Convergence Analysis and Preliminary Experimental Feasibility for

MRI-Localized Region Spectroscopy with NIR Diffuse Optical

Tomography Jia Wanga, Scott Davisb, Shudong Jiangb, Brian W. Pogueb, Keith D. Paulsenb

aDepartment of Physics and Astronomy, bThayer School of Engineering Dartmouth College, Hanover NH 03755

Email: jia.wang@dartmouth.edu

ABSTRACT NIR tomography image reconstruction can be improved by incorporating spectral constraints and prior spatial

information. The convergence of scattering power was studied based on the distribution of projection error with different parameters. The reason that scattering properties are harder to recover than chromophore concentrations was discussed. Using “hard prior” spectral reconstruction, the role of stopping criteria was found to be important. Multiple wavelength simulations were used to choose suitable stopping criteria. Preliminary tests using a wavelength tunable Ti-Sapphire laser shows promise for frequency domain measurements covering a wide range of wavelengths. Keywords: near infrared, spatial priors, spectral constraints

INTRODUCTION Near infrared (NIR) imaging has demonstrated a promising ability to quantify tissue chromophore concentrations and scattering properties ([1],[2],[3],[4]). Spectroscopy can be performed by illuminating tissue with multiple wavelengths and reconstructing concentrations of Oxy-Hemoglobin (HbO2), DeoxyHemoglobin (Hb), water. The inverse problem is ill-posed and under-determined because of the light propagation properties in tissue. One way to overcome this problem is to incorporate spatial information from other modalities such as Magnetic resonance imaging (MRI)([5,6,7]). The prior structural information can be built into the reconstruction algorithm either through regularization or by mapping the Jacobian matrix onto a spatial matrix. This study used the latter method, also termed “hard prior”. Analysis of the convergence problem has been discussed in this work based on spectral reconstruction with “hard prior” spatial information. Experimentally, a new laser source system using a tunable Ti-Sapphire laser was tested to demonstrate frequency domain acquisition throughout the NIR spectrum.

METHODS In the frequency domain, the approximated diffusion equation is used to simulate light propagation in the tissue,

),(),(])([),()( 0 ωωωμωκ rqrc

irrr a =Φ++Φ∇•∇−

where 0 ( , )q r ω is an isotropic light source at position r, ( , )r ωΦ is the isotropic fluence at modulation frequency

ω , and c is the speed of light in tissue. Two parameters in the equation are the absorption coefficient )(raμ and

diffusion coefficient , where is the reduced scattering

coefficient, , is the anisotropy factor g(r) is the mean cosine of the single scatter

function. Our goal is to recover the tissue properties

'( ) 1/[3( )]a srκ μ= + μ ' ( )s rμ

)](1)[()(' rgrr ss −= μμ ( )g r

)(raμ and ( )rκ . The finite element method was used to solve

this inverse problem. Parameters are updated iteratively by minimizing the objective function which is defined by the

difference between measured boundary data ( , )M r ωΦ and calculated data ( , )C r ωΦ . The solution is assumed to be

obtained when the projection error decreases slower than a predetermined threshold value. A

Levenberg Marquard algorithm is used to update the optical parameters, ,

where y* is the measurement data, J is the Jacobian matrix which gives the sensitivity of measurement data to the optical properties. J

2

1(

NMC Mi i

iχ

=

= Φ −Φ∑ 2)

)],(*[)( 1 κμλδμ aTT FyJIJJ −+= −

TJ is an ill-conditioned matrix so regularization λ is added on the diagonal terms. Multi-wavelength measurement can be used simultaneously to solve for the concentration of chromophores and scattering parameters using the known spectral behavior of tissue chromophores and the Mie scattering approximation. The concentrations of

chromophores are fitted based on Beer’s Law, , where N is the total number of chromophores in

the tissue and C is the concentration of each chromophore. Scattering properties (scattering amplitude a and scattering

power b ) are fitted to an empirical approximation to Mie Scattering theory , .

i

N

iia C)()(

1λελμ ∑

=

=

bs a −= λμ '

Implementing prior structural information of the tissue in the reconstruction has been shown to be very useful in improving the accuracy of the reconstructed images and decreasing the ill-posedness of the inverse problem. There are different methods to incorporate the spatial information into the reconstruction as constrains[[8],[9],[10]]. In this work, the “hard prior” approach was used, which segments the image space into several homogeneous regions based on Magnetic Resonance images (MRI). The iterative update is simplified from updating nodes individually to updating the parameters for several homogenous regions.

Among the five parameters (concentration of Hb, HbO and water, scattering amplitude and power), scattering power is the only non-linear coefficient. The convergence problem of scattering power is discussed here. The strategy of choosing suitable stopping criteria was studied based on simulation of frequency domain measurements with data sets of 6 wavelengths and 31 wavelengths. The six wavelengths case was chosen to match an existing NIR tomography system and the 31 wavelengths were chosen uniformly between 650nm and 950nm to investigate a larger data set which may be acquired with a newly developed Ti: Sapphire based laser source. Random Gaussian distributed noise of 1% in amplitude and 1 degree in phase was added to the simulated data.

RESULTS Figure 1 and 2 show the distribution of projection error as a function of two varying parameters. As shown here, there

is a global minimum in the projection error for most cases, however, several local minima appear when Hb and

Scattering Power are varied over the given values. Multiple local may minima introduce difficulties in estimating scattering power.

Figure.1. The projection error of amplitude with different values of Hb and HbO, Scattering amplitude and Scattering power.

Figure.2. The projection error of phase with different values of Hb and HbO, Scattering amplitude and Scattering power.

Another factor which also has significant effect on scattering power lies in the process of mapping from absorption

and scattering coefficient to spectral parameters, as shown below.

22

2

1 and '3( ')

ln 3 'ln (1)3( )

so change of scattering power depends on change of diffusion coefficient1 (2)

3 'lnbecause it is r

bs

a s

b

sba

s

A

Ab A

bb

κ μ λμ μ

κ λ λ κ μ λμ λ

κ κκ κ μ λ

−

−

−

= =+

∂→ = =

∂ +

∂Δ = Δ = Δ

∂easonable to assume

1 3 (3)3 ' 3 ln

from (2) and (3), we can get3ln

b

bs

b

b AA

Ab

λκμ κ λ λ

κλ λ

∂= = → =

∂

Δ = Δ

From results above, the amplification of noise in scattering power due to diffusion coefficient can be estimated.

Assuming a scattering amplitude and power of A=1.2 and b =1.2, 20bb κ κκ∂

Δ = Δ = − ∗Δ∂

This results

demonstrates the difficulty in estimating scattering power in spectral reconstructions. In the frequency domain measurement, both amplitude and phase data are acquired. The projection error is the sum of

the difference between measurement and calculate data including both amplitude and phase. Typically, the algorithm is stopped if the change of projection error between iterations is below some value (i.e. 2%). The amplitude mainly depends on the absorption of light in the tissue while phase is dominated by the scattering process. Due to the different physics origin, it is therefore reasonable to consider these two parameters separately during the reconstruction. To investigate the effect of stopping criteria on the reconstruction, two stopping criteria are compared here: 1.The projection

error includes both amplitude and phase together; 2.Separate the phase and amplitude projection errors and iterate until both reach the defined stopping criteria. The choice of stopping criteria is shown to be critical in “hard” prior reconstruction, especially for measurement with a large number of wavelengths.

Figure 3. Comparison of different stopping criteria for frequency domain reconstruction (a) true values. Reconstruction results of 6

wavelengths with (b) stopping criteria1 and (c) stopping criteria2, reconstruction results of 31 wavelengths with (d) stopping criteria1

and (e) stopping criteria2.

Figure 3 depicts a simulated inclusion (HbO=0.02M, Hb= 0.02M, water=90%, A=1.2,b=1.2) in a two layer breast domain, composed of an outer fatty layer and a homogeneous fibro-glandular layer (HbO=0.01M, Hb= 0.01M, water=60%, A=1.0,b=1.0). The spatial information was assumed to be known and reconstructions were completed based on frequency domain data at 6 wavelength and 31 wavelengths. For each case, stopping criteria1 and 2 were compared. Reconstruction results with 6 wavelengths are shown in figure 3 (b) , (c) with stopping criteria1 and 2 respectively. The results of stopping criteria2 are slightly better than 1. This is also shown in the cross sectional comparisons in figure 4 (a) and (b). The advantage of stopping criteria 2 is more dramatic for the 31-wavelength case as shown in figure 3 (d) and

(e). In this case, stopping criteria 2 resulted in more accurate parameter estimation. Mapping the projection error directly indicated that amplitude data dominated the objective function and the algorithm stops while the phase stopping criteria is still relatively high. As the number of wavelengths increases, this dominance is more pronounced. If the phase and amplitude error is considered separately, the error in phase data is also minimized, producing more accurate images.

Figure 4. The cross section values of the different chromophores and scattering parameters. The cross-section line is shown in figure 1

(e). The dot line is true values and the solid line is reconstructed values. (a) and (b) are reconstruction results of 6 wavelengths with

stopping criteriaⅠandⅡ, respectively. (c) and (d) are reconstruction results of 31 wavelengths with stopping criteriaⅠandⅡ,

respectively.

A new experimental system which uses the 80MHz pulse signal from a Ti: Sapphire laser to achieve frequency

domain measurements at a full range of NIR wavelengths is currently under development. To achieve stable measurements using the Ti-Sapphire laser, the measured light signal must be locked with the reference signal. In figure 5, the yellow line is the reference signal generated by signal generator which is stimulated by the electronic signal (80MHz) from Ti-Sapphire laser. The laser light passed through a solid phantom and the light signal was measured by Photomultiplier Tubes (PMTS) detector. The angle between source and detector is about 180 degree. The detected (blue

line below) and reference signals ( yellow line above) are shown.

Figure.5. Phase locked detected and reference signals of Ti-Sapphire laser light passing through a solid tissue phantom.

CONCLUSIONS

Incorporating spectral constraints and prior spatial information can improve the NIR tomography image reconstruction. The convergence of scattering power was discussed based on the distribution of projection error as a function of different parameters. The potential difficulty in estimating of scattering power may originate from the multiple local minimum and amplification of noise during the spectral mapping process. Suitable stopping criteria plays an important role in “hard prior” spectral reconstruction, especially for data sets with a lager number of wavelengths. Preliminary tests using a Ti-Sapphire laser shows promise for acquiring frequency domain measurements over a large range of wavelengths. Acknowledgements This work was funded by the National Institutes of Health grants NCI PO1CA80139 and U54CA105480.

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