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# Tracing the Light—Introduction to the Mathematics of BioLuminescence Tomography

Communicated by *Notices* Associate Editor Reza Malek-Madani

## 1. BioLuminescence Tomography

On a tranquil summer night, the air carries the sweet fragrance of blooming flowers, mingled with the soft rustle of grass in a meadow. The moonlight casts a gentle glow, illuminating the serene ambiance. Amidst this nocturnal scene, fireflies are dancing and playing. These tiny, luminous creatures orchestrate a mesmerizing spectacle, transforming the darkness into an ethereal glow.

This natural light show has captivated the imagination of poets and writers for its aesthetic beauty in various literary works. Interestingly, the phenomenon goes beyond aesthetics. The principle behind fireflies’ illumination has inspired researchers to harness and manipulate this natural phenomenon for scientific benefits. The idea has led to the development of a cutting-edge technology in biomedical imaging known as *BioLuminescence Tomography (BLT)*.

Bioluminescence is the process of light emission in living organisms. The phenomenon occurs widely in nature, with typical examples including fireflies, jellyfish, and certain types of fungi (Figure 1). These creatures illuminate as they carry DNA that encodes luminescent proteins, and these proteins emit visible light when they undergo specific biochemical reactions. In 2018, the Nobel Prize in Chemistry was awarded to three researchers: Osamu Shimomura, Martin Chalfie, and Roger Tsien for their discovery and development of a glowing jellyfish protein known as the green fluorescent protein. The green light has since played crucial roles in biomedical research, enabling scientists to track how cancer tumors form new blood vessels, how Alzheimer’s disease kills brain neurons, and how HIV-infected cells produce new viruses.

**Figure 2**.

The electromagnetic spectrum.

The term “tomography” is derived from the Greek words *tomos* meaning “slice” or “section,” and *graphia* meaning “writing” or “drawing.” Tomography is an imaging technique that enables non-destructive visualization of objects by acquiring cross-sectional images. In a tomographic imaging process, the object is positioned within an imaging device that can capture multiple cross-sectional images of the object from different angles. These images are processed by computers to reconstruct three-dimensional representations of the internal structures.

BLT operates on the fundamental idea of utilizing bioluminescent sources to trace and visualize biological processes at the cellular level. Bioluminescent sources are typically cells that have been genetically engineered to express bioluminescent proteins. In a BLT experiment, researchers inject bioluminescent sources into biological tissue. Following the injection, the tissue is placed in a dark environment to minimize external light interference. Optical detectors or cameras are positioned around to capture the bioluminescent light emitted from within. When the sources are excited, they undergo biochemical reactions and illuminate. The light illumination is recorded by the optical detectors and utilized for computing the spatial distribution of the bioluminescence source. The distribution serves as effective biomarkers for biological processes of interest. BLT shows significant potential in the context of cancer diagnostics: By introducing bioluminescent sources specific to cancer cells, this technology enables dynamic imaging to monitor the progression of malignant cells.

This introductory essay endeavors to guide readers through some fascinating mathematics within BLT. The focus lies on formulating the problems and outlining key ideas for their solution without delving into technical details. The exploration encompasses modeling light propagation in biological tissue using an integrodifferential equation in Section 2, analyzing the well-posedness of the forward modeling in Section 3, investigating inverse problems arising in the imaging process of BLT in Section 4, as well as discussing several related contemporary research areas in Section 5.

## 2. Light Propagation in Biological Tissue

The illumination utilized in BLT generally falls within the visible light spectrum. Visible light is a narrow band of the entire electromagnetic (EM) spectrum. EM radiation is a form of energy that travels through space in the form of waves. Examples include radio wave, microwave, visible light, X-ray, and gamma ray (Fig. 2). The elementary particles of EM radiation are *photons*. Each photon carries energy that is inversely proportional to the wavelength of the EM radiation, with shorter wavelengths corresponding to higher photon energy (Fig. 2). Photons interact with biological tissue primarily through two processes: absorption and scattering (Fig. 3).

Absorption is the process in which photons are absorbed by atoms, molecules, or particles in a medium. The absorbed photons elevate electrons in the absorber from ground states to excited states. In the meanwhile, loss of photons causes graduatal reduction of the intensity of EM radiation as it propagates in the medium. A medium’s ability to absorb photons is quantitatively characterized by its *absorption coefficient* which is defined as the probability of photon absorption per unit path length. The representative value of , in biological tissue is *absorption mean free path*.

Scattering is the process in which photons change the direction of propagation after interacting with small particles in a medium. The amount of scattering depends on wavelength of the EM radiation as well as size and structure of the medium. Scattering redirects photons, causing a diffused spread of EM radiation. A medium’s ability to scatter photons is quantitatively characterized by its *scattering coefficient* which is defined as the probability of photon scattering per unit path pength. The representative value of , in biological tissue is *scattering mean free path*.

**Figure 3**.

Light absorption and scattering.

### 2.1. Radiative Transfer Equation

We henceforth denote biological tissue by a bounded open convex subset with smooth boundary The dimensions of practical significance are . (2D) and (3D), but the mathematical framework works equally well in other dimensions.

The fundamental quantity for light propagation is *radiance*, defined as the photon energy per unit normal area per unit solid angle. It is a measure of the photon intensity at a point in a particular direction. The distribution of radiance is generally dynamic as photons propagate. However, the processes of interest in BLT typically evolve on timescales much longer than the rapid propagation of light. It is thus valid to simply consider a stationary distribution of radiance. We denote the radiance at in the direction by where , is the unit sphere. As a result, the rate of change of the radiance in the -dimensional is naturally modeled by the directional derivative -direction where is the spatial gradient.

The variation in radiance at in a particular direction primarily results from three factors: 1. loss of photons by absorption and scattering; 2. gain of photons by scattering; and 3. gain of photons by bioluminescent emission. These effects are modeled as follows: 1. Loss of photons by absorption and scattering is proportional to the photon density, with the proportional factors and respectively. If we write , then the total loss is , 2. Gain of photons by scattering is proportional to the photon density that is scattered to the direction . from other directions. Let be the probability of photons in the being scattered to the -direction then the gain of photons due to scattering is -direction, where is the (normalized) spherical measure on 3. Gain of photons by bioluminescent emission is due to the presence of the bioluminescent source . All these factors combined lead to the following .*Radiative Transfer Equation (RTE)* on that dictates light propagation in biological tissue:

Here is known as the *scattering kernel*. As a probability density function, it is non-negative and satisfies We will refer to . as *optical parameters*. These parameters along with the bioluminescent source are allowed to be spatially varying (that is, in the model to capture inhomogeneity of the medium. In applications, the following spatially-invariant Henyey-Greenstein scattering kernel has proven to be useful in approximating the angular scattering dependence of single scattering events in biological tissue: -dependent)

where the constant

### 2.2. Boundary condition

In BLT, the bioluminescent sources are cells that have been genetically engineered to express bioluminescent proteins. No external light source is imposed to prevent contamination of the internal light source. As a result, there is no radiance flowing into the tissue from the boundary. Note that the RTE holds on *outgoing boundary* and *incoming boundary*, respectively. The fact that no radiance flows into the tissue translates to the boundary condition

**Figure 4**.

A numerical solution to the boundary value problem 1 2 on the 2D unit square

( a)

( b)

## 3. The Forward Problem and Structure of Solutions

The RTE provides a mathematical framework for understanding light propagation in biological tissue. In particular, if all the optical parameters

Introduce

Here

Therefore, solving for the RTE solution

### 3.1. Non-scattering media

When photon scattering in a medium is relatively weak, we can neglect the scattering effect by taking

In a non-scattering medium, we have

where

**Figure 5**.

Definition of

### 3.2. Scattering medium

The calculation above shows that

where

This representation immediately implies that the integral equation 5 admits a unique solution, and the solution depends continuously on the source *sub-critical conditions*

**Figure 6**.

Selected photon trajectories in the presence of scattering. The black/blue/red dotted lines represent photon trajectories that experience no/single/multiple scattering before contributing to the radiance at

## 4. The Inverse Problem and Integral Transforms

Given the optical parameters

The operator *source-to-data map*. As a result, the imaging problem in BLT can be mathematically formulated as inverting the source-to-data map. This type of problem, where the goal is to recover the cause (i.e., the bioluminescent source) from an observed effect (i.e., the outgoing radiance), is common in science and engineering. Such problems are known as *inverse problems*, in contrast to “forward problems” where the cause is given and the task is to simulate or predict observations. For example, solving the RTE boundary value problem with specified optical coefficients and bioluminescent sources, as discussed in Section 3, is a forward problem. This section focuses on the inverse problem in BLT. Along the journey of investigation, we will make brief detours to explore mathematical models for a few other notable medical imaging methods. These models, which can be viewed as simplified versions of the BLT model, hold both practical significance and mathematical interest on their own. Throughout this section, we assume the optical parameters

### 4.1. Non-scattering media

#### Non-absorbing non-scattering media

Let us begin with the idealized scenario where neither absorption nor scattering occurs, that is

Here *X-ray transform*. The inverse problem in BLT in a non-absorbing, non-scattering medium reduces to inverting the X-ray transform.

**Figure 7**.

The line of integration in the definition of

The term “X-ray transform” might seem strange in the context of BLT, where no X-rays are involved. Indeed, this name originates from another medical imaging modality *Computed Tomography*, or simply CT. CT images biological tissue utilizing X-rays, a type of EM radiation with wavelength in the range of *sinogram*, see Fig. 8(b) for an example. Given that the drop in X-ray intensities is solely attributed to absorption, the sinogram consists precisely of line integrals of the absorption coefficient ^{1} Radon’s investigation of the inverse problem was earlier than the invention of CT scanners and was mostly driven by mathematical considerations. Nevertheless, his mathematics turns out to play a crucial role in the theory of CT imaging. Interested readers are referred to the monographs

^{1}

The definitions of these two transforms diverge in 3D or higher dimensions, where the X-ray transform refers to integration over lines while the Radon transform refers to integration over hyperplanes.

We sketch the idea to invert the X-ray transform. First, it is easy to show using the Cauchy-Schwarz inequality that

In words, *back-projection*.

Now that we have two operators:

where

This is the inversion formula obtained by Radon. From the perspective of signal processing, the operator *filtered back-projection*.

This filtering process includes the differentiation *ill-posed*, meaning that the solution

**Figure 8**.

Numerical inversion of the X-ray transform

( a)

( b)

( c)

( d)

#### Absorbing non-scattering media

This case corresponds to

This gives rise to another integral transform *attenuated X-ray transform*. The BLT inverse problem in an absorbing, non-scattering medium thus requires inversion of the attenuated X-ray transform.

This inverse problem, although motivated by the discussion of BLT here, rises earlier in another medical imaging modality *Single Photon Emission Computed Tomography (SPECT)*. In SPECT, small doses of radioactive tracers that are able to emit single photons (e.g., Xenon-133) are injected into patients. The choice of the tracers depends on the specific organ or function being studied. These tracers accumulate in target tissues and emit gamma rays as they undergo radioactive decay. The detectors are rotated around the tissue to record the outgoing gamma rays. This data is then processed to create cross-sectional images of the distribution of the tracers. The spatial distribution of the gamma-ray radiance in SPECT can be effectively modeled by the same RTE 1 and the boundary condition 2, where the internal source

Inversion of the attenuated X-ray transform presents greater challenges compared to its counterpart for the classical X-ray transform. In particular, the inversion formula must account for the exponential decay of X-ray intensity. Various inversion formulae have been derived in the literature by building insightful connections with other mathematical theories. Notably,

### 4.2. Scattering media

We are ready to invert the full source-to-data map

The first term

We now view the operator

This is an explicit inversion formula that recovers the bioluminescent source

While the inversion holds mathematical validity, the illposedness remains due to the application of the unbounded operator

Ill-posedness is ubiquitous in inverse problems, and various methods have been developed to mitigate it. A generic class of techniques is *regularization*. Regularization stabilizes solutions of ill-posed problems by incorporating prior knowledge about the solution into the mathematical formulation. The integration of prior knowledge is achieved by adding a penalty term to discourage unlikely or unrealistic solutions. Typical penalty terms include constraints on the solution, assumptions about its smoothness, or expectations regarding characteristics of the solution. This additional penalty contributes to a more well-posed problem, reducing sensitivity to noise and guiding the inversion process towards more stable solutions. The idea of regularization has also found broad applications in machine learning for model complexity reduction and overfitting prevention.

## 5. Related Topics

The preceding sections provide an overview of both the forward and inverse problems in BLT, with a concise presentation summarizing the main ideas and results. However, there are crucial topics that are not covered in the outline. In this section, we delve into three topics concerning simplification and generalization of the BLT model. These topics remain vibrant research areas and give rise to even more intriguing mathematical questions.

### 5.1. Diffusion approximation

The RTE provides an accurate modeling of light propagation when the transport mean free path of photons is at the same order as the characteristic length of the medium. Nevertheless, the mean free path of photons in biological tissue is typically much shorter than the characteristic length. Light propagation in biological tissue is thus predominantly governed by diffusion, allowing for an effective approximation of the RTE using a diffusion equation

subject to a homogeneous Robin boundary condition *diffusion coefficient*, and *absorption coefficient*. The diffusion approximation is derived using the ansatz that the radiance is linear in the angular variable

The diffusion approximation offers remarkable advantages in terms of computational efficiency and simplicity. For instance, the 3D RTE depends on three spatial variables and two angular variables, making numerical computation prohibitively expensive due to its high dimensionality. In contrast, the 3D diffusion approximation is a standard second-order elliptic equation for which various fast numerical solvers are readily available. However, the diffusion approximation also suffers a clear weakness: the BLT inverse problem becomes under-determined as the unknown source

### 5.2. Ultrasound modulated BLT

In BLT, photons emitted from a bioluminescent source undergo significant scattering in biological tissue, which blurs directional information and makes it challenging to accurately locate the source. Consequently, conventional BLT often suffers from limited spatial resolution. An emerging approach to overcome this limitation, known as *Ultrasound Modulated BLT (UMBLT)*, integrates BLT with ultrasound modulation. In UMBLT, ultrasound waves are used to manipulate the optical properties of the medium. The frequencies in use are approximately 1–5 MHz, which provide a good compromise between axial resolution and penetration depth. The interaction of ultrasound with biological tissue alters the paths of the photons, leading to more controlled redirection and increased spatial resolution.

Asymptotic analysis

where

### 5.3. Riemannian RTE

Biological tissues often have complex structures that cannot be effectively modeled by Euclidean geometry. For instance, the scattering and absorption coefficients in structured tissue may not only vary spatially but also depend on direction. In such cases, Riemannian geometry allows for the incorporation of directional information, providing a more comprehensive representation of the optical properties of tissues. The BLT formulation on an

subject to the the boundary condition 2. Here,