Parametric level set reconstruction methods for hyperspectral diffuse optical tomography.

Abstract

A parametric level set method (PaLS) is implemented for image reconstruction for hyperspectral diffuse optical tomography (DOT). Chromophore concentrations and diffusion amplitude are recovered using a linearized Born approximation model and employing data from over 100 wavelengths. The images to be recovered are taken to be piecewise constant and a newly introduced, shape-based model is used as the foundation for reconstruction. The PaLS method significantly reduces the number of unknowns relative to more traditional level-set reconstruction methods and has been show to be particularly well suited for ill-posed inverse problems such as the one of interest here. We report on reconstructions for multiple chromophores from simulated and experimental data where the PaLS method provides a more accurate estimation of chromophore concentrations compared to a pixel-based method.

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Parametric level set reconstruction
methods for hyperspectral diffuse optical
tomography
Fridrik Larusson,1,∗Sergio Fantini,2and Eric L. Miller1
1Department of Electrical and Computer Engineering, Tufts University, Medford, MA 02155
USA
2Department of Biomedical Engineering, Tufts University, Medford, MA 02155 USA
∗fridrik.larusson@tufts.edu
Abstract:
image reconstruction for hyperspectral diffuse optical tomography (DOT).
Chromophore concentrations and diffusion amplitude are recovered using
a linearized Born approximation model and employing data from over 100
wavelengths. The images to be recovered are taken to be piecewise constant
and a newly introduced, shape-based model is used as the foundation
for reconstruction. The PaLS method significantly reduces the number of
unknowns relative to more traditional level-set reconstruction methods and
has been show to be particularly well suited for ill-posed inverse problems
such as the one of interest here. We report on reconstructions for multiple
chromophores from simulated and experimental data where the PaLS
method provides a more accurate estimation of chromophore concentrations
compared to a pixel-based method.
A parametric level set method (PaLS) is implemented for
© 2012 Optical Society of America
OCIS codes: (170.3010) Image reconstruction techniques; (170.6960) Tomography;
(100.3190) Inverse problems; (170.3830) Mammography; (170.3880) Medical and biological
imaging; (170.3660) Light propagation in tissues; (170.5280) Photon migration; (290.1990)
Diffusion; (290.7050) Turbid media.
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1. Introduction
Near-infrared light has proven to be useful for imaging the human body and providing func-
tional information for applications including breast cancer detection and characterization [1–5].
In recent years significant improvements have been made in diffuse optical tomography (DOT)
by using data from multiple wavelengths. The use of this kind of hyperspectral information
has moved DOT away from the recovery of space and time varying maps of absorption and
scattering properties of the breast to the recovery of chromophore concentrations as well as
diffusion amplitude. Inverting for multiple chromophores such as hemoglobin, lipids and water
has been shown to provide an improved ability to localize tumours or other objects of interest
in the breast cancer application [6–8]. While these and other advancements have been critical
for moving DOT from the lab into the clinic, there remain significant obstacles to be addressed
so that the method can be used to stably recover these quantities.
The imaging of chromophore concentration and diffusion amplitude from DOT data requires
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the solution of an ill-posed and non-linear inverse problem [3]. The physics associated with
the photons travelling through turbid media, such as breast tissue, makes the problem ill-posed
and therefore introduces significant challenges for the DOT problem. The ill-posedness make
the reconstruction highly sensitive to noise and un-modeled effects which can reduce accuracy
in the recovered images [7]. Additional problems are encountered when recovering multiple
chromophores such as crosstalk, e.g. when two spatially disjoint chromophores pollute the
images of each other.
The ill-posedness of the DOT problem is traditionally solved by putting the recovery of
images in a variational context. To achieve this, numerous regularization techniques [9] can
be used to reduce artifacts and improve the overall accuracy. Traditionally, methods such as
Tikhonov and total variation (TV) regularization have been employed along with L-curve meth-
ods to optimally choose regularization parameters [10–12]. Increasing the amount of data used
to solve the DOT problem has also proven effective in generating high accuracy images [8,13].
To this end hyperspectral information is employed, which along with regularization has been
shown to reduce the non-uniqueness of the solution to the image formation problem [8,10,11].
In this paper, we expand on our previous work in [10] where we employed hyperspectral
information for recovery of chromophore concentrations. Implementing the Born approxima-
tion along with Tikhonov regularization we demonstrated that hyperspectral information was
shown to have utility for a pixel-based reconstruction of chromophores [10]. The large data set
introduced significant challenges in constructing and computing with the forward model and
great care had to be taken in choosing the regularization parameters. The known limitations
of the Born approximation and the high number of unknowns encountered in the pixel-based
approach, made reconstructions sensitive to the choice of regularization parameters [14]. This
motivated us to move to a geometric reconstruction approach based on a low-order parametric
model that has been shown to perform well in the context of ill posed inverse problems [15–17].
The primary contribution of this work is the development of a new approach to the multi-
chromophore inverseprobleminwhichthemediumisorcanbewellapproximated aspiecewise
constant. Piecewise constant DOT problems have been considered in the past. In [18] level sets
were used in a two-step method for shape estimation assuming that prior information of the
absorption parameter was known. Schweiger et al. [19] and Kilmer et al. [20] employed level
sets for the DOT problem estimating parameter distributions using a piecewise basis. Arridge
et al. investigated shape based methods by estimating level-sets, specifically investigating an
explicit method using basis functions and an implicit shape reconstruction to recover absorption
and diffusion assuming a known background [21]. For a detailed review of the use of level sets
in inverse scattering problems we point the reader to [22].
The approach we consider in this paper is significantly different from those in [18,19,21].
In addition to the fact that none of these papers have considered the fully hyperspectral case,
some of these methods [18–20] require the recovery of unknown quantities defined on a fine
scale pixelated discretization of the region of interest. More specifically in [21] absorption and
scattering is estimated using level sets assuming the background known. With the Born ap-
proximation we assume the absorption and diffusion is known in the background but here we
estimate chromophore concentrations and diffusion of the object of interest as well as chro-
mophore concentration in the background. TV reconstructions recover images while traditional
level set methods work with a level set function defined on a pixel-based grid. In both cases,
regularization is required to obtain adequate results and one is faced with the corresponding
challenge of choosing regularization parameters [11,19].
In this paper we consider the use of a shape-based approach to the hyperspectral DOT prob-
lem based on a newly-developed parametric level set (PaLS) formulation. In [15], a basis func-
tion expansion was used to provide a low order representation of the level set function and
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yielded very strong results for a number of highly ill-posed inverse problems including a re-
stricted form of the DOT problem where a single wavelength was employed to determine only
optical absorption. The method in [15] required no explicit regularization and, due to the low-
order nature of the model (number of parameters much, much less than number of pixels) was
amenable to Newton-type inversion algorithms known to converge more rapidly than gradient-
based schemes. Moreover, in [15] our experiments indicated a surprising lack of sensitivity to
the initialization of the inversion algorithm.
Although the breast is a highly heterogeneous medium, in the level set method we assume the
images to be recovered to be piecewise constant. This approximation is supported in the litera-
ture. For example Schweiger et al. assumed anatomical prior information to derive a piecewise
constant region basis [4]. The co-located structures of different species of chromophores is dis-
cussed further in Section 5. In the future, where we move to more complicated simulations and
experiments the goal would be to implement multiple characteristic functions that require the
estimation of multiple level sets, discussed further in Section 8. To prove the accuracy of our
method, we use simulation data to show parametric level-set reconstructions for multiple chro-
mophores and diffusion amplitude. Experimental data are used to show the improvement of the
PaLS method over pixel based methods using previously reported data sets [10].
The remainder of this paper is structured as follows. In Section 2 we discuss our forward
model and derive in detail the Born model used. In Section 3 we discuss the PaLS method and
in Section 4 we derive the Levenberg Marquardt algorithm used for the inverse problem. In
Section 7.1 we present simulation results with multiple chromophore and diffusion amplitude
reconstructions, and in Section 7.2 we present experimental results that show the improvement
of the PaLS method over pixel based methods.
2.Forward problem
In this paper we restrict our attention to problems in which the transport physics [2] associated
with the propagation of light at wavelength λ through tissue can be adequately approximated
using a diffusion model [23,24] of the form
∇·D0(r,λ)∇Φ(r,λ)+vµ0
a(r,λ)Φ(r,λ) = −vS(r,λ)
(1)
where Φ(r,λ) is the photon fluence rate at position r due to light of wavelength λ injected
into the medium, v is the electromagnetic propagation velocity in the medium, µ0
spatially varying absorption coefficient, and S(r,λ) is the photon source with units of optical
energy per unit time per unit volume. For the work in this paper the sources are considered to
be delta sources in space and can be written as S(r,λ) = S0(λ)δ(r−rs) with S0(λ) the source
power at wavelength λ. For spatially varying scattering we assume that the diffusion coefficient
D0(r,λ) follows Mie scattering theory where a scattering prefactor Ψ depends on the size and
density of scatterers while a scattering exponent b depends on the size of the scatterers. Using
this, we write the perturbation in the diffusion coefficient as
a(r,λ) is the
∆D(r,λ) =
v
3∆Ψ
?λ
λ0
?b
= v∆Ψ′d(λ).
(2)
The arbitrarily chosen reference wavelength λ0is introduced to achieve a form of the Mie
model where Ψ has the units of mm−1and Ψ′has units of mm. In this paper, for simplicity we
consider the case where the background medium is infinite and homogeneous. Generalization
to the more practical case where there are boundaries is straightforward in theory though some-
what more complex in terms of implementation [25]. As the primary objective of this work is
the demonstration of a new approach to inversion, we prefer to consider the simpler physical
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