APPENDIX

This Appendix outlines the methodology o f the deconvolution technique used in the study

presented, i.e. singular value decomposition (SVD). Further details can be found in

0stergaard et al. (1996a), and references therein.

From Chapter 3 (section 3.2.3.1), the contrast agent time-course within a region o f interest

can be written:

C (0 = S - C B F ■ {AIF(t) ® R{t)) = - ^ C B F \ A I F { z ) R ( t - z)dT [A5.1]

where R(t) is the residue function, i.e. the fraction o f contrast agent remaining in the ROI at time t following the injection o f an instantaneous bolus at t=0, Ca i f( 0 is the arterial input

CHAPTER 5 : BOLUS TRACKING PERFUSION MEASURMENT IN INFANTS 156

flow, p is the density o f brain tissue, and kn accounts for the difference in haematocrit between capillaries and large vessels.

If the arterial and tissue concentration curves are sampled at time points where

At=tn-tn-i, and if it is assumed that the functions Q /XO and R{t) are constant over the At

intervals, then:

i=l

[A5.2]

Eq. [A5.2] can then be expressed as a matrix equation:

" C ( l S 'A IF (t,) 0 C ( t,) = At A IF(t,) AIF(t,) ,A IF (t,)

\ 'r(hŸ

r{t^)

A ‘nh

where r{t) = { p ! k ^ ) - CBF • R {t) . This matrix equation can be written:

c = A . r [A5.4]

Solving this matrix equation by inverting A to find r is extremely sensitive to noise (0stergaard et al., 1996a). To overcome this problem, the matrices V, W and U are constructed such that

A =

u.w.r

where ^ is a diagonal matrix, containing the singular values o f A, and V and U are orthonormal matrices; this is the singular value decomposition o f A. The superscript ‘T ’ indicates a matrix transpose. The inverse o f matrix A is therefore given by

CHAPTER 5 : BOLUS TRACKING PERFUSION MEASURMENT IN INFANTS 157

where the elements o f W are simply the reciprocals o f the elements in W (since W is diagonal). Combining this with Eq. [A5.4] gives

r = V .W - \( U ^ .c ) [A5.7]

which can be used to find the vector o f interest, r. It can be shown, analytically, that

(Hansen 1992):

T

UkO.

H [A5.8]

k ^ k

where <Jk is the singular value (i.e. the A:* element o f W), and vectors y* and ujc are the columns in matrices V and U respectively. The expansion o f Eq. [A5.8] is analogous to a Fourier series, and can be thought o f as a weighted sum o f terms, which correspond to

increasing frequency elements (y* and ujç are functions o f whose frequency increases as k

increases). However, this approach to calculating r is very sensitive to small values o f such that small errors in the measured values o f c (caused, for instance, by noise contributions) are amplified and lead to large errors in the calculated r. 0stergaard et al. (1996a) proposed that setting crU^ values to zero, for values o f o* below a certain threshold,

produces more reliable solutions. Thus, the calculated W matrix is truncated; the use o f this truncated matrix in Eq. [A5.7] is equivalent to multiplying Eq. [A5.8] by a step-

function (or filter), such that the contribution to r at high k values is set to zero. This truncation does introduce errors o f its own, notably oscillations in the solution for r (Liu et al., 1999).

CHAPTER 6

Implementation of the FAIR ASL technique

CONTENTS

6.1 INTRODUCTION... 158 6.2 FAIR SEQUENCE IMPLEMENTATION... 159 6.2.1 Sequence details... 160 6.2.2 Stability testing... 164 6.2.3 Inversion slice w idth... 168 6.2.4 In vivo test data... 169 6.2.5 Intravascular signal... 172 6.3 DATA ANALYSIS CONSIDERATIONS... 174 6.3.1 Image artefacts... 175 6.3.2 Image subtraction...176 6.4 IN VIVO D A T A ... 178 6.4.1 Data acquisition... 178 6.4.2 Data analysis... 179 6.4.2.1 Fitting for transit time, coil inflow time, and perfusion... 184 6.4.2.2 Perfusion calculations using single TI data...187 6.5 SUM M ARY... 189

6.1 INTRODUCTION

The problems associated with bolus tracking make it desirable to implement an additional,

alternative technique for perfusion measurement. This is particularly important for cases in

which the deconvolution data analysis procedure is inappropriate (see Chapter 4), as well as

offering an alternative for patients with an intolerance o f contrast agent injections due to

allergies or difficult vascular access (e.g. some sickle cell disease patients). Arterial spin

labelling (ASL) approaches have the inherent advantages o f being totally non-invasive,

unrestrictedly repeatable and potentially able to quantify absolute perfusion values, and can

CHAPTER 6: FAIR IMPLEMENTATION________________________________________ 1 ^

As discussed in Chapter 3, many different ASL schemes have been proposed. The

appropriate sequence choice for implementation at any site depends upon the specifics o f

the hardware available (field strength, gradient capabilities, etc), as well as the patient

population and planned clinical uses. One o f the most important considerations within

Great Ormond Street Hospital is the fact that the patient population is almost exclusively

paediatric. The labelling efficiency o f continuous ASL (CASL) techniques is poor because

restrictions are imposed upon the train o f radioffequency (RF) pulses used to produce the

continuous inversion, due to power deposition limits. These restrictions lead to non-ideal

conditions (Utting et al., 2003), which are exaggerated in paediatric studies, where power

deposition limits can easily be exceeded. For this reason, a pulsed ASL (PASL) method

was selected. One o f the most widely used, simple and well-tested PASL techniques is

flow-sensitive alternating inversion recovery (FAIR) (Kwong et al., 1995, Kim 1995), and

this was the approach chosen for implementation.

This chapter describes the implementation o f the FAIR sequence at our institution. In

Chapter 7, FAIR data collected in healthy volunteers are presented, in the context o f an

investigation o f a novel PASL sequence. The limited amount o f in vivo data presented in

this chapter serve to illustrate the effects o f some o f the sequence parameters and

components, and to demonstrate some o f the problems associated with PASL sequences.