## Fluorescence lifetime optical tomography with Discontinuous Galerkin discretisation scheme |

Biomedical Optics Express, Vol. 1, Issue 3, pp. 998-1013 (2010)

http://dx.doi.org/10.1364/BOE.1.000998

Acrobat PDF (9880 KB)

### Abstract

We develop discontinuous Galerkin framework for solving direct and inverse problems in fluorescence diffusion optical tomography in turbid media. We show the advantages and the disadvantages of this method by comparing it with previously developed framework based on the finite volume discretization. The reconstruction algorithm was used with time-gated experimental dataset acquired by imaging a highly scattering cylindrical phantom concealing small fluorescent tubes. Optical parameters, quantum yield and lifetime were simultaneously reconstructed. Reconstruction results are presented and discussed.

© 2010 OSA

## 1. Introduction

8. V. Y. Soloviev, J. McGinty, K. B. Tahir, M. A. A. Neil, A. Sardini, J. V. Hajnal, S. R. Arridge, and P. M. W. French, “Fluorescence lifetime tomography of live cells expressing enhanced green fluorescent protein embedded in a scattering medium exhibiting background autofluorescence,” Opt. Lett. **32**, 2034–2036 (2007).

## 2. Methodology

### 2.1. Inverse problem

*θ*. This approach can be easily generalized if necessary. Then, the variational problem is formulated as a minimization problem of the cost functional ℱ:

*u*and

_{θ}*v*are the model excitation and fluorescence energy densities, respectively, corresponding to the projection angle

_{θ}*θ*; the functions

*e*and

_{θ}*h*are experimentally measured excitation and fluorescence energy densities at the surface of the light scattering object, respectively. The function

_{θ}*ξ*(

*θ*) is introduced for convenience and for emphasizing similarity with the backprojection operator. It represents the source distribution, which for the case of point sources, as used in this paper, is

*N*is the number of source-camera positions. Similarly, the functions

*χ*and

_{θ}*ς*represent sampling of measurements in space and frequency

*M*is the number of discrete points on the imaged phantom's surface;

*L*denotes the number of samples in the Fourier domain (

*ω*); the vector

**r**

*denotes the surface points visible by the CCD camera corresponding to the excitation source at*

_{θm}**r**

*. Factors*

_{θ}*a*are surface areas around

_{m}**r**

*such that ∫ χ*

_{θm}*(*

_{θ}**r**)

*d*

^{3}

**r**gives the total visible area. The form of ℑ

*is chosen in order to simplify a variational procedure. Thus, the function χ*

_{θ}*allows to replace a sum over surface points visible by the CCD camera with a volume integral. Analogously, the function*

_{θ}*ς*replaces a sum over samples in the Fourier domain with an integral.

*and explicitly given by*

_{θ}*ψ*and

_{θ}*φ*are Lagrange multipliers satisfying the same boundary conditions as the energy densities. The Helmholtz operator Λ is given by

_{θ}*c*is the speed of light in the medium;

*ρ*is the power of the excitation source at

_{θ}**r**

*. The unknown reduced scattering and absorption coefficients are denoted as*

_{θ}*μ*′

*and*

_{s}*μ*, respectively. The function

_{a}*q*depends on unknown quantum yield,

*η*, the absorption coefficient and lifetime,

*τ*, in the medium as

*ημ*as “fluorescence efficiency”. The quantum yield,

_{a}*η*, is understood here as a fraction of two energy densities: the energy density of re-emitted fluorescent photons over the energy density of absorbed excitation ones. The energy loss due to the Stokes shift is already included into this definition.

*κ*, on the frequency,

*ω*. For low frequencies ∣

*ω*/

*cμ*′

*∣≪1 and*

_{s}*μ*′

*≫*

_{s}*μ*this dependence is negligible. However, for ∣

_{a}*ω*∣ above a few gigahertz and in presence of regions with relatively low values of

*μ*′

*this dependence becomes important.*

_{s}**x**= (

*μ*′

*,*

_{s}*μ*,

_{a}*ημ*,

_{a}*τ*)

*at every point of the domain and choose a dynamic form of the regularization term, ϒ(*

^{T}**x**), depending on

*k*-th iteration as

*x*is

^{i}_{k}*i*-th component of

**x**

*; and*

_{k}*α*are Tikhonov regularization parameters.

_{i}*f*(

_{i}*θ*,

**x**

*) are given by*

_{k}*θ*. Insertion of the function

*ξ*(

*θ*), Eq. (3), into Eq. (14) allows us to rewrite this equation in the form

*x*

^{i}_{0,k}=

*x*, and define a subsequence of

^{i}_{k}*k*-th iteration by letting

**x**

*depend on the projection angle*

_{k}*θ*as

_{s}*k*with samples in the Fourier domain. In this form Eq. (20) presents a variant of the Landweber-Kaczmarz method [29].

*with conditions ℒ*

_{θ}*[30]. Equations satisfied by adjoint energy densities,*

_{θ}*θ*, and frequencies,

_{n}*ω*. Parameters 1/

_{l}*α*in Eq. (20) are computed at each iteration step as described in [17]. Iterations are terminated when ℑ

_{i}*+ ϒ attains its global minimum. Notice, that this approach can be used even for one frequency. For example, for the time independent case (*

_{θ}*ω*= 0) the quantum yield and optical parameters can be reconstructed.

### 2.2. Implementation

*f*by solving Helmholtz and adjoint Helmholtz equations numerically. The direct solver employs the DG method, which is outlined below. Let us start with the Helmholtz equation written as a system of two first order equations:

_{i}*I*(

**s**·

**n**<0)=

*γI*(

**s**·

**n**>0) at the open boundary of the scattering domain, where

**n**is the surface normal;

**s**is the unit vector in a particular direction;

*c*

**q**is the energy flux; and

*γ*is a constant depending on the refractive index mismatch [33]. Furthermore, the computational domain is divided into cells, where the solution of Eqs. (21)–(22) is expanded over shape functions

*ϕ*(

_{i}**r**) satisfying the completeness condition Σ

*= 1 inside a cell. For the sake of computational performance we represent optical parameters as piecewise constant functions, following the Finite Volume framework, while all other functions are expanded over piecewise linear basis. Therefore, the energy density and the source term are represented as*

_{i}ϕ_{i}*ϕ*and integrating over the cell's volume we arrive at a weak formulation of the direct problem in the local form

_{j}*V*denotes the cell's volume having an outward normal

**n**, and

*∂V*denotes cell's interface. At this stage the local form, Eq. (25), consists of a system of uncoupled equations. Coupling between cells is provided by replacing

**q**with the interface flux, which is derived below.

*u*′ and

**q**′, denotes corresponding quantities in a neighboring cell. A sum of fluxes at cells' interfaces, Eqs. (26), together with the observation that

**n**= −

**n**′ results in

**q**] ·

**n**= 0 is satisfied for the exact solution and taking into account the flux equation

**q**= −

*κu*∇

_{i}*ϕ*Eq. (31) simplifies to

_{i}*w*, Eq. (26), in the form

_{j}*j*-th row and

*i*-th and

*i*′-th columns of the system matrix is

**q̂**= −{

*κu*∇

_{i}*ϕ*} in Eq. (32), which replaces

_{i}**q**in Eq. (26), is the so-called Bassi-Rebay interface flux [34]. The scheme with the Bassi-Rebay flux is stable for the polynomial interpolation of shape functions

*ϕ*of degree higher then 1. For a linear interpolation, the scheme is unstable and must be regularized. For this purpose we impose a set of constraints that the solution is continuous across the cell's interfaces, i.e. [

_{i}*u*] = 0, and add the following “zero-term” to the right hand side of Eq. (32):

*β*= {−1,0,1} and

*δ*∈ ℝ

^{+}are penalty parameters. In the same way as before we identify

*v*in this expression as:

_{j}**r**

*. Shape functions are chosen in the trilinear form*

_{i}*ξ, η, ζ*} ∈ [−1, 1]. In order to compute matrix elements we map the cell's interior and interfaces into reference domains, which are a cube and square, respectively. Jacobians of these transformations can be computed by employing quadrature formulae. However, semi-analytical expressions are used in our implementation, which involve computation of the following tensors

*u*and

_{i}*ρ*in Eq. (25) with their arithmetic averages

_{i}**q**becomes

**0**inside each cell and, therefore, is defined only on cells' interfaces, where, according to Eq. (22),

**n**·

**q**becomes a

*δ*-function. In the weak form

**n**·

**q**is found in terms of a jump of the energy density

*ū*across the cell's interface [35, 36], see Eq. (29). Both, FV and DG, belong to the family of so-called shock capturing schemes. DG solution space belongs to broken Sobolev's spaces [14] while the solution space of FV is the space of piecewise constant functions. Hence, both methods are well suited for handling discontinuities in parameters as well as in the solution itself [16].

## 3. Instrumentation

*et al*[39]. Our mapping procedure is different from those and, therefore, we briefly outline it in Appendix. As an illustration of our mapping method, recorded, corrected and computed images for the first projection angle at the excitation wavelength are shown in Fig. 2.

## 4. Results and Discussions

*f*from Eqs. (15)–(18). An integration of these functions over all projection angles serves as a diffusion analogue of the backprojection operator. However, unlike our previous approach [17], where this backprojection operator was computed for every frequency, we update fluorescent and optical parameters for every projection angle, Eq. (20). This noticeably improves the performance of the algorithm without significant difference in reconstruction results. The drawback of this improvement, however, is that the operator Λ, Eq. (6), must be updated together with optical parameters, i.e. for each projection angle, which was not needed previously. To avoid repeating computation of tensors [Eq. (40)–(43)], we store them in a file. Reconstruction starts with some initial guess where background values of

_{i}*μ*′

*and*

_{s}*μ*were chosen. However, any physically meaningful values of the quantum yield and lifetime can serve as the initial guess. Here we set

_{a}*η*= 0.001 and

*τ*= 0.01

*ns*initially everywhere in the computational domain except for the boundary, where

*η*= 0 and

*τ*= 0. Then, all parameters are updated iteratively according to Eq. (20). As an illustration of the algorithm, functions

*f*are shown in Fig. 3 for 3 projection angles at

_{i}*ω*= 500

*MHz*. Slices are taken at the middle of the phantom.

*O*(

*n*log

_{8}

*n*), where

*n*is number of terminal tree nodes, which is much cheaper operation than solving four linear systems.

*μ*′

*in*

_{s}*mm*

^{−1}; (ii) the absorption coefficient

*μ*in

_{a}*mm*

^{−1}; (iii) the fluorescence efficiency

*ημ*in

_{a}*mm*

^{−1}, and (iv) the lifetime

*τ*in nanoseconds. Each column displays slices at three different heights

*y*= 40, 50, and 60

*mm*. Two frequencies were used in reconstruction: 500

*MHz*and 750

*MHz*.

*η*can be estimated by dividing reconstructed fluorescence efficiency by

*μ*. It is seen that the reduced scattering coefficient was reconstructed relatively well. Its minimal value in the tube

_{a}*A*is about 0.45

*mm*

^{−1}at height

*y*= 40

*mm*, which is slightly higher than the true value 0.415

*mm*

^{−1}. The value of

*μ*′

*slightly increases with height achieving 0.52*

_{s}*mm*

^{−1}at

*y*= 60

*mm*. As usual, reconstruction artifacts are also present. Reconstruction of the absorption coefficient

*μ*is far less accurate. Its value in the tube

_{a}*B*is roughly 1.5 lower than it should. Thus, its maximum reconstructed value is 0.027

*mm*

^{−1}while the true value is 0.04

*mm*

^{−1}. Tubes

*A*and

*C*, which were filled with fluorophore, also appear in reconstruction. This is an expected results due to absorbing properties of fluorophores. Localization of the fluorescent efficiency appears to be relatively good. It is clearly seen that two tubes appear at height

*y*= 40

*mm*and only one at heights 50 and 60

*mm*. The value of the quantum yield is lower than expected. Thus, dividing

*ημ*by reconstructed

_{a}*μ*we obtain approximately [0.18;0.2] at the the center of the tube

_{a}*A*, while the true value is about [0.26;0.27]. The lifetime distribution

*τ*, the last column in Fig. 4, has quite high contrast and is well localized. Slices showing the lifetime have background value almost 0. Reconstructed lifetime values are close to the true value of Nile Blue fluorophore, which is 1.2

*ns*. Thus, (i) at

*y*= 40

*mm*the maximum lifetime is 1.17

*ns*; (ii) 1.29

*ns*at

*y*= 50

*mm*; and (iii) 1.14 at

*y*= 60

*mm*. Reconstruction in Fig. 4 shows reasonable errors in optical and fluorescent parameters. The reconstruction error in fDOT must be attributed to the ill-conditioning nature of the inverse problem and can be much higher than it is presented here. It is well-known that it is notoriously difficult to obtain perfectly correct values of parameters in turbid media. The same arguments apply for localization and, especially, for shapes of targets.

### 4.1. Finite volume method

*mm*

^{−1}at the middle of the tube

*A*at height 40, 50 and 60

*mm*, respectively. The absorption coefficient is almost correct for the tube

*B*at

*y*= 40

*mm*but decreases to 0.03

*mm*

^{−1}at

*y*= 60

*mm*. Two tubes

*A*and

*C*are well separated at height 40

*mm*and only one tube

*A*is present at heights 50 and 60

*mm*on fluorescence efficiency, as it should. The value of the quantum yield varies in the range [0.20;0.28], which corresponds to reality. Lifetime images does not show separation of tubes

*A*and

*C*at

*y*= 40

*mm*. However, the lifetime value varies in the range [1.0; 1.28]

*ns*, which is a reasonable result.

*μ*′

*,*

_{s}*μ*,

_{a}*ημ*and

_{a}*τ*, we have 4 distinct datasets: (i) real and imaginary parts of the excitation energy density; (ii) real and imaginary parts of the fluorescent energy density. If we consider distinct optical parameters for excitation and fluorescent light transport, then the non-uniqueness problem will appear again. The same argument applies to the case of multiple lifetimes. Therefore, our assumption is quite reasonable even though values of

*μ*′

*and*

_{s}*μ*can be noticeably different at different wavelengths, as happens in the visible part of the light spectrum. Finally, we would like to emphasize the importance of knowledge of the excitation amplitude, which is a complex number coming from the excitation source term

_{a}*ρ*, Eq. (21). As it was suggested previously [17], we use the least squares fit for estimating the absolute value of the amplitude and its phase. Fitting works well when the background optical parameters are known. However, in the case of unknown background values the non-uniqueness problem appears. That is because two additional unknown scalar values are introduced. Thus, starting with any physically meaningful values for background parameters it is always possible to find a reasonably good fit for the excitation amplitude. This obviously affects reconstruction. For instance, higher background value of

*μ*′

*results in increased value of reconstructed*

_{s}*ημ*, while incorrect phase affects reconstruction of

_{a}*μ*′

*and*

_{s}*τ*due to non-linearity of the inverse problem.

## Appendix

*I*is the intensity,

*ρ*is the optical thickness, and

*θ*is the polar angle. At the boundary we have

*I*(0,

*θ*) = 0 with

*θ*>

*π*/2. Next, average intensities,

*I*

_{1}and

*I*

_{2}, are introduced by integrating

*I*(

*ρ,θ*) sin

*θ*over

*θ*from 0 to

*π*/2 and from

*π*/2 to π, respectively. Therefore, Eq. (47) becomes

*θdθ*and integrating over the interval from 0 to

*π*/2 and, consequently, from

*π*/2 to

*π*we obtain a system of two coupled ordinary differential equations for

*I*

_{1}and

*I*

_{2}. This system is solved resulting in

*F*is a constant of integration. Knowledge of the function

*B*(

*ρ*) provides a possibility to compute the intensity leaving the surface. Solving Eq. (46) we arrive at

*F*, we find that it is expressed by the flux of radiation. Equation (50) gives an approximate angular dependence of radiation leaving the surface. Thus, the quantity

*I*(0,

*θ*) defines the distribution of brightness over the image. Let us denote the ratio of brightness at the center of the image to that at the edge by

*r*=

*I*(0,0)/

*I*(0,

*π*/2). In accordance with Eq. (50), this method gives

*r*= 3.

*I*(0,

*θ*) cos

*ψ*is proportional to the energy collected by a pixel of the CCD camera, where

*ψ*is an angle between pixel's normal and the line connecting this pixel to the emitting surface [7]; (ii) the quantity

*F*is the outgoing energy flux

*c*

**n**·

**q**; and (iii) cos

*θ*=

**n**·

**s**

_{0}, where

**s**

_{0}is the unit vector pointing to the CCD camera. Application of boundary conditions, Eq. (23), allows to replace

**n**·

**q**in Eq. (50) with (1/3)

*u*and results in the expression for the energy density on the cylinder surface in terms of the observed intensity

*I*(0,

*θ*) is converted to the image brightness as described in [7]. Obviously, other approximate methods for image correction can be employed as well. However, some of them are too complex while the others are less accurate at the surface. For instance, Eddington's method [41] provides the ratio

*r*= 2.5. The method of Chandrasekhar [42] with two discrete directions gives

*r*= 2.73. Of course, the proper treatment of this problem requires solving the radiative transfer equation with an appropriate phase function.

## Acknowledgments

## References and links

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**OCIS Codes**

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(290.0290) Scattering : Scattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: August 19, 2010

Revised Manuscript: September 7, 2010

Manuscript Accepted: September 12, 2010

Published: September 20, 2010

**Citation**

Vadim Y. Soloviev, Cosimo D'Andrea, P. Surya Mohan, Gianluca Valentini, Rinaldo Cubeddu, and Simon R. Arridge, "Fluorescence lifetime optical tomography with discontinuous Galerkin discretisation scheme," Biomed. Opt. Express **1**, 998-1013 (2010)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-3-998

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### References

- R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14, 024004 (2009).
- M. A. O'Leary, D. A. Boas, X. D. Li, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21, 158–160 (1996).
- A. T. N. Kumar, J. Skoch, B. J. Bacskai, D. A. Boas, and A. K. Dunn, “Fluorescence-lifetime-based tomography for turbid media,” Opt. Lett. 30, 3347–3349 (2005).
- A. T. N. Kumar, S. B. Raymond, G. Boverman, D. A. Boas, and B. J. Bacskai, “Time resolved fluorescence tomography of turbid media based on lifetime contrast,” Opt. Express 14, 12255–12270 (2006).
- L. Zhang, F. Gao, H. He, and H. Zhao, “Three-dimensional scheme for time-domain fluorescence molecular tomography based on Laplace transforms with noise-robust factors,” Opt. Express 16, 7214–7223 (2008).
- F. Gao, J. Li, L. Zhang, P. Poulet, H. Zhao, and Y. Yamada, “Simultaneous fluorescence yield and lifetime tomography from time-resolved transmittances of small-animal-sized phantom,” Appl. Opt. 49, 3163–3172 (2010).
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