# The Optical Inverse Problem in Quantitative Photoacoustic Tomography: A Review

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Fundamentals

#### 2.1. Generation of Initial Pressure Rise

^{−1}). Combined with the photothermal conversion, the above equation can be rewritten as [11,28]:

^{−1}), and $\Phi $ is local optical fluence (J· cm

^{−2}). H is the absorbed energy density, derived from the product of the fluence and the absorption coefficient. $\Gamma $ is the Grüneisen parameter (dimensionless) and ${\eta}_{th}$ denotes the efficiency (dimensionless) of converting light into heat energy, both of which are nearly uniformly distributed within biological media. Hence, for simplicity, a common assumption is that the energy density obtained by photon absorption is considered equivalent to the initial pressure and then used as initial data for the optical inversion, which implies the parameters are negligible. This assumption is widely adopted in the later-mentioned studies. Notably, it is found in Equation (2) that ${p}_{0}$ is not only determined by ${\mu}_{a}$ but also depends on $\Phi $, which provides a theoretical insight for the spectral coloring problem mentioned in the next section.

#### 2.2. Photoacoustic Tomography-Based Concentration Measurement

#### 2.3. Spectral Coloring

## 3. Methods for the Optical Inverse Problem

#### 3.1. Forward Model-Based Methods

#### 3.1.1. Fluence Correction Based on Prior Knowledge

#### 3.1.2. Model Fitting Methods

#### 3.1.3. Fixed-Point Iteration Methods

#### 3.1.4. Minimization-Based Methods

**(i) Research for implementing the minimization framework**. To perform the forward model-based minimization, optimization algorithms play a key role within the framework, as illustrated in Figure 4, which offers the update vector of iterative variables. Two types of algorithms that merit mention are the gradient-based and Jacobian-based approaches. The former primarily refers to the limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm (L-BFGS) [25,59,60,61,62], which is a quasi-Newton method. Quasi-Newton methods aim to provide a super-linear convergence speed close to Newton’s method with less computational effort by using the secant method to approximately update the Hessian matrix, instead of calculating it from scratch [63]. On this basis, the popular L-BFGS further utilizes gradient vectors of successive iteration points to directly update the inverse of the Hessian matrix in a two-loop recursion manner, which, hence, is less computationally intensive and highly memory-efficient due to only necessitating the calculation and storage of gradients, especially for large-scale problems. On the other hand, the Jacobian-based optimization strategy pertains primarily to the Gauss–Newton method [26,64,65], which requires the explicit computation and storage of the Jacobian matrix. Although this approach is more storage-intensive, it has fewer computation processes when compared to gradient-based methods.

**(ii) Research for enhancements for the minimization framework.**While the minimization framework can be conducted based on the above approaches theoretically, a successful optical inversion with desired results demands further research. One of the most significant obstacles is the non-uniqueness problem that arises from the highly stochastic nature of optical interactions. Hence, tissues with distinct optical properties might result in the same simulated distribution, leading to the possibility of converging to a wrong solution. A commonly adopted strategy for addressing the non-uniqueness is to incorporate a regularization term into the objective function. Tikhonov regularization is frequently employed to achieve solutions with desirable smoothness properties [75]. Alternatively, total variation regularization has been demonstrated to effectively remove false details while preserving sharp contrast at tissue edges or boundaries, making it particularly suited for the piece-wise constant characteristic of biological tissues [22,76,77]. Some more efforts have been made for the improved performance of total variation by featuring it with directional sensitivity [78,79,80], which mitigates the excessive smoothness and recovers directional textures.

#### 3.2. Fluence Correction with Assisted Techniques

#### 3.2.1. Fluence Correction with Diffusion-Based Techniques

#### 3.2.2. Fluence Correction with Acousto-Optic Theory

#### 3.2.3. Fluence Correction with Passive Ultrasound

#### 3.3. Data-Driven Methods

#### 3.3.1. Methods Based on U-Net

#### 3.3.2. Dataset Acquisition

#### 3.4. Decomposition-Based Methods

## 4. Discussion

**Forward Model-Based Methods.**The first reviewed forward model-based methods rely on mathematical models to describe the underlying physical process of data acquisition and generate simulated counterpart data. For the fluence correction approaches that only require computing the forward operator once, it can yield improved quantitative results with less time consumption, especially suitable for cases with lower accuracy demands. However, the effectiveness of the methods is highly dependent on the prior knowledge of optical and anatomical parameters that cannot be determined for in vivo scenarios, which hinders practical applications. In the future, the research focus is supposed to lie in iterative modeling frameworks, especially minimization-based methods. Albeit with a compromised imaging speed, the minimization-based framework has demonstrated great potential for clinical translation due to its broad applicability and minimal restrictions. However, before that, the non-uniqueness involved, which presents a significant obstacle, necessitates more research efforts. Moreover, there are two common challenges for all forward model-based methods. Firstly, there is a trade-off in the light propagation model selection, as an improvement in modeling accuracy may lead to an increase in time consumption. In this regard, the high-accuracy Monte Carlo simulation technique, which can leverage the computational power of rapidly evolving graphic processing units (GPUs) for high-speed computation, holds significant potential [135,136]. Furthermore, the accuracy of forward model-based methods is largely contingent on a thorough understanding of the experimental setup, e.g., the specific type and location of light sources. In practical applications, acquainting and reproducing these conditions in sufficient detail for simulations can be complicated and exceedingly time-consuming, especially when experimental conditions are subject to change during measurements. Consequently, there is a pressing need for research aimed at improving the robustness of such methods in the face of incomplete or imperfect knowledge of the experimental configurations.

**Fluence Correction with Assisted Techniques.**Section 3.2 provides an overview of methods that utilize alternative techniques to produce fluence distributions. Although the inherent complexities associated with the PA field are circumvented, several limitations still exist. The effect of diffusion-based techniques suffers from low spatial resolution, because of the high light scattering [101,104]. As for the acousto-optic theory, the scanning point density determines the fluence resolution. However, an increased scanning point density comes with the cost of a proportional reduction in imaging speed [110]. It is anticipated that the evolution of these techniques will further facilitate addressing these issues. In addition, due consideration ought to be given to emerging theories. On the other hand, this category of methods commonly requires additional devices in a standard PA imaging system. Hence, it is crucial to pay enough attention to integrating these devices into a compact and effectively coupled system. Furthermore, Nykänen et al. [90] have proposed a promising approach that warrants exploration, wherein the data acquired through PAT and DOT are combined in a joint minimization problem. This integration appears to generate a synergistic effect that can ameliorate the inherent limitations associated with each technique.

**Data-Driven Methods.**The emerging data-driven approaches are mentioned in Section 3.3. The advantage of these methods is that they do not rely significantly on prior knowledge and do not require constructing a specific physical model. Classical machine learning methods require human-designed feature extraction algorithms that are intractable and inefficient, especially for highly complex problems such as the optical inverse problem. Thus, preeminent deep learning algorithms are underscored which adopt deep neural networks containing multilevel nonlinear mappings that enable adaptive feature extraction and representation learning without human intervention. The U-Net has exhibited exceptional performance in its initial applications, thereby attracting significant research attention and boosting a flurry of related results. It is reasonable to conjecture that the U-Net will continue to be a dominant architecture for some time. Hence, there is a feasible and valuable research direction that involves generating improved variant networks based on the U-Net. Two promising frameworks that merit consideration in this regard are the multi-input U-Net and the multi-task U-Net. Networks fed with multiple inputs, such as the Y-Net [127] and the ultrasound-enhanced U-Net [126], have demonstrated the ability to outperform the U-Net with a single input (the PA image) because the multifaceted information ensures that the network performs a more comprehensive feature extraction. As for multiple-task U-Net, two U-Nets are frequently utilized to address the optical inverse problem alongside related tasks, e.g., vessel segmentation [118,123] and fluence estimation [124]. Through such an integrated framework, synergies between multiple tasks can be harnessed to attain superior outcomes. Notably, the practical implementation of deep learning techniques still faces several challenges that necessitate further consideration. The first and most important point is that the absence of dependable in vivo optical property measurement techniques poses a significant challenge in generating abundant labeled data required for network training. Numerical biological phantoms can serve as a viable alternative to produce sufficient data, but their utility is hindered by the domain gap issue [128]. Recent research using GAN-based simulated data realism enhancement has demonstrated promising potential in addressing the domain gap problem, though further comprehensive validation is necessary to ascertain its efficacy [124]. In addition, there is an urgent need to establish standardized datasets to facilitate the uniform validation of proposed networks in studies, which makes it intuitive for the concerned researchers to conduct performance comparisons [116]. Moreover, a trained network is typically tailored to a specific training dataset that corresponds to a particular situation, thereby limiting its applicability when system settings or scenarios differ. In this context, augmenting the generality of the network represents an essential research direction, where the transfer learning technique is a possible solution for achieving this goal.

**Decomposition-Based Methods**. The last classification of methods takes advantage of decomposition techniques to express related variables by a collection of basis functions founded on simplified assumptions. These intelligent methods often enable satisfactory results with less computational effort. Nonetheless, their practicality is rather restricted owing to underlying assumptions. Therefore, the theoretical validity of the linear representation and the adequacy of the incomplete collection of basis functions need further research. Specifically, the eigenspectral theory operates under the assumption that the light propagation within media is chiefly affected by several constituent chromophores, such that a finite number of basis spectra associated with these chromophores can be extracted to represent the transmitted spectrum [132]. However, the assumption may not hold for in vivo applications, where the interactions between light and tissues are intricate. In addition, the basis spectra are derived from the spectra sampled at discrete grids on the image via PCA. The grid selection determines the applicability of the basis spectrum to the whole image. Therefore, the efficient selection of sampled spectra warrants further research.

**From a global perspective.**A general issue is that most of the findings and conclusions in this area are based on simulation results. More experimental validations should be performed to examine the clinical applicability of methods with the tissue property heterogeneity and all other measurement factors taken into consideration [114,116]. For the entire process of qPAT, it is worth noting that the acoustic inversion and spectroscopic inversion also play a key role in achieving a complete concentration estimate for qPAT, despite being beyond the scope of this paper. The high-quality PA image is a prerequisite for accurate qPAT. In most of the methods discussed in this paper, the PA image is assumed to be well-established and error-free. However, in practical applications, factors such as finite characteristics of detection and system noise can cause artifacts and errors in the PA image [27,91,137]. Zuo et al. [94] recently proposed the concept of spectral crosstalk that indicates a mutual interaction exists between the reconstructed spectra of two arbitrary pixels in PA images, inducing spectral distortions. Based on this observation, methods that can process multiple wavelength images at once may be more effective because of the ability to consider information on spectral profiles. On the other hand, even after performing the optical inversion, a certain level of fluence-induced residual error exists. Therefore, the commonly used linear unmixing algorithm might still produce undesirable results. In this context, some sophisticated unmixing algorithms can be utilized to process the results from the optical inversion, e.g., independent component analysis that shows robustness to the residual spectral error [18]. Furthermore, if computing power is sufficient, it seems more reasonable to solve the three inversion problems in one step, so that all the comprehensive information of the whole process can be considered together, which can effectively avoid the errors arising from the individual steps and eventually superimposed on the quantitative results through error propagation.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Diagram of the entire process for quantitative photoacoustic tomography. Colors indicate processes at different wavelengths. Obviously, three successive inversions are included.

**Figure 3.**Schematic diagram of the complete procedure for optical forward modeling. Circles with a cross indicate a pixel-wise multiplication operation.

**Figure 4.**Schematic diagram of the minimization procedure of Cox et al., as a typical framework in this category of methods.

Category | Number of Forward Modeling | Advantages | Major Limitations |
---|---|---|---|

Fluence correction methods [9,41,42,43,44,45,46,47,48,49,50,51] | Single | Easy implementation; little computational load. | Extremely high dependence on predefined tissue properties, both geometrical and optical. |

Model fitting methods [23,52,53] | Multiple | Certain applicability to unknown simple media; good computational efficiency. | Low accuracy due to the unrealistic optical homogeneity assumption. |

Fixed-point iteration methods [54,55,56,57,58] | Iterative | High accuracy; high capability for unknown absorption distributions. | Requiring specified scattering distributions. |

Minimization-based methods [59] | Iterative | Highest accuracy; high capability for all unknown optical property distributions. | Computationally intensive and time- consuming. |

Category | Key Processes | Advantages | Limitations |
---|---|---|---|

Forward modeling-based methods | Utilizing a forward model to generate simulated counterparts of related variables. | Abundant choices of available frameworks with distinct features; a logically simple understanding due to the high conformity to the underlying physical process. | High dependence on the performance of the used light propagation model; a strict requirement of adequate knowledge of the experimental configurations. |

Fluence correction via other techniques | Resorting to other techniques to measure the fluence map and correcting its impact. | Avoiding inherent complexity and limitations in the PA field. | Inherent drawbacks from the used assisted techniques; incorporating additional devices and procedures; compromising the system’s compactness. |

Deep learning methods | Training deep neural networks to produce desired distributions in an end-to-end manner. | Significantly less dependence on prior knowledge of the object tissue and related physics; high computational efficiency in the implementation phase. | The extensive demand for training data labeled with true values and the lack of reliable in vivo measurement techniques; low generality of trained networks to system configurations and target scenario. |

Decomposition-based methods | Decomposing related variables into a linear combination of a set of prescribed basis functions. | Producing acceptable results at a relatively less computational cost. | Very limited applicable cases due to the use of strong assumptions and the incomplete collection of basis functions. |

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Wang, Z.; Tao, W.; Zhao, H.
The Optical Inverse Problem in Quantitative Photoacoustic Tomography: A Review. *Photonics* **2023**, *10*, 487.
https://doi.org/10.3390/photonics10050487

**AMA Style**

Wang Z, Tao W, Zhao H.
The Optical Inverse Problem in Quantitative Photoacoustic Tomography: A Review. *Photonics*. 2023; 10(5):487.
https://doi.org/10.3390/photonics10050487

**Chicago/Turabian Style**

Wang, Zeqi, Wei Tao, and Hui Zhao.
2023. "The Optical Inverse Problem in Quantitative Photoacoustic Tomography: A Review" *Photonics* 10, no. 5: 487.
https://doi.org/10.3390/photonics10050487