Morphology-dependent resonance enhanced nonlinear photoacoustic effect in nanoparticle suspension: a temporal-spatial model

Abstract

The morphology-dependent resonances (MDRs) hotspot, ubiquity formed between the pairs of nanoparticles in close vicinity, has garnered considerable recent attention. By extending this phenomenon to pulse-laser irradiated nanoparticle suspension, we demonstrate that such collective optical/thermal enhancement can give rise to the nonlinear photoacoustic (PA) generation. In this study, a temporal-spatial analytical expression is derived to quantitatively describe the nonlinear PA signal generation from nanoparticles, incorporating the Grüneisen increase at the microscopic individual particle level and MRDs enhancement at the macroscopic suspension level. The dependence of PA nonlinearity on the critical contributors, including the laser pulse width, the particle size, and the statistical interparticle spacing, is quantitatively discussed. The theory is well validated with the finite element method (FEM) and experimentally proved with semiconducting polymer nanoparticles (SPN) suspension. This work may pave a new direction towards effective MDR based nonlinear PA contract agent design.

1. Introduction

Photoacoustic (PA) technique has become the focus of intensive research in noninvasive biomedical imaging during the last two decades [1,2]. In most optical imaging applications, the strong optical diffusion coefficient in biological tissues imposes severe challenges on deep penetration imaging. PA technique breaks the optical limit by effectively harnessing both high ultrasonic penetration and excellent optical specificity; it successfully demonstrates optical contrast imaging at several centimeters of biological imaging depth [3–5]. The basics of the PA technique is the conversion of absorbed photons into ultrasound via the transient thermoelastic expansion, where the contrast mainly results from the variation in the absorber's optical absorption properties.

Among major endogenous biological absorbers, hemoglobin has a superior optical absorption in the visible (400-700 nm) and the first near-infrared (NIR) spectrum (NIR-I,700–1000 nm) [6]. Therefore, the PA modality can utilize visible and NIR-I light to detect biodistribution of hemoglobin-rich regions such as regions with tumors in the advanced stage [7,8]. While for the detection of the early-stage tumor, where the quantity of hemoglobin is insufficient to serve as the efficient contrast agents, PA imaging can utilize tailored exogenous contrast agents that conjugate selectively with tumor [9–22]. By endowing the nanomaterials with NIR-II photothermal conversion ability, recent studies have reported exogenous organic and inorganic agents in the NIR-II region (1000–1700 nm) [19,23–29] for further improvement of imaging performance in terms of imaging depth and signal to noise ratio (SNR). These exogenous agents are designed to enhance the contrast or obtain complementary information at malignant cells without affecting the surrounding normal cells.

However, it remains challenging to discriminate between the overwhelming background absorption by endogenous tissue chromophores and the contrast agents. The PA nonlinearity, describing the phenomena where the PA signal amplitude nonlinearly increases as a function of laser fluence, can be utilized to counter this issue [30]; it provides an approach for selective detection of the contrast agents from a linear absorbing background. A variety of nonlinear mechanisms have been investigated, including optical saturation [31,32], temperature dependence of the Grüneisen parameter [33–39], photobleaching [40], or nano/micro-bubble generations [41,42]. Among them, laser-induced bubbles may involve potential hazards to tissue and hinder its applications for in vivo imaging.

The temperature-dependent Grüneisen parameter is at the core of this work as the thermoelastic-correlated mechanism will be our primary focus. The mechanism behind this has been explored analytically and numerically in recent studies [33,35,43–46]. [44] provides a computational approach with an isotropic model comprising a single gold nanosphere, where a homogeneous temperature value is assigned over the entire nanosphere to compute the nonlinear PA signal amplitude. In this model, only the temperature variation induced thermoelastic stress in the single nanosphere and the surrounding liquid environment account for the PA generation; it means that only the individual particle contribution is considered for nonlinear PA generation while the interparticle interaction is totally neglected. However, the previous experimental studies reported the nonlinear phenomena within aggregated plasmonic metal nanoparticles, presumably caused by interparticle localized surface plasmon resonances (LSPR) among clustered metallic cells [10,34,47–49]. Both the experimental results and simulations proved that the collective interparticle interaction also contributes to the PA generation [43]. Unlike the LSPR, which is only typically reserved for metallic materials clusters, the so-called morphology-dependent resonances (MDR) effect can generalize the interpretation of the collective mechanism beyond the metallic nanomaterials [50–54]; In this context, the MDR is a result of various diameters of nano-dimers in contact, and it demonstrates that the resonant interactions of light with subwavelength-scale objects can also result in an intensive hotspot at the junction of the absorbing dimer, significantly enhancing the local temperature, regardless whether the objects are conductive or dielectric and absorptive or transparent as shown in Fig. 1. Although [33] showcases an analytical solution for the generation of PA wave from a point-absorber, assuming that the stress and thermal confinements are satisfied within the whole particle suspension; in fact, the validity of the two conditions may no longer hold for the individual particle within the nanosecond scale [36,44]. Hence the previously reported models [33,44] assuming uniform temperature within the single nanoparticle may not be adequately extended to the system of nanoparticle suspension. Consequently, the laser heating profile within the suspension is paramount in the MDR-induced nonlinear PA modelling, since the amount of change in the thermal expansion coefficient depends on the spatial heating profile. Nevertheless, the compact quantitative temporal-spatial modelling of MDR-induced nonlinear PA signal generated from the nanoparticle suspension lacks by far.

Fig. 1.

The MDR enhancement of PA nonlinearity exists among nanoparticles in close proximity.

The motivation of the theoretical work is to construct a general and comprehensive temporal-spatial analytical model, growing from individual particle level to whole suspension clusters, to make quantitative predictions regarding the thermoelastic-originated nonlinear PA generation, incorporating the effects of several physical parameters, including the thermal confinement, nanoparticle concentration, and the size etc., thus piloting the rational design of high-performance nonlinear contrast agents. Furthermore, the Finite element method (FEM) simulation with COMSOL Multiphysics was adopted to simulate the complete process and investigate the effectiveness of MRDs enhancement to the nonlinear PA generation. Finally, the experiments were performed with tailored semiconducting polymer nanoparticles (SPNs) of various sizes and concentrations to validate the feasibility of our proposed model.

2. Theories and methods

In part 1, we first consider the PA generation from the individual system comprising a single nanoparticle and its surrounding PA generating layer. Then, the analytical solution for the nonlinear PA of this system will be derived incorporating the time-domain Grüneisen saturation and thermal diffusion effect. Finally, in part 2, FEM simulation with COMSOL Multiphysics will be used to investigate the nonlinear contribution in the spatial domain from the interparticle MDR enhancement effect and extend the solution from a single nanoparticle system to the nanoparticle suspension at different concentrations.

2.1. PA power-dependent nonlinear PA generation in single nanoparticle system–time-domain study

[44] reports that the PA nonlinearity originates from the nanoparticle together with its surrounding medium. Here we consider the overall nonlinear PA generated from this system consisting of the single nanoparticle and effective PA generating layer as shown in Fig. 2(a).

Fig. 2.

the unmet stress & thermal confinement leads to the two-pulse nonlinear PA generation. (a) The single nanoparticle system consisted of one nanosphere and the thin water layer, which partially contributes to the PA generation; (b) Thermal confinement for water sphere of different sizes; (c) Second nonlinear pulse generated from long pulse illumination; (d) The nonlinear PA amplitude increases with nanoparticle diameter.

In conventional short pulse induced linear PA imaging, under the stress and thermal confinements where the temporal pulse width $\mathrm{\Delta }t$ is much shorter than the stress and thermal confinements constant ${\tau }_{th}$ , the analytical expression for PA initial pressure is:

$$p_0={\mathrm{\Gamma }}_0{\eta }_{th}{\mu }_a\Phi \mathrm{\Delta }t,$$ (1)

where ${\mathrm{\Gamma }}_0$ is the effective ambient Grüneisen parameter, ${\eta }_{th}$ is the effective conversion efficiency from thermal energy to acoustic power, ${\mu }_a$ is the effective optical absorption coefficient of the system and $\Phi$ is the light irradiation with the unit $(W/c{m}^2)$ .

However, the stress and thermal confinements both fail at the individual particle level within the nanosecond timeframe.

When the laser pulse illuminates the single nanoparticle system, the nanoparticle absorbs the energy and quickly heats the surrounding medium, i.e., water in this scenario; they together generate the PA signal. According to [44], the PA signal is predominately generated from the surrounding medium, with particle diameter-scale effective thickness, rather than just from the particle, given that the laser pulse width is sufficiently long. The thickness of the water layer accounting for PA generation typically has the same order of magnitude as the nanoparticle radius. Here is an estimation of the magnitude thermal confinement of the water layer. Taking the thermal diffusivity of water at ambient temperature and pressure to be $0.146\times {10}^{-6}{m}^2/s$ and the effective water layer diameters $d_{water}$ in the typical range of tens to hundreds of nanometer, the thermal confinement of the water layer hence can be calculated to be ${\tau }_{th}=\frac{d_{water}^2}{{\alpha }_{water}}\approx {10}^{-9}s-{10}^{-8}s$ as in Fig. 2(b), which is comparable to the nanosecond scale laser pulse width (∼ ns for the conventional pulse laser). Hence (1) does not apply to the current setting.

Unlike the conventional setup, as the incident laser pulse width exceeds the system stress relaxation time, a dual-edged PA signal will exhibit right after the rising and falling edges of the laser pulse, as depicted in Fig. 2(c) [55]. Since they are indistinguishable in waveform with the limited bandwidth ultrasound transducer (typically several to a few tens of $MHz$ ) at the nanosecond time scale, the nonlinear PA amplitude collected from the single nanoparticle system can be simply represented with the summation of the positive and negative pulses; consequently, the total PA amplitude can be reorganized as follows by taking account of the Grüneisen increase due to heat accumulation and releasing process (Supplementary Note A.):

$$p_{nonlinear,single}=b{\eta }_{th}^2{\mu }_a^2\mathrm{\partial }t\cdot T({\tau }_{th})\cdot {\Phi }^2+\left(2\mathrm{\partial }t+\mathrm{\Delta }t\cdot e^{-\frac{\mathrm{\Delta }t}{{\tau }_{th}}}\right){\mathrm{\Gamma }}_0{\eta }_{th}{\mu }_a\cdot \Phi ,$$ (2)

where b is the coefficient indicating the conversion of the absorbed heat to the change in Grüneisen coefficient, the short transient time (rising and falling) in the laser profile is denoted as $\mathrm{\partial }t$ , $\mathrm{\Delta }t$ is the laser pulse-width and $T({\tau }_{th})$ is defined in this article as the thermal confinement factor, which can be expressed as:

$$T({\tau }_{th})=\left[1-\left(1+\frac{\mathrm{\Delta }t}{{\tau }_{th}}\right)e^{-\frac{\mathrm{\Delta }t}{{\tau }_{th}}}\right]\cdot {\tau }_{th}^2.$$ (3)

To further explore the relationship PA amplitude with nanoparticle diameter $d_{nano}$ , (2) can also be expressed as:

$$\begin{array}{c}{p}_{nonlinear,single}(d_{nano})\\ ={\displaystyle \frac{b{\eta }_{th}^2{\mu }_a^2\mathrm{\partial }t{\mathrm{\Phi }}^2}{{\alpha }_{system}^2}\cdot \left[1-\left(1+{\displaystyle \frac{{\alpha }_{system}\mathrm{\Delta }t}{d_{nano}^2}}\right)e^{-{\displaystyle \frac{{\alpha }_{system}\mathrm{\Delta }t}{d_{nano}^2}}}\right]\cdot d_{nano}^4}\\ +{\mathrm{\Gamma }}_0{\eta }_{th}{\mu }_a\mathrm{\Phi }\cdot \left(2\mathrm{\partial }t+\mathrm{\Delta }t\cdot e^{-{\displaystyle \frac{{\alpha }_{system}\mathrm{\Delta }t}{d_{nano}^2}}}\right),\end{array}$$ (4)

where ${\alpha }_{system}$ is the system's effective thermal diffusivity. When the PA nonlinearity is significant, the first term in (4) dominates the equation, and such that the PA amplitude would be nonlinearly increased with $d_{nano}$ (Fig. 2(d)). (2) indicates the quadratic dependence of PA signal upon the light irradiation $\Phi$ . Since $\mathrm{\partial }t$ is generally much shorter than $\mathrm{\Delta }t$ , the linear relationship between the laser fluence F and the irradiation $\Phi$ can be approximated as:

$$F=\Phi \cdot \mathrm{\Delta }t.$$ (5)

Given a fixed $\mathrm{\Delta }t$ , we can also conclude the quadratic dependence of the PA amplitude on the laser fluence, which is consistent with the past studies [30,44,45]. It also predicts that the $2^{nd}$ order nonlinear PA coefficient would further increase with the thermal confinement factor $T({\tau }_{th})$ ; as the ${\tau }_{th}$ is related to the dimension of the system, the effect of ${\tau }_{th}$ will be investigated by adjusting the particle size in the following section.

FEM analysis:

To validate the proposed theoretical framework, a finite element numerical study in COMSOL Multiphysics is carried out to determine the magnitude of the nonlinear PA signal stemmed from the photothermal expansion of the particle and the surrounding water; Moreover, the effect of illuminating pulse width on the heat accumulation as the source of PA nonlinearity is also investigated. The simulation is constructed upon the following coupled physical phenomena [56]: a. EM properties extracted from simulation of EM field interaction within the nanoparticle system; b. Heat transfer modelling to calculate transient temperature field and heat transfer; c. Structural mechanics analysis for thermal expansion, stress and strain distribution and followed by d. Acoustic propagation in the fluid.

A nanosecond plane electric wave of amplitude $E_0$ with a Gaussian temporal profile is introduced to irradiate the whole geometry in our model. The peak irradiance can be calculated according to $\Phi =\frac{n_{medium}{\epsilon }_0{c}_0}{2}|E_0^2|$ , where $n_{medium}$ is the index of refraction in the medium of propagation, ${\epsilon }_0$ is the vacuum permittivity, $c_0$ is the light speed in the vacuum, and $E_0$ is the electric field amplitude. The laser irradiation is provided at the wavelength of $750nm$ within the NIR-I window.

As shown in Fig. 3(a), the simulated geometry consists of one sphere (diameter $d=40nm$ ) representing the SPN nanoparticle floating in the center of the simulation box, which was immersed in water as a surrounding medium. Table 1 summarizes the physical properties of SPN material and the surrounding medium involved in this computation, the values for all these properties (assumed to be constant) were those taken at room temperature. As a particular case, the dependence of thermal expansion coefficient on temperature is depicted in Fig. 3(b),

Fig. 3.

The FEM results. (a) The temperature profile. The nanosphere and the surrounding water layer together heat up and give rise to PA generation; (b) The water thermal expansion coefficient is a function of temperature and contributes to the PA nonlinearity [58]; (c) The simulated nonlinear PA generated from the system vs irradiance. (d) The PA amplitudes plotted against the nanoparticle radius, in equivalent to increasing the system thermal relaxation time.

Table 1. The parameters for the nonlinear PA FEM simulation [57].

	$\mathbf{D}ensity$ $\rho (\mathbf{k}g/{\mathbf{m}}^{-3})$	$\mathbf{S}pecific\mathbf{h}eat$ ${\mathbf{c}}_{\mathbf{p}}(\mathbf{J}/(\mathbf{k}g\cdot \mathbf{K}))$	$\mathbf{T}hermal$ $\mathbf{c}onductivity$ $\kappa (\mathbf{W}/(\mathbf{m}\cdot \mathbf{K}))$	$1^{\mathbf{s}t}\mathbf{L}ame$ $\mathbf{c}oefficient$ $\lambda (\mathbf{N}/{\mathbf{m}}^2)$	$2^{\mathbf{n}d}\mathbf{L}ame$ $\mathbf{c}oefficient$ $\mu (\mathbf{N}/{\mathbf{m}}^2)$	$\mathbf{S}peed\mathbf{o}f$ $\mathbf{s}ound$ $\mathbf{c}(\mathbf{m}/\mathbf{s})$	$\mathbf{R}elative$ $\mathbf{p}ermittivity$ $\epsilon$
SPN	1000	2000	0.2	$147\times {10}^9$	$27.8\times {10}^9$	1000	3.13
Water	1000	4200	0.598	$2.25\times {10}^9$	0	1500	1.77

The equivalent incident irradiance is swept from $0mW/u{m}^2$ to $5mW/u{m}^2$ with a step size of $0.5mW/u{m}^2$ to investigate the PA nonlinearity. [30] suggests that increasing heat deposition in nanoparticle and diffusion to the surrounding medium will enhance ambient temperature elevation. The second-order coefficient in (2) represents the temperature-sensitive component of the Grüneisen factor, which is proportional to the local temperature change; it confirms that the PA nonlinearity is originated from the temperature dependency of thermodynamics properties, i.e., the absorbed energy will elevate the temperature and increase the value of the temperature-sensitive component in Grüneisen parameter, especially the thermal expansion efficient; for example, Fig. 3(b) shows the thermal expansion coefficient of water as a function of temperature. Figure 3(c) from the FEM result illustrates how the nonlinear PA amplitude grows with increasing light irradiance, which agrees with the proposed expression.

The complex expression of the thermal confinement factor (3) can be approximated to be a polynomial function, and it is positively correlated with the thermal relaxation time ${\tau }_{th}$ . To further validate the effect of ${\tau }_{th}$ on the nonlinear PA amplitude, the nanoparticle diameter d is tuned to $20nm$ , $40nm$ and $60nm$ , respectively. The PA amplitudes obtained from these systems are shown in Fig. 3(d). The FEM simulation confirms that by increasing the nanoparticle diameter, the nonlinear PA amplitude also increases due to enhanced ${\tau }_{th}$ ; the result is consistent with the analytical predictions in Fig. 2(d).

By observing (3), the thermal confinement factor can be enhanced from the following two aspects. Firstly, we can ensure that the laser pulse width is sufficiently wide compared with the ${\tau }_{th}$ such that the first multiplier of expression, $\left[1-\left(1+\frac{\mathrm{\Delta }t}{{\tau }_{th}}\right)e^{-\frac{\mathrm{\Delta }t}{{\tau }_{th}}}\right]$ can be boosted towards saturation. Secondly, with a bit of contradiction, we can improve the thermal confinement property within the single-particle system level to enhance the second multiplier ${\tau }_{th}^2$ . In fact, the thermal relaxation time ${\tau }_{th}$ , indicating how well the system can trap the heat inside, is reversely proportional to the system's effective thermal diffusivity ${\alpha }_{system}$ . Recall that the single-particle system defined in this work comprises the nanoparticle sphere and its surrounding thin PA generating layer; hence the effective ${\alpha }_{system}$ can be formulated as:

$${\alpha }_{system}=\beta \times {\alpha }_{nanoparticle}+(1-\beta )\times {\alpha }_{medium},$$ (6)

where ${\alpha }_{nanoparticle}$ and ${\alpha }_{medium}$ are the thermal diffusivity of nanoparticle and the surrounding medium respectively, and the $\beta$ is the proportional contribution factor indicating the percentage of PA contribution by the nanoparticle, which will increase with laser pulse width up to nearly $100\mathrm{\%}$ [44]. The overall effective diffusivity can be adjusted to achieve optimized time-domain nonlinear PA contrast agent design by altering the particle's or the surrounding medium's thermal property. Therefore, it is worthy of paying attention to the surrounding medium's thermal confinement (solvent, coating material, etc.) than focusing on the nanoparticle itself. As the nonlinearity is sensitive to the surrounding environment, the nonlinear performance of the nanoparticle may be easily affected by external factors. For instance, the identical nanoparticle would perform differently depending on whether it is immersed in the water or buried in the tissues. For in vivo application, when the nanoparticle travels from watery tissue like blood to oily environment like cholesterol, the ${\alpha }_{system}$ will reduce with the thermal diffusivity of the surrounding medium ${\alpha }_{medium}$ decreased from $0.146\times {10}^{-6}{m}^2/s$ to $0.074\times {10}^{-6}{m}^2/s$ ; consequently, the PA nonlinearity would be enhanced according to (4). To reduce environmental disturbance in biomedical application, the proper utilization of encapsulation can provide robust environmental stability in harsh physiological conditions.

2.2. PA power-dependent nonlinear signal generation in nanoparticle suspension–spatial domain study

This section will consider the spatial characteristic of whole nanoparticle suspension by investigating the effect of concentration on the PA nonlinearity. If each dispersed particle system described in the previous section is completely isolated without any interparticle reaction occurs, the overall PA generated from nanoparticle suspension should be the linear summation of all individual system; it can be simply expressed as:

$$\begin{array}{c}{p}_{nonlinearsuspension}(C_{nano})\propto p_{nonlinear,single}\cdot C_{nano}\\ =B_0{\eta }_{th}^2{\mu }_a^2\mathrm{\partial }tT({\tau }_{th}){\Phi }^2\cdot C_{nano}+B_1\left(2\mathrm{\partial }t+\mathrm{\Delta }t\ast e^{-\frac{\mathrm{\Delta }t}{{\tau }_{th}}}\right){\mathrm{\Gamma }}_0{\eta }_{th}{\mu }_a\Phi \cdot C_{nano},\end{array}$$ (7)

where the $C_{nano}$ is the nanoparticle concentration, $B_0$ and $B_1$ are constants.

Besides the time domain thermal confinement constraints due to heat deposition and diffusion, the spatial interparticle distance l distribution, related to the nanoparticle concentration C, also affects the Grüneisen parameter and influence the nonlinear second-order coefficient.

As the concentration increases, the mean interparticle distance l will decrease. According to [50], when the laser hits the subwavelength-scale nanodimer, an intense electric field near the point of contact is formed. Therefore, a resonance hotspot will be created at the nanodimer gap due to MDR, as shown in Fig. 4(a). The electric field enhancement therein is proved to be greatly dependent on the interparticle distance l. Although a collection of complicated electromagnetic field modes could coexist within the dielectric dimer, the radially symmetric internal mode dominates others when considering the full absorptive behavior of the dielectric dimer, which is associated with the imaginary part k from the index of refraction $\tilde{n}=n+ik$ , i.e., material extinction coefficient. Although other MDR modes will also affect the spatial heating profile, only the most dominant radially symmetric mode is analyzed for simplification purposes while preserving the modelling accuracy. In this way, the enhanced electric field localized to the proximal point of contact allows more heat to be generated and concentrated around the gap. Compared to the previously reported model [33,44], where the surrounding medium temperature elevation results from the isotropic heat conduction from the nanosphere, the distribution of highly localized and enhanced resonance hotspot can heat the surrounding medium around this region much more efficiently. The effect of localized heating from the surrounding medium on PA generation will be investigated quantitively. Here the MDR enhancement term $G(C)$ is introduced, as a function of concentration, to describe its overall impact on the Grüneisen parameter. We consider the following FEM analysis to simulate the MDR enhancement vs the interparticle distance $F(l)$ using COMSOL.

Fig. 4.

The nanoparticle dimer simulation result. (a) The resonance spot created from MDR will further enhance the PA nonlinearity. (b) The simulation setup consists of the nanosphere dimer and the surrounding medium (water). (c) The dissipated power density around the dimer of different sizes. The MDR enhancement shows a strong dependence on particle size and interparticle distance l. (d) For dimer of size d = 60 nm, the spatial nonlinearity enhancement indicator, the integrated square of energy, is plotted against the interparticle distance l. (e) Comparison of the nonlinear enhancement among the dimers of different size.

FEM analysis:

In this simulation, the involved coupled physics and parameter setting are identical to those in the previous section. The polarization of the incident plane wave excitation was along the long axis of the dimer floating in the simulation box. Second-order scattering conditions were used at the boundaries of the simulated region. A gap width of l nm was introduced between the surface of the spheres, as shown in Fig. 4(b).

Here the gap width is swept from $1nm$ to $9nm$ . Figure 4(c) illustrates that more power will be concentrated and dissipated in the center gap region as l reduces. It is already known that the PA nonlinearity is correlated to the square of the laser pulse energy, which is equivalent to the spatial integral of the square of dissipated power density $|P_{dis}{|}^2$ ; hence it is a good indicator for the $2^{nd}$ order coefficient, and it can be used to describe the MDR enhancement quantitatively. The enhancement term is integrated over the dotted region of interest (ROI), i.e., $\int \int {\int }_{ROI}{|P_{dis}|}^{2}dV$ , and plotted against l as shown in Fig. 4(d), the MDR enhancement vs interparticle distance can be empirically retrieved in the reciprocal form as:

$$F(l)=\frac{\int \int {\int }_{ROI}{|P_{dis}(l)|}^{2}dV}{\int \int {\int }_{ROI}{|P_{dis}(\mathrm{\infty })|}^{2}dV}=1+\frac{a}{l},$$ (8)

where the constant a indicates the enhancement sensitivity of the dimer.

Simulating the nonlinear enhancement for nanosphere dimer with different size, Fig. 4(e) also shows that the nonlinear enhancement will become more intensive with the diameter increase. For example, compared with 20 nm nanoparticles, 50 nm and 300 nm ones can achieve 3.5- and 76.5-times enhancement, respectively.

The statistical study is conducted to formulate the overall spatial enhancement factor, here assume that the mean value of the nearest 12 interparticle distance l follows the standard normal distribution $l\sim N(\mu ,{\sigma }^2)$ [59,60], where the mean interparticle distance l can be expressed as a function of concentration $C_{nano}$ [61] (more details in Supplementary Note B):

$$\mu =l_0=\frac{\gamma }{\sqrt[3]{C_{nano}}},$$ (9)

where $\gamma$ is a constant. The probability density function (pdf) of the interparticle distance l can be represented as:

$$f(l,l_0)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{{(l-l_0)}^2}{2{\sigma }^2}}.$$ (10)

Hence the expectation of MDR enhancement with mean interparticle distance $l_0$ can be calculated via integration:

$$E[F(l)]{|}_{u=l_0}=\int F(l)f(l,l_0)dl.$$ (11)

Moreover, the total number of dimer pairs can be calculated as:

$$D_{total}=C(N_{nano},2)=\frac{N_{nano}\cdot (N_{nano}-1)}{2},$$ (12)

where $N_{nano}$ is the number of nanoparticles, which is proportional to $C_{nano}$ , hence, (12) can be modified as:

$$D_{total}(C_{nano})={\kappa }_1\cdot C_{nano}\cdot (C_{nano}-{\kappa }_2),$$ (13)

where ${\kappa }_1$ and ${\kappa }_2$ are constants.

Finally, the MDR enhancement term $G(C)$ can be obtained by substituting (9) into (11) and then multiplying it with (13):

$$\begin{array}{c}G(C_{nano})={E\left[F\left(\frac{\gamma }{\sqrt[3]{C}}\right)\right]|}_{u=\frac{\gamma }{\sqrt[3]{C_{nano}}}}\cdot D_{total}(C_{nano})=\\ \int F\left(\frac{\gamma }{\sqrt[3]{C}}\right)f\left(\frac{\gamma }{\sqrt[3]{C}},\frac{\gamma }{\sqrt[3]{C_{nano}}}\right)d\left(\frac{\gamma }{\sqrt[3]{C}}\right)\cdot {\kappa }_1\cdot C_{nano}\cdot (C_{nano}-{\kappa }_2).\end{array}$$ (14)

Up to this point, (14) shows that the MDR enhancement term is nonlinearly dependent on the nanoparticle concentration. Therefore, incorporating the contribution from concentration-dependent MDR, the nonlinear PA equation should be modified as a function of the particle concentration and the laser pulse fluence:

$$\begin{array}{c}{p}_{nonlinear}(C_{nano},\Phi )=\\ B_0{\eta }_{th}^2{\mu }_a^2\mathrm{\partial }t\cdot T({\tau }_{th})\cdot G(C_{nano})C_{nano}\cdot {\Phi }^2+B_1\left(2\mathrm{\partial }t+\mathrm{\Delta }t\cdot e^{-\frac{\mathrm{\Delta }t}{{\tau }_{th}}}\right){\mathrm{\Gamma }}_0{\eta }_{th}{\mu }_a{C}_{nano}\cdot \Phi \end{array}$$ (15)

Up to this point, the analytical solution for nonlinear PA signal generating from nanoparticle suspension at different concentrations is derived, where the Grüneisen saturation and MRD enhancement terms are considered as the concrete contributors to the nonlinearity. The effect of various physical properties, including the laser pulse width, concentration, and particle size etc., were thoroughly discussed and simulated. This proposed analytical model can be extended to predict and describe nonlinear relations between the PA amplitude and the laser fluence exhibited from other types of nanoparticles.

3. Experimental section

3.1. Multi-spectra PA setup

The experimental setup is with a dark-field reflection mode PA microscopy system, as shown in Fig. 5. PA excitation is provided by a tunable laser (spectrum ranging from $680nm$ to $2600nm$ , Radiant 532LD, Opotek Ltd) with a $10ns$ pulse width $\mathrm{\Delta }t$ . A wavelength at $750nm$ within the NIR-I window is picked according to the absorption spectrum for maximum absorption. The collimated laser beam is expanded by a conical lens into ring-shaped then weakly focused by a homemade condenser on the sample with a spot size of $\sim 1mm$ . Meanwhile, a beam splitter (BSF10-B, Thorlabs) and photodiode (DET10A, Thorlabs) are used to record the laser intensity variation in the oscilloscope (WaveRunner 640Zi, LeCroy). A focus ultrasound transducer ( $NA=0.25$ , V303-SU, Olympus) with $20MHz$ central frequency is utilized to record the PA signals from the phantom placed near the focus. Both the ultrasound transducer and nanoparticle phantom are immersed in water for optimum optical and acoustic coupling. The water tank containing the phantom is fixed on the linear stages (LIMES 64, OWIS) for raster scanning at a range of $15mm$ with a step size of $0.05mm$ in both x and y directions. The transducer output is connected to a low-noise amplifier (5662, Olympus) with $40dB$ of amplification, then recorded by the oscilloscope with $500Msps$ sampling rates. The laser pulse energy is sweeping from 0.5 to 5 $mJ/pulse$ . C-scan over the entire regions of interest (ROI) is performed to obtain the raw data with $300\times 300\times 5000$ pixels and repeated ten times for each laser fluence level.

Fig. 5.

The dark-field reflection mode PA microscopy system.

3.2. Experiment results

The solutions of NIR SPNs with five different concentrations were prepared for the experiment and encapsulated in quartz tubes for PA imaging, as shown in Fig. 6(d). The TEM image in Fig. 6(a) reveals a spherical morphology for the SPNs (Supplementary Note C for more morphology characterization). As observed in the absorbance plot (Fig. 6(b)), the material has a NIR-I absorption peak at 750 nm, where the strong absorption results from a highly electron-delocalized backbone [27] and gives enhanced NIR PA contrast. The solutions of various SPN sizes, with DLS characterization shown in Fig. 6(c), are also prepared for investigating the size effect. The solutions with averaged SPN sizes of 20.29 nm, 52.44 nm and 306.78 nm are labeled as group A, B and C, respectively.

Fig. 6.

The characterization of the SPNs encapsulated in quartz tubes for PA imaging and the respective results. (a) Representative TEM image; (b) UV-Vis-NIR absorption spectra of the SPN; (c) Representative particle size distribution profiles of SPN of three size groups; (d) The photograph of the phantom and the PA image series obtained for different fluence levels; (e, f, g) Nonlinear PA amplitudes against the laser pulse energy of different SPN sizes and concentration; (h, i, j) The retrieved second-order coefficient vs SPN concentrations for group A, B and C.

To better visualize the data, the region containing the interference and ultrasound echoes in the raw data PA waveform is removed. Then the 2D image sequences are reconstructed by mapping the maximum envelope amplitudes (Fig. 6(d)) to the pixel values within the imaging window. Furthermore, the ROIs enclosed within the dotted boxes were defined, and the means and standard deviations of row maximums within every ROI were retrieved at each concentration in every graph in Fig. 6(d). After that, the PA signal's nonlinearity is plotted against laser fluences in Fig. 6(e), (f) & (g). It can be observed that with the identical absorbance (i.e., same SPN concentration), the PA signal is growing with the SPN size. The results are consistent with (15) and the FEM result in Fig. 3(d): the thermal confinement time ${\tau }_{th}$ is longer for the bigger SPN, therefore according to (3), the thermal confinement factor $T({\tau }_{th})$ is also enhanced to boost PA amplitude.

(15) shows that the quadratic coefficient can represent the PA nonlinearity of the target. To access the nonlinearity from the experiment data, the method provided by [30] was utilized to fit a second-order polynomial from the fluence-dependence of PA image intensity according to (16):

$$p(\Phi )=a_2{\Phi }^2+a_1\Phi ,$$ (16)

where $a_1$ and $a_2$ are the $1^{st}$ and $2^{nd}$ -order fitting coefficients, respectively.

The fitted values for the nonlinearity indicator $a_2$ are recorded and plotted against the nanoparticle concentrations for each group as shown in Fig. 6(h), (i) & (j).

The results show that for group C, the value of $a_2$ nonlinearly increases with the concentration. In contrast, they are more likely linearly proportional to the concentration for both groups A and B. These results are again consistent with our proposed model and FEM results. In (15), the analytical expression of second-order coefficient $a_2$ with respect to irradiance $\Phi$ is :

$$a_2(C_{SPN})\propto G(C_{SPN})C_{SPN}.$$ (17)

The MDR enhancement term $G(C_{SPN})$ is gradually decreasing towards one with the nanoparticle size for the following two reasons. Firstly, it can be explained with Fig. 4(e) that as the SPN diameter increase, the nonlinear enhancement will become more intensive. Secondly, from a statistical perspective, the mean inter-particle distance is smaller for nanoparticles with a larger diameter such that MDR is more likely to happen. However, increasing particle size will impede biomedical application, and a trade-off must be made.

The MDR enhancement $G(C_{SPN})$ can also be quantitatively assessed with another approach. Three groups of SPN with an identical concentration of $200ug/mL$ are irradiated with laser pulses with an energy level ranging from $0.5mJ$ to $4.5mJ$ and the corresponding PA amplitudes are recorded in Fig. 7(a-c) (orange color). Here we define the following term, Relative Standard Deviation (RSD), to quantify the MDR contribution to the PA nonlinearity:

$$RSD=\frac{s_{power}}{{\overline{x}}_{power}},$$ (18)

where $s_{power}$ is the PA sample standard deviation at a given power level and ${\overline{x}}_{power}$ is the PA sample mean per power level.

Fig. 7.

The PA vs Laser pulse energy and the RSD curve for group A, B and C, respectively. The SPN concentration (200 µg/mL) is identical for the three groups. (a) For group A, the RSD remains constant over the irradiation energy; (b) For group B, the RSD increases slowly over the irradiation energy; (c) For group C, the RSD increases quickly over the irradiation energy.

The RSD is the normalized standard deviation of the PA amplitude at a particular power level. The RSD plots for group A, B and C are presented in Fig. 7 in blue color. The RSD trend mainly consists of a DC component and an AC component. The DC component stands for the normalized experiment random error, and the MDR enhancement induces the AC component. The MDR enhancement from a single measurement is unstable and strongly depends on the distribution of SPNs: if the SPNs are well dispersed, the overall MDR is weak; however, it will be much improved if SPNs aggregate.

Consequently, the motion of the SPNs due to laser heating will contribute to the uncertainty in MDR enhancement. In Fig. 7(a), the AC component is lacking, which means that the MDR enhancement is insignificant in small SPNs. On the other hand, the RSD is slightly going up with the laser pulse energy for group B (Fig. 7(b)), and the increase becomes remarkable for group C with the largest SPN sizes. In short, the results demonstrate that the analytical expression agrees with the experiment results and well expounds on the functional relation between the second-order coefficient and nanoparticle concentration.

4. Conclusion

In conclusion, we report a comprehensive temporal-spatial analytical solution for the fluence dependent nonlinear PA signal generated from nanoparticle suspension in thermoelastic regime from macroscopic to microcosmic perspectives, especially on the particle thermal properties and morphology. This framework provides new insights into our understanding of the critical physical process involved in the nonlinear PA generation from different perspectives. Firstly, from the micro-level, the analytical solution illustrated in the time domain that the synergistic effect of thermal confinement and diffusion, as a function of thermal relaxation time in the single nanoparticle system attributes to the PA nonlinearity. Secondly, the equation calculates the overall spatial morphology-dependent nonlinear contributions from the interparticle MDR effect in the nanoparticle suspension with the statistical model derived on the macro level. Furthermore, the model's validity is verified with FEM simulations and experiments with tailor SPNs suspension. Instead of linearly summing up the MDR contribution from the individual pairs, further studies can be conducted to investigate the collective MDR enhancement in a more complex setting, involving a system of aggregation with more than two nanoparticles. Finally, the theory can be applied to advance nanoparticles’ design and practical application as a nonlinear PA nanoprobe.

Funding

Ministry of Education - Singapore10.13039/501100001459 (AcRF Tier 2: MOE2019-T2-2-179).

Disclosures

The authors declare that they have no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 ^{(1.3MB, pdf)} for supporting content.