th th th

APPLICATION OF PHYSICS AND MATHEMATICS I N CLINICAL NUCLEAR MEDICINE

ST. JOHN ' S

by

© Lin Ling

A thesis submitted to the School of Graduate Studies

in partial fulfillment of the requirements for th e degree of

Doctor of Philosophy .

Division of Biomedical Sciences Memorial University of Newfoundland

September 2013

NEWFOUNDLAND AND LABRADOR

Abstract

This PhD research project consisted of two p arts . In the first part a n ew pharmacoki- netic model was introduced to improve t he accuracy of kidney function estimation b ased on a sampling schedule of 2 and 4 blood sample measurements. Previous models such SETJ and SET2 have b een shown t o b e unreliab le with resp ect to the choice of sampling schedule and in some cases provide physiologically imp ossible values of t he glomerular filtrat ion rate ( G F R). The new model called Tk-GV uses a T ikhonov reg- ularized gamma variate function to fit the plasma clearance data. Based on a group of 46 patients, a comparison of four ph armacokinetics models (SETJ , SET2, OLS-GV, and Tk-GV) revealed that the Tk-GV model was the most robust with respect to sample size and sampling schedule and provided no physiologically impossible values of G FR. When compared to constant infusion results in t he literatures, the Tk-GV model was shown to eliminate t he proportional overestimation of G F R produ ced by the SETJ model given by Chantler's co rrection and the con st ant overestimation produced by the SET2 mo del.

Application of the Tk-GV model for estimating a patient 's GF R requires t he collection of 4 blood samples. In cases where it is not p ossible to collect four blood sam ples, two b lood samples a re collected and the patient 's GFR is calculated using the SETJ model. A crit erion called t he ren al sufficient index (RSI) was developed by others using a group of children to correct the overestimation of G F R as obt ained

11

from the SET1 model. In this work t he R S I was applied to a mixed gro up of 26 patients (adults, children , males, and females) and shown , based on addit ional medical information , to b e able t o accur ately distinguish b etween patients wit h norm al an d abnormal G FR.

Bone mineral density (BMD) is important for evaluating bone health , esp ecially for elderly p eople such as post-men opause women and m en with prost ate disease.

However, analyzing a p atie nt 's BMD change is difficu lt because of naturally occur- ring s hort-term and long-t erm fluctu ations in a p atient 's BMD , and also the sh ort- term and long-term errors provided by the equipment used to measure BMD. The World Healt h Organization h as provided a least sign ificant ch ange (LS C) criterion for distingu ishing b etween normal and abnormal ch an ges in a p atient's BMD value.

Unfortunately t he LSC criterion only accounts for short-term machine error. In t he second part of this t hesis a new criterion called the tot al detectable difference (TDD) is introduced for analyzing changes in BMD in consideration of naturally occurring changes in th e pat ient's BMD, and instrumentation error occurring on both short and long time scales. Based on t he analysis of a group of 8,800 patients , a T DD valu e of approximately 0.045 g/cm

is obtained for the hip and 0.060 g j cm

for the lumbar spine. Based on t he study of 9,379 patients , it is shown t hat t he LSC criterion has a p otential o f over-diagnozing BMD change by approximately 30% comp ared to t he TDD criterion.

iii

Acknowledgments

I would like to t hank Rick Scanlan from t he Dep artment of Nuclear Medicine for helping me with t he lab oratory equipmen t and radiation safety training . I also would

like to thank Stan, Kelly, T ara, Maureen , Nina, and Tammy from t he Departmen t of

Nuclear Medicine, who have taught me a great deal about clinical practices.

I greatly appreciate the support given to me by t he Faculty of Medicine at Memo- rial University and t he d epart ment of Nuclear Medicine at Eastern Health.

I would like to th ank my M.Sc. supervisor Dr. J olanta Lagowski from the De- partment of Physics and P hysical Oceanography at Memorial University for her help in providing me wit h th e necessary skills for computer modelin g and for conducting basic research.

I would like to t hank Dr. Christopher Kovacs for helping me access t he CaMos data and the support from the CaMos Research Group .

Finally I would like to t hank my husb and Luc Beaulieu , my parents Ling Chuan- ming and Liu Jing in China a nd my beautiful baby boy Patrick for all t heir love and support .

lV

Structure

This thesis includes two research projects. 60% of t his thesis focuses on th e plasma clearance model which includes an introduction , literature review, d at a and analysis methods, results and discussion and possible applications. The remain ing 40% of this t hesis introduces a new criterion for analyzing bone mineral density change as described above. The outline of the t hesis is listed as follows:

1. Chapter 1 gives a general introduction of Nuclear Medicin e. The topics include radioisotopes, radiation detection , instrument ations, and tracer kinetics.

2. Chapters 2 to 7 present the first project of t his thesis. Ch apters 8 t o 10 present the second proj ect of this thesis.

3. Chapter 2 provides a t heoretical background and motivation of the p lasma clear- ance project. This chapter provides a brief overview of the clinical assessment of kidney function and a literat ure review of the major pharmacokinetic models along with a d escription of their p erforman ce.

4. Chapter 3 outlines the data and an alysis methods used to investigate t he pro- posed model. The uncertainty of t he measurement and numerical metho ds such as the Nelder Mead minimization method are discussed .

v

5. Chapter 4 present s the results of the calculations applied to 46 p atients using th e SET1, SET2, OLS-GV and Tk-GV models using the full samples and subsets.

6. Chapter 5 compares t he performance of t he SET1 , SET2, OLS-GV and Tk-GV models including the estimated glomerular filtraltion rate ( G F R) , the effects of subset, case study and a cross comparison with other plasma clearance method s.

7. Chapter 6 applies t he Tk-GV mod el to 24 h data.

8. Chapter 7 presents a validation study of a kidney function correction method to compensate for the inaccuracy of the two-sample SET1 method.

9. Chapter 8 introduces t he con cept of bone mineral density (BMD ). The proposed method of an alyzing time-b ased BMD results is introduced after a literature review. The motivation and the theory of the new criterion , total detectable difference (TD D), are provided.

10. Chapter 9 lists t he d ata and an alysis methods used for investigating t he new TDD criterion.

11. Chapter 10 presents the results and discussion of th e TDD criterion based on 8 ,800 p atients d ataset. Details about the patient selection , data analysis an d clinical impact are listed.

12. C hapter 11 summarizes the major results and implications of both research projects and describes future research projects.

13. Appendix A shows the clinical protocol implemented for collecting 4 blood sam- p les at t he General Hospital in St . John's, NL and the mathematical formulae for t he t heory of the T k-GV model.

Vl

Contents

Abstract

Acknowledgments

Structure

List of Tables

List of Figures

Abbreviations

1 Introduction

1.1 Nuclear Medicin e 1. 2 R adioisotopes . .

1.2.1 Decay an d Half-life 1.2.2 I nte ract ions with Matter 1.3 R ad ioph armaceut icals

1. 3.1 C he lat ion 1.4 Ga mma Count er 1.5 T heor y of SPECT .

Vll

11

IV

v

xvi

xxiii

XXIV

1 1

2

3

5

6

9

10

13

1.6 Tracer Kinetic Modeling 16

1.7 Bone Densitometer 17

1.8 My Contrib ut ions . 23

2 Glomerular Filtration Rate ( GFR) Determination 27

2.1 Glomerular Filtration Rate . .. . . 27

2.1.1 Imp act of Kidney Function . 30

2.2 G F R Measurements . 32

2.2. 1 Biomarkers 32

2.2.2 Estimated GFR ( eGFR) 36

2.2.3 R enography 38

2.3 Plasma Clearance . 39

2.3 .1 Ph armacokinetic Models 41

2.3.2 One-compartment Model . 45

2.3 .3 Two-compartment Model . 48

2.4 Limitations of the Compartment Models 50

2.5 Motivation . 55

2.6 T heory .. . 55

2.6.1 Gamma Variate ( GV) Model 55

2.6.2 R at e of Exchan ge, GFR and Vol 58

2.6.3 Ill-p osed Problem . . . . 60

2.6.4 Tikhonov Regularization 61

3 Data and Analysis M ethods 68

3.1 D ata .. . . . . . . . .. . 68

3.2 Measurement Uncertainty 70

3.3 Numerical Met hods . . . . 71

Vlll

3.4 Regression Methods . 72

4 Results 74

4.1 SETJ Results 74

4.2 SET 2 Results 83

4.3 OLS-GV R esults 88

4.4 Tk-GV Results 92

5 Discussion 96

5.1 GFR Estimat ion . . . . . . 96

5.1.1 Non-physical Results 96

5. 1.2 Variations wit hin the Model 98

5.2 Effect of Subsets . . .. 101

5.2.1 4-sample Subsets 101

5.2.2 Hump Subsets . 109

5.3 C ase Study . . . . 110

5.4 Cross Comparison . 113

5.5 Best Sampling Schedule 115

5.6 C linical Impac t 117

5.7 Summary . . . 120

6 Applications to 24 h Data 121

6.1 D ata . . . . .. .. . 121

6.2 SETJ versus Tk-GV 123

6.2. 1 GFR Results 123

6.2 .2 Curve Fit . . 124

6.2.3 4 h versus 24 h 127

lX

6.3 Effect of Subsets . . . . . . 6.4 Exploring the GV Function 6.5 Summary . . . . . . . . . .

7 G F R Correction

7.1 Introduction . . . . . . . . . 7.2 Data and Analysis Methods 7.3 Results and Discussion 7.4 Conclusion . . . . . .

8 Bone Mineral D ensity

8. 1 Introd uction .. ..

8.2 Literature Review . 8.3 Motivation . . . . .

8.4 T otal Detectable Difference

9 Data and Analysis Methods

9.1 Dat a . .. .. . . 9.2 Analysis Methods

9.2.1 Half-normal Distribution 9.2.2 Small Sample Correction 9.3 TD D Calculation ..

10 Results and Discussion

10.1 Results . 10. 2 Gender . 10.3 Age . ..

10.4 Examination Time

X

129 135 137

138

138 140 1 41 145

146

1 46 149 152 153

156

156 157 158 161 161

164

165

166

170

172

10.5 GE and Hologic 10.6 CaMos d ata . . 10.7 Impact on Diagnosis 10.8 Conclusion .

11 Conclusions

11.1 Plasma Clearan ce 11 .2 Bone Mineral Density . 11.3 Future Directions

Appendix A

Bibliography

A

A .1 C linical Protocol for Measur ing G F R A .2 Mathematical Formulae . . . . . . .

Xl

174 177 179 180

182

182 183 184

188 188

208

208

210

List of Tables

2.1 The stages of chronic kidney disease (CKD ). All GF R values are in units of ml/ min / 1.73 m

(1.73 m

represents the stan dard body surface area (BSA) of a healthy young adult) . . . . . . . . . . . . . . . . . . 32

4.1 GF R (ml/ min) and Vol (L) values calculated from the SETJ m od el for the 46 p atients using the full samples (denoted as " full") and subsets:

(60 , 180)min and (10, 180)min. . . . . . . . . . . . . . . . . . . . . . 75

4.2 The effects of the 1,148 2-sample subsets on t he values of A , a , G F R (ml/ min) a nd Vol (L) using the SETJ model. GF R (ml/ min) is the mean GF R, and SDaFR (ml/ min ) is t he standard deviation of t he GF R values. Vol (L) is the mean Vol, and SDvo l (L) is t he standard d eviation of the Vol values. . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 The effects of t h e 3-sample subsets (2,296 for all 41 patients) on the val- ues of A , a , ^{GF R (m l/ min) , and Vol} (L ) u sing SETJ. GF R (ml/ min) is the mean GFR and SDaFR (ml/ min) is the standard deviation of the

GF R values. Vol (L) is t he mean Vol and S Dvoz (L) is the standard d eviation of the Vol values. . .

Xll

79

4.4 The effects of the 4-sample subsets (2,870 for all41 patients) on values of A, a, GFR (ml/min) and Vol (L) using SET1. GFR (m l/ min ) is the mean GF R and SDaFR (ml/ min) is t he standard deviation of t he GF R values. Vol (L) is t he mean Vol and SDval (L) is the standard deviation of the Vol values. . . . . . . . . . . . . . . . . . . . . 80 4.5 The GF R results of t he SET1 model for the 41 patients using the full S

samp les and four subsets: (10, 20 , 60 , 180) min, (10, 30 , 120, 240)min, (10 , 20, 30 , 45)min and (60 , 120, 180, 240)min. Q st ands for quartile. 80 4.6 Estimated GF R (ml/ min) and Vol (L) values of the SET2 model for

the 46 patients using t he full samples and the (10, 20 , 60 , 1 80)min subset .

4.7 Estimated GFR (ml/ min) calculated with t he SET2 model using t he 4-samp le subsets: (10, 20 , 60 , 180)min, (10, 30, 120, 240)min, (10 , 20 , 30, 45)min and (60, 120, 180, 240)min from the 41 patients. Q stands

84

for quartile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.8 The effects of 4-sample subsets (2,870 subsets) using th e SET2 model

on values of .\

and .\

GFR (ml/ min) and Vol (L) from t he 41 pa- tients. SDaFR (ml/ min) is t he stan dard deviation of t he GF R values.

SDva t (L) is the standard deviation of the Vol values .

4.9 Quartiles of t he mean estimated parameters K , a, and /3, ^{and the}

values of GF R (ml/ min) , and Vol (L) for the OLS-GV model from the 41 patients using t he 2,870 4-sample subsets. GF R is t he mean GF R and SDaFR is t he standard deviation of the GF R values. Vol (L) is the mean Vol and S Dvat (L) is the standard deviation of the Vol values.

Xlll

86

89

4.10 The G F R and Vol results of t he OLS-GV model for the 41 patients using the full 8 samples and four chosen subsets: (10 , 20 , 60 , 180 )min, (10, 30, 120 , 240)min, (10, 20 , 30, 45)min and (60, 120, 180, 240)min.

Q stands for quartile.

4.11 The GF R and Vol results of the Tk-GV model for the 41 p atients using the full 8 samples and four different subsets: (10 , 20, 60 , 180 )min, (10 , 30, 120 , 240)min , (10, 20 , 30, 45)min and (60 , 120, 180, 240)min. Q

stands for quartile. . . . . .

4.12 Estimated values of>., lnK, a , (3, GFR , Vol and the standard d evia- tions SDcFR (ml/ min) and SDval (L) values using the 2,870 4-sample subsets from the 41 patients using the Tk-G V model.

5.1 The slope and R

values obtained fr om t he correlation between t he results from t he SET1, SET2, OLS-GV and Tk-G V mo dels using t he 4-sample subsets (shown from Figures 5.1 to Figure 5.4) and t he fu ll samples from OLS linear regression .

5.2 Estimated GF R values for Pt15 and Pt19 using the 4-sample subsets (70 subsets) using t he SET1 , SET2, OLS-GV and Tk-GV models . The GFR

(in ml/ min) and Vol

(in L) are the estimated results using 8 samples. The GF R

and Vol

are the mean estimated results of GF R and Vol using 4 samples.

6.1 Estimated GFR (ml/min) values from the 10 patients with 4 h , 12 h 89

92

93

107

112

and 24 h sampling schedules, using the SETJ and Tk-GV model s. 123

XIV

6.2 The values of the slope and R

from the linear regression of the esti- mated GFR values obt ained from the SETJ, SET2 and Tk-GV models using 4 h , 6 h, 8 h and 12 h sampling schedules to those under the same models using the 24 h sampling schedule for the 10 patients. . . . . . 134

9.1 The probability density function (PDF) of t he folded-normal distribu- tion (FND ), half-normal distribution (HND) , and normal distribution (ND) , where J-L is the m ean value and

is t he standard deviation. . . 160

10.1 TDD results (g/cm

of approximately 5,500 p atients analyzed using the GE Lunar Prodigy densitometer at the General Hospital using t he HND and FND. The R

* (see the definition below) value is for both the HND and FND. Area indicates the fo lded area of the FND from the ND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

10.2 The TDD (g/cm

values of the f em ale and male groups, for HND and

FND , using a GE Lunar Prodigy densitometer. 169

10.3 The T DDHND results (g/cm

of four age groups from the 5,500 pa- tients, using a GE Lunar Prodigy densitometer. . . . . . . . . . . . . 170

10.4 T he TDD results (g/cm

using the HND for 10 examination t ime intervals from the 5,500 patient data, using a GE Lunar Prodigy den- sitometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

10.5 The TDD values (g/ cm

from t he GE and Hologic densitometers in

St.John's, NL. 175

10.6 TDD results (g/ cm

from NL and t he CaMos database using the GE Lunar densitomet ers , with number of patients in the parentheses . . . 177

XV

10.7 TDD results (g/ cm

from NL and the CaMos database using t he Ho- logic QDR 4500 densitometers, with number of patients in the paren- theses.

XVl

177

List of Figures

1.1 Schematic representation of the transition from

Mo to

Ru . . . . . 7

1.2 The single-well gamma counter a nd thyroid uptake camera (manufac- t ured by Laboratory Technologies, Inc.) used at the G en eral Hospital in St.John's , Newfoundland and Labrad or. . . . . . . . . . . . . . . . 12

1.3 Diagram of a typical gamma camera. This diagram was ad apted wit h p ermission from the Journal of RadioGraphies [23]. . . . . . . . . . . 14

1.4 (a) Demonstr ation of a BMD image of t he lumbar spine a nd (b) t he hip , obtained using a GE Lunar Prodigy Dual-energy X-ray absorptiometry densitometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1 Sc h ematic representation of the a natomy of a human kidney.

2.2 Sc h ematic representat ion (light ly s had ed area) of the Area Under t he Curve (AUC) used for calculating the elimination rate of a tracer.

The p lasma conce ntration of the tracer is plotted as a funct ion of t he time (wit h time=O set at the fini shing point of t he injection ) after the administration. T he three p hases in the renog raph are labeled as I, II

28

and III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

XVll

2.3 Schematic representation of the one-compartment model used to model the plasma clear an ce. T he circle represents the plasm a as the compart- ment. The two a rrows indicate the flow of the r adiopha rmaceu ticals in a nd out of the co mpartment. . . . . . . . . . . . . . . . . . . . . . . 46

2.4 The two-compartment model used to describ e plasm a cleara nce. The two circles cl and c2 r epresent compartments for the plasma and sec- ondary volume of distribution of the t racer respectively. The flow in a nd out of the two compartments is indicated with arrows showing t he ra te of exchan ge

K out ,

a nd

4.1 Results of the mean of the differ en ce (GFRhump- GFRfull ) between the G F Rhump a nd G F Rfu ll using the SET1 model from th e 41 patients.

The hump subse ts are 1 to 3, 1 to 4, . .. , ⁵ to 8 a nd 6 to 8.

4.2 Results of the mean of the d iffer ence (GFRhump - GFRfull) between the GF Rhump a nd GF Rfull using t he SET2 model from th e 41 patients.

49

82

The hump subsets a re 1 to 5, 1 to 6, 1 to 7, 2 to 8, 3 t o 8 a nd 4 to 8. 8 7

4.3 R esults of the mean of t he differ ence (GFRhump - GFRfull) between t he GFRhump a nd GFRfull using the OLS-GV model from th e 41 pa- tients. The hump subsets ar e 1 to 5, 1 to 6, 1 to 7, 2 to 8, 3 to 8 a nd

4 to 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4 R esults of the mean of the differ en ce (GFRhump - GFR1uu) between t he GF Rhump and GF Rfu ll using the Tk-GV model from the 41 pa- tients. T he hump subsets ar e 1 to 5, 1 to 6, 1 to 7, 2 to 8, 3 to 8 a nd

4 to 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

XV111

5.1 Comparison of the G F Rser

values obtained using t he 4-sample s ub- sets to t he GF R full obtained using t he full samples. a) (10, 20 , 30 , 45)min , b) (10, 20 , 60, 180)min, c) (10 , 30, 120, 240)min a nd d) (60, 120, 180, 240) min. The solid line indicates a fit of t he GF R values and the dashed line is the identity line Y = X. . . . . . . . . . . . . . . . 103 5.2 Comparison of t he GF Rser

values obtained using the 4-sample sub-

sets to the GF Rfull obtained using the full samples. a) (10, 20, 30, 45) min, b) (10, 20 , 60, 180)min, c) (10 , 30, 120, 240)min a nd d) (60, 120, 180, 240)min. The solid line indicat es a fit of t he GF R values and the das hed line is the identity line Y = X . . . . . . . . . . . . . . . . 104 5 .3 Compa rison of the G F

values obtained using th e 4-sample sub-

sets to the GF Rfull obtained using t he full samples. a) (10 , 20, 30, 45) min, b) (10, 20, 60, 180)min, c) (10 , 30, 120, 240)min and d ) (60, 120, 180, 240)min . The solid line indicat es a fit of the G F R values and the das hed line is the identity line Y = X. . . . . . . . . . . . . . . . 105 5.4 Comparison of t he G F

values obtained using t he 4-sample sub-

sets to t he GF Rfull obtained using the full samples. a) (10 , 20, 30, 45)min , b) (10, 20 , 60, 180)min, c) (10 , 30, 120, 240)min a nd d) (60, 120, 180 , 240)min. The solid line indicates a fit of t he GF R values and the das hed line is the identity line Y = X. . . . . . . . . . . . . . . . 106 5.5 Comparison of performance of the concent r ation time (with time = O

set at the finishing point of the inject ion ) curve fitt ing us ing t he SET1 (fin e d ashed line) , SET2 (solid line) , 0 LS-G V (dash-dot-dash line ) and Tk-GV (dashed line) models for Ptl us ing the full samples. The original 8 samples are r epresented with dark circles.

XlX

111

5.6 Comparison of t he 46 estimated GFR values using SET1 (d ark cir- cles) and SET2 (gray rhombus) compared wit h Tk-GV , using t he full samples . The d ash ed line is the identity line Y = X.

5.7 Comparison of (a) (GFRseTl - GFRTk-Gv) / GFRTk-GV from SET1 with Tk-GV and (b) (GFRsET2- GFRTk-Gv )/GFRTk-Gv from SET2 with the Tk-GV model for t he 46 pat ients. The vertical dashed lines

114

indicate t he CKD stages from CKD1 to CKD5. . . . . . . . . . . . . 118

6.1 Logarit hm of the concentration versus t ime (with t ime=O set at t he finishing point of the injection) using the SET1 and Tk-GV models

for Pt3. The original data is shown wit h hollow circles, the Tk-GV fit is indicated by a solid line while t he SET1 fit is indicated by a dashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.2 Logarithm of the concentration versus t ime (with time=O set at the finishing point of the injection) using the SET1 model for Pt3. The

original data are shown as crosses ( +) . T he straig ht lines indicate the fits obtained using t he SET1 model with different sampling times as shown in t he legend. The insert on t he bottom left shows how the SET1 model overestimates t he con centration d ata for earlier time. 126

6.3 (a) Estimated GF R results (ml/ min) f or t he 10 patients with a 24 h

sampling schedule using the Tk-GV model (black bars) and t he SET1 model (white bars) using the 4 h sampling schedule. (b) The relative

. . G F R se Tl - G F RTk-GV

d1fference m G F R between t he two models G . 128 FRse r 1

XX

6.4 Correlation between the estimated G F R values using the SET1 model and the Tk-GV model from a) the 4 h and b) t he 24 h samples. T he d as hed line indicates the ident ity line Y = X while the solid line is t he fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.5 Comparison of t he 10 estimated GF R values from t he Tk-GV model using 4 h, 6 h, 8 h , and 12 h samples with t hose using t he full 24 h samples. The symbols for each subset are shown in the legend. 131

6.6 Comparison of the 10 estimated GF R values from t he SET1 model using 4 h , 6 h, 8 h , and 12 h samples with those u sing t he fu ll 24 h samples. The symbols for each subset are shown in t he legend. 132

6. 7 Comparison of t he 10 estimated G F R values from t he SET2 model using 4 h , 6 h , 8 h , and 12 h samples with th ose u sing t he full 24 h samples. The symbols for each subset are sh own in the legend. 133

6.8 Percentage of the power fun ction (hollow circles) and exponential fu nc- tion (solid circles) in the whole GV model as a fun ction of sampling time t (with t ime=O set at t he finishing point of the injection) in min for a) Pt3 and b) Pt13. . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.1 a) Estimated GFRsETl results using the 2-sample SET1 model (ml/ min) . b) Estimated results of GFRHaycock (ml/ min/ 1.73 m

(dark bars) and

RSIHaycock (white bars). The solid line indicates the Haycock crite- rion . c) Estimated results of GFRJ(Vol,W) (ml/ min) (d ark b ars) and RSI J(Vol, W) (white bars). b) and c) T he d ashed lin e represent the 85.89% RSI t hreshold. . . . . . . . . . . . . . . . . . . . . . . . . . 142

XXl

7.2 Correlation of RSIHaycock and RSIJ(Vol,W) results using the linear re- gression. The solid line indicates the best fit using a second order polynomial from the 26 p atients , and the dashed line indicates the fit from the initial study by Wesolowski et al. based on study of 133 children.

8 .1 D emonstration of the composition of total BMD changes including

short-term and long-term machine errors, as well as patients ' BMD variations . The short-term and long-term machine errors were esti- mated from local study while the short-term and long-term patients '

144

BMD variations were estimated from the reference [155]. . . . . . . . 155

9.1 Demonstr ation of the generation of t he folded-normal distribut ion (F ND) from the normal distribution (ND). . . . . . . . . . . . . . . . . . . . 159 9.2 A flow ch art indicating how the TDD value is calculated from t he initial

patient's d ataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

10.1 The HND fit ting for p atients with three sequential BMD studies at the examination site of (a) the lumbar spine 11-1 4 and (b ) t he lumbar spine 1 2-1 4, using a GE densitometer. . . . . . . . . . . . . . . . . . 167 10.2 The HND fitting for patients with three sequential BMD stud ies at the

examination site of (a) the femoral neck and (b) the tot al hip , using a G E densitometer.

10.3 The TDD values (g/cm

for the four exami nation sites of the four age gro ups ( < 50 , (50, 59) , (60 , 69) and 70 < ) . The co lor scheme indicating each examinat ion site is shown in the legend.

XXll

168

171

10.4 The TDD values from the 10 t ime intervals for the four examination sites: lumbar spine 12-14, lumbar spine 11-14, total h ip and femoral neck. The color scheme indicating each examination site is sh own in the legend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 10.5 a) Comparison of the TDD values (g/ cm

from the GE (white bar) and

Hologic QDR 4500 d ensitometer (gray bar) for the lu mbar spine 12-14, 11-14, femoral neck and total hip. b ) Relative difference (TDDGE- TDDHo!ogi c)/TDDGE of the TDD values b etween t he GE and Hologic d en sitometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 10.6 P ercentages of t he d etectable BMD differen ces using t he LSC (dark

b ars) and TDD (white bars) from the local GE Lunar dens itometer. 179

XXlll

Abbreviations

• A Age

• AUC Area Under the Curve

• BMD Bone Mineral Density

• BSA Body Surface Area

• C L Clearance

• CT Computed Tomography

• CV Coefficient of Variance

• DEXA Dual Energy X-ray Absorptiomet ry (DXA)

• DMSA Dimercaptosuccinic Acid

• DTPA Diethylene Triamine P entaacetic A cid

• ERG Relative Error of the Estimated GFR

• ERV Relative Error of the Estimated Vol

• EDTA Ethylen ediaminetetraacetic Acid

• FND Folded-normal Distribution

XXIV

• GFR Glomerular Filtration Rate

• GV Gamma Variate

• H Height

• HND Half-normal Distribution

• LSC Least Significant Change

• MRI Magn etic Resonance Imaging

• NM Nuclear Medi cine

• ND Normal Distribution

• OLS Ordinary Least Square

• PE Precision Error

• Q Quartile

• QCT Quantitative Computed Tomography

• ROI R egion of Interest

• R

Coefficient of Determination

• R

* Adjusted Coefficient of Determin ation

• RSI Renal Sufficient Ind ex

• SD Standard Deviation

• SE Standard Error

XXV

• SET1 Sum of 1 Exponential Term

• SET2 Sum of 2 Exponential Terms

• SETn Sum of n Exponential Terms

• SD Standard Deviation

• Tc Technetium

• TDD Total Detectable Difference

• Tk T ikhonov (Regularization)

• US Ultrasonography

• Vol Volume of Distribution

• W Weight

XXVI

Chapter 1

Introduction

This chapter presents a brief introduction of Nuclear Medicine , basic nuclear physics , radiopharmaceuticals, and radiation detectors.

1.1 Nucle ar M e dicine

Nuclear medicine is a branch of medicine and medical imaging t h at uses the nuclear properties of matter in diagnosis and therapy. Nuclear medicine is uniqu e (differ- ent from other medical imaging modalities) in that it provides both anatomical and functional information. Clinical information can be derived from observing t he radi- ation emitted from radiopharmaceuticals (or radionuclide alone) administered orally, or int ravenously. Radiopharmaceuticals are radioactive pharmaceuticals made up of a radionuclide and a carrier molecule. T he carrier molecule d elivers t he rad ionuclide to the specific physiological area to be examinated or treated [ 1]. Measurements in nuclear medicine can be performed either in vitro or in vivo. In vitro measurements are based on samp les (e.g. blood, urine) taken from patients after administering th e radioph arm aceuticals. The radioactivity in t hese samples can help determine th e

1

CHAPTER 1. INTRODUCTION 2

physiological fun ctions of t he body or organs. In vivo measurements are performed on p atients directly with an external detector (e.g. gamma camera) for measuring the radiation emitted from the radiopharmaceuticals inside the body. In general, in vivo measurements are more commonly used than in vitro measurements.

Although natural radioisotopes were discovered b efore 1900 [2], t he fi rst nuclear medicine study was n ot performed until the late 1940s using radioactive iodine for thyroid cancer treatment. With the development of pharmacology and biochemistry, more radiopharmaceuticals were designed and used for d et ecting various d iseases. By the 1970s , most organs could be visualized using nuclear medicine techniques. Con- currently, radiation detection technology improved the quality of th e nuclear medicine images. The first gamma camera for recording r adiation counts and produ cing pla- nar images was invented by Hal Anger in t he 1950s [ 3]. In 1963 , t he first Single P hoton Emission Computed Tomography (SPECT) study was performed by Kuhl and Edwards [ 2]. SPECT is a tomographic technique of nuclear medicine which uses gamma rays to scan the body at different angles and combines the acquired two- dimensional (2D) images to reconstruct a t hree-dimensional (3D) representation of t he body [4] . By the 1990s, Positron Emission Tomography (PET) was invented [5].

P ET h as an advantage over other imaging modalities such as Magnetic Resonance Imaging (MRI) [6] or ultrasonography [7] in that it can detect metabolic abnormal- ities. Recent developments in imaging techniques consist of t he fusion of several existing imaging techniques s uch as SPECT / CT, PET / CT , PET / MRI, etc.

1. 2 Radioisotopes

A radioisotope is an ato m with an unstable nucleus that can release energy when

it und ergoes rad ioactive decay. Commonly used r adioisotopes in nuclear medicine

CHA PT E R 1. I NTRODUCTIO N 3

are fluorine 18 (1

F )t, Phosphorus 32 (3

P ), G allium 67 (

Ga), R ubidium 82 (

Rb ), Stront ium 89 (

Sr), Technetium 99 (

T c), Indium 111 (min), Iodine 123 (1

I), Iodine 131 e

^I), Thallium 201 e

^{T l), etc [ 8].} Besides their use in medical diagnosis, radioisotopes a re also used for t reating certain diseases by delivering high radiation dosage to regio ns of interest . For example, hyperthyroidism an d thyroid cancer can be t reated using

I, and palliative b one pain can be managed using

Sr [ 9].

1.2 .1 D ecay a nd Half-life

Most radioisotop es decay by on e or several of t he following ways: 1) alpha decay, 2) beta- minus decay, 3) beta -plus d ecay, or 4) gamma d ecay. The following list shows some examples of t hese decay processes [10]:

a decay:

(3- decay:

(3+ decay:

I' decay:

A X --tA-4 y +4 ^He

Z Z-2 2

n -t p + e- +"De

en ergy+ p -t n + e+ +

~X -+~ ^X + /'-

Alpha (a) decay occurs when an atom ic nucleus (e.g. ~~

U) emits an alpha (a) particle (a ~ He helium nucleus) yieldin g another nucleus (e .g. ~5

Th) in the process.

Because of t heir mass, a lpha p art icles lose most of their energy within a relatively short distance (short penetration dept h ) making it difficult to detect them with an external detection camera [1 0]. T herefore, alp ha decay is not commonly used in n uclear med icine.

T here are two types of beta ((3 ) decay: beta-minus decay and beta-plus decay [10]. Beta- minus ((3- ) decay occurs wit h the conversion of a neutron (n) into a

tThe full notation is §

F where 18 is the mass num ber and 9 is t he atomic number.

CHAPTER 1. INTRODUCTION 4

proton (p) along with t he ejection of a negatively ch arged b eta p art icle (e-, electron) and an antineutrino (ve) - On t he other hand , bet a-plus (,8+) decay results in t he conversion of a proton (p) into a neutron ( n) accompanied by t he ejection of a p ositron (e+ ) as well as an electron neutrino (ve) · There is a relevant decay mode called electron capt ure, also known as inverse bet a decay, which occurs when t he nucleus capt ures an orbital (e .g. K -shell or £-shell) electron, wit h t he co nversion of a prot on (p ) into a n eutron (n ) accompanied by t he ejection of a neutrino (ve) d escribed as p + e- -+ n +

Gamma ( 1) d ecay occurs when an atomic nucleus decays to a lower energy state from an excited state. The energy differen ce between the two st at es is released by the emission o f a gamma ray photon (t) [ 10]. Gamma rays a re very important in nuclear medicine b ecause of their relatively long p enetration range th at allows information (e.g. location and amount ) from t he administ ered radiopharmaceut icals to be collect ed .

Oth er useful transit ion processes include: isomeric transition where t he decay pro- cess y ields gamma radiation without emitting or capt uring a p art icle from t he nucleus, and electron/ positron annihilation (the basis of PET scanning) where an electro n (e- ) and a p ositron ( e+) collide (an nihilate) resulting in a photon p air (gamma rays, 1) e- + e+ -+ 1 + 1, etc.

R adioact ive decay is a random process where t he p robability of a single event (su ch as whether a specific atom will decay or not) cannot be determined precisely.

However, p robability t heory en ables t he radioactive decay pro cess to be d escribed using t he d ecay equat ion:

(1. 1)

where A(t ) is t he radioactivity at t ime t (t ~ 0 ), A

is t he initial radioactivity at

CHAPT ER 1. I NTRODUCTION 5

time t= 0, and ). is t he decay const ant given by ). = (I n 2)/T

= 0 .693/ T

where T

is t he half -life. Half-life is defined as the time for a radioact ive sample to decay to half of it s initial rad ioact ivity [1] . The h alf -lives of t he radiopharmaceut icals used in nuclear med icine range from seconds to years. T h ose radiopharmaceut icals with relatively short half -lives ensure that medical scanning can b e p erformed in a timely fashion and t hat patient s are not exposed to prolonged radiation.

1.2.2 Inte ractions with Mat t er

Gamm a rad iation ioni zes matter th rough three major processes: t he photoelectric effect , Compton scattering, and pair production [ 10].

The photoelectri c effect transfers an incident photon 's energy to an electro n caus- ing it to be ejected from the atom as a photoelect ron. The phot oelectron 's kinetic energy is the difference between t he incident p hoton 's energy h v (where h is th e P lanck co nstant and v is t he frequency of t he photon ) and t he electron 's b in ding energy. This photoelectron usu ally trave ls in a different direction from t he incident gamma photon [10].

Compto n scattering occurs when an incident photon (with energy hv) interacts wit h matter, causing t he ejection of an orbit al electron (known as a Compton electron) and t he scattering of t he photon at a reduced energy hv' . T he difference between th e photoelectric effect and Compton scattering is t hat in t he latter case t he electron is ejected via a scattering process while in the fo rmer case th e electron is emitted after absorbing energy from the gamma photon .

P air product ion can occur when the energy of a gam ma photon exceeds 1.022

MeV [10] . A gamma photon (with energy of hv ) can be converted into a n elect ron-

p ositron pair ( e+ and e- ) by interacting wit h a nucleus. Since each of the part icles

CHAPTER 1. I NTRODUCTION 6

has a rest mass energy of 0.511 MeV, t he remaining gamma photon energy (greater t han 1.022 MeV) appears as th e kinetic energy of the electron-positron pair. Other pairs can also be produced including a tau and anti- tau or muon and ant i-muon p air.

When photons pass through matter, the gradual loss in the intensity of the photons is called attenuation. Attenuation reduces the intensity of the incident beam due to absorption or scattering. The inten sity, I , of a beam of photons going through an absorber is given by:

(1.2)

where !

is the original intensity of the beam , l is the distance traveled in the absorb er material and 1-Lt is the attenuation coefficient (also called the linear attenuation coef- ficient) [10] . The minus sign (-) in the argument of the exponential indicates t hat the intensity decreases with increasing distance l. The qu antity 1-Lt can be expressed

as /-Ll = 1-LmP where p is the density of th e absorber and /-Lm is t he mass attenuation

coefficient of the absorber. 1-Lm depends on the atomic numb er of the absorber and the photon energy. The mass attenuation coefficient 1-Lm includes three parameters as follows /-Lm =

+

+

where

is the part due to the photoelectric effect ,

is the part due to Compton scattering and

is t he part due to pair production .

1.3 Radiopharmaceuticals

Our study of kidney function (Ch apters 2 to 7) involved the use of the metastable

(m) isotope technetium 99 (

m T c) as a tracer for measuring plasma clearance.

m Tc

is the metastable isomer of

Tc. Technetium 99 has an atomic number of 43 and

a mass numb er of 99. Its half-life is approximately 2.1 2 x 10

years . On the other

hand ,

mTc h as a half-life of approx imately 6 .02 hours which is suitable for medi cal

CHAPTER 1. INTRODUCTION 7

imaging and b lood sample collection techniques. Moreover , t he relat ively short half- life of

mT c limits the radiation exposure to patients.

m T c has m any ad vantages for kidney function t ests such as: 1) Its h alf-life (6.02 hours) is comparable to t he du- rat ion of many diagnostic studies; 2) It emit s gamma rays within the detectable range of diagnostic equipment ; 3) It is chemi cally suitable for chelating with certain phar- maceuticals [11]. In clinical nuclear med icine applications,

m T c-lab eled compounds occupy approximately 85% of all radiopharmaceutical usage.

~-decay

(87.5%)

~-decay

(12.5%)

....,..,.-Tc-99m,6.02 hours

~ '

~-decay

Figure 1.1: Schematic represent ation of the transition from

Mo t o

Ru.

Moly bdenum 99 (

Mo) is a radiopharmaceu tical wit h a half -life of 66. 02 hours .

The beta-minus decay from

Mo to

T c has a probability of 12.5% as shown in

Figure 1.1. The second bet a-minus decay (wit h a p robability of 87.5%) results in

the formation of

mT c through

Mo

m T c + e- + ^'De where 'De is the emitted

antineutrino.

mTc undergo es an isomeric t ransit ion to

Tc th rough t he emission

of a gamma photon . It is the gamma phot on from this t ransition t h at is detected

CHAPTER 1. I NTRODUCTION 8

during measurements .

A generator is a system which holds a parent/ daughter (herein

Moj9

m Tc ) mix- t ure in a transient equilibrium. The generator is used for transporting rad ioisotopes whose parents' half-lives are relatively long compared with the transportation time.

Because of t heir relatively short half-lives, certain radioisotopes cannot be shipped by commercial carriers and hence must be made locally or on-site.

Mo can easily be stored and transported between medical institu tions.

A

Mo to

mTc generator is based on an ion-exchange column [1] . The working process of a generator can be described by t he following . Firstly,

Mo exists as

99

Mo sodium molybdate (Na

Mo0

and is init ially bounded to an alumin a (Alz0

column. When

Mo in the molybdate ion decays, it is transformed into

m Tc as a p ertechnetate ion

mTc04 . Passing a saline eluant solution th rough the alumina column removes the

mTc04 and leaves

Moo~- on the alumin a column . The

99

mTc04 reacts with t he sali ne solution formin g sodium pertechnetate (Na

m T c0

After the

mTc isotope is extracted fr om the sodium pertechnetate in one elution, the

mTc 's activ ity in the generator will regenerate from zero. The total amount of

99

mTc extracted depends on the time interval between the elutions, the quantity of

99

Mo and the efficiency of t he elution.

There are several types of chemical impurities produced by a

Moj

m Tc gen-

erator eluate such as: 1)

Mo impurities contained within the product

mTc, 2)

the radiochemical impurity introduce by hydrolyzed technetium, and 3) the chemical

impurity of free Al

+ ions [ 1]. Therefore, qu ality control is required.

CHAPTER 1. INTRODUCTION 9

1.3.1 Chelation

Chelation is the form ation of chemical bonds b etween a ligand and a single atom [1 2].

The ligands used in r adiopharmace uticals are usually organic compounds. Chelants react with metal atoms therefore preventing the latter from reacting with other ele- ments. Chelants are used in chelation therapy for h eavy metal detoxification.

One of the most commonly used chelants of

m T c is dieth ylenetriamine penta- acetate (DTPA).

mTc-DTPA is used for kidney function tests because it is mostl y eliminated from the kidneys after an intravenous administrat ion [ 13].

mTc-DTPA was the radiopharmaceut ical used in this study. Other co mmonly used chelants for

99

mTc for performing kidney function tests are Mercapto Acetyl Tri G lycine (MAG3) and Dimercaptosuccinic acid (DMSA ) [14].

mT c-MAG3 is used to d etect scarring or necrosis of the renal cortex, as well as pyelonephritis (an infection in the kidneys) [1 5].

99

mTc-DMSA is usu ally used for ren al cortical imaging.

Besides

mTc, there are several other radioisotopes used for kidney function tests.

One of the altern ative radiopharmaceutical s is

Chromium- ethylenediaminetetra- acetic acid (

Cr-EDTA) which is freely fil trated by t he glomerulus (see defini tion in Chapter 2) in the kidneys [16]. However ,

Cr-EDTA is less often used than

99

mT c-DTPA in Canada. Another radiopharmaceutical is

1 ort ho-iodohippurate

(1

1 -0IH) which is cleared by t ubular secretion in the kidneys.

One thing to be c onsidered while using

mTc-labeled radiopharmaceu ticals is t he

possibility of protein binding of the radiopharmaceuticals with the plasma after t he

administration. Protein binding affects kidney function test results because it is t he

unbounded fraction of

mTc-DTPA in the body that produces the pharmacological

effect. The normal range of protein-binding of the

mTc-DTPA is between 3.7% and

13.5% [17]. Dual injections (inject the same tracer twice over a certain t ime period)

CHAPTER 1. INTRODUCTION 10

of

mTc-DTPA introduces a larger protein binding rate compared to using a single injection [ 18]. To reduce the inaccuracy of the kidney function test due to protein binding, one can use an ultr afilt rated solut ion of plasma (centrifu ge the plasma for an additional 10 to 20 minutes) rather than a standard filtrated p las ma (centrifuge the b lood for 10 to 15 minutes until t he plasma is sep arated from the blood) and use chromatography (a laboratory technique for separating mixtures) to d etermine t he b inding fraction [12].

1.4 Gamma Counter

The convent ional instrument for count ing gamma rays is the Geiger Counter [19] . The Geiger counter is composed of a tube filled with an inert gas and two electrodes (anode and cathode). When radiation ent ers the t ube, the gas is ionized resulting in the form ation of positively ch arged ions (moving towards the cathod e) and electrons (moving towards the anode) . During this process, additional ion pairs are a lso gen- erated . The net effect is t he generation of a measurable electrical current . A more precise device for counting radioactivity is the scint illation d etector.

Scintillation is a physical phenomenon whereby certain materials can emit light

(luminescence) when struck by radiat ion [10] . Radiation emitted from the b ody

enters the d evice through a window and travels to the scintillator crystals. In th e

scint illator crystal, t he incident radiation produces p hotoelectrons which move aro und

the crystal excit ing orbital electrons. The excitation of these orbital electrons results

in orbital vacancies, which are then filled by high er en ergy electrons . The excess

energy of the decaying electrons is released as photons. Some of t he photons strike th e

photocathod e and p roduces electrons. T his process is repeated in a p hotomultiplier

tube (PMT) where th e signals are magnified to a measurable electric sign al.

CHAPTER 1. INTRODUCTION 11

The inorganic scintillator , thallium-activated so dium iodine (Nai(Tl)) , is the most widely used scintillator in nuclear medicine. Nai (Tl) has several advantages such as:

1) Its dense structure (p = 3 .67 gjcm

which makes it a good absorb er, 2) Nai is transparent to lig ht, 3) The output signal is a pproximately linear over a wide range of energy (for a relatively thick crystal , e.g. 25 mm) which is suit able for counting, and 4) It has a high light output b ecause its emission spectrum matches well wit h t h e sensitivity of photomultiplier tubes. A maximum effi ciency (> 90%) can be achieved for incident wavelengt hs between 400 and 450 nanometers (nm); and 5) T he cost is relatively low [1] .

The Nai (Tl) counters used in this work were commercial well counters where t he radioactive materials were placed inside a container consisting of wells of various sh ap e

such as 1.6 em diameter x 3.8 em deep or 13 em d iameter x 25 em deep. Nai (Tl) counters are very suitable for count ing samples with r elatively small rad ioactivity (on the order of micro curies) due to their high geometric effi cien cy.

Figure 1.2 shows the single-well gamm a counter used in t he d ep artm ent of nuclear

medicine at the General Hospit al in St . John 's, NL. T he cylinder at the b ottom is

the sing le well, surrounded by a shielding lid. T h at is where the test sam ple, plasma

containing a radiopharmaceutical, was located when measuring t h e activity. The

single-well gamma counter is connected to a computer f or displaying and analyzing

the d at a. The accuracy of the single-well counter can b e affected by factors such

as dead time correction , detector effi ciency or geomet ric efficiency [1 , 20]. A ro ut ine

quality cont rol for ensuring th e performance of t h e single-well co unter should include

steps such as d aily tests for the calibration of peaking and sensit ivity, monthly test

using the Chi-squ are t ests, occasionally (quarterly) test for th e energy resolu tion and

an annual t est for t he d etector effi ciency [ 21]. T here is also an adjustable probe for

CHAPTER 1. I NTRODUCTION 12

Figure 1.2: The single -well gamma counter and thy roid uptake camera (manufac-

tured by Laboratory Technologie s, Inc.) used at the General Hospit al in St . J ohn 's,

Newfound land and Labrador.

CHAPTER 1. ^{INTROD UCTION} 13

performing a thy roid uptake test using

1 or

I. To perform such meas urements, the patient sits in a ch air facing the st ationary probe positioned over their t hyroid gland in the neck.

1.5 The ory of SPECT

The vast majority of nuclear medicine investigations are based on the usage of SPECT imaging . SPECT imaging is used to provide a three-dimensional (3D) image of regions of int erest in the body. In order to construct the image t he p atient is injected wit h a solution containing a radio active tracer that moves throughout the patient t hrough the bloodstream. G amma cameras are used to measure the gamm a rays emitted from the tracer to form two-dimensional (2D ) images. Using a computer assisted tomographic technique, the two-dimensional images are combined t o form a three- dimensional representation of the regions of interest under study. SPECT imaging is commonly u sed for performing kidney function tests . By providing dynamic images during a live scan , SPECT images can show how the kidneys are clearing t he t racer out in real time. At the heart of a SPECT machine a re gamma cameras. The most commonly used gamma cameras are scintillation cameras which were first d eveloped by Hal 0. Anger in 1950s [ 22] .

Figure 1.3 shows a schematic representation of a gamma camera and how it is

used to convert captured gamma rays into an image [23] . Gamma rays emitted from

the radiopharmaceutical in the p atient exit the body (labeled as subject shown in

Figure 1.3 ) and reach t he collimators. The co llimat ors which are usually mad e of

tungsten or lead are used for refining t he direction of the gamma rays by blocking

or ab sorbing gamma rays which arrive at an unwanted angle. Once t he photons

pass the collimat ors, they strike t he Nai (Tl) scintillation cryst al wh ich converts t he

CHAPTER 1. I NTROD UCT ION 14

Image X-signal

PHA

PMT Array Nai(TI) Collimator

y-rays Subject

Figure 1.3: Diagram of a typical gamma camera. This diagram was ad apted with

p ermission from t he Journal of R adioGrap hies [ 23].

CHAPTER 1. I NTRODUCTION 15

energy of an incident gamma-ray photon to a lower ene rgy photon (light) usu ally in the v isu al r a nge. The light enters a photomultiplier t ubes (PMTs) which converts t h e incident photon into a n electron via the photoelectric effect which is t hen amp lified into a measurable signa l. This signa l is split to produce information regarding t h e X and Y positions of t he initial gamma r ay along with a sign al evaluated by a pulse height an a lyzer (PHA) which selectively counts pulses from t h e PMTs that fall within certai n voltage amplit ude inter vals, known as cha nnels . The combined information is used to produce a 2D image as shown on top of Figure 1.3. The 2D images can be acquired eit her by a single-head rotating gamma camera (such as an arc with 180 degree rotation or a multi-h ead gamma camera where multiple gamma camer as rotate at the same t ime capturing gamma rays). The multi-head SPECT is more commonly used at t h e present because it takes less examination time and less r adiation dose.

The 2D image, also known as a pla nar image , has no information on the depth and structure at different depths. In order to obtain a 3D image using a gamma camer a, multiple 2D proj ections are required taken from different directions , which is the principle behind SPECT. Using sophisticated tomogr aphic reconstruction al- gorithms such as an analytical a lgorit hm or iterative algorithm [24], 2D images taken from multiple angles are then reconstructed to 3D images of the patient . Image r e- construction methods used in SPECT also help improve t he quali ty of the 3D im ages by reducing the effect of noise factors.

When analyzing the quality of the images , th ere are, in gener al, three aspects to be considered. The first one is the unsharpnesst, seen as blurring or fuzziness. Factors s uch as the geometric distance between the detector and subj ect, the subj ect's indis- tinguishable boundaries, involuntary or voluntary motions a nd receptor limitations

tunsha rpness is a term used in medical imaging processing, which ref ers to the loss of edge detail

of t he geometric properties of the object or image.

CHAPTER 1. I NTRODUCTION 16

in displaying the images, can contribute to the unsharpness. T he second aspect is t he contr ast of an image which allows subtle f eat ures to be distinguished . The distinction can b e a difference in physical density or chemical comp osition (atomic number). To improve this, some substances can be used as co ntrast agents to enhance the intrinsic

contr ast. This technique is usually used in x-ray imaging, medical ult rasonography and MRI imaging. The third asp ect is image noise. Any irrelevant information in the image can be defined as noise . These could be structure noise (e.g. unimportant structures), radiation noise (e.g. nonuniform intensity of the b eam or scattered radi- ation) , receptor noise (e.g. unb alance detectors in the gamma camera) and qu antum noise (e.g. fluctuations of electric power supply or the t hermal noise due to Browni an motion t)

1.6 Tracer Kinetic Modeling

Nuclear medicine diagnosis based on gamma counter measurements can be p erformed by pharmacokinetic analysis. One of the most important t heories in pharmacokinetics is t racer kinetic modeling.

The ob ject of t he tracer kinetic model is called the tracer (herein a radioisotope or radiopharmaceutical) which is the substan ce that follows a physiological or bio- chemical process [ 1]. The kinetic process of how t he body handles the tracer can be

mathematically described using parameters su ch as: volume of distribution Vol (e.g.

in ml or liter) , transit time T (e .g. in min or sec), and clearance C L (e.g. in ml/ min).

Vol is calculated at transient equilibrium by Vol = D / C wh ere D is the amount of tracer administered (e.g. in mg or Bq (for describing radioact ivity)) and C is t he

tBrownian motion is the random movement of particles suspended in a medium.