that calibration can be achieved in a number of ways, but that the use of a reference object is
most effective in eliminating systematic errors from measured data. Chapter 2.3 demonstrated
that, in limited cases, use of a reference object may cancel model errors including the effect of
using a 2D reconstruction for 3D data. However the use of a reference object imposes a
number of assumptions on image reconstructions. If a homogenous reference phantom is
available, only the optical properties of the (well matched) phantom are required to produce
absolute images of absorption and scatter. However practical limitations such as geometry
and matching optical properties to clinical subjects mean that manufacture of a homogenous
reference phantom is not always likely to be possible.
The other form of difference imaging is where the object being imaged itself is used as
the reference phantom. Image data are acquired before and after a change in optical properties
within an object that almost invariably has heterogeneous optical properties. This change may
be invoked, be a deterioration or change over time, or be the difference in measurement at
multiple wavelengths. However the resulting image reconstructions from such difference data
can be incorrect. The problems are a direct result of the inherent non-linearity of optical
tomographic image reconstruction. This chapter describes the perturbation approximations
intrinsic in both linear and non-linear image reconstruction schemes. We then explore how
these assumptions affect images of changes in absorption in the presence of a) heterogeneous
background scatter, and b) heterogeneous background absorption using simulations.
Comparisons of results using non-linear (TOAST) and linear image reconstruction techniques
(see section 1.3.2.2), along with experimental studies are included throughout. The origin and
dependence of the error are then investigated. We then present a method found to improve
results which uses estimates of background structure from baseline images. This is shown to
improve quantitation and object localisation in simple images. The significance of this error is
discussed in context with similar studies and in relation to successful, reliable clinical imaging
as well as implications for other forms of NIR spectroscopy.
Much of the validation of optical tomography has been performed on very simple
phantoms with discrete inclusions ((Hebden et al, 1999), (Eda et al, 1999), (Schmitz et al,
2000), (Pogue et al, 1995)). Yet many researchers make the assumption that methods found to
be effective on simple phantoms can be directly extended to clinical imaging subjects, in
E. M . C. H illm a n . P h D th e sis 2 0 0 2 ______________________________________________________________________C h a p te r 2 .4 — 1 4 8
calibration, data acquisition and image reconstruction techniques. Subject variability and the
complex structures involved require consideration of situations where structures do not
conform to the assumptions required for imaging methods shown to be successful for simple
phantom studies. This chapter demonstrates the limitations of using linear and non-linear
image reconstructions, when modelling the exact initial state of the object is not possible.
(Hillman et al, 2001c).
2.4.1 Linear and non-linear reconstruction sc h e m e s .
As introduced in chapter 1.3, image reconstruction in optical tomography is generally
achieved by determining a distribution of optical properties within an object that corresponds
to a set of measurements made on the object’s surface. In our case the diffusion
approximation (DA) to the radiative transfer equation is used as the basis of our light
propagation model.
As shown in relation to derivation of a PMDF in section 1.3.2.1, the sensitivity o f a
measurement yn.m(x(r),(o) made using source m and detector n, to changes in optical properties
Ax(r) can be evaluated from a Taylor series expansion:
yn,m (^, (r), CO) = y„ (%o (r), co)-h [%, {r) - (r)] + ...
dx{r)
where x(r) =
VJC{r) ^
[ 2 . 4 . 1 ]
We can express the first derivative in terms of the Green’s function solution to the DA,
using the assumption of reciprocity, where Gm(r,co) and Gm(r,co) are the forward and adjoint
Green’s function solutions (Arridge, 1999):
- n.m(^o(c)yCO) -
VG+(r,w) VGn(r,w)
dx(r)
V /
[ 2. 4. 2 ]
For simple geometries and distributions o f optical properties, G(r,co) and thus
^m.n(xdr},câ) can be derived analytically (Arridge et al, 1992), (Arridge, 1995) using, for
example the 3D infinite space, the Green's function solution to the DA ([ 1.3.18 ]). For more
complex geometries, FEM can be used to derive a discrete representation of the Green’s
operator (Arridge et al, 1995a) and thus Jn.m(xo(r)) (see section 1.3.1.3.1). Equation [ 2.4.2 ] can be generalised for any measurement type(M„ „,), such as integrated intensity or mean-time
by applying a suitable operator (Ai[ ]) to both sides: Mn,m(xo(f)) = M \yn,m(xo(r),t)\, such that
A l {Jn.m(xd(r),t)] = J„_j^ (xo(r)) (see section 1.3.3 ). Evaluating and plotting Jn,j^ (x(r)) onto r
E. M . C . H illm a n . P h D th e s is 2 0 0 2 C h a p t e r 2. 4— 149
for a single sou rce-d etecto r pair, d ataty p e M and state x{r), will yield a photon m easu rem en t
den sity function (P M D F ) as show n in F igure 1.3.5 in section 1 .3 .2 .1.
2.4.1.1 Linear inversion
C o n v ertin g to d ataty p es and rearran g in g [ 2 .4 .1 ] (neglecting h ig h er order term s) w e get:
[ 2.4.3 ] w here M den o tes m easured data. P ertu rb atio n s in the optical pro p erties o f an object can be
calcu lated from the differences betw een experim ental m easu rem en ts Mn.m''' - u sin g
linear inversion o f [ 2.4.3 ], assum ing som e given xo{r) is know n to co rresp o n d to a referen ce state Mn.m'"' and that xj{r) - xo{r) is sm all.
Figure 2.4.1 show s how [ 2.4.3 ] assu m es that the ch ange in m easu rem en t is linearly related to optical property changes, as long as th e changes are sm all. T he im p o rtan ce o f calcu latin g Jn.r/^(-^o(r)) as a function o f the initial state Xo(r) is also clear:
Non-linear function: M ( x ( r ) ) = M [ y ( x ( r ) ) \ Measurement ■aO) Optical Properties 4 / ( linear)
F ig u re 2 .4 .1 I f th e ch a n g e in o p tic a l p r o p e r tie s is sm all, a n d o v e r th is .small ra n g e w e can a ss u m e o u r fu n c tio n is lin ear, b y eva lu a tin g the d e r iv a tiv e Jn.,n '(xd(f)) of Xq, w e w ill c a lc u la te Xnnneao r a th e r than th e tru e x , S ection 1.3.2.2 details the im p lem en tatio n o f linear im age reco n stru ctio n via T ik h o n o v rég u larisatio n , a one-step linear inversion tech n iq u e that solves [ 2.4.3 ] ior x f r ) - xo(r) using:
ax = [J’'j+ Ii]“'j"a m,
I 2.4.4 ]