Optimization of ultrasound heating with Pickering droplets using core–shell scattering theory

Graphical abstract

Highlights

• •Ultrasound scattering theory was used to study agar phantoms doped with the Pickering droplets.
• •The theoretical ultrasound heating efficiency was affected by the physical properties of Pickering droplets.
• •The adjustment of the core radius and shell thickness of the droplets gave improvement to the modeled ultrasound heating.

Abstract

Nanoparticles find widespread application in various medical contexts, including targeted nanomedicine and enhancing therapeutic efficacy. Moreover, they are employed to stabilize emulsions, giving rise to stabilized droplets known as Pickering droplets. Among the various methods to improve anti-cancer treatment, ultrasound hyperthermia stands out as an efficient approach. This research proposes Pickering droplets as promising sonosensitizer candidates, to enhance the attenuation of ultrasound with simultaneous potential to act as drug carriers. The enhanced ultrasound energy dissipation could be, therefore, optimized by changing the parameters of Pickering droplets.

The ultrasound scattering theory, based on the core–shell model, was employed to calculate theoretical ultrasound properties such as attenuation and velocity. Additionally, computer simulations, based on a bioheat transfer model, were utilized to compute heat generation in agar-based phantoms of tissues under different ultrasound wave frequencies. Two types of phantoms were simulated: a pure agar phantom and an agar phantom incorporating spherical inclusions. The spherical inclusions, with a diameter of 10 mm, were doped with various sizes of Pickering droplets, considering their core radius and shell thickness.

Computer simulation of these spherical inclusions incorporated within agar phantom resulted in different enhancement of achieved temperature elevation, which depending on the core radius, shell thickness, and the material properties of the system. Notably, spherical inclusions doped with Pickering droplets stabilized by magnetite nanoparticles exhibited a higher temperature rise compared to droplets stabilized by silica nanoparticles. Moreover, nanodroplets with a core radius below 400 nm demonstrated better heating performance compared to microdroplets. Furthermore, Pickering droplets incorporated into agar phantom could allow obtaining a similar effect of local heating as sophisticated focused ultrasound devices.

1. Introduction

Hyperthermia is a treatment approach that refers to the procedure of raising tissue temperature to 40 °C – 45 °C for specific medical treatment purposes [1], [2]. The effectiveness of thermal enhancement in therapy depends on both the temperature and the duration of exposure within biological cells [3]. Subsequently, hyperthermia exhibits direct cytotoxic effects in microenvironments characterized by low blood supply, hypoxia, and low pH, such as those commonly found in tumor tissues [4]. Clinically, hyperthermia has been used for radiosensitization, enhancing the effectiveness of radiation therapy [5]. Over the past decades, numerous techniques have been employed to perform thermal therapies. Among others, a high-frequency alternative electromagnetic fields [2] as well as the magnetic nanoparticles under an alternating magnetic field [6], [7] are used as a source of heat for hyperthermia treatment. Using mechanical ultrasound waves, operating within the frequency range of 1 MHz to 10 MHz, represents an alternative method for inducing heat in biological tissues [8], [9], [10].

Ultrasound offers advantages over other treatment techniques, primarily in its non-invasive ability to generate heat in deep-seated tissues. Additionally, the ultrasound technique stands out for its cost-effectiveness, simplicity, and energy efficiency. The direction of ultrasound waves can be controlled through the design of the transducer. Specifically, the application of a focused ultrasound beam, emitted from the curved surface of the transducer, involves converging multiple ultrasound waves onto a single point. The local temperature obtained using ultrasound is crucial in medical processes such as tumor ablation [11], [12], [13] and targeted drug delivery [14]. The heat generated by ultrasound waves depends on several parameters, including acoustic intensity, frequency, and the absorption rate of ultrasound waves in the medium. Different wavelengths of the ultrasound wave contribute to distinct relaxation times. Lower frequencies correlate with reduced fluctuations in the medium, featuring longer wavelengths associated with lower absorption and scattering of the wave. These factors collectively influence heat generation and penetration depth [15].

There are various ways to optimize the value of heating produced by ultrasound waves. For instance, ultrasound wave frequency and intensity have a direct effect on effective treatment plans [16]. The size of the treatment area can be optimized by splitting the ultrasound beam to achieve a larger focal point [17]. On the other hand, the properties of the medium have also an effect on heat value, which might change depending on the human body parts. The medium properties modification such as sonosensitizers, i.e., additional materials incorporated into the medium, offer an alternative means to modify ultrasound absorption, facilitating rapid temperature increases for lower intensity and shorter sonication durations. The ultrasound wave scattered by nanoparticles as sonosensitizers has been used to control the ultrasound attenuation [18], [19]. The particle radius is commonly comparable with wavelength, and the physical properties of the nanoparticles provide different scattering and absorption contributions. For instance, the comparison of the magnetite and silica nanoparticles showed different attenuation results theoretically and experimentally, as the physical properties such as density contrast between phases and agglomeration rate had a direct effect on the ultrasound attenuation [20]. This variability extended to different temperature increases observed in the agar phantom enriched with nanoparticles under the ultrasound application [21], where ultrasound velocity and attenuation were affected by the presence of nanoparticles. In general, the addition of magnetic nanoparticles into tissue-mimicking material resulted in a significant increase in the specific absorption rate compared to the pure tissue-mimicking material [22]. Potentially, the Pickering droplet stabilized by nanoparticles can also alter the properties of the tissue phantoms and function as sonosensitizers. The droplets can be stabilized by several types of solid particles such as hydroxyapatite, silica, magnetic nanoparticles, polymers, and carbon nanotubes [23]. Also, the droplet stability can be enhanced by cationic surfactants such as quaternary ammonium salts [24]. Similarly to the nanoparticles, the presence of the Pickering droplets in the system was showed to lead to a change in the attenuation of the medium [25]. The influence of the Pickering droplet on ultrasound hyperthermia was not presented yet in the literature. However, there are studies investigating the thermal effect of the Pickering droplet under the influence of an AC magnetic field [26], [27], [28]. In the case of Pickering droplets, we could have many more parameters to control for optimization of the outcome of ultrasound hyperthermia such as the size of the liquid core, the size of the stabilizing particles, their mechanical and thermal properties, and even the shape, as non-spherical particles have been also investigated for Pickering emulsions [29].

In this research, we investigated theoretically the effect of utilizing the Pickering droplets in agar phantom as sonosensitizers. The ultrasound attenuation and velocity of the silicone oil droplet stabilized with magnetite and silica nanoparticles incorporated in pure agar phantom were calculated based on core–shell model proposed by Anson and Chivers [30]. Recently, this model was successfully used to analyze the experimental data of ultrasound spectroscopy in the Pickering emulsion. The output results theoretically and experimentally indicated the changes in the absorption of the medium with different concentrations, Pickering droplet sizes, and other physical properties of the system such as density, specific heat, and thermal conductivity [25], [31]. Consequently, in the current research, the difference in physical properties and the resulting different value absorption of the whole system (agar phantom doped with Pickering droplets) led to different amounts of generated heat. The maximum value of the scattering and thermal absorption varied with various wavelengths and object sizes as well as carrier fluid properties. In this research, both a pure agar phantom and an agar phantom with a spherical inclusion were simulated using COMSOL Multiphysics. Heat transport within the agar phantom was modeled by employing the classic Pennes bioheat equation [32]. The simulation incorporated the physical properties of inclusions doped with either magnetite or silica Pickering emulsion, and examined the impact of varying droplet radius and shell thickness on the final heating effect. This research lays the groundwork for understanding ultrasound wave propagation in agar phantom doped with Pickering droplets, setting the stage for future experimental investigations. The simulation results offer insights into the potential of Pickering nano- and microemulsions, comparing favorably to the current use of nanoparticles, to enhance thermal therapy.

2. Theoretical background

2.1. Ultrasound scattering theory based on core–shell model

The core–shell model is a model proposed by Anson and Chivers that describes the scattering of the ultrasound wave by the core–shell objects that consist of a spherical core (solid or liquid) covered with another type of material as a shell and dispersed in continuous phase [30]. The Pickering droplet has a similarity to such core–shell structure which allows for using the Anson and Chivers model to calculate the ultrasound parameters of the Pickering droplets.

As ultrasound wave travels through emulsion and suspension, it is attenuated. The reason for the attenuation is the absorption and scattering of the wave. The particles and droplets dispersed in continuous phase constituting suspensions or emulsions have a direct effect on the ultrasonic properties and make the ultrasound attenuation and velocity dependent on the size distribution and volume concentration. The Epstein–Carhart–Allegra–Hawley (ECAH) theory [33], [34] describes the ultrasound scattering theory by solid or liquid suspension in the continuous phase caused by the viscous-inertial and thermal transport mechanism. The viscous mechanism occurs due to the densities are different for both phases. The thermal mechanism occurs due to pressure–temperature coupling resulting from the thermal contrast between phases. In the core–shell model, the additional physical parameters of the shell material such as its attenuation coefficient are taken into consideration in comparison to ECAH theory [30].

Similarly to the ECAH model, in the core–shell model, the wave equation for the propagation of compressional, shear, and thermal wave are derived as follows:

$$\left({\mathrm{\nabla }}^2+k_c^2\right){\phi }_c=0,$$ (1a)

$$\left({\mathrm{\nabla }}^2+k_t^2\right){\phi }_t=0,$$ (1b)

$$\left({\mathrm{\nabla }}^2+k_s^2\right)A_{\mathrm{\Psi }}=0.$$ (1c)

The main part of our implementation of the core–shell model was to calculate the partial scattering amplitude for the compression wave in the continuous phase, which further was used for the calculation of the ultrasound velocity and the attenuation of the compression wave in the continuous phase. The formula below shows the potential of compression ultrasound wave as series expansions that contain the scattering coefficient:

$${\phi }_{c1}={\sum }_{n=0}^{\infty }{i}^n\left(2n+1\right)A_n{h}_n\left(k_{c1}r\right)P_n\left(cos\mathrm{\Theta }\right).$$ (2)

${\phi }_{c1}$ is the potential scattering wave equation for the compression wave in the continuous phase,$A_n$ is the partial scattering amplitude in the continuous phase, $h_n$ is the Hankel function, $k_{c1}$ is the complex wave number of the compression wave, $r$ is the object radius, $P_n$ is the Legendre polynomials of the order n, and $\mathrm{\Theta }$ is the scattering angle. Similarly, the potential of thermal and shear wave were calculated by considering the thermal and shear wavenumbers $k_{t1}$ and $k_{s1}$. Therefore, in Equation (3), ${\phi }_{t1}$ is related to the potential of the thermal wave, and the $B_n$ is related to the partial amplitude of the thermal wave. In turn, in Equation (4), $A_{\mathrm{\Psi }1}$ is related to the potential of the shear wave in the continuous phase, and $C_n$ is related to the partial wave amplitude of the shear wave.

$${\phi }_{t1}={\sum }_{n=0}^{\infty }{i}^n(2n+1)B_n{h}_n(k_{t1}r)P_n(cos\mathrm{\Theta })$$ (3)

$$A_{\mathrm{\Psi }1}={\sum }_{n=0}^{\infty }{i}^n\left(2n+1\right)C_n{h}_n\left(k_{s1}r\right)P_n^1\left(cos\mathrm{\Theta }\right)$$ (4)

The wave scattered forward (the boundary between the droplet shell and agar phantom) and backward (the boundary between the droplet shell and core − silicone oil) can be expressed as the following equations for the compressional, thermal, and shear wave:

$${\phi }_{c2}={\sum }_{n=0}^{\infty }{i}^n\left(2n+1\right){[D}_n{j}_n\left(k_{c2}r\right)+G_n{h}_n\left(k_{c2}r\right)]P_n(cos\mathrm{\Theta })$$ (5a)

$${\phi }_{t2}={\sum }_{n=0}^{\infty }{i}^n\left(2n+1\right){[E}_n{j}_n\left(k_{t2}r\right)+H_n{h}_n\left(k_{t2}r\right)]P_n\left(cos\mathrm{\Theta }\right)$$ (5b)

$$A_{\mathrm{\Psi }2}={\sum }_{n=0}^{\infty }{i}^n\left(2n+1\right){[F}_n{j}_n\left(k_{s2}r\right)+I_n{h}_n\left(k_{s2}r\right)]P_n^1(cos\mathrm{\Theta })$$ (5c)

The $D_n$, $E_n$, and $F_n$ are the partial wave amplitudes of the backward compressional, thermal, and shear modes, respectively. The $G_n$, $H_n$, and $I_n$ are the partial wave amplitudes for the three modes of the forward wave.

The wave scattered in the core (in our implementation in silicone oil droplet) is a backward wave, i.e., the wave propagating back inside the core material, and can be represented as following equations:

$${\phi }_{c3}={\sum }_{n=0}^{\infty }{i}^n(2n+1)J_n{j}_n(k_{c3}r)P_n(cos\mathrm{\Theta })$$ (6a)

$${\phi }_{t3}={\sum }_{n=0}^{\infty }{i}^n(2n+1)K_n{j}_n(k_{t3}r)P_n(cos\mathrm{\Theta })$$ (6b)

$$A_{\mathrm{\Psi }3}={\sum }_{n=0}^{\infty }{i}^n(2n+1)L_n{j}_n(k_{s3}r)P_n^1(cos\mathrm{\Theta })$$ (6c)

The 12 boundary equations and 12 unknown coefficients from the core–shell model ($A_n,B_n,C_n,D_n,E_n{F}_n,G_n,H_n,I_n,J_n,K_n,L_n$) were solved using a 12 x 12 matrix equation [30]. In the boundary equations, the additional contribution of the shell phase was considered in core–shell model. This model was recently used for comprehensive theoretical and experimental characterization of oil-in-oil magnetic Pickering emulsions [25]. To calculate the ultrasound attenuation coefficient, the imaginary part of the partial compression wave amplitude $A_{\mathrm{n}}$ had to be determined. Furthermore, the real part of the compression wave amplitude $A_{\mathrm{n}}$ was used to calculate the ultrasound velocity. In Table 1, the physical properties of the phases required for the core–shell model calculation are presented.

Table 1.

The physical properties of agar phantom as continuous phase, silicone oil as dispersed droplets, and magnetite nanoparticles as stabilizing particles at 25 °C.

Parameters	Agar phantom	Silicone oil	Magnetite shell	Silica shell
Viscosity, $\eta$ (Pa∙s)	–	50 × 10-3 (measured)	–	–
Density, $\rho$ (kg/${\mathrm{m}}^3$)	1040 [37]	960 (measured)	5180 [38]	1970 [39]
Thermal conductivity coefficient, $\kappa$ (W/m∙K)	0.616 [37]	0.15 (data sheet)	52 [38]	1.6 [39]
Specific heat, $c_{\mathrm{p}}$ (J/kg∙K)	3900 [40]	1460 (data sheet)	653 [38]	728 [39]
Thermal expansion coefficient, ${\beta }_{\mathrm{T}}$ (1/K)	3 × 10-4[41]	9.5 × 10-4 (data sheet)	11.8 × 10-6[38]	1.35 × 10-6[39]
Ultrasound velocity, $c$ (m/s)	1547 [42]	1004 [36]	7157 [38]	5968 [39]
Ultrasound attenuation coefficient, $\alpha$ (Np/m)	4.6 × 10-6f + 8.7 × 10-14f2[42]	3.79 × 10-13f2.02 (measured)	10-17f2[38]	2.6 × 10-22f2[39]
Shear modulus, $\mu$ (N/m2)	40 × 103[43]	–	6.03 × 1010[38]	2.79 × 1010[39]

The frequency (f) is expressed in Hz unit.

The complex wavenumber equation was used to calculate the ultrasound attenuation and velocity of the Pickering droplets when placed in an agar phantom:

$$\beta =\sqrt{k_c^2-\frac{3i{\varphi }_{\mathit{Si}}}{k_c{a}^3}(A_0+{3A}_1+{5A}_2)}.$$ (7)

Here, $\beta =\frac{\omega }{c_{\mathit{Agar}\mathit{phantom}}}+i{\alpha }_{\mathit{Agar}\mathit{phantom}}$ is the complex wavenumber of the scattered wave; the real part, $\frac{\omega }{c_{\mathit{Agar}\mathit{phantom}}}$, is related to the ultrasound velocity and $i{\alpha }_{\mathit{Agar}\mathit{phantom}}$ is related to the ultrasound attenuation. ${\varphi }_{\mathit{Si}}$ is the volume fraction of the dispersed phase, ${\varphi }_{\mathit{Si}}=\frac{4}{3}\pi b^{3}N$, where $N$ is the number of silicone oil droplets (dispersed phase) distributed in the continuous phase (agar gel), $a$ and b are the outer and inner radii of the Pickering droplet, respectively. $A_0$ and $A_1$ are partial scattering amplitudes that depend on the thermal (monopole effect) and viscous mechanism (dipole effect), respectively. In turn, $A_2$ is related to the object resonance mechanism (quadrupole effect) that occurred due to the relationship between the wavelength and surface wave that is produced from the interface of compression and shear wave [35]. The mentioned mechanisms occurred due to the different physical properties of the components of core–shell objects.

It is well-known that the shear wave of ultrasound wave has a higher contribution in the solid-state material [36]. However, the existence of shear waves is negligible in soft tissues because soft tissues are approximated and considered liquid [9]. That is the reason why we did not take into account the influence of shear waves propagating in agar phantoms. Table 1 presented the physical properties of the phases required for the core–shell model calculation. The physical properties of the magnetite and silica phases presented in Table 1 represent the shell phase properties.

The values of ultrasound attenuation coefficient and phase velocity for Pickering emulsions with assumed different core size, shell thickness, and volume concentration in relation to continuous phase (agar gel) were calculated using the Mathematica 13.2 software. These were then expressed in the function of frequency for the range of 1 MHz to 12 MHz. These frequencies were chosen based on the requirements for the frequencies used in the ultrasound heating procedures [13], [44], [45].

2.2. Ultrasound heating model implemented in computer simulation

Numerical calculations of ultrasound heating were carried out using COMSOL Multiphysics 6.1. Considering the high symmetry of the studied structure, 2-dimensional axisymmetric rotational geometry was used to reduce calculation time. Pressure Acoustics, Frequency Domain together with Bioheat Transfer modules were used to model the ultrasound heating of the samples. In our simulation, we assumed that the inclusion is filled evenly with Pickering emulsion droplets. Ultrasound was generated by the normal displacement of the transducer end with a diameter of 10 mm, providing a dynamic source for the model. The cylindrical phantom (with a diameter of 30 mm and a height of 30 mm) is placed 1 mm away from the transducer. The intervening space between the phantom and the transducer is filled with water, which serves as the coupling medium for the simulation. Additionally, to account for wave reflections and ensure accurate results, the whole system was enclosed by a Perfectly Matched Layer (PML), which indicates the acoustic boundary condition. The scheme of the designed geometry for simulations is shown in Fig. 1.

Fig. 1.

The scheme of the simulation design for (a) pure agar phantom and (b) agar phantom with an inclusion filled with Pickering droplets.

In simulations, the transducer was driven at several frequencies from 1 MHz to 10 MHz with an application time of 180 s for each frequency. The output intensity of the ultrasound wave was determined by changing the displacement amplitude of the transducer with different frequencies to achieve the acoustic intensity of 2.5 W/cm². This value was chosen to be similar to the intensity that was produced by the ultrasound transducer used commercially for medical purposes and in the research on ultrasound hyperthermia in tissue-mimicking phantoms [32] as well as the ultrasound-triggered release of active substances from Pickering capsules [46]. The acoustic pressure implemented in COMSOL Multiphysics was used for the simulation of ultrasound wave propagation and acoustic pressure change in an agar phantom. The classic Pennes bioheat equation was used in the simulation to model heat transfer in the tissue [32]. This is the mathematical model that describes the heat transfer in the biological tissue, expressed as follows:

$$\rho C_p\frac{\partial T}{\partial t}-\kappa {\mathrm{\nabla }}^{2}T-w_b{c}_b\left(T_a-T\right)=Q+Q_m$$ (8)

In the Equation (8), $\rho$ and $C_p$ are the density and the specific heat of the tissue, T and t are the temperature and the time, respectively. $\kappa {\mathrm{\nabla }}^{2}T$ is the thermal conductivity multiplied by the Laplacian operator that represents the spatial temperature gradient. $w_b$ and $c_b$ are the perfusion rate and specific heat of the blood, respectively. In turn, $Q$ and $Q_m$ are the heat source of the absorbed ultrasound energy and heat of the metabolism, respectively. The effect of blood perfusion and metabolic processes on heating effect is omitted ($Q_m=0$) as the simulation was performed in the agar phantom system where there was no vasculature.

In our simulation, as it was used elsewhere [47], the heat is generated by ultrasound waves that propagate perpendicularly through a cylindrical agar phantom. As an axisymmetric cylindrical coordinator, the temperature distribution in the phantom is along the axis of the beam:

$$\rho C_p\frac{\partial T(r,z,t)}{\partial t}-\kappa \left[\frac{\partial }{\partial t}{r}^2\frac{\partial T\left(r,z,t\right)}{\partial r}+\frac{{\partial }^{2}T\left(r,z,t\right)}{\partial z^2}\right]=Q.$$ (9)

The r, z, and t represent the distance from the center, the vertical distance from the top, and the application time.

For the initial condition, the time is equal to 0 and then, the temperature in the phantom is $T_0$:

$$\partial T\left(r,z,0\right)=T_0$$ (10)

The following boundary conditions for simulating the experimental conditions:

$$\frac{\partial T(r,z,t)}{\partial r}{|}_{r=0}=0$$ (11a)

$$\frac{\partial T(r,z,t)}{\partial r}{|}_{r=R}=-\frac{h}{\kappa }(T-T_0)$$ (11b)

$$\frac{\partial T(r,z,t)}{\partial z}{|}_{z=0}=-\frac{h}{\kappa }(T-T_0)$$ (11c)

$$\frac{\partial T(r,z,t)}{\partial z}{|}_{z=L}=-\frac{h}{\kappa }(T-T_0)$$ (11d)

In Equations (11a)-d), R and L are the cylindrical phantom radius and height respectively and h is the convective heat coefficient. The heat map of the temperature distribution of height of the cylindrical agar phantom (L) as well as the different application time (t) was simulated for several frequencies (1 MHz, 2 MHz, 5 MHz, and 10 MHz).

3. Results and discussion

3.1. Attenuation spectra of Pickering droplets stabilized by magnetic nanoparticles in an agar phantom

First, we investigated the acoustic properties of the Pickering droplets stabilized by magnetic nanoparticles dispersed in agar phantom. This was further used to optimize ultrasound heating, by determining the ultrasound attenuation and velocity across various radii of silicone oil droplets, magnetic shell thicknesses, and volume concentrations of the dispersed droplets in an agar phantom. The results in Fig. 2 and Fig. 4 show the effect of acoustic properties of Pickering nano- and microdroplets with different shell thicknesses. In one of our previous studies, we showed that this parameter plays a crucial role in attenuation of the magnetic Pickering emulsion [25]. Additionally, Fig. 3 reveals the differences resulting from the use of ultrasound scattering theory based on core–shell model with and without the thermal contribution. The thermal penetration length, which is several hundred nanometers, is much shorter than the core radius in micrometer size, leading to negligible differences between the existence of the thermal term or its absence. The neglected thermal term in the theoretical calculation results in a reduction of matrix elements by ignoring the thermal contribution in the theoretical calculation of the Anson and Chivers model to 8 × 8 [30]. This allows a direct theoretical comparison between the thermal and non-thermal contribution models. For instance, Kanamori et al. [31], provide the calculation details of the core–shell model for studying the Pickering emulsion stability. The researchers stated that the thermal contribution is negligible with a larger droplet radius. However, the consideration of the contribution of thermal effects to the overall acoustic energy loss in the medium provides important knowledge of the difference in attenuation coefficient and ultrasound velocity between Pickering nano- and microdroplets that we simulated as doped in agar phantom.

Fig. 2.

The calculated ultrasound attenuation spectra based on core–shell model for the Pickering droplets stabilized by magnetic nanoparticles and distributed in agar phantom for (a) different liquid core radii, (b) different particle shell thicknesses, and (c) different volume concentrations.

Fig. 4.

The calculated ultrasound velocity in the function of frequency based on core–shell model for the Pickering droplets stabilized by magnetic nanoparticles and placed in agar phantom for (a) different core radii (b) different shell thicknesses, and (c) different volume concentrations.

Fig. 3.

The calculated ultrasound attenuation spectra based on core–shell model including thermal contribution and without including thermal contribution in the core–shell model for Pickering (a) nanoemulsion, and (b) microemulsion.

Moreover, in this theoretical calculation, we used a monodisperse system with a fixed size of Pickering droplet radius in the system without including the distribution factor. However, when studying the polydisperse system and comparing the theoretical results to the results from the experiment, the polydisperse distribution may need to be included in the complex wavenumber calculation as calculated, for instance, by Challis et al. [39] for better accuracy.

The results in Fig. 2 show the attenuation value strongly depends on the different parameters such as silicone oil droplet radius, particle shell thickness, and concentration. Then, controlling these parameters might optimize the ultrasound heating. Interestingly, the Pickering nanodroplets with a size of 200 nm show higher ultrasound attenuation compared with larger droplets (1 μm) at constant concentration and shell thickness (Fig. 2a) for frequencies higher than 2.5 MHz. However, the ultrasound scattering theory based on ECAH showed no significant increase in the ultrasound attenuation with different size of silicone droplets when were dispersed in castor oil [48]. Additionally, the core–shell model for oil-in-oil magnetic Pickering emulsion, when silicone droplets were stabilized with magnetic particles, showed a maximum for ultrasound attenuation increase for 500-nm radius of the droplet [25]. Here, the physical properties of the agar phantom contributed to the calculation as well as the nano-size of the Pickering droplet. We expected that the thermal contrast contribution to ultrasound attenuation is higher in the case of nanoscale objects. It is visible as an increase in the ultrasound attenuation above 4 MHz for 200-nm core radius. This is the reason why the non-thermal core–shell model needs to be investigated (Fig. 3).

The ultrasound attenuation is also affected by the shell thickness of magnetic particles around the silicone droplets. There is a significant increase in the attenuation between 100-nm and 400-nm size of the shell as presented in Fig. 2b. Additionally, the theoretical result showed no significant differences between the 400-nm and 900-nm shell thicknesses in ultrasound attenuation in this range of frequency from 1-12 MHz. Fig. 2c shows the change in the ultrasound attenuation with different ranges of volume concentration. The higher volume concentration had a larger contribution to the ultrasound attenuation, with around 200 Np/m difference between 1 % and 2 %.

The results in Fig. 3 show that thermal contribution has a higher impact on ultrasound attenuation for Pickering droplets with a core radius of 200 nm and shell thickness of 100 nm compared with the micrometer-sized Pickering droplets (1 μm droplet radius) when dispersed in an agar gel. Therefore, thermal contribution was not dominating within micrometer size, which led to the appearance of different ultrasound attenuation trends between nano and micrometer-size Pickering droplets. For nano-size droplets, the calculations of core–shell model should be done including thermal contrast between phases to provide better accuracy of the results. The thermal and viscous loss did not make linear changes in the ultrasound attenuation to corresponding particles or droplets sizes.

3.2. Phase velocity of Pickering droplet stabilized by magnetic nanoparticles in an agar phantom

The values of ultrasound velocity in the function of frequency also depended on the Pickering droplet sizes and were required in further COMSOL simulation. The velocity increased with both the increasing radius of droplets and the thickness of the particle shell as shown in Fig. 4a-b. Interestingly, the velocity of agar phantom doped with Pickering droplets was lower than the ultrasound velocity in pure agar which was 1546 m/s (Table 1). The ultrasound velocity was also influenced by the volume concentration of droplets. The change in the compressibility contrasts with different radii and shell thicknesses influences the velocity value resulting in these differences shown in Fig. 4c. In a magnetic fluid, the ultrasound velocity also exhibited a similar trend depending on the concentration and temperature [49].

3.3. Attenuation spectra of Pickering droplets stabilized by silica nanoparticles in an agar phantom

One of the most important feature of the Pickering droplets is that, in practice, they could be prepared with a variety of materials used as particle stabilizers. Here, we provide the comparison between the Pickering droplets stabilized with magnetic (Section 3.1) and non-magnetic (silica) nanoparticles to show how they change the ultrasound attenuation based on core–shell model when distributed in the agar phantom. The formation and functionalization of silica particles are quite well investigated and they have been widely used in medical applications such as drug delivery [50] and bioimaging [51]. Using them as stabilizers could extend the Pickering droplet application [52], [53].

The ultrasound attenuation value of the Pickering droplets distributed in the agar phantom and stabilized by silica nanoparticles (Fig. 5) was lower compared to the ultrasound attenuation of Pickering droplets stabilized by magnetite nanoparticles as shown in Fig. 3. Furthermore, the value of ultrasound attenuation decreased for the micro-droplets (core diameter of 1 μm) when comparing to those with a core of 200 nm as presented in Fig. 5a. Attenuation coefficient values were generally lower when compared to magnetic nanoparticles with the exact same conditions such as core and shell sizes, volume concentration, and the range of frequencies. The main reason was the difference in physical properties between silica and magnetic nanoparticles such as density and thermal contrast (Table. 1). This made Pickering droplets coated by magnetic nanoparticles more attenuating material compared to those stabilized with the silica nanoparticles. That was even more clear in lower range of frequencies (e.g., for 2 MHz), which was important as this range of frequency is used in various medical applications [54].

Fig. 5.

The calculated ultrasound attenuation spectra based on core–shell model for the Pickering droplets stabilized by silica nanoparticles and distributed in agar phantom for (a) different liquid core radii, (b) different particle shell thicknesses, and (c) different volume concentrations.

Additionally, the influence of shell thickness on ultrasound attenuation records significant changes between 200 nm and 400 nm (Fig. 5b). Interestingly, the values of attenuation coefficient are quite similar for thicknesses of particle layer between 400 nm and 900 nm with constant droplet radius of 1 μm and concentration of 1 %. Similarly to the results obtained for magnetic nanoparticles, the increase in the volume concentration also provided higher attenuation in the ultrasound wave as presented in Fig. 5c.

3.4. Phase velocity of Pickering droplets stabilized by silica nanoparticles in an agar phantom

The Pickering emulsions stabilized with silica nanoparticles showed a change in the velocity values for various droplet sizes (Fig. 6a) and shell thicknesses (Fig. 6b). These values were higher than those calculated for magnetic nanoparticles stabilizing the droplets due to the difference in the compressibility contrast between phases when these two types of particles were considered (their shear modulus and ultrasound velocity). Furthermore, the higher volume concentrations revealed a lower value of ultrasound velocity in the frequency function (Fig. 6c), which reflects the same trend as for the results for the magnetic Pickering droplets.

Fig. 6.

The calculated ultrasound velocity in the function of frequency for the Pickering droplets stabilized by silica nanoparticles and distributed in agar phantom for (a) different core radii (b) different shell thicknesses, and (c) different volume concentrations.

3.5. Computer simulation

3.5.1. Ultrasound heating of the pure agar phantom

In our approach, as shown in the scheme in Fig. 1, the ultrasound waves generated by the transducer propagate through the water first and then entering the agar phantom. As an entry point, the simulation of ultrasound propagation was performed for pure agar phantom with different frequencies (1, 2, 5, and 10 MHz). The chosen frequencies are within the typical ultrasound diagnostic range (1–15 MHz). It should be noted that the level of accuracy of a wave propagation model is influenced by mesh density or, in other words, the number of elements per wavelength. The result for 5 elements per wavelength was chosen for the acoustic and thermal study [47].

Fig. 7a-d presents the change in temperature within pure agar phantom for different vertical positions (10 mm, 15 mm, and 20 mm) and different application times (10 s, 60 s, and 180 s) for frequencies of 1 MHz, 2 MHz, 5 MHz, and 10 MHz. Moreover, the heat maps, i.e., the spatial distribution of the temperature elevation, were presented for the 180th second of ultrasound application. In the heat maps, the Z=0 mm represents the phantom in the transducer side, and Z=30 mm represents the end of the agar phantom (see, Fig. 1). The physical properties of the water and agar phantom used for simulation were as listed in Table 2. As one can see, the relative temperature (i.e., the temperature rise related to the room temperature of 25 °C) in the center (Z=15 mm) of the phantom model was higher for higher ultrasound frequency; the temperature increase was of 6 °C for 10 MHz compared to 3.5 °C as shown in Fig. 7d. Subsequently, temperature distribution was more homogenous for lower frequencies as shown both in temperature versus time and temperature versus height graphs depicted in Fig. 7a-b. Generally, it indicates higher depth with better temperature homogeneity can be achieved within the lower frequency range (Fig. 7a-b). One can see that the ultrasound waves of higher frequencies such as 10 MHz and 5 MHz can be used to raise the temperature two times more compared with the lower frequencies over a short distance. In general, each frequency range can be used in different applications depending on the penetration depth required. The provided simulation results indicate that using ultrasound of different frequencies improved control of the penetration depth by changing frequency of the ultrasound wave. The range of ultrasound in the tissue can be changed from 1 cm to 12 cm by changing the frequencies from 1 MHz to 10 MHz as well as tissue properties [44]. Nevertheless, as we will demonstrate in the next section, achieving more precise control over the temperature increase in the phantom is feasible when the properties of the sonosensitizers incorporated into the phantom are taken into account.

Fig. 7.

The COMSOL simulation results of ultrasound heating for the pure agar phantom. The heat map illustrates the temperature change in the middle of the cylindrical agar phantom after 180 s. Furthermore, the variations in temperature over time are showcased for three distinct Z positions (10 mm, 15 mm, and 20 mm), and the temperature changes are presented in relation to Z for selected sonication times (10 s, 60 s, and 180 s) at frequencies of (a) 1 MHz, (b) 2 MHz, (c) 5 MHz, and (d) 10 MHz.

Table 2.

The physical properties of the pure agar phantom and water used in the COMSOL simulation.

Parameters	Agar phantom	Water
Density, $\rho$ (kg/m3)	1040 [37]	997 [39]
Thermal conductivity coefficient, $\kappa$ (W/m∙K)	0.616 [37]	0.5952 [39]
Specific heat, $c_{\mathrm{p}}$ (J/kg∙K)	3900 [40]	4179 [39]
Ultrasound velocity, $c$ (m/s)	1547 [42]	1496.7 [39]
Ultrasound attenuation coefficient, $\alpha$ (Np/m) (1 MHz)	4.69 [42]	0.025 [55]
Ultrasound attenuation coefficient, $\alpha$ (Np/m) (2 MHz)	9.55 [42]	0.1 [55]
Ultrasound attenuation coefficient, $\alpha$ (Np/m) (5 MHz)	25.2 [42]	0.63 [55]
Ultrasound attenuation coefficient, $\alpha$ (Np/m) (10 MHz)	54.75 [42]	2.5 [55]

3.5.2. Ultrasound heating of agar phantom with inclusion agar phantom doped with magnetite Pickering droplets

The agar phantom with a spherical inclusion doped by magnetic Pickering droplets was simulated using COMSOL. Various physical properties were considered for the spherical inclusion, including thermal and acoustic properties. The ultrasound scattering theory based on the core–shell model provided the ultrasound attenuation and velocity values in the function of frequency for different core radii and shell thicknesses. The 2 MHz frequency and 1 % volume concentration of Pickering droplets were set as constant in the simulation. The focus was on evaluating the different droplet radii and shell thickness of Pickering droplets for more efficient heat generation within the inclusion inside agar phantom.

MG 1 and MG 2 were samples of magnetite-stabilized Pickering droplets. They had a consistent shell thickness of 100 nm, but their core radii differed, with MG 1 having a core radius of 200 nm and MG 2 having a core radius of 1 μm. MG 3 and MG 4 were samples with a constant Pickering droplet radius of 1 μm, but with varying shell thicknesses: 400 nm for MG3, and 900 nm for MG4. The physical properties of these samples implemented in the simulation are presented in Table 3.

Table 3.

The physical properties of agar phantom with magnetic Pickering droplets for samples MG 1 and MG 2 (constant shell thickness of 100 nm and core radii 200 nm and 1 μm, respectively) and MG 3 and MG 4 (constant core radius of 1 μm and different shell thicknesses of 400 nm and 900 nm, respectively).

Parameters of phases	Density, $\rho$(kg/m3)	Thermal conductivity coefficient, $\kappa$ (W/m∙K)	$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}$, $C_{\mathrm{p}}$ (J/kg∙K)	Ultrasound velocity, $c$ (m/s)	Ultrasound attenuation coefficient, $\alpha$ (Np/m)
Agar phantom	1040 [37]	0.616 [37]	3900 [40]	1547 [42]	9.55 [42]
Water	997 [39]	0.5952 [39]	4179 [39]	1496.7 [39]	0.1 [55]
MG 1	1080.6	0.975	3843.13	1484.8	96.21
MG 2	1080.6	0.975	3843.13	1528.8	42.8
MG 3	1080.6	0.975	3843.13	1529.4	94.28
MG 4	1080.6	0.975	3843.13	1533.4	82,18

The parameters are provided for a constant temperature of 25 °C.

The additional materials dispersed in the phantom change its properties that are need for the simulation of the temperature distribution. Apart from the changed attenuation and velocity depending on frequency determined based on core–shell model (Section 3.1), also the effective density and the effective specific heat were determined based on a weighted average of respective values for phases. Effective density can be calculated using the formula:

$${\rho }_{\mathit{eff}}=\left({\varphi }_1\times {\rho }_1\right)+\left({\varphi }_2\times {\rho }_2\right)+\left({\varphi }_3\times {\rho }_3\right),$$ (12)

where ${\rho }_1$, ${\rho }_2$, and ${\rho }_3$ refer to the density of agar phantom, silicone oil, and particle stabilizer, respectively, and are multiplied by the volume concentration of each phase $\varphi$.

Similarly, the weighted average of specific heat in agar phantom doped with Pickering droplets was calculated as:

$$C_{\mathit{eff}}=\left({\varphi }_1\times C_{p1}\right)+\left({\varphi }_2\times C_{p2}\right)+\left({\varphi }_3\times C_{p3}\right).$$ (13)

The $C_{p1}$, $C_{p2}$, and $C_{p3}$ are related to the specific heat of agar phantom, silicone oil, and particle stabilizer, respectively, and multiplied by the volume concentration of each phase $\varphi$.

The weighted average of the thermal conductivity of two phases can be calculated from Equation (14a) (as used in [56] to calculate the thermal conductivity of an oil-in-water emulsion):

$${\kappa }_{\mathit{eff}}={\kappa }_1\frac{1+2\varphi \chi -2(1-\varphi )\xi {\chi }^2}{1-\varphi \chi -2(1-\varphi )\xi {\chi }^2},$$ (14a)

where

$$\chi =\frac{{\kappa }_2-{\kappa }_1}{{\kappa }_2-{2\kappa }_1},$$ (14b)

$$\xi =0.21068\varphi -0.04693{\varphi }^2.$$ (14c)

Here, $\kappa$ and $\varphi$ represent the thermal conductivity and volume concentration, respectively.

In real medical applications, the ultrasound wave generated from the transducer must travel through multiple tissue layers such as skin, fat, and muscle. Therefore, controlling low changes in the temperature in these layers is important to leave healthy tissue untouched. A frequency of 2  MHz can be used for this purpose due to the homogeneous temperature change in the system with a longer penetration depth as shown in Fig. 7b.

Fig. 8 shows the dependence of the temperature on the duration of the ultrasound wave application with the different positions in the agar phantom (10 mm, 15 mm, and 20 mm). Additionally, the dependence of the relative temperature at different heights within the agar phantom for different durations of ultrasound exposition (10 s, 60 s, and 180 s) for MG 1 and MG 2 samples. These samples showed the influence of the different core radii 200 nm and 1  μm of magnetic Pickering droplets, at a constant shell thickness of Pickering droplet of 100 nm and volume concentration of 1 %.

Fig. 8.

The COMSOL simulation results of ultrasound heating for pure agar phantom with spherical inclusion filled with magnetic Pickering droplets for (a) MG 1 (magnetite–stabilized Pickering droplets with a shell thickness of 100 nm and core radius 200 nm) and (b) MG 2 (magnetite–stabilized Pickering droplets with a shell thickness of 100 nm and core radius 1 μm). The 2 MHz frequency and 1 % volume concentration were set as constant in the simulation. The spherical inclusion on the heat map represents a doped agar phantom at 180th seconds from sonication. The blue area represents the coupling medium (water).

Fig. 8a presents the results for the agar phantom doped with nano-size of Pickering droplets with a core radius of 200 nm and shell thickness of 100 nm. The part of agar phantom with distributed Pickering droplets (spherical inclusion) was placed in the center of the pure agar phantom, 10 mm away from the transducer position (see, Fig. 1b). The temperature change in the agar phantom after 180 s above the spherical inclusion (Z=2 mm) was 1.8 °C and below the inclusion (at the position of Z=28 mm) was around 1.4 °C, which corresponded to the temperature rises at 2 MHz frequency. However, as shown in Fig. 7b, the maximum change in the temperature was around 1.6 °C for Z=2 mm and 1.6 °C for Z=28 mm. Moreover, the insertion of magnetic Pickering droplets in the spherical inclusion resulted in a reduction of the penetration depth below the spherical inclusion and the creation of a shadow effect. The temporal evolution of the heat map between 0 and 180th second of sonication for MG 1 sample is also presented in Supplementary Materials (Movie S1).

Furthermore, at Z=8 mm (2 mm above the inclusion), there are rising temperature changes (around 5.3 °C) due to the heat transfer from the inclusion itself as shown in Fig. 8a. Inside the spherical inclusion, the gradient of the temperature change appeared, which was higher at the beginning of spherical inclusion till achieved the maximum at Z=13.5 mm with a change in the temperature of 9.6 °C, and then started to decrease as presented in the change of the temperature in the function of Z in Fig. 8a. In the case of the MG 2 (Fig. 8b), the temperature increase in spherical inclusion was lower compared to the results for phantom doped with smaller droplets due to the reduction in ultrasound absorption value presented in Fig. 2a. The highest change in the temperature of 6.5 °C was recorded at Z=14 mm. This led to lower heat transfer to the pure agar phantom around the inclusion compared to Fig. 8a. Therefore, the temperature record at Z=8 mm was 4.4 °C and for Z=22 mm was 3.8 °C. As it was for MG 1 sample, the temporal evolution of the heat map between 0 and 180th second of sonication for MG 2 sample is presented in Supplementary Materials (Movie S2). It is also worth noting that, when ultrasound heating efficiency was experimentally investigated for tissue-mimicking materials doped with magnetic micro- and nanoparticles instead of Pickering droplets, nanoparticles exhibited also higher attenuation and a greater temperature rise [57].

The influence of different shell thicknesses in Pickering emulsions stabilized with magnetic particles was also investigated for a constant core radius of 1 μm and concentration of 1 %. The simulation results showed no significant changes in the relative temperature when different shell thicknesses were considered for MG 3 (Fig. 9a) and MG 4 (Fig. 9b). The Pickering droplets with shell thickness of 900 nm showed a bit lower attenuation compared to those with the 400-nm particle layer as shown before in Fig. 2b. However, the temperature rise was lower for thinner shells, e.g., when 100-nm 400-nm size were compared, as corresponding to ultrasound attenuation in Fig. 2b. This is related to the monotonous change in the ultrasound attenuation with the size of scattered objects. Therefore, determining the relation between the object size and wavelength of ultrasound waves could help to improve enhancement of the thermal efficiency during sonication provided by sonosensitizers.

Fig. 9.

The COMSOL simulation results of ultrasound heating for pure agar phantom with spherical inclusion filled with magnetic Pickering droplets for (a) MG 3 (magnetite–stabilized Pickering droplets with a shell thickness of 400 nm and core radius 1 μm) and (b) MG 4 (magnetite–stabilized Pickering droplets with a shell thickness of 900 nm and core radius 1 μm). The 2 MHz frequency and 1 % volume concentration were set as constant in the simulation. The spherical inclusion on the heat map represents a doped agar phantom at 180th seconds from sonication. The blue area represents the coupling medium (water).

3.5.3. Ultrasound heating of agar phantom based on silica Pickering droplet

As mentioned in Section 3.2, silica nanoparticles are one of the most attractive nanoparticles due to their biocompatible and ease of functionalizing in terms of surface chemistry [58]. That is why, as an example of non-magnetic material, silica nanoparticles were used to stabilize silicone oil Pickering droplets incorporated into the agar phantom with spherical inclusion. The weighted average of the density, specific heat, and thermal conductivity of the doped agar phantom was calculated based on the physical properties of both pure agar and silica nanoparticles using Equations 12–14. The ultrasound scattering theory based on the core–shell model was used to calculate ultrasound attenuation and velocity with different core radii and shell thicknesses. As it was for Pickering emulsions stabilized with magnetic nanoparticles, the 2 MHz frequency and 1 % volume concentration were set as constant in the simulation.

SI 1 and SI 2 were samples of silica-stabilized Pickering droplets. They had a consistent shell thickness of 100  nm, but their core radii differed, with SI 1 having a core radius of 200 nm and.

SI 2 having a core radius of 1 μm. In turn, SI 3 and SI 4 were samples of Pickering droplets with constant droplet radius (1 μm) and different shell thicknesses: 400 nm for SI 3, and 900 nm for.

SI 4. The physical properties of these samples are presented in Table 4.

Table 4.

The physical properties of agar phantom with silica-stabilized Pickering droplets for SI 1 and SI 2 samples (constant shell thickness of 100 nm and core radii 200 nm and 1 μm, respectively) and SI 3 and SI 4 samples (constant core radius of 1 μm and different shell thicknesses of 400 nm and 900 nm, respectively).

Parameters ofp hases	Density, $\rho$ (kg/m3)	Thermal conductivity coefficient, $\kappa$ (W/m∙K)	$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}$, $C_{\mathrm{p}}$ (J/kg∙K)	Ultrasound velocity, $c$ (m/s)	Ultrasound attenuation coefficient, $\alpha$ (Np/m)
Agar phantom	1040 [37]	0.616 [37]	3900 [40]	1547 [42]	9.55 [42]
Water	997 [39]	0.5952 [39]	4179 [39]	1496.7 [39]	0.1 [55]
SI 1	1048.5	0.975	3842.8	1521.3	38.50
SI 2	1048.5	0.975	3842.8	1538,5	15.86
SI 3	1048.5	0.975	3842.8	1535.05	33.79
SI 4	1048.5	0.975	3842.8	1537.25	36.24

The parameters were provided for at a constant temperature of 25 °C.

The simulation results in Fig. 10 show the effect of inserting nano- and micrometer-sized Pickering droplets with a constant shell thickness of silica nanoparticles and volume concentration of 1 %. Fig. 10a indicates a higher change in the temperature observed for Pickering droplets with a core radius of 200 nm and shell thickness of 100 nm (sample SI 1).

Fig. 10.

The COMSOL simulation results of ultrasound heating for pure agar phantom with spherical inclusion filled with Pickering droplets for (a) SI 1 (silica–stabilized Pickering droplets with a shell thickness of 100 nm and core radius 200 nm) and (b) SI 2 (silica–stabilized Pickering droplets with a shell thickness of 100 nm and core radius 1 μm). The 2 MHz frequency and 1 % volume concentration were set as constant in the simulation. The spherical inclusion on the heat map represents a doped agar phantom at 180th seconds from sonication. The blue area represents the coupling medium (water).

The temperature increased gradually in the spherical inclusion filled with silica-stabilized Pickering droplets. The highest relative temperature of 6.3 °C was recorded close to the center of the inclusion at Z=14.2 mm. This is correlated with the increase in the ultrasound attenuation of 200 nm core radius and shell thickness of 100 nm (Fig. 5a). The decreasing ultrasound attenuation with a core radius of 1 μm and shell thickness of 100 nm (Fig. 5b) led to the reduction of the temperature elevation in the spherical inclusion as shown for the SI 2 sample in Fig. 10b. The overall temperature of the SI 2 sample was close to this shown for pure agar phantoms Fig. 7b as well as the ultrasound attenuation presented of the SI 2 sample in Table 4 was close to the ultrasound attenuation of pure agar phantom presented in Table 1.

Fig. 11 presents the simulation results with different shell thicknesses of silica nanoparticles with a constant droplet core radius of 1 μm at a constant volume concentration of 1 %. Both samples SI 3 and SI 4 show comparable results. The maximum change in the relative temperature for SI 3 was 5.9 °C at Z=14.2 mm. In contrast, the maximum change in the temperature of SI 4 was 6.09 °C at Z=14.2 mm. Moreover, there is also no significant change in the ultrasound attenuation between SI 3 and SI 4 samples presented in Table 4.

Fig. 11.

The COMSOL simulation results of ultrasound heating for pure agar phantom with spherical inclusion filled with Pickering droplets for (a) SI 3 (silica–stabilized Pickering droplets with a shell thickness of 400 nm and core radius 1 μm) and (b) SI 4 (silica–stabilized Pickering droplets with a shell thickness of 900 nm and core radius 1 μm). The 2 MHz frequency and 1 % volume concentration were set as constant in the simulation. The spherical inclusion on the heat map represents a doped agar phantom at 180th seconds from sonication. The blue area represents the coupling medium (water).

The results show the different object sizes as well as different physical properties of the system have a direct impact on the ultrasound wave scattering and absorption. This influence is even more apparent when the materials are used for coating the droplet interface, which was shown previously in the calculation of the core radius and shell thickness of magnetic Pickering emulsion [25]. In the literature, nano-size emulsions with a mean diameter of 260 nm have been used to enhance the bio-effect of high-intensity focused ultrasound [59]. The nano-emulsion also provided efficient and controllable thermal ablation by using short pulses of high-intensity ultrasound wave 19 W/cm² tested in vitro [60]. Furthermore, the ultrasound wave application to the system doped with microbubbles was used recently for enhanced ultrasound thermal therapy [61], [62]. The heat generation biologically changed the oxidation and the blood flow in the body leading to better treatment performance by following the radiotherapy with lower wave intensity. From practical point of view, one should expect that the penetration of sonosensitizing material will be higher at the nanoscale compared to the microscale. For this reason, the application of the microbubbles in thermal therapy was limited. However, there still remains the path for using nano-sized Pickering droplets that, as we showed, could enhance the thermal effect of applied ultrasound.

The simulation results showed the possibility of using magnetite and silica Pickering droplets as sonosensitizers to enhance the thermal therapy as an alternative to the nanoparticles. The temperature elevation depends on several parameters including core radius, shell thickness, and the material properties of the system. The material properties are one of the crucial parameters for obtaining an optimal temperature elevation. For instance, the percentage temperature elevation between MG 1 and SI 1 samples was recorded as 47.3 %, which indicates the strong influence of the shell phase properties on the ultrasound heating. The capsulated structure of Pickering droplets can be beneficial to combine two types of treatment when the droplet contains specific medical drugs. As mentioned in the Introduction, the control to release the cargo with additional temperature rise could be achieved by ultrasound wave [14]. The efficiency of such process may be controlled by determining specific Pickering droplet size. The Pickering droplet has unique features such as the possibility of stabilizing with different types of particles and the ability to control the outer radius of the droplet which has a direct contribution to ultrasound heating as shown in simulation results, which leads to good nomination as sonosensitizing material. Additionally, the localized or general hyperthermia treatment could be achieved using different types of transducers. The local sphere doped by nano- or microdroplets showed an increase in the temperature locally by using a non-focused ultrasound beam with a constant intensity of 2.5 W/cm² and a frequency of 2 MHz within penetration depth of 1 cm (Fig. 8, Fig. 9). Using specific physical parameters of Pickering emulsions i.e., droplet size distribution that can be obtained by optimization of the process of preparation, for instance, by using different concentrations of stabilizing particles, it is possible to optimize both the maximum temperature elevation and the distribution over the tissue (or tissue-like phantom). Different particle materials provide different contrasts between phases that, along with the size of droplets, contribute mostly to the overall acoustic energy dissipation according to core–shell model of Anson and Chivers.

4. Conclusion

In this study, the computational approach provided a way to effectively investigate the heat generation inside the pure agar phantom including spherical inclusion doped with magnetite and silica-stabilized Pickering droplets induced by ultrasound wave of the intensity of 2.5 W/cm². The ultrasound attenuation and velocity of the agar phantom with Pickering droplets were calculated theoretically based on core–shell model. The influence of different radii and shell thicknesses of Pickering droplets on the acoustic properties was investigated. Significantly reduced ultrasound attenuation was observed with silica-based Pickering droplets compared to those stabilized with magnetite from 96.2 Np/m of magnetite (sample MG 1) to 38.5 Np/m of silica (sample SI 1) at the frequency of 2 MHz. The main reason was the varying contribution of density contrast between phases to overall attenuation. Interestingly, the calculation showed the ultrasound attenuation of the nanoemulsion (sample MG 1) was above twice higher than microemulsion (sample MG 2) in the case of magnetite particles used as stabilizers.

Numerical simulations further illustrated the impact of parameters of Pickering droplets on temperature changes within the agar phantom. Both magnetite and silica Pickering droplets dispersed in spherical inclusions exhibited higher ultrasound attenuation compared to the pure agar phantom. Notably, a greater temperature change was recorded for nanodroplets (the highest temperature change of 9.6 °C close to the center of the inclusion) than microdroplets when simulated droplets had a constant shell thickness of 100 nm, volume concentration of 1 %, and ultrasound frequency was 2 MHz.

The droplet radius and shell thickness emerged as crucial factors influencing the simulated ultrasound heating efficiency. The precise control of these parameters, aligned with the ultrasound frequency and physical properties of the system, holds the key to enhancing thermal performance. This study emphasizes the importance of tailoring droplet characteristics to optimize the efficiency of ultrasound-induced thermal effects.

CRediT authorship contribution statement

Bassam Jameel: Writing – original draft, Visualization, Investigation, Formal analysis. Yaroslav Harkavyi: Writing – review & editing, Investigation. Rafał Bielas: Writing – review & editing, Supervision, Methodology. Arkadiusz Józefczak: Supervision, Methodology, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data

The following are the Supplementary data to this article: