as the spatial illumination profile of a Gaussian beam.²² However, we associate this parameter with the thermal size of the object. This hypothesis is justified from the assumption that a NP only can absorb radiation, due to plasmonic effect;⁸ meanwhile, the surrounding fluid medium (water) does not. According to this, a minimum value for d₀ is the NP diameter (d_NP); and as maximum the quantity (d_NP + d_th), where

Equation (2) is related to the thermal diffusion length,³⁸ where χ_w is the water diffusivity (0.143 × 10⁻⁶m²/s) and τ_l is the laser pulse (FWHM of 10 ns); therefore, d_th = 150 nm. This analytical approximation is equivalent to solve the coupling heat and pressure equations considering the laser time profile.

A code was written in the software Wolfram Mathematica™ to emulate the experiments performed. Considering Eq. (1) and the optical attenuation, the numerical expression employed was

with D_ij = s – r_ij. Here, ϵ corresponds to the extinction coefficient and ζ is the sample concentration, which were measured for each NPs aliquot by UV-VIS spectroscopy. Likewise, s corresponds to the sensor position. Counters i and j are used to label each NP and the cylinder section, respectively, the length of the sections is Δz. The complete methodology details used to perform the simulations can be found in the supplementary material.

In Figs. 3(a)–3(c), simulated PA signals for stock concentration and d₀ = d_NP are presented. A statistical study shows that all signals are symmetric with well-defined maximum and minimum peaks, which always appear near the central temporal range. To obtain the P-P PA amplitude, it was sought the higher and lower peaks values for each individual simulation. The same behavior was observed when the BL law is considered; nonetheless, the maximum amplitude is diminished, as expected, being 60% less when compared to the case without attenuation.

In Figs. 3(d)–3(f), a comparison between the experimental and simulated P-P PA amplitudes as a function of the NPs concentration is shown; all data were normalized to the respective maximum amplitude. Simulations for d₀ = d_NP without BL law predict completely the experimental trend, but when the optical attenuation is considered, it fails; contrary to the expected results, the inclusion of BL law in the model did not predict the experiments. Trying to understand this discrepancy, the numerical Fourier transform can be performed to the simulated signals, and when doing this, broad spectra are predicted with high central frequencies for d₀ = d_NP. The frequencies are at 100 GHz for the 5 nm and 10 nm samples and at 10 GHz for 100 nm; however, if this is not in agreement with the actual spectral response of the sensors that we used in our experiments, then the d₀ = d_NP assumption must be discarded.

In Figs. 4(a)–4(c), simulations supposing d₀ = d_NP + d_th are shown for the stock concentration of NPs. There are two aspects that must be highlighted: First, amplitude-shapes for 5 and 10 nm samples are well-defined for all simulated concentrations; however, for 100 nm NPs, it remains noisy, but its shape is more defined than for the case d₀ = d_NP. Second, opposite to the above case, PA signals are asymmetric for all samples, like the experimental results and the reported literature. Consider that BL law still decreases the PA amplitude approximately at 40% of the non-attenuated value. Comparison between experimental and simulated P-P PA amplitudes as a function of the NPs concentration is presented in Figs. 4(d)–4(f). The simulated PA signals without BL law are linear. When light attenuation is considered, the behavior of experimental signals is well predicted; as in the previous cases, the spectra for the simulated signals can be calculated numerically, and the obtained frequencies now are between 1 and 100 MHz. For both sets of simulations, optical attenuation, through BL law, defines the amplitude of pressure wave and its effect is only to decrease the total PA amplitude.

To better understand the consequences for choosing a d₀ value, three aspects must be considered. First, from the PA power spectrum of a single NP, it is found that the maximum frequency value occurs at $νmax=c/2π(d0/2)$⁠. From this value, the spatial region where pulses of one NP interact with each other can be calculated, and it corresponds to $λ≡2πd0$⁠. Second, from the specific volume of the NPs suspension, an average distance between NPs L can be determined. Third, Eq. (1) is proportional to the time derivative of a Gaussian function, which has a bipolar temporal profile, i.e., it is composed of a compression and rarefaction cycle. When summation over two individual PA signals is performed at the measuring point, there are three extreme possible situations for the time delay (or acoustical path difference Δl), namely: (i) if it is equal to zero, then the PA pulses match exactly and only constructive interference appears; (ii) if it is equal to λ/2c, then the rarefaction of one pulse corresponds exactly to the compression of the other pulse, and therefore partial destructive interference is produced; (iii) if it is greater than λ/c, they cannot superimpose. These cases are displayed in Figs. 5(a)–5(c). Using the above information, the ratios L/λ were calculated for all d₀ values and are shown in Fig. 5(d). When d₀ = d_NP, L/λ ≫ 1 for all NPs diameters and all NPs concentrations, a high number of NPs cannot be superimposed; therefore, the sum of the individual signals, at the measurement point during a time interval, looks noisy and symmetric. For d₀ = d_NP + d_th, we can see in Fig. 5(d) that L/λ < 2 for d_NP = 5 nm and d_NP = 10 nm, respectively; now summation over individual signals produces well defined shape and asymmetric PA signals with linear behavior of the P-P PA amplitudes as a function of the NP's number. This is because at the measurement point, in the time interval, the superposition of individual pressure waves occurs. For d₀ = 100 nm + d_th, if the corresponding ratio is in the range of 5 < L/λ < 12.5, then the interference is more probable than the case d₀ = 100 nm, but less when d₀ = d_NP + d_th for 5 nm and 10 nm. The consequence for adding individual PA pulses, considering the heat diffusion from the NP volume to their surroundings, gives a PA signal with high signal-to-noise ratio, asymmetric shape peaks, and a linear behavior in the P-P PA amplitude as a function of the NP's number. When the optical attenuation takes into account the simulated PA amplitudes loss, its linear dependence with the NPs concentration and the signal saturates, predicting properly the experimental results.

It is important to remark that, if the results are displayed as a function of the NPs OD instead of their concentrations, the nonlinear behavior of the P-P PA amplitudes is given for OD ≥ 0.5. A meticulous review of Refs. 3, 9, and 11–21 is in concordance with this threshold (see the supplementary material).

In summary, when heat propagates beyond the individual NP's volume and the optical attenuation of the sample is ignored, the P-P PA amplitude as a function of NP's concentration is linear; this extended PA source can improve the interference between the single US pulses. The simulations showed that the asymmetric shape of the PA signal is obtained under this condition. When the heat is confined inside the NP dimensions, symmetric signals and a nonlinear P-P PA amplitude are obtained; the NP thermal confinement can also be discarded through the frequency spectrum. The saturated behavior of P-P PA amplitude for the extended thermal source is correctly explained when the optical attenuation is considered. Finally, our simulations and experimental results showed that no linear behavior appears for an OD ≥ 0.5. This threshold was well-matched with previous experimental reports. This value can be taken as a point of departure to obtain linear PA amplitudes as a function of the concentration for NP samples.

See supplementary material for the complete NPs characterization, the simulation methodology, and for a complete review of the references.

The computation for this work was performed on the high-performance computing infrastructure provided by Research Computing Support Services and in part by the National Science Foundation under Grant No. CNS-1429294 at the University of Missouri. The experimental part of this work was financed with the 2nd edition of UG-CIO research grants and CONACyT (Grant No. 5215708). The authors thank to Martin Olmos and Enrique Noe Arias for their support in this research and CONACyT.