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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
