Bimonthly Meeting on Dec. 5, 2008

Absolute Metabolite Concentrations on Brain Tissue

by Gaussian and Lorentzian Functions

Amarjeet Bhullar

How to get absolute signal?

Absolute Signal = Raw data - Noise

Raw data = Real Spectrum without any manipulation

Noise = Draw a Baseline using few anchor points on Spectrum

Anchor Points Real Spectrum Baseline

Noise=Baseline is determined by interpolating anchor points on spectrums.

Absolute Metabolite Concentrations

• Create baseline using few anchor points on spectrum.

• Find metabolite peaks.

• Fit Mathematical function on metabolite peaks.

• Integrate peaks between the limits to calculate absolute metabolite concentrations.

Mathematical Model: Gaussian Function

2

2)(2

2/)( w

xx c

ew

Axf

)4ln(

1ww 2/w

Ah

60 62 64 66 68 70 72 74 76 78 800

50

100

150

200

250

300

1w

Adxew

A w

xx c

2

2)(2

2/

?2/

max

min

2

2)(2

dxe

w

Ax

x

w

xx c

dxe

w

Ax

x

w

xx cmax

min

2

2)(2

2/

w

xxErf

w

xxErf

A cc minmax 22

2

Integral of Gaussian Function : Error Function

Numerically: Codes developed in C and Mathematica 6.0

x

t dtexErf0

22)(

Error Function

dxe

w

Ax

x

w

xx cmax

min

2

2)(2

2/

Integral of Gaussian Function : Gamma Function

2

2min

2

2max 2

,2

12,

2

1

2

12

2 w

xx

w

xxA cc

dtetaFunctionGamma ta

0

1

2

1

dtetxaFunctionGammaIncompleteUpper t

x

a

1,

dtetxaFunctionGammaIncompleteLower txa

0

1,

xaxaa ,,

Mathematical Model: Lorentz Function

22)(4

2)(

wxx

wAxf

c

Adxwxx

wA

c

22)(4

2

?)(4

2max

min

22

dxwxx

wAx

x c56 58 60 62 64 66 68 70 72 74

0

50

100

150

200

250

300

350

400

Arb

itra

ry U

nit

Image Number

FWHMwh

w

Ah

2

Integral of Lorentzian Function : ArcTan

w

xxArcTan

w

xxArcTan

A cc )(2)(2 minmax

dxwxx

wAx

x c

max

min

22)(4

2

50 55 60 65 70 75 800

100

200

300

400

500

Gaussian Function

x

f(x)

Lorentzian Function

Difference Between Lorentzian and Gaussian Function

50 100 150 200 250-250

0

250

500

750

1000

1250 Voxel # 32

Sig

nal

(M

R U

nit

s)

Image Number

Manipulated Spectrum

0 50 100 150 200 250-250

0

250

500

750

1000

1250 Voxel # 32

Sig

nal

(M

R U

nit

s)

Image Number

Anchor Points Real Spectrum Baseline

Voxel #32 Gaussian

Cho/Cre 1.58

Cho/NAA 0.34

Metabolite ratios by Gaussian function

0 50 100 150 200 250-250

0

250

500

750

1000

1250 Voxel # 32

Sig

nal

(M

R U

nit

s)

Image Number

Anchor Points Real Spectrum Baseline

50 100 150 200 250-250

0

250

500

750

1000

1250 Voxel # 32

Sig

nal

(M

R U

nit

s)

Image Number

Manipulated Spectrum

Metabolite ratios by Lorentzian function

Voxel #32 Lorentzian

Cho/Cre 1.54

Cho/NAA 0.33

Voxel #32 Gaussian Lorentzian Average

Cho/Cre 1.58 1.54 1.55

Cho/NAA 0.34 0.33 0.33

Conclusion:

Both mathematical models have produced the same ratios.

Suggestions are welcome