OSA's Digital Library

Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 4, Iss. 8 — Apr. 12, 1999
  • pp: 231–240
« Show journal navigation

Diffuse optical 3D-slice imaging of bounded turbid media using a new integro-differential equation

Deva N. Pattanayak and A. G. Yodh  »View Author Affiliations


Optics Express, Vol. 4, Issue 8, pp. 231-240 (1999)
http://dx.doi.org/10.1364/OE.4.000231


View Full Text Article

Acrobat PDF (125 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

A new integro-differential equation for diffuse photon density waves (DPDW) is derived within the diffusion approximation. The new equation applies to inhomogeneous bounded turbid media. Interestingly, it does not contain any terms involving gradients of the light diffusion coefficient. The integro-differential equation for diffusive waves is used to develop a 3D-slice imaging algorithm based on the angular spectrum representation in the parallel plate geometry. The algorithm may be useful for near infrared optical imaging of breast tissue, and is applicable to other diagnostics such as ultrasound and microwave imaging.

© Optical Society of America

1. Introduction

Tomographic imaging of deep tissues with diffusive waves makes possible functional imaging based on novel optical contrasts derived from tissue chromophores, structure and metabolism [1–3

1. B. Chance, Q. M. Luo, S. Nioka, D. C. Alsop, and J. A. Detre, “Optical investigations of physiology: a study of intrinsic and extrinsic biomedical contrast,” Phil. Trans. Royy. Soc. London B. 352, 707 (1997). [CrossRef]

]. The near-infrared spectroscopic properties of breast tumors for example, have been found to differ from adjacent normal tissues. Such spectroscopic signatures hold promise for increased tumor detection sensitivity and specificity [4–10

4. J. H. Hoogenraad, M. B.van der Mark, S. B. Colak, G.W. ’t Hooft, and E.S.van der Linden, “First Results from the Philips Optical Mammoscope”, Proc.SPIE / BiOS-97 (SanRemo, 1997 ).

]. In order to take full advantage of these new contrasts it is critical to develop high fidelity three-dimensional optical imaging algorithms for diffusing light in turbid media such as the breast. To this end a number of image reconstruction schemes have been developed [11–13

11. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M.S. Patterson, “Optical Image reconstruction using frequency-domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996). [CrossRef]

]. Many of these schemes use integral equations based on Born or Rytov approximations to generate a set of linear equations which are then solved to update the absorption and scattering coefficients associated with each voxel in the reconstructed volume. The integral formulation is attractive because of its speed. However, to our knowledge most of these formulations ignore terms involving gradients of the light diffusion coefficient [14

14. An integro-differential equation valid for infinite medium,but containing no gradient of Diffusion constant was used for image reconstruction by M. A. O’Leary, D. A . Boas, B. Chance, and A.G. Yodh, “Experimental Images of heterogeneous turbid media by frequency-domain diffusing-photon tomotography,” Opt. Lett. 20, 426–428 (1995). [CrossRef]

,15

15. An anonymous reviewer has brought to our attention a recent paper by S.R. Arridge and W.R.B. Lionheart, “Non-uniqueness in diffusion based optical tomography,” Opt. Lett. 23, 882–884 (1998), which deals with the second order derivative of the diffusion coefficient. [CrossRef]

] Although this approximation is often reasonably accurate, a recent paper [17

17. J. L. Ye, R.P. Millane, K. J. Webb, and T. J. Downar, “Importance of the gradD term in frequency resolved optical diffusion imaging,”Opt. Letters ,23,1998. [CrossRef]

] has established the importance of this gradient term, showing that its neglect is responsible for cross coupling between absorption and scattering images.

In this paper, we derive a new integro-differential equation for diffuse photon density waves (DPDW) within the diffusion approximation. The new equation, which is developed for bounded turbid media does not explicitly contain any terms involving gradients of the light diffusion coefficient. We then use this integro-differential equation and develop a theoretical inverse scattering algorithm for three-dimensional image reconstruction. The theoretical framework follows the principles of near-field diffraction tomography based on the angular-spectrum representation; in particular we show how to develop three-dimensional slice images of a breast compressed between two parallel plates. The technique employs a series of plane diffuse photon density waves with different modulation frequencies. For this set of plane incident diffusive waves, the algorithm requires two-dimensional FFT operations, and a one-dimensional matrix inversion. Thus the method is non-iterative in two space dimensions and is therefore computationally fast.

2. Integro-differential equation for diffuse photon density waves

We begin with the equations satisfied by the diffuse photon density, Φ and the diffuse photon flux density, J [18

18. K. M. Case and P. F. Zweifel, in Linear Transport Theory (Addison -Wesley,MA,1967).

]

J(r,ω)=Φ(r,ω)S1(r,ω)3μ's(r,ω)
(1)
Φ(r,ω)=.J(r,ω)S0(r,ω)(iωc+μa(r,ω))
(2)

Here, ω denotes the modulation frequency of the source intensity, μα and μs are the absorption and transport scattering coefficients of the bounded turbid media, c is the velocity of light in the medium and S 0 (r,ω), S1 (r, ω) are the monopole and dipole moments of the intensity modulated optical source respectively. From these equations we eliminate the photon flux density and obtain, after a lengthy calculation, the following wave equation involving only the photon density function

(2+kd2)Φ(r,ω)=3μ¯sδμaΦ(r,ω)+·(δμsμsΦ(r,ω))
·(3δμsμsS1(r,ω))3μ¯s(r)S0(r,ω).
(3)

Here

kd2=3μ¯s(iωcμ¯a)
(4)

is the square of the complex wavenumber kd. The bar over a physical parameter denotes spatial average of that parameter. Notice that for unmodulated source intensity (the continuous dc domain in contrast to the frequency domain), the photon density is a non-propagating damped wave. The wave number in this case is purely imaginary. For frequency modulated source intensity, the wave number is complex and the photon density acquires the characteristics of a damped or attenuated wave.

Because of its differential nature Eq. (3) will admit many solutions and boundary conditions are necessary to obtain the physical solution. The problem of finding the proper boundary condition is complex and is generally further exacerbated because the diffusion approximation is not valid near the boundary. Inspite of this basic difficulty, diffusion theory is used widely due to its simplicity and the applicability of numerous analytical and numerical techniques for its solution in finite domains of arbitrary geometry. Approximate boundary conditions are generally adopted which have been shown to be fairly accurate. Here, we adopt the general boundary condition [19

19. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978) Vol.. 19.

]

n̂·J(r,ω)=αΦ(r,ω).
(5)

We follow the standard Green’s function technique for the derivation of an integral equation from a partial differential wave equation [20

20. P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill,1953).

]. However, we deliberately choose a simple Green’s function corresponding to the operator on the left hand side (L.H.S.) of Eq. (3) that satisfies the equation

(2+kd2)G(r,r,ω)=4πδ(rr),
(6)

and is given by

Gd(r,r,ω)=exp(ikdrr)rr,
(7)
withkd=3μ¯s(iωcμ¯ a).
(8)

Here Im(kd) > 0. Notice that our choice of an infinite space Green’s function will not limit in any way when we consider finite domains; finite medium effects are properly treated by the boundary integrals. If we now multiply both sides of Eq. (3) and (6) by Gd (r, r′ , ω) and Φ;(r, ω) respectively and subtract the resulting equations from one another and use a number of standard Green’s identities, we obtain

Φ(r,ω)=3μ¯s4πVS0(r,ω)Gd(r,r,ω)d3r+34πVS1(r,ω).Gd(r,r,ω)d3r
+14πS(3αμ¯sGd(r,r,ω)+Gd(r,r,ω).n̂)Φ(r,ω)dS
3μ¯s4πV(μa(r)μ¯a)Φ(r,ω)Gd(r,r)d3r
+14πVμs(r)μ¯sμs(r)Φ(r,ω).Gd(r,r)d3r.
(9)

For a homogeneous diffusing media, the integro-differential equation becomes

Φ(r,ω)=3μ¯s4πVS0(r,ω)Gd(r,r,ω)d3r+34πVS1(r,ω).Gd(r,r,ω)d3r
+14πS(3αμ¯ sGd(r,r,ω)+Gd(r,r,ω).n̂)Φ(r,ω)dS.
(10)

Equations (9) and (10) are the basic equations of our paper. They are derived using the diffusion approximation to photon transport and without any other approximations. Eq. (9) is an intro-differential equation for the photon density wave and involves only the absorption and transport scattering coefficients. Notice that Eq. (9) does not explicitly contain gradients of light diffusion coefficient. Our derivation is exact and does not assume any condition on the gradient of the light diffusion coefficient. As has been pointed out recently [17

17. J. L. Ye, R.P. Millane, K. J. Webb, and T. J. Downar, “Importance of the gradD term in frequency resolved optical diffusion imaging,”Opt. Letters ,23,1998. [CrossRef]

] neglect of the gradient term leads to cross coupling between scattering and absorption images. Eq. (10) describes the homogeneous or the background photon density field that includes the source contributions (the first two terms on the R.H.S.), as well as contributions from the boundary (the last term of the R.H.S.). Note that Eq. (10) contains only average values of the scattering parameters and the surface integral includes the total field.

Eq. (10) relates the value of the diffuse photon density wave at any point inside the turbid medium to the monopole and dipole moment of the source and to the photon density on the surface bounding the medium. A surface integral equation for the boundary value of the photon density is obtained if we take the point r to be on the surface. This limiting operation must be done carefully because the integrals involving the gradient of the Green’s function are not continuous when the interior point approaches a point on the surface. The other integrals involving the Green’s function however are continuous [21

21. Claus Muller, Foundations of the Mathematical theory of Electromagnetic Waves (Springer-Verlag,New York,1969).

]. From an experimental point of view, if the photon density on the surface surrounding the turbid medium is measured, then one can use Eq. (10) to determine the diffuse photon density wave anywhere inside the medium by simply carrying out the surface integrals in Eq. (10). If the value of this estimated photon density matches that of the measured value at all points on the surface, then an absence of tissue heterogeneity is indicated. In practice there will never be perfect cancellation, but a larger difference indicates the presence of stronger heterogeneities inside the tissue.

Another approximate way to evaluate the background field is to evaluate the surface integrals in Eq. (10) iteratively. First ignore the third surface term involving the boundary values of the photon density in Eq. (10) and use only the first two terms involving the source terms to calculate the first order background field everywhere. Then update the value of the background field by using the first order values in the last integral in Eq. (10). It is important to devise a way to estimate a reasonable background field, because when this field is subtracted from the total measured diffuse photon density field, then the remainder is the field directly attributable to the presence of the heterogeneity.

3.Inversion algorithm for 3D-slice imaging

We make the assumption that the heterogeneities are weak and do not perturb the background field substantially. Then we can approximate the photon density within the integral operator with the background field. With this first order Born approximation, Eq. (9) becomes

ΦP(r,ω)=ΦM(r,ω)ΦH(r,ω)=3μ¯s4πV(μa(r)μ¯a)ΦH(r,ω)Gd(r,r,ω)d3r
+14πVμs(r)μ¯sμs(r)ϕH(r,ω).Gd(r,r,ω)d3r.
(11)

The superscript M implies measured field and the superscript P denotes the processed (scattered) field values. In Eq.(12), we have used the identity

ΦH(r,ω)3μ¯s4πVS0(r,ω)Gd(r,r,ω)d3r+34πVS1(r,ω).Gd(r,r,ω)d3r
+14πS(3αμ¯ sGd(r,r,ω)+Gd(r,r,ω).n̂)ΦM(r,ω)dS.
(12)

The quantity ΦH(r, ω) often identified as the background field can be determined from the knowledge of the source and the measured field at the boundary from Eq. (12).

Although our integro-differential equation is valid for arbitrary geometry, from now on we shall restrict ourselves to the case of a plane parallel plate configuration. The plane parallel plate is an accepted clinical diagnostic configuration for breast imaging. It offers the possibility of compressing the breast to decrease its thickness for increased signal strength. We also take the transverse dimension of the plate to be sufficiently larger than the plate separation. Because of the negligibly small value of the field at the periphery of sufficiently large plates, in practice the transverse plate extent can be taken to be infinite. In essence, Eq. (11) is an integral equation for μa(r) , and μs(r) with all the other variables such as the background field, the Green’s function and the background optical parameters either given or estimated.

In the new integro-differential equation, we use the Weyl representation [22–23

22. H. Weyl and Ann. Phys. 60, 481, (1919).

] of Green’s function

G(r,r,ω)=dpdqi2πmexp(ip(xx)+iq(yy)+imzz),
(13)

where m=kd2p2q2,withIm(m)>0.

The direction z is chosen perpendicular to the surface normal and the x and y directions are chosen along the transverse directions. There is a singularity in the z direction as can be seen from the absolute value operator in the exponent of the Weyl representation. This representation is also known as the angular spectrum representation and is fundamental to the theory and algorithm now commonly known as Diffraction tomography [24–25

24. E. Wolf, “Principles and Development of Diffraction Tomography” in Trends in Optics, ed. A. Consortini (Academic Press, San Diego, 1996). [CrossRef]

]. Diffraction tomography was applied to far field coherent optical imaging systems, but recently this technique has been extended to the near field diffuse photon density wave based imaging [26–34

26. For an application of diffraction tomography to near field diffusion wave imaging and an analytical expression relating theoretical resolution and tissue thickness see, GE Class I Technical report (publicly available on request) by D. N. Pattanayak, “Resolution of Optical Images Formed by Diffusion Waves in Highly Scattering Media,” GE Tech. Info. Series 91CRD241 (1991).

]. With this representation of Green’s function the integrals in x and y become convolution integrals which turn into algebraic products in p, q space. Taking the two-dimensional Fourier transform of Eq. (12) at the detector plane z = zd we find that

Φ̂P(p,q,zd,ω)=zszddzT̂a(p,q,z,ω)K̂(p,q,zdz,ω)
+zszddzT̂s(p,q,z,ω).kK̂(p,q,zdz,ω),
(14)

where

Ta(r,ω)=3μ¯ s4π(μa(r)μ¯ a)ΦH(r,ω),
(15)
Ts(r,ω)=14π(1D(r,ω)D)ϕH(r,ω),
(16)

and

andK̂(p,q,zdz,ω)=i2πmexp(im(zdz)).
(17)

In Eqs.(14)–(17), k = x̂p + ŷq + ẑm is the wave vector and zs, zd denote the positions of the source and detector plates respectively. The caret (̂) over functions signifies the Fourier transform of a function. Several things are worth noting at this point. First, the absorption coefficient contribution depends on a scalar function, while the transport scattering coefficient depends on the product of two vectors. Second, the gradient of the diffuse photon density wave interacts with the scattering heterogeneity, while the diffuse photon density wave interacts with the absorption heterogeneity. Finally, the inverse of the transport scattering coefficient has been explicitly replaced by the photon diffusion coefficient D=c3μs.

Eq. (14) is a very simple looking equation, but in fact a lot of physics is hidden. The Fourier transform of the so-called tumor functions, Ta and Ts which are products of the scattering parameters and the background field in real space, will be convolution integrals in the Fourier space. This convolution in the spatial frequency space between the tumor parameters and the background field couples temporal frequency, ω with spatial frequency (p,q). This implies that the use of a set of modulation frequencies does not in any easy manner let us solve for the spatial Fourier amplitudes of the tumor functions. This coupling is eliminated if the background field is spatially in the form of a plane wave rather than a spherical wave. The usefulness of plane wave sources is well known in the field of diffraction tomography [24

24. E. Wolf, “Principles and Development of Diffraction Tomography” in Trends in Optics, ed. A. Consortini (Academic Press, San Diego, 1996). [CrossRef]

].

We will use a series of plane waves whose central modulation frequencies are stepped to span some frequency bandwidth. For the plane wave case the tumor functions, T̂a, T̂s, can be written as a product of two functions. One function, t̂a or t̂D , contains information about the heterogeneities and depends upon (p, q, z) and the other function depends upon the background plane waves and modulation frequency (see Eq. (18) below). One can then utilize this temporal frequency bandwidth to obtain resolution in the longitudinal (z) direction. The key advantage of using the series of stepped central modulation frequencies is that the detection system is still narrow band and one can use maximum permissible signal strength at all discrete frequencies. We plan to publish in a later paper the details of this technique with simulated data, but here we point out the salient features of the method. For the plane wave case we find

T̂a(p,q,z,ω)=ta(p,q,z){Φ+exp(+im0z)+Φexp(im0z)},
T̂s(p,q,z,ω)=td(p,q,z){k0+Φ+exp(+im0z)+k0Φexp(im0z)},
t̂a(p,q,z,ω)=34π[μ¯aδ(pp0)δ(qq0)μa(pp0,qq0)],
t̂D(p,q,z)=14D¯[D¯δ(pp0)δ(qq0)D(pp0,qq0)],
andk0+=x̂p0+ŷq0+ẑm0,k0=x̂p0+ŷq0ẑm0.
(18)

Here p 0 = Re(kd)cos(θ)cos(ϕ) ,q 0 = Re(kd)cos(θ)sin(ϕ) where θ is the angle the plane wave makes with respect to the z direction, and ϕ is the corresponding azimuthal angle. Care must be exercised in the angle interpretations as we are dealing with attenuating plane waves. For the case of a normal incidence, p 0 = 0 and q 0 = 0. The parameter m0=kd2p02p02withIm(m0)>0. The general mode structure of the background plane wave for the parallel plate slab geometry can be shown from the solution of the homogeneous equation ( Eq. (10)) to be a sum of both forward and backward propagating waves [33

33. J. Ripoll and M. Nieto-Vesperinas, “Reflection and Transmission Coefficients of Diffuse Photon Density Waves, (to be published).

]

Φ0(x,y,z,ω)=Φ+exp(ip0x+iqoy+im0z)+Φexp(ip0x+iq0yim0z).
(19)

Here, Φ+ , and Φ- are complex constants obtained from solution or measurements as discussed earlier in the section on the background field.

We approximate the integration in Eq. (14) in the z variable by means of a sum over N slices between the parallel plates (the slice thickness can be different and is denoted by ∆zj with j ranging from 1: N), and use Eqs. (15)–(19) to obtain the following result

Φ̑(zd,ωj)=i=1N[t̂a(zi)f̂1(zi,ωj)+t̂d(zi)f̂2(zi,ωj)]forj=1:2N.
(20)

where

f̂1(zi,ωj)=Δzi{Φ+exp(+im0zi)+Φexp(im0zi)}K̂(zdzi,ω),
f̂2(zi,ωj)=Δzi{(pp0+qq0+mm0)Φ+exp(+im0zi)+
(pp0+qq0mm0)Φexp(im0zi)}K̂(zdzi,ω).
(21)

Here, we suppressed the (p,q) arguments for the sake of brevity. Eq. (20) provides 2N linearly independent complex equations that can be solved for the two dimensional Fourier transform of the absorption and scattering coefficients at the N selected slice positions in the z direction. This can be seen more clearly when Eq. (20) is written in a matrix form

Φ̲̂=f̳̂·t̲̂ .
(22)

Where Φ̂ is the vector of 2N dimension consisting of the measurements of the processed (scattered) field at the 2N modulation frequencies, f̳̂ is a 2NX2N matrix consisting of two 2NXN matrices f̂1̳ , and f̂2̳ defined in Eqs. (21a)–(21b). Eq. (22) can be solved by direct matrix inversion as shown below or iteratively by following methods such as SIRT, ART [13

13. S. R. Arridge, “Forward and inverse problems in time-resolved infrared imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, ed. G. Muller, B. Chance, Rl. Alfano, S. Arridge, J. Beuthan, E. Gratton, M Kaschke, B. Masters, S. Svanberg, and P. van der Zee, Proc SPIEIS11, 35–64 (1993).

] . t̂ consists of the sum of the two N dimensional vectors t̂a and t̂D.

t̲̂=(f̳̂)1·Φ̲̂.
(23)

t̂ in Eq. (23) is the two-dimensional Fourier transform of the absorption and scattering coefficients at each of the N number of slices in the direction normal to the plates (the z direction). The values of the scattering coefficients in the direction transverse to the surface normal (the x and y directions) are obtained by applying the two-dimensional inverse Fourier transform, i.e.

t̲(x,y,zi)=14π2dpdq(f̳̂(p,q))1·Φ̲̂(p,q)exp(ipxiqy)fori=1:N
(24)

For the Breast imaging case, Eq. (24) provides one with 3D-slice images of the absorption and scattering coefficients at the selected “virtual” slice positions in the interior of the breast compressed between the two plates. These images may be reconstructed with light of different wavelengths within the near- infrared window. Such measurements improve the usefulness of nearfield optical imaging modality employing diffuse photon density waves.

4. Conclusion

We presented a rigorous derivation of a new integro-differential equation for the diffuse photon density wave in an inhomogeneous bounded turbid medium within the diffusion approximation. The resulting equation contains only the absorption and scattering coefficients and does not include gradients of the light diffusion coefficient. We also derived a novel surface integral equation for the diffuse photon density waves for the case of a bounded homogeneous turbid medium. We used the theory to develop a detailed imaging algorithm that provides one with 3D-slice images of the absorption and scattering coefficients of tissue (breast) that is compressed between two parallel plates. We believe this algorithm will be useful in increasing the specificity of the near infrared optical imaging modality. Our algorithm being based on inhomogeneous wave equation of the Helmholtz type may also be applicable to other imaging modalities such as ultrasound.

Acknowledgements

We gratefully acknowledge useful conversations with Britton Chance, Joe Culver, Leonid Zubkov, Monica Holboke, Turgut Durduran, Xingde Li, Regine Choe. We acknowledge partial support from US Army RO DAMD17-97-1-7272 and NIH 1-RO1-CA75124-01 grants respectively.

References and links

1.

B. Chance, Q. M. Luo, S. Nioka, D. C. Alsop, and J. A. Detre, “Optical investigations of physiology: a study of intrinsic and extrinsic biomedical contrast,” Phil. Trans. Royy. Soc. London B. 352, 707 (1997). [CrossRef]

2.

A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain functions,” Trends. Neurosci. 20, 435 (1997) . [CrossRef] [PubMed]

3.

Y. Hoshi and M. Tamura, “Near-Infrared Optical Detection of Sequential Brain Activation in The Prefrontal cortex during mental tasks,” Neuroimage. 5, 292 (1997). [CrossRef] [PubMed]

4.

J. H. Hoogenraad, M. B.van der Mark, S. B. Colak, G.W. ’t Hooft, and E.S.van der Linden, “First Results from the Philips Optical Mammoscope”, Proc.SPIE / BiOS-97 (SanRemo, 1997 ).

5.

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, and B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” Appl. Opt. 36, 10 (1997). [CrossRef] [PubMed]

6.

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag, and M. Kaschke. “Frequency-domain instrumentation techniques enhance optical mammography: Initial clinical results” Proc. Natl. Ac ad. Sci. USA 94, 6468–6473 (1997). [CrossRef]

7.

S. K. Gayen, M. E. Zevallos, B. B. Das, and R. R. Alfano, “Time-sliced transillumination imaging of normal and cancerous breast tissues,” in Trends in Opt. And Photonics, ed. J. G. Fujimoto and M. S. Patterson.

8.

B. W. Pogue, M. Testorf, T. McBride, U. Osterberg, and K. Paulsen ,“Instrumentation and design of a frequency-domain diffuse optical tomography imager for breast cancer detection,” Opt. Express 1, 391 (December 1997). http://epubs.osa.org/oearchive/source/2827.htm [CrossRef] [PubMed]

9.

D. Grosenick, H. Wabnitz, H. H. Rinneberg, K. T. Moesta, and P. M. Schlag, “Imaging and Characterization of Breast tumors using a laser-pulse mammograph,” SPIE3597 (1999).

10.

A. G. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (March 1995). [CrossRef]

11.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M.S. Patterson, “Optical Image reconstruction using frequency-domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996). [CrossRef]

12.

Y. Yao, Y. Wang, Y. Pei, W. Zhu, and R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions using a born iterative mehod,” J. Opt. Soc. Am. A 14, 325–342 (1997). [CrossRef]

13.

S. R. Arridge, “Forward and inverse problems in time-resolved infrared imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, ed. G. Muller, B. Chance, Rl. Alfano, S. Arridge, J. Beuthan, E. Gratton, M Kaschke, B. Masters, S. Svanberg, and P. van der Zee, Proc SPIEIS11, 35–64 (1993).

14.

An integro-differential equation valid for infinite medium,but containing no gradient of Diffusion constant was used for image reconstruction by M. A. O’Leary, D. A . Boas, B. Chance, and A.G. Yodh, “Experimental Images of heterogeneous turbid media by frequency-domain diffusing-photon tomotography,” Opt. Lett. 20, 426–428 (1995). [CrossRef]

15.

An anonymous reviewer has brought to our attention a recent paper by S.R. Arridge and W.R.B. Lionheart, “Non-uniqueness in diffusion based optical tomography,” Opt. Lett. 23, 882–884 (1998), which deals with the second order derivative of the diffusion coefficient. [CrossRef]

16.

D. N. Pattanayak and E. Wolf , “ Resonance States as Solutions of the Schrodinger Equation with a Non-Local Boundary Condition, ” Phys. Rev. D13, 2287(1976).

17.

J. L. Ye, R.P. Millane, K. J. Webb, and T. J. Downar, “Importance of the gradD term in frequency resolved optical diffusion imaging,”Opt. Letters ,23,1998. [CrossRef]

18.

K. M. Case and P. F. Zweifel, in Linear Transport Theory (Addison -Wesley,MA,1967).

19.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978) Vol.. 19.

20.

P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill,1953).

21.

Claus Muller, Foundations of the Mathematical theory of Electromagnetic Waves (Springer-Verlag,New York,1969).

22.

H. Weyl and Ann. Phys. 60, 481, (1919).

23.

See, also , A.J. Banos, in Dipole Radiation In the Presence of a Conducting Half-Space (Pergamon Press, New York, 1966).

24.

E. Wolf, “Principles and Development of Diffraction Tomography” in Trends in Optics, ed. A. Consortini (Academic Press, San Diego, 1996). [CrossRef]

25.

A. J. Devaney, “Reconstructive tomography with diffracting Wavefields,” Inv. Probl. ,2,161–18391986).

26.

For an application of diffraction tomography to near field diffusion wave imaging and an analytical expression relating theoretical resolution and tissue thickness see, GE Class I Technical report (publicly available on request) by D. N. Pattanayak, “Resolution of Optical Images Formed by Diffusion Waves in Highly Scattering Media,” GE Tech. Info. Series 91CRD241 (1991).

27.

C. L. Matson, N. Clark, L. McMackin, and J. S. Fender, “Three-dimensional Tumor Localization in Thick Tissue with The Use of Diffuse Photon-Density Waves,” Appl. Opt. 36, 214–219 (1997). [CrossRef] [PubMed]

28.

X.D. Li, T. Durduran, A.G. Yodh, B. Chance, and D.N. Pattanayak, “Diffraction Tomography for Biomedical Imaging With Diffuse Photon Density Waves,” Opt. Lett. 22, 573–575 (1997). [CrossRef] [PubMed]

29.

BQ Chen, JJ Stamnes, and K Stamnes, “Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium.” Pure Appl. Opt. 7, 1161–1180 (1998). [CrossRef]

30.

X. Cheng and D. Boas, “Diffuse Optical Reflection Tomography Using Continous Wave Illumination,” Opt. Express 3, 118–123 (1998); http://epubs.osa.org/oearchive/source/5663.htm [CrossRef] [PubMed]

31.

S. J. Norton and T. Vo-Dinh, “Diffraction Tomographic Imaging With Photon Density Waves: an Explicit Solution, J. Opt. Soc. Am. A 15, 2670–2677 (1998). [CrossRef]

32.

J. C. Schotland, “Continuos Wave Diffusion Imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997). [CrossRef]

33.

J. Ripoll and M. Nieto-Vesperinas, “Reflection and Transmission Coefficients of Diffuse Photon Density Waves, (to be published).

34.

J. Ripoll and M. Nieto-Vesperinas, “Spatial Resolution of Diffuse Photon Density Waves, ” J. Opt. Soc. Am.A (to be published).

35.

C.L. Matson and H. Liu, “Analysis of the forward problem with diffuse photon density waves in turbid media by use of a diffraction tomography model,” J. Opt. Soc. Am. A 16, 455–466 (1999). [CrossRef]

36.

For the angular spectrum representation in a slab field and for an excellent account of the angular spectrum representation see, L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York,1995).

OCIS Codes
(170.3010) Medical optics and biotechnology : Image reconstruction techniques
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.3830) Medical optics and biotechnology : Mammography
(170.5270) Medical optics and biotechnology : Photon density waves
(170.5280) Medical optics and biotechnology : Photon migration

ToC Category:
Focus Issue: Biomedical diffuse optical tomography

History
Original Manuscript: March 5, 1999
Published: April 12, 1999

Citation
Deva Pattanayak and Arjun Yodh, "Diffuse optical 3D-slice imaging of bounded turbid media using a new integro-differential equation," Opt. Express 4, 231-240 (1999)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-8-231


Sort:  Journal  |  Reset  

References

  1. B. Chance, Q. M. Luo, S. Nioka, D. C. Alsop and J. A. Detre, "Optical investigations of physiology: a study of intrinsic and extrinsic biomedical contrast," Phil. Trans. Royy. Soc. London B. 352, 707 (1997). [CrossRef]
  2. A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain functions," Trends. Neurosci. 20, 435 (1997). [CrossRef] [PubMed]
  3. Y. Hoshi and M. Tamura, "Near-Infrared Optical Detection of Sequential Brain Activation in The Prefrontal cortex during mental tasks," Neuroimage. 5, 292 (1997). [CrossRef] [PubMed]
  4. J. H. Hoogenraad, M. B.van der Mark, S. B.Colak, G.W.t Hooft, E.S.van der Linden, "First Results from the Philips Optical Mammoscope," Proc.SPIE / BiOS-97 (SanRemo, 1997 ).
  5. J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner and B. J. Tromberg, "Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject," Appl. Opt. 36, 10 (1997). [CrossRef] [PubMed]
  6. M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag and M. Kaschke. "Frequency-domain instrumentation techniques enhance optical mammography: Initial clinical results" Proc. Natl. Ac ad. Sci. USA, 94, 6468-6473 (1997). [CrossRef]
  7. S. K. Gayen and M. E.Zevallos, B. B. Das, R. R. Alfano, "Time-sliced transillumination imaging of normal and cancerous breast tissues," in Trends in Opt. And Photonics, ed. J. G. Fujimoto, M. S. Patterson.
  8. B. W. Pogue, M. Testorf, T. McBride, U. Osterberg and K. Paulsen ," Instrumentation and design of a frequency-domain diffuse optical tomography imager for breast cancer detection ," Opt. Express 1, 391 (December 1997). http://epubs.osa.org/oearchive/source/2827.htm [CrossRef] [PubMed]
  9. D. Grosenick, H. Wabnitz, H. H. Rinneberg, K. T. Moesta and P. M. Schlag, "Imaging and Characterization of Breast tumors using a laser-pulse mammograph," SPIE 3597 (1999).
  10. A. G. Yodh, B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (March 1995). [CrossRef]
  11. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue and M.S. Patterson, "Optical Image reconstruction using frequency-domain data: Simulations and experiments," J. Opt. Soc. Am. A 13, 253-266 (1996). [CrossRef]
  12. Y. Yao, Y. Wang, Y. Pei, W. Zhu and R. L. Barbour, "Frequency-domain optical imaging of absorption and scattering distributions using a born iterative mehod," J. Opt. Soc. Am. A 14, 325-342 (1997). [CrossRef]
  13. S. R. Arridge, "Forward and inverse problems in time-resolved infrared imaging," in Medical Optical Tomography: Functional Imaging and Monitoring, ed. G. Muller, B. Chance, Rl. Alfano, S. Arridge, J. Beuthan, E. Gratton, M Kaschke, B. Masters, S. Svanberg, P. van der Zee, Proc SPIE IS11, 35-64 (1993).
  14. An integro-differential equation valid for infinite medium,but containing no gradient of Diffusion constant was used for image reconstruction by M. A. OLeary, D. A . Boas, B. Chance, A. G. Yodh, "Experimental Images of heterogeneous turbid media by frequency-domain diffusing-photon tomotography," Opt. Lett. 20, 426-428 (1995). [CrossRef]
  15. An anonymous reviewer has brought to our attention a recent paper by S. R. Arridge and W. R. B. Lionheart, "Non-uniqueness in diffusion based optical tomography," Opt. Lett. 23, 882-884 (1998), which deals with the second order derivative of the diffusion coefficient. [CrossRef]
  16. D. N. Pattanayak and E. Wolf, " Resonance States as Solutions of the Schrodinger Equation with a Non-Local Boundary Condition," Phys. Rev. D13, 2287(1976).
  17. J. L. Ye, R. P. Millane, K. J. Webb and T. J. Downar, "Importance of the gradD term in frequency resolved optical diffusion imaging," Opt. Lett. 23, (1998). [CrossRef]
  18. K. M. Case and P. F. Zweifel, in Linear Transport Theory (Addison -Wesley, MA ,1967).
  19. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978) Vol. 19.
  20. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill,1953).
  21. Claus Muller, Foundations of the Mathematical theory of Electromagnetic Waves (Springer-Verlag,New York,1969).
  22. H. Weyl, Ann. Phys. 60, 481, (1919).
  23. See, also, A. J. Banos, in Dipole Radiation In the Presence of a Conducting Half-Space (Pergamon Press, New York, 1966).
  24. E. Wolf, "Principles and Development of Diffraction Tomography" in Trends in Optics, ed. A. Consortini (Academic Press, San Diego, 1996). [CrossRef]
  25. A. J. Devaney, "Reconstructive tomography with diffracting Wavefields," Inv. Probl. 2,161-1839 (1986).
  26. For an application of diffraction tomography to near field diffusion wave imaging and an analytical expression relating theoretical resolution and tissue thickness see, GE Class I Technical report (publicly available on request) by D. N. Pattanayak, "Resolution of Optical Images Formed by Diffusion Waves in Highly Scattering Media," GE Tech. Info. Series 91CRD241 (1991).
  27. C. L. Matson, N. Clark, L. McMackin and J. S. Fender, "Three-dimensional Tumor Localization in Thick Tissue with The Use of Diffuse Photon-Density Waves," Appl. Opt. 36, 214-219 (1997). [CrossRef] [PubMed]
  28. X. D. Li, T. Durduran, A. G. Yodh, B. Chance and D. N. Pattanayak, "Diffraction Tomography for Biomedical Imaging With Diffuse Photon Density Waves," Opt. Lett. 22, 573-575 (1997). [CrossRef] [PubMed]
  29. B. Q. Chen , J. J. Stamnes, K. Stamnes, "Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium," Pure Appl. Opt. 7, 1161-1180 (1998). [CrossRef]
  30. X. Cheng and D. Boas, "Diffuse Optical Reflection Tomography Using Continous Wave Illumination," Opt. Express 3, 118-123 (1998); http://epubs.osa.org/oearchive/source/5663.htm [CrossRef] [PubMed]
  31. S. J. Norton, T. Vo-Dinh, "Diffraction Tomographic Imaging With Photon Density Waves: an Explicit Solution, J. Opt. Soc. Am. A 15, 2670-2677 (1998). [CrossRef]
  32. J. C. Schotland, "Continuos Wave Diffusion Imaging," J. Opt. Soc. Am. A 14, 275-279 (1997). [CrossRef]
  33. J. Ripoll, M. Nieto-Vesperinas, "Reflection and Transmission Coefficients of Diffuse Photon Density Waves," (to be published).
  34. J. Ripoll and M. Nieto-Vesperinas, "Spatial Resolution of Diffuse Photon Density Waves, " J. Opt. Soc. Am.A (to be published).
  35. C. L. Matson and H. Liu, "Analysis of the forward problem with diffuse photon density waves in turbid media by use of a diffraction tomography model," J. Opt. Soc. Am. A 16, 455-466 (1999). [CrossRef]
  36. For the angular spectrum representation in a slab field and for an excellent account of the angular spectrum representation see, L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York,1995).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited