A mathematical model for simulating photoacoustic signal generation and propagation in biological tissues

Abstract

Photoacoustic (PA) medical imaging is a crossbred technique relying on light-induced ultrasonic waves due to the PA effect phenomenon recorded primarily in 1880 by A. G. Bell. Numerical simulation, also known as in-silico, studies assist scientists in minimizing incorrect PA experiments in both in-vitro and in-vivo. Numerical modeling techniques help to achieve a fast simulation process in contrast to pure mathematics alone. However, if a suitable facilitated mathematical model can be established prior to applying numerical modeling, it will be of great interest to the whole numeric model. Numerous mathematical equations, theorems, and propositions have been proposed to model the whole PA signal generation and propagation process in biological media. However, most of them are complicated and difficult to be understood by researchers, especially beginners. That’s why this paper was introduced. Our paper aims to simplify the understanding of the generation and propagation process of biomedical PA waves. We have developed a facilitated mathematical model for the entire process. The introduced developed mathematical model is based on three steps: (1) pulsed laser stimulation, (2) light diffusion, and (3) PA stress wave generation and propagation. The developed mathematical model has been implemented utilizing COMSOL Multiphysics, which relies on the finite element method (FEM) numerical modeling principle. The in-silico time-dependent study's results confirmed that the proposed mathematical model is a simple, efficient, accurate, and quick starting point for researchers to simulate biomedical PA signals' generation and propagation process utilizing any suitable software such as COMSOL multiphysics.

1 Introduction

Photoacoustic imaging (PAI) is a hybrid imaging technique merging optical illumination and ultrasound detection in one modality by utilizing the laser-induced ultrasound signals due to the photoacoustic (PA) effect phenomenon (Wang 2008; Wang et al. 2023; Attia et al. 2019; GadAllah et al. 2022; Wang and Song 2012; Lapierre-Landry et al. 2018; Chu and Chen 2023; Nyayapathi and Xia 2019; Beard 2011; Hui and Cheng 2019; Chen and Tian 2021; Zhang et al. 2023; Liu et al. 2018; Wang and Wang 2018). PAI has been able to produce images almost free of speckle artifacts (Wang 2008; GadAllah et al. 2022; Wang and Song 2012; Guo et al. 2009; Joseph 2013; Narayan, et al. 2014), in contrast to traditional ultrasound imaging (Kirk Shung et al. 1992; Kirk Shung 2006; Wagner 1983; Burckhardt 1978; GadAllah 2015; Najarian and Splinter 2012; Dickinson and Nassiri 2004; Szabo 2014; Webb 2003) and optical coherence tomography (OCT) (Wang and Hsin-i 2007a; Fercher 1996; Huang et al. 1991; Tuchin 2016; Hendon and Rollins 2016). PAI has gained a variety of implementation techniques and applications through the last three decades (Wang et al. 2023; GadAllah et al. 2022; Nyayapathi and Xia 2019; Liu and Zhang 2016; Sung-Liang Chen 2015). For instance: photoacoustic tomography (PAT) (Wang 2008, 2011; Zhou et al. 2016; Attia et al. 2019; Lin et al. 2023; Ntziachristos et al. 2005), multispectral optoacoustic tomography (MSOT) (GadAllah et al. 2022) or multispectral PAT, light emitting diode (LED) based PAI (Chandramoorthi and Thittai 2020), optical detection PAT (OD-PAT) or also named non-contact laser ultrasound (NC-LUS) (Zhang et al. 2019; Haupt et al. 2019; Dong et al. 2017), photoacoustic microscopy (PAM) (Sung-Liang Chen 2015; Zhang et al. 2023), optical resolution photoacoustic microscopy (OR-PAM) (Cao et al. 2023), acoustic resolution photoacoustic microscopy (AR-PAM) (Sung-Liang Chen 2015), raster-scan optoacoustic mesoscopy (RSOM) (GadAllah et al. 2022), LED-based PA microscopy (LD-PAM) (Li et al. 2021), Second generation PA remote sensing based microscopy (Ecclestone et al. 2022), PA endoscopy which is providing molecular contrast at considerable depths allowing for simultaneous imaging of structural and functional information (GadAllah et al. 2022; Yang et al. 2015; Liang et al. 2022), PA topography through an ergodic relay (PATER) (Li and Li 2020), and photoacoustic elastography (PAE) (Hai et al. 2016a, b; Suheshkumar Singh and Thomas 2019). PAI has a big contribution to cancer diagnosis (GadAllah et al. 2022; Bai et al. 2023; Chen et al. 2014) for instance in breast cancer (Attia et al. 2019; GadAllah et al. 2022; Nyayapathi and Xia 2019; Oraevsky et al. 2001), which has a high record in threatening human health, mostly females (Ayana et al. 2022; Zhang et al. 2018), the second most frequent cancer worldwide following lung cancer and the fifth leading cause of cancer death and a major cause of cancer death among women (Mohamed et al. 2022; Badawy et al. 2021a, 2021b). Imaging Alzheimer’s disease would be sensible by PAI of the brain (Park et al. 2019). PAI has been able to help in a real-time assessment of tissue hypoxia (Gerling et al. 2014). PAI’s guidance plays a strong role in more applications of image-guided surgery (IGS) and high intensity focused ultrasound (HIFU) (GadAllah et al. 2022).

It is merit to mention here the main difference between optoacoustic imaging (OAI) and thermoacoustic imaging (TAI). Commonly in the PA scientific community, PAI is a general term due to the PA effect basic phenomenon including two overlapped terms: optoacoustic and thermoacoustic imaging (Kellnberger 2013). The first, OAI, is mentioned to reflect the absorption of optical energy (optical wavelengths range from 400 to 700 nm) by tissue to generate the PA signal. The second, TAI (Aliroteh et al. 2016), is more general and is mentioned to reflect the absorption of any form of energy (popularly it is related to the whole electromagnetic spectrum’s radiation bands including but not restricted to the radiofrequency (RF) and microwave (MW) bands) by biological tissue to induce PA signal (Kellnberger 2013). So, in our opinion, the term thermoacoustic imaging (TAI) could be considered the more general term including basically PAI and OAI.

The basic principle for PAI or OAI (Wang 2008; Zhou et al. 2016; Oraevsky and Karabutov 2003) is the PA effect, been reported firstly in 1880 by Bell (1880, 1881), which can be described in biological tissues (Duck 1990) by two energy exchange processes through three forms of energy: (1) electromagnetic (light) energy, (2) heat energy, and (3) kinetic energy (generated ultrasound mechanical waves). Figure 1 represents the basic idea for PAI or OAI, where; the “target” tissue in the figure is the region of interest (ROI) to be investigated.

Fig. 1

Illustrative diagram for the basic idea of the three energy forms’ exchange in biological tissues due to the PA phenomena

Numerical simulation for the generation process of PA signals in biological tissues helps researchers decrease error experiments in vitro, hence increasing safety rates (Sliney and Trokel 1993; Smalley 2013) in vivo. Numerical modeling methods are fast when compared to pure mathematics. If a suitable simplified mathematical model can be obtained to be an easy base for the aimed numerical model, it will be a great advantage.

A variety of scientific theories, equations, and assumptions through the literature of PA imaging have been introduced aiming to completely model the biomedical PA signal generation and propagation process. Starting from light-tissue interaction, heat-tissue interaction, and ending by the mechanical ultrasound pressure wave generation and propagation (Cox and Beard 2009, 2005; Cox et al. 2004; Treeby and Cox 2010; Treeby et al. 2023; Wang et al. 2012a; Wang et al. 2012b; Wang et al. 2012c; Wang et al. 1995; https://www.comsol.com; Metwally and El-Gohary 2014; Wilson and Adam 1983; https://www.ansys.com; El-Gohary et al. 2014; Metwally et al. 2014; Grahn 2017, 2015; Chandramoorthi and Thittai 2017;Liu et al. 2018; ElGohary et al. 2020; Song et al. 2021). However, more of them are complicated and are difficult being understood by researchers, especially beginners. So, this paper was proposed.

In this paper, we have attempted to simplify the modeling for the biomedical PA wave’s generation and propagation process, obtaining an easy simplified deduced mathematical model for the overall process to be a base for any numerical simulation software. This research paper is a conclusion of about two years of research into the deep literature of photoacoustic sensing and imaging. The proposed model here aims to shortcut the way for all beginners to simply but accurately model and simulate the photoacoustic wave generation and propagation through biological tissues. The introduced deduced model is based on three main stages. The first stage is applying pulsed laser irradiation. The second stage is light diffusion through biological tissue. The third stage is the PA pressure wave generation and propagation from the target tissue to the ultrasound transducer surface. The third stage contains inherently two energy transformation processes through three forms of energy: (1) electromagnetic (light) energy to (2) deposited heat energy to (3) kinetic energy (ultrasound pressure mechanical wave).

The proposed deduced mathematical model has been validated numerically by COMSOL Multiphysics. The simulation started with a simulated laser pulse input of about 10 ns duration on a tissue model with an inherent tumor. The produced simulated PA signal has been drawn vs. the time for more than one point containing a point in the center of the modeled target tumor. The results of the numerical simulation study assure that the proposed mathematical model may be considered as a simplified, easy, and fast startup base for scientific researchers to numerically model and simulate PA signals generation and propagation in biological tissues utilizing any numerical simulation platform such as COMSOL Multiphysics.

In the subsequent second section of this paper “Related Work”, we present some of the related research along the PA modeling history. In the following third section “Methods”: we present the mathematical model deduced in this work from the PA literature. Followed by the fourth section “Numerical simulation by COMSOL multiphysics”: illustrates the application of the proposed model on a simulated biomedical tissue including a tumor inside. The subsequent fifth section presents the “Qualitative and analytical results”. Finally, in the following sixth section, we concluded that this research paper may be considered as a simplified, but accurate base for the scientific research of PA signals’ generation and propagation studies using any numerical modeling software.

2 Related work

Cox and Beard (2009), have presented Modeling PA propagation in tissue utilizing frequency-wavenumber or k-space techniques, utilizing fast Fourier transform (FFT) and inverse FFT algorithms. Their model was based on the assumption that the produced PA waves distributed within the biomedical tissue often be considered instantaneous, allowing their model to be cast as an initial value problem with an easy k-space solution. Their work may be considered as an extension to their previous k-space based work, in 2005 (Cox and Beard 2005), for a Fast calculation of pulsed PA fields in fluids. An experimental validation for k-space based PA propagation models was introduced in 2004 by Cox et al. (2004), developing propagation models for temporal output prediction of a sensor due to an arbitrary PA generated initial stress distribution.

Treeby and Cox (2010), have introduced a new and freely available MATLAB toolbox to simulate and reconstruct PA wave fields, which they called: the “k-Wave” toolbox. After the birth of the K-wave toolbox, they both and Jiri Jaros (Treeby et al. 2023), have developed the K-Wave open-source toolbox for MATLAB and C+ +.

Wang et al. (2012a), have introduced an evaluation of a finite element method (FEM) (Logan 2023; Zienkiewicz et al. 2005; Reddy 2019) based numerical simulation model for PA signals in biological tissues using COMSOL Multiphysics (https://www.comsol.com). Their model has been built basically on four steps: (1) light energy irradiance described by a diffusion equation, (2) conversion from light to heat energy using a bio-heat transfer equation, (3) conversion from thermal to kinetic energy (PA wave generation) by a stress–strain model, (4) propagation of the induced PA wave by the homogeneous acoustic wave propagation equation. In the same year, they proposed their paper (Wang et al. 2012b) for discussing the applicability of their method introduced in Wang et al. 2012a using COMSOL to develop a new illumination scheme, a light catcher, for improving light penetration into deep tissues, in silico. At almost the end of the same year 2012, they introduced their study (Wang et al. 2012c), based on their last two papers (Wang et al. 2012a, b), improving their new light irradiance approach (the light catcher) for achieving more imaging depths in PA imaging, evaluating their study not only in silico but also by in-vitro experiments applied on different targets at various locations.

Metwally and El-Gohary (2014), introduced their investigation study about the influence of the light fluence distribution on PA back-propagation imaging using: (1) Monte Carlo technique (Wang et al. 1995; Wilson and Adam 1983) to estimate the propagation and distribution of the light fluence, and (2) FEM for numerically simulating the PA wave generation utilizing ANSYS (https://www.ansys.com).

El-Gohary et al. (2014), proposed a design Study for a PA Probe for Prostate Cancer Detection utilizing 3D Monte Carlo Simulation and FEM.

Metwally et al. (2014), presented their study about the influence of the anisotropic mechanical characteristics (elasticity) of Breast Cancer on PAI-based back-propagation technique utilizing FEM (The numerical simulation has been achieved by ANSYS (https://www.ansys.com), while the fluence distribution has been evaluated by light propagation resolving through a tissue model using Monte Carlo technique (Wang et al. 1995; Wilson and Adam 1983).

Grahn (2017), introduced his research article on PA Modeling with COMSOL, simulating a cantilever-based PA sensor popularly utilized in gases analysis and detection. His study’s governing equations have been derived by linearizing the Navier–Stokes equations, continuity equation and energy equation. This study is partially based on his previous study, in 2015 (Grahn 2015), which has been for modeling thermoacoustics of loudspeakers by COMSOL.

Chandramoorthi and Thittai (2017), have introduced a simulated PAT system utilizing: (1) COMSOL (in a four steps scheme such as in: Wang et al. (2012a, 2012b, 2012c)) for a FEM-based numerical simulation study, and (2) MATLAB (https://www.mathworks.com/products/matlab.html) for reconstruct an initial pressure distribution (image reconstruction) from the aggregated raw data from COMSOL with comparing the performance of two popularly utilized image reconstruction techniques called: filtered back projection (FBP) and synthetic aperture (SA) beamforming, resulting in a lateral resolution largely depends on the aperture parameters obtained by FBP versus SA.

Liu et al. (2018), have proposed their study for a handheld real-time PAI system for animal neurological disorder models. In their study, COMSOL has been utilized in numerical simulation (in a four-stage manner like in: Wang et al. (2012a, 2012b, 2012c)) for estimating both light propagation and hence the simulated PA wave generation to investigate the significant factors affecting the generation of PA signals.

El Gohary et al. (2020), introduced a numerical study for PA signals from eye models to discover diabetic retinopathy. Their study has been based on (two steps methodology like in: (Metwally and El-Gohary 2014; El-Gohary et al. 2014; Metwally et al. 2014)): (1) Monte Carlo method utilized to simulate the interaction of a 0.8 ns duration laser pulse with eye tissues at 750 nm wavelength, and (2) thermal structural and acoustical analyses have been carried out utilizing a commercial FEM based analysis program: ANSYS 19 (https://www.ansys.com).

Song et al. (2021), have introduced their work for a spectrum analysis of PA signal based on COMSOL. Their study has utilized COMSOL in studying the relevance between the spectral characteristics of the PA signal and the properties of the absorber, concluding that the spectral characteristics of the PA signal are very related to the size of the absorber (Song et al. 2021).

3 Methods

The proposed model is based on three modules as illustrated in the following framework presented in Fig. 2.

Fig. 2

The proposed model framework

3.1 (A)—Pulsed laser irradiance on the tissue’s surface:

The first operation for the PA signal generation is the pulsed laser irradiance. The measure of laser light intensity, (also called: flux rate or irradiance), is in Joule per second per unit area or (W/m²). The deposited light energy rate per unit volume (lightning function) or the laser light source function is described in W/m³ which is equivalent to Kg. m⁻¹. s⁻³ in the standard international system of units (SI system) (https://en.wikipedia.org/wiki/International_System_of_Units; Newell and Tiesinga 2019). The unit of W/m³ is the unified unit here for the deposited energy rate per unit volume through the whole model starting from the light energy input rate per unit volume in W/m³, to the deposited heat energy rate per unit volume in W/m³, and ending with the source term for the compact acoustic pressure wave equation for generation (from the deposited heat energy rate per unit volume) and propagation, which can be described as a pressure source generation rate in Pa/s which is equivalent to Kg. m⁻¹. s⁻³ which is also leading to W/m³.

The input laser pulse can be modeled as a Gaussian (bell shape) signal with a pulse width τ_p in seconds and a maximum amplitude of Ipmax in W/m³ which represents the energy rate (joule/second) of the input laser pulse per unit volume. Figure 3 illustrates an example of the input laser pulse with τ_p ≈ 10 ns, that’s means that the full-width half maximum (FWHM) is about 10 ns, at pulse center t₀ = 30 ns. The peak value of the pulse was normalized to one and multiplied by the maximum value of the input light signal’s power Ipmax = 400,000 W/m3 (Ipmax = 400 mW/cm³) as shown in Fig. 3.

Fig. 3

Example of the input laser pulse shape

The analytical input laser pulse (for example the curve shown in Fig. 3) utilized to irradiate a biological tissue surface at a position r (x, y, z) can be described mathematically as:

$$S \left(r,t\right)=\mathrm{ Ipmax}* {e}^{- 2.77 {\left(\frac{t-{t}_{0}}{{\tau }_{p}}\right)}^{2}}$$

(1)

where:

$\mathrm{S }(\mathrm{r},\mathrm{t})$: represents the input light pulse of laser irradiance’s energy per unit volume per unit time at location r (x, y, z) and at time t (s), it will be the source term directly W/m³ for the light irradiance diffusion equation in the next step in (B).

$\mathrm{Ipmax}$: is representing the peak value of the input laser pulse S (r, t), here in Fig. 3, Ipmax = 400,000 W/m³ (i.e.: Ipmax = 400 mW/cm³).

t₀: is representing the center time value of the laser input pulse shape, here in Fig. 3: t₀ = 30 ns.

τ_p: represents the pulse width, where: τ_p ≈ FWHM = $2\upsigma \sqrt{2ln2 } \approx 2.355\upsigma$; where: σ is representing the standard deviation of the signal [016], as shown in Fig. 2 (τ_p $\approx$ 10 ns).

In (1), we utilized a peak value normalization for the Gaussian bell shape signal so that the peak of the input laser pulse is equal to one multiplied by the maximum power for the signal per unit volume (W/m³) as assumed here in this model to be Ipmax = 1 * 400 mW/cm³. For obtaining an integral normalization for the Gaussian distribution, the term (Ipmax) should be equal to (1/ σ $\sqrt{2\pi } )$ multiplied by a value related to the signal power or energy to obtain a complete term representing the peak value for the pulse signal (COMSOL Multiphysics Reference Manual 2023). The (1/ σ$\sqrt{2\pi }$) term is only used to make the total area under the pdf curve equal to one (Smith 1997–1999).

A detailed explanation of deriving Eq. (1) is found in: Appendix-A.

3.2 (B)—Diffusion of light through biological tissues:

Modeling light interaction with biological tissues, an optically complex scattering media (Cheng et al. 2023), has different scientific theories, assumptions, and techniques. Some of these are modeling the light as a function in diffusion distance only and are not concerned with the time distribution of the light pulse irradiance (time-independent) as Beer-Lambert law (https://en.wikipedia.org/wiki/Beer-Lambert_law; https://www.comsol.com/blogs/modeling-laser-material-interactions-with-the-beer-lambert-law/; Ishimaru 2017) and Helmholtz equation (https://en.wikipedia.org/wiki/Helmholtz_equation; Goodman 1996; https://www.atomic.physics.lu.se/fileadmin/atomfysik/Biophotonics/Education/MultiphysicsExercise.pdf), such that stationary studies are suitable into the frequency domain and steady-state studies only. Some others are time-dependent such as the basic time-varying diffusion equation (Ishimaru 1978; Morse and Feshbach 1953; Star 2011), which is originally derived based on approximating the basic equation of transfer or the basic radiative transfer equation (RTE) (Ishimaru 1978). The origin of the RTE returns to the basic radiative transfer theory (RTT) or transport theory which was initiated by Schuster (1905). The RTE (Ishimaru 1978; Manning 1989) is equivalent to the Maxwell–Boltzmann collision equation (Sommerfeld 1956) or Boltzmann’s equation utilized in the kinetic theory of gases and neutron transport theory (Williams 1971).

The RTE is difficult to solve since it has more independent variables, so the RTE is commonly simplified in the diffusion approximation assuming that the radiance in a high-albedo (${{\varvec{\mu}}}_{{\varvec{a}}}\ll {{\varvec{\mu}}}_{{\varvec{s}}}^{\mathrm{^{\prime}}}$; i.e., the absorption is very less than the scattering) scattering medium is almost isotropic after sufficient scattering (Wang and Hsin-i 2007b).

In our study, the following time-dependent diffusion Eq. (2), concluded and deduced from Wang et al. 2012c; Liu et al. 2018; Morse and Feshbach 1953; Star 2011; Ishimaru 1978; Wang and Hsin-i 2007b; Arridge 1999; Arridge et al. 2011), has been utilized to describe exactly the light fluence rate $\mathbf{\varphi }\left(\mathbf{r},\mathbf{t}\right)$ (Named also: the light flux rate or the irradiance or the light intensity) of the incident light pulse beam exactly not only at each location r but also at each time $\mathbf{t}$ in W/m².

$$\frac{\mathrm{n}}{\mathrm{C}} \frac{\partial \mathrm{\varphi }(\mathrm{r},\mathrm{t})}{\partial \mathrm{t}}+ \nabla \cdot \left(-\mathrm{D}\nabla \mathrm{\varphi }\left(\mathrm{r},\mathrm{t}\right)\right)+ {\upmu }_{\mathrm{a}}\mathrm{\varphi }\left(\mathrm{r},\mathrm{t}\right)=\mathrm{S }(\mathrm{r},\mathrm{t})$$

(2)

where:

n is the refractive index of the medium.

C is the light velocity in vacuum = 3 × 10⁸ m/s.

${\upmu }_{\mathrm{a}}$ is the absorption coefficient for the medium in 1/m at a certain wavelength.

${\upmu }_{\mathrm{s}}$ is the scattering coefficient for the medium in 1/m at a certain wavelength.

${\mu }_{s}^{\prime}$ is the reduced scattering coefficient for the medium in 1/m at a certain wavelength.

D is the diffusion coefficient [in m]; D = $\frac{1}{3\left({\mu }_{a}+{\mu }_{s}{\prime}\right)}$;${\mu }_{s}{\prime}=(1-\mathrm{g }){\mu }_{s}$; g is anisotropy coefficient. The anisotropy factor g is the average of the cosine value of the deflection angles [048, 10], so it characterizes the average amount of scattering in a medium (Jacques and Wang 1995), ranges from − 1 (backwardly peaked scattering) via 0 (isotropic scattering) to 1 (forwardly peaked scattering) (Wang et al. 2012b). The meaning of isotropy of a medium is that its properties are independent of the direction of the light wave’s polarization i.e., the direction of the electric and magnetic fields vectors (Goodman 1996). The anisotropy coefficient g, an important optical property of biomedical tissues (Jacques 1996, 2013; Bashkatov et al. 2011), is the parameter that determines the asymmetry of the well-known scattering phase function utilized in examining the anisotropy’s influence in light scattering (Peters et al. 1990), originally stated by Henyey and Greenstein (Henyey and Greenstein 1941) (H-G function) for galactic scattering, and verified for single scattering by Jacques et al. with only a slight modification (Jacques et al. 1987).

$\mathrm{\varphi }\left(\mathrm{r},\mathrm{t}\right)$ is the Light Intensity/Flux rate/ Irradiance/Fluence rate in W/m². Where: r denotes the location and t is representing time. $\mathrm{\varphi }\left(\mathrm{r},\mathrm{t}\right)$ is the solution for Eq. (2) and it represents the light energy diffusion through the tissue media per unit time per unit area).

$\nabla$ is “del” or “napla” differential operator representing the gradient (grad), $\nabla$ = $\frac{\partial }{\partial x} i+ \frac{\partial }{\partial y} j+ \frac{\partial }{\partial z} k$,where: i, j, k are unit vectors along x, y, z, respectively (Schey 2005; Kreyszig 2011).

$\nabla$. is the divergence “div”, for a function [f (x, y, z)]: $\nabla$. f = ($\frac{\partial }{\partial x} {f}_{x}+ \frac{\partial }{\partial y} {f}_{y}+ \frac{\partial }{\partial z} {f}_{z}$Schey 2005).

3.3 (C)—Acoustic pressure wave generation and propagation

The basic idea behind the photoacoustic wave generation and propagation as illustrated in Fig. (3), is three energy forms’ exchange (Two transformations of energies) in the target (light to heat to kinetic) to obtain the final acoustic pressure wave generation and propagation which can be induced with ultrasonic transducers (GadAllah et al. 2022).

To model the two-energy exchanges illustrated before in Fig. 1, there are two steps have been implemented as follows:

3.3.1 From Light energy to heat or thermal energy:

The solution of (2) is $\mathbf{\varphi }\left(\mathbf{r},\mathbf{t}\right)$: a lightening function representing the light energy per unit time per unit area in W/m² (at a location r and at a time t). To model the transfer operation from light energy to thermal energy, we will utilize the following equation (Cox and Beard 2009; Wang et al. 2012a; Chandramoorthi and Thittai 2017):

$$H\left(r,t\right)= {\mu }_{at}* \varphi \left(r,t\right)*Y$$

(3)

where;

$H\left(r,t\right)$ represents the heating function induced due to the irradiance from the pulsed laser beam. $H\left(r,t\right)$ is equivalent to the thermal energy deposited per unit time per unit volume in W/m³ (at a location r and at a time t through the tissue).

${\mu }_{at}$ is representing the absorption coefficient for the target tissue in m.⁻¹

Note: if the target tissue you need to image is the same tissue that the laser pulse irradiated on its surface, in that case only: ${\mu }_{at}$ will be equal to ${\mu }_{a}$ mentioned in the previous Eq. (2). Otherwise ${\mu }_{at}$ is equal to the absorption coefficient for the target tissue in m⁻¹.

$Y$ is representing the yield. In other words: it represents the ratio of the deposited light energy that has been converted into thermal energy (Wang et al. 2012a). Here it is assumed that Y = 1 for simplification as in Cox and Beard (2009); Chandramoorthi and Thittai 2017).

3.3.2 From thermal energy to kinetic energy or ultrasound pressure wave (PA wave):

The ultrasound pressure wave (P) resulting from the energy exchange processes due to the photoacoustic phenomena is a function both in location r and time t ( $P= P\left(r,t\right)$). The common photoacoustic equation describing the photoacoustic wave generation and propagation in an inviscid medium as biological tissue is (Wang 2008; Zhou et al. 2016; Cox and Beard 2009; Wang et al. 2012a, 2017; Diebold et al. 1990 Oct 5; Wang and Anastasio 2011; Westervelt and Larson 1973; Diebold 2009; Sigrist 1986):

$$\left(\frac{1}{{v}_{s}^{2}}\frac{{\partial }^{2}}{\partial {t}^{2} }{ - \nabla }^{2} \right)P\left(r,t\right)= \frac{\beta }{{C}_{P}} \frac{ \partial H(r, t)}{\partial t}$$

(4)

where;

${v}_{s}$ Is the speed of sound in the target biological tissue (m/s); ${v}_{s}= \sqrt{\frac{1}{\rho k }}$; where: $\rho \, and \, k,$ are representing the density (in: Kg/m³) and the compressibility (in: Pa⁻¹), respectively (Morse and Uno Ingard 1968).

${\nabla }^{2}$ is the Laplacian operator, also may be noted as: “$\Delta$”, it is equivalent to the divergence $\nabla$. of the gradient $\nabla ,$ for a function f (x, y, z): ${\nabla }^{2} f=\Delta f= \nabla .(\nabla f)$ = ($\frac{{\partial }^{2}}{\partial {x}^{2}} f+ \frac{{\partial }^{2}}{\partial {x}^{2}} f+ \frac{{\partial }^{2}}{\partial {x}^{2}} f$Schey 2005; Kreyszig 2011).

$\beta$ is the thermal coefficient of volume expansion for the target biological tissue (K⁻¹).

${C}_{P}$ is the specific heat capacity at constant pressure for the target biological tissue (J/ Kg· K).

Note 1: the three variables ${v}_{s}^{2}, \beta , {C}_{P}$ mentioned in Eq. (4) are characteristic variables for each biological tissue, and the three are forming the Gruneisen parameter ${\varvec{\Gamma}}$ (Gamma), where: $\Gamma = \frac{\beta }{{C}_{P}} {v}_{s}^{2}$. The Gruneisen parameter ${\varvec{\Gamma}}$ is a dimensionless, temperature-dependent factor proportional to the fraction of thermal energy converted into mechanical stress (Oraevsky and Karabutov 2003; Oraevsky et al. 1882; Shah et al. 2008; Gusev and Karabutov 1993; Oraevsky et al. 1997). Hence, the Grueneisen parameter is a temperature-dependent factor (due to the velocity of sound and the volume expansion coefficient), thus the photoacoustic signal P (r, t) is directly related to temperature (Shah et al. 2008).

Note 2: if the right-hand side of (4), the source term, is equaled to zero, the equation is returned to the familiar three dimensions, linear, lossless, and homogeneous acoustic wave propagation equation (Morse and Uno Ingard 1968; Kinsler et al. 2000). That may happen if the heating function $H\left(r,t\right)$ is constant (time invariant), so that its first-time derivative $\frac{\partial H(r, t)}{\partial t}$ will be equal zero. So, time-invariant heating doesn’t turn out acoustic wave (Wang 2008).

Equation (4) can be rearranged in terms of ${\varvec{\Gamma}}$ as follow (Cox and Beard 2009; Ammari et al. 2010):

$$\left(\frac{1}{{v}_{s}^{2}}\frac{{\partial }^{2}}{\partial {t}^{2} }{ - \nabla }^{2} \right)P\left(r,t\right)= \frac{\Gamma }{{v}_{s}^{2}} \frac{ \partial H(r, t)}{\partial t}$$

(5)

Finally: From (3) and (4) and assuming Y = 1, then:

$$\left(\frac{1}{{{\varvec{v}}}_{{\varvec{s}}}^{2}}\frac{{\partial }^{2}}{\partial {{\varvec{t}}}^{2} }{ - \nabla }^{2} \right){\varvec{P}}\left({\varvec{r}},{\varvec{t}}\right)= {{\varvec{\mu}}}_{{\varvec{a}}{\varvec{t}}}* \frac{{\varvec{\beta}}}{{{\varvec{C}}}_{{\varvec{P}}}} \frac{ \partial \boldsymbol{\varphi }({\varvec{r}}, {\varvec{t}})}{\partial {\varvec{t}}}$$

(6)

Now, the complete mathematical model has been built and can be concluded in the following Fig. 4:

Fig. 4

The three steps of the proposed mathematical model for simulating PA signal generation and propagation in biological tissues

4 Numerical simulation by COMSOL multiphysics

A numerical simulation model is introduced here for applying the proposed developed mathematical model in simulating the generation and propagation of a breast cancer’s PA signal stimulated by pulsed laser irradiation. In this model: COMSOL Multiphysics (https://www.comsol.com), a numerical simulation software based on the finite element method (FEM) (Logan 2023; Zienkiewicz et al. 2005; Reddy 2019), has been utilized for applying numerical modeling of the proposed mathematical model shown in Fig. 4 to simulate the generation and propagation of the PA signal generated from a point inside a simulated breast cancer by a sphere (100 µm in diameter) inherent in a simulated biological tissue (represented by a cylinder measuring 1 mm in diameter and height) as illustrated in Fig. 5.

Fig. 5

The simulation model geometry for the illustrative example to verify the application of the proposed developed mathematical model presented in Fig. 4. Where: P1:(0,0,0.5): the laser irradiance point source on the surface of the tissue, P2:(0,0,0.4): a point inside the tumor, P3: (0,0,0): a point away from the tumor and is about 0.5mm away from P1

Table 1 presents the definitions of the parameters utilized in the model. The majority of numerical simulation parameter values used are largely compatible with the laser wavelengths in the near-infrared region (NIR) as extracted from reference sources (Metwally et al. 2014) and (Chandramoorthi and Thittai 2017). The references of the parameters' values are listed in the right column notes, while any unmentioned references imply that the values were added by the authors.

Table 1 Definition for the simulation parameters through the numerical simulation example of the proposed mathematical model

A clarification tutorial for programming the proposed mathematical model on COMSOL is mentioned in Appendix-B.

The proposed COMSOL-based numerical simulation program (a COMSOL application file ) is attached as a supplementary material.

5 Qualitative and analytical results

In Fig. 6, six slices are presented to depict the spatial distribution of the PA pressure signal. The different shapes of the PA signal propagation are shown at six different time snapshots: 30, 40, 50, 60, 80, and 100 ns.

Fig. 6

Six slices showing the spatial distribution of the produced PA pressure signal (in: Pa) through six different time snapshots: 30, 40, 50, 60, 80, and 100 ns

In the following Fig. 7, the simulated laser light intensity’s distribution $\mathbf{\varphi }\left(\mathbf{r},\mathbf{t}\right)$ in W/m² along time is demonstrated from (0–150) ns for a three sample measuring points: P1, P2, and P3 showing the difference in amplitude of each point making a sense that the diffused flux is decreasing by increasing the measuring depth.

Fig. 7

Three samples from the simulation output solution for light intensity distribution $\mathbf{\varphi }\left(\mathbf{r},\mathbf{t}\right)$ in W/m² for Eq. (1), where: r: P1, P2, and P3 as illustrated in Fig. 5, and t: from (0–150) ns

Figure 8 displays the distribution of the induced PA signal over time for the three points, P1, P2, and P3, from (0–150) ns. This graph illustrates the difference in amplitude for each point, indicating that the major pressure originated from the simulated tumor at P2, with the highest amplitude. On the other hand, the pressure at P1 on the surface decreased due to attenuation from the simulated breast tissue. Figure 9 presents the frequency spectrum distribution for the three output PA signals shown in Fig. 8. The output spectrum for the three points ranges from less than 100 MHz.

Fig. 8

Three samples from the simulation output photoacoustic pressure signal output $\mathbf{P}\left(\mathbf{r},\mathbf{t}\right)$ in Pa (solution of (6) in the model presented in Fig. 4), where: r: P1, P2, and P3 as illustrated in Fig. 5, and t: from (0–150) ns

Fig. 9

Frequency spectrum of the three samples from output photoacoustic pressure signal solution $\mathbf{P}\left(\mathbf{r},\mathbf{t}\right)$ shown in Fig. 8

In Fig. 10, a trial is presented that explores both light and acoustic signals at a specific point within a simulated tumor. This is particularly helpful for new researchers who want to simulate the real production of the PA signal with pulsed laser irradiation. Figure 11 displays the frequency spectrum of Fig. 10.

Fig. 10

comparing the light intensity diffusion $\mathbf{\varphi }\left(\mathbf{r},\mathbf{t}\right)$ simultaneously vs the photoacoustic pressure generation and propagation $\mathbf{P}\left(\mathbf{r},\mathbf{t}\right)$, inside the target tumor model at r = P2 (illustrated in Fig. 5), and t: from (0–150) ns

Fig. 11

comparing the frequency spectrum for the light intensity diffusion $\mathbf{\varphi }\left(\mathbf{r},\mathbf{t}\right)$ vs the frequency spectrum of the photoacoustic pressure generation and propagation $\mathbf{P}\left(\mathbf{r},\mathbf{t}\right)$, inside the target tumor model at r = P2 (illustrated into Fig. 5) into one graph

6 Conclusion

A mathematical model has been created using information from previous research on photoacoustic (PA) imaging to simulate how PA signals are generated and travel through biological tissues. The model was examined using a time-dependent study in COMSOL Multiphysics, a simulation platform. The results of this simulation can be used to understand better PA signal generation and propagation in a virtual environment, which can assist with both in-vitro and in-vivo studies. The study's analytical and theoretical findings confirm that the proposed mathematical model is valid for the numerical simulation of biomedical PA signals using platforms such as COMSOL Multiphysics. This research will be beneficial for researchers who are new to studying, modeling, and simulating different biomedical photoacoustic algorithms, prototypes, and systems.

Appendices

Appendix-A:

1.1 Deriving Eq. (1):

We can drive Eq. (1) from the basic gaussian distribution formula (https://en.wikipedia.org/wiki/Full_width_at_half_maximum; Smith 1997–1999) as follow:

$$f \left(r, t\right)=f \left(t\right)=\frac{1}{\sigma \sqrt{2\pi }}\mathrm{exp }\left[-\frac{{\left(t-\mu \right)}^{2}}{2{\sigma }^{2}}\right]$$

(A-1)

where; $\sigma$ is representing the standard deviation.

r is representing the location in Cartesian coordinates x, y, z.

t is representing the time (x-axis).

$\mu$ (Or: t₀) is representing the mean of the full bell shape’s actual data distributed on the x-axis (here: time axis), in other words it is equivalent to the center time value of the pulse shape (t₀).

The variable (r) in equation (A-1) representing the location in the three Cartesian coordinates x, y, z, the main importance of r is that it determines exactly the point source that the tissue will be irradiated from.

Starting from equation (A-1):

From (A-1), we can notice that the maximum value of f (r, t) = f_max = $\frac{1}{\sigma \sqrt{2\pi }}$ (at: t = $\mu$  = t₀), and f (r, t) = f_max/2 at time t = t₀ ± FWHM/2; then at time t = t₀ + FWHM/2, f (r, t) = $0.5*\frac{1}{\sigma \sqrt{2\pi }}$, hence:

$$0.5*\frac{1}{\sigma \sqrt{2\pi }}=\frac{1}{\sigma \sqrt{2\pi }}\mathrm{exp }\left[-\frac{{\left({t}_{0}+ \frac{FWHM}{2}-{t}_{0}\right)}^{2}}{2{\sigma }^{2}}\right]$$

Leading to:

$$0.5 =\mathrm{exp}\left(-0.125*{\left[\frac{FWHM}{\sigma }\right]}^{2}\right)$$

(A-2)

Taking the natural logarithm for the two sides for the equation (A-2) leading to:

$$\mathrm{ln}\left(0.5\right)= - 0.125\mathrm{ * }{\left[\frac{FWHM}{\sigma }\right]}^{2}$$

(A-3)

From the logarithmic rules (Stroud and Booth 2013):

ln (0.5) = ln ($\frac{1}{2}$) = ln (1) – ln (2) = 0 – ln (2) =—ln (2).

then equation (A-3) will become:

$$-\mathrm{ln}\left(2\right)= - 0.125\mathrm{ * }{\left[\frac{FWHM}{\sigma }\right]}^{2}$$

(A-4)

From (A-4), then:

$$FWHM = \sqrt{8ln2}* \sigma=2 \sigma \sqrt{2ln2}\approx 2.355 \sigma$$

(A-5)

Leading to: τ_p ≈ FWHM = 2 $\sigma \sqrt{2ln2}$ $\approx 2.355 \sigma$, then:

$${\rm So}, \sigma \approx \frac{1}{2 \sqrt{2ln2}}{\varvec{\uptau}}\mathbf{p}\approx \frac{1}{2.355} \boldsymbol{\tau_{p}}$$

(A-6)

For a peak value normalization (making the peak value equal to one) for equation (A-1), we will divide the right-hand side on f_max (= $\frac{1}{\sigma \sqrt{2\pi }}$), and substituting for $\sigma \approx \frac{1}{2.355}$ τ_p from (A-6) then; the peak value normalized f (r, t) in τ_p instead of $\sigma$ will be Sn (r, t) where:

$$\begin{gathered} {\mathbf{Sn}}\user2{ }\left( {{\varvec{r}},{\varvec{t}}} \right) = \exp - { }\left[ {\frac{{\left( {t - t_{0} } \right)^{2} }}{{2\left( {\frac{{{\mathbf{\tau_p}}}}{{2 \sqrt {2ln2} }}} \right)^{2} }}} \right] = {\text{exp }} - 4\ln 2{ }\left[ {\frac{{\left( {t - t_{0} } \right)^{2} }}{{\left( {{\mathbf{\tau_p}}} \right)^{2} }}} \right] \hfill \\ {\mathbf{Sn}}\user2{ }\left( {{\varvec{r}},{\varvec{t}}} \right) \approx {\text{exp }} - 2.77{ }\left( {\frac{{t - t_{0} }}{{{\mathbf{\tau_p}}}}} \right)^{2} \hfill \\ \end{gathered}$$

(A-7)

From (A-7) and multiplying by the maximum input laser light energy rate per unit volume $\mathrm{Ipmax}$, hence: $\mathbf{S}\mathbf{n}\boldsymbol{ }\left({\varvec{r}},{\varvec{t}}\right)\boldsymbol{ }\mathrm{will leading to }(\mathbf{A}-8)$ which is identical to Eq.(1):

$${\varvec{S}} \left({\varvec{r}},{\varvec{t}}\right)= \mathbf{I}\mathbf{p}\mathbf{m}\mathbf{a}\mathbf{x}* {{\varvec{e}}}^{- 2.77 {\left(\frac{{\varvec{t}}-{{\varvec{t}}}_{0}}{{{\varvec{\tau}}}_{{\varvec{p}}}}\right)}^{2}}$$

(A-8)

Appendix-B:

2.1 Clarification Tutorial on COMSOL programming of the proposed mathematical model presented in Fig. 4:

Coefficient Form PDE and Wave Equation modules in COMSOL have been utilized in programming (2), and (6), respectively; while, (1) has been programmed to be input point source to (2). Note: the output solution (dependent variable) of (2) is the intensity distribution in location and time $\boldsymbol{\varphi }\left({\varvec{r}},{\varvec{t}}\right)$ W/m² and it is utilized to complete the input source formula of the next last stage of the proposed mathematical model (6).

In COMSOL, there is the following equation:

$${e}_{a }\frac{{\partial }^{2}u}{\partial {t}^{2}}+ {d}_{a }\frac{\partial u}{\partial t}+ \nabla \cdot \left(-c\nabla u- \alpha u+ \gamma \right)+ \beta \cdot \nabla u+au=f$$

(B-1)

This equation (B-1) called in COMSOL: “Coefficient Form PDE (c)”, found under “PDE Interfaces”, found under “Mathematics”, found under “Add Physics” section, found in “Physics” tab found into COMSOL Multiphysics 6.0.

To program (B-1) in COMSOL to apply (2), the terms.

${e}_{a }, {d}_{a }, c, \alpha , \gamma , \beta , a, and f$ must be defined as follow:${e}_{a }=0; \alpha = \gamma = \beta =0, 0, 0$; ${d}_{a }=$ n/C; c = D;$a= {\mu }_{a}$; and finally the source term from (1) was applied as a “Point Source” at position P1 as shown into Fig. 5, (Right click on Coefficient Form PDE (c) founded under the model builder section, then: “Points”, then: “Point Source”): ${\varvec{f}}=$ $\mathbf{I}\mathbf{p}\mathbf{m}\mathbf{a}\mathbf{x}\mathbf{*}\mathbf{e}\mathbf{x}\mathbf{p}(-2.77\mathbf{*}((\mathbf{t}-\mathbf{t}0)/\mathbf{t}\mathbf{p})^2)$ Where: t0 = t₀, tp = ${\tau }_{p}$, and t is the study time instants from 0 to 150 ns.

Also, in COMSOL, there is the following equation:

$${e}_{a }\frac{{\partial }^{2}p}{\partial {t}^{2}}+ \nabla \cdot \left(-c\nabla p\right)=f$$

(B-2)

This equation (B-2), called in COMSOL: “Wave Equation (waeq)”, found under “Classical PDEs”, found under “Mathematics”, found under “Add Physics” section, found in “Physics” tab found into COMSOL Multiphysics 6.0.

To program (B-2) in COMSOL to apply (6), the terms.

${e}_{a }, c, and f$ must be defined as follow: ${e}_{a }=1/ ( {v}_{s} (\mathrm{target}) ) ^2;$ c = 1; and finally the source term f was applied as a “Source” for the tumor sphere domain founded inside the breast cylinder as shown into Fig. 5, (Right click on Wave Equation (waeq) founded under the model builder section, then: “Source”), where, from (14), $f=$ ( ${\mu }_{at}$ (target) $*$ ut $* \beta$ (target)) / ${C}_{P}$ (target)). Where: ut (in COMSOL language) = $\frac{\partial u}{\partial t}$, and t is the study time instants from 0 to 150 ns.

 The time step, relative tolerance, and mesh size applied in the numerical simulation are as follow:

Start time: 0 ns.

Step time: 1 ns.

Stop time: 150 ns.

Relative tolerance: 0.01.

Maximum element size: 0.15 mm.

Minimum element size: 0.028 mm.

Maximum element growth rate: 1.6

Curvature factor: 0.7

Resolution of narrow regions: 0.4