Abstract

An explicit formula is derived for reconstructing tissue inhomogeneities from scattered photon-density waves. When intensity-modulated light is used as a source, such waves obey a Helmholtz equation with a complex wave number. A process of analytical continuation in the spatial-frequency domain is used to make this wave number real, whereupon inversion formulas based on conventional diffraction tomography are applied to invert the data to yield an image. In this way an explicit solution to the inverse problem can be derived within the Born approximation. Here the spatially varying quantity imaged is a linear combination of the transport scattering and absorption coefficients. Data are assumed measured at one modulation frequency, but multiple modulation frequencies could in principle be used to generate independent images of the scattering and absorption coefficients.

© 1998 Optical Society of America

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References

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  1. See, for example, two review articles: J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997); and S. R. Arridge, J. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
    [CrossRef] [PubMed]
  2. T. Vo-Dinh, M. Panjehpour, B. F. Overholt, P. Buckley, “Laser induced fluorescence for cancer diagnosis without biopsy,” Appl. Spectrosc. 51, 58–62 (1997).
    [CrossRef]
  3. See, for example, the special issue of J. Opt. Soc. Am. A 14(1), (1997) on Photon Migration through Turbid Media.
  4. J. C. Schotland, “Continuous-wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997).
    [CrossRef]
  5. X. D. Li, T. Durduran, A. G. Yodi, B. Chance, D. N. Pattanayak, “Diffraction tomography for biomedical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
    [CrossRef] [PubMed]
  6. C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express 1, 6–11 (1997).
    [CrossRef] [PubMed]
  7. A. Ya. Polishchuk, S. Gutman, M. Lax, R. R. Alfano, “Photon-density modes beyond the diffusion approximation: scalar-wave diffusion equation,” J. Opt. Soc. Am. A 14, 230–234 (1997).
    [CrossRef]

1997 (7)

Alfano, R. R.

Arridge, S. R.

See, for example, two review articles: J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997); and S. R. Arridge, J. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Buckley, P.

Chance, B.

Delpy, D. T.

See, for example, two review articles: J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997); and S. R. Arridge, J. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Durduran, T.

Gutman, S.

Hebden, J. C.

See, for example, two review articles: J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997); and S. R. Arridge, J. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Lax, M.

Li, X. D.

Matson, C. L.

Overholt, B. F.

Panjehpour, M.

Pattanayak, D. N.

Polishchuk, A. Ya.

Schotland, J. C.

Vo-Dinh, T.

Yodi, A. G.

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Figures (3)

Fig. 1
Fig. 1

Recording geometry.

Fig. 2
Fig. 2

Two point inhomogeneities.

Fig. 3
Fig. 3

(a) Image reconstruction in the absence of noise, (b) image with 2% noise, (c) image with 5% noise.

Equations (60)

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-c232ψ+α0 2ψt2+(νs+β0νa) ψt
+νa(νs+γ0νa)ψ=S,
2ψ-α 2ψt2-β ψt-γψ=-S,
β(r)=3c2[νs(r)+β0νa(r)],
γ(r)=3c2νa(r)[νs(r)+γ0νa(r)].
2ψ(r)+k2(r)ψ(r)=-S(r),
k2(r)ω2α+iωβ(r)-γ(r).
k2(r)=iωβ(r).
k02iωβ0,
2ψ(r)+k02ψ(r)=-iωδβ(r)ψ(r)-S(r).
2g(r|r)+k02g(r|r)=-δ(r-r),
g(r|r)=i4π- dkxγexp[ikx(x-x)+iγ|z-z|],
γk02-kx2.
ψ(r)=ψi(r)+iωδβ(r)ψ(r)g(r|r)d2r,
ψi(r)=S(r)g(r|r)d2r
ψs(rd, rs)=iωIsδβ(r)g(r|rs)g(rd|r)d2r.
g(r|rs)=i4π- dkxγexp[i(kxxˆs-γzˆs)·(r-rs)],
g(r|rd)=i4π- dkxγexp[i(kxxˆd+γzˆd)·(r-rd)],
ψs(rd, rs)=-iωIs(4π)2d2rδβ(r)- dkxγ×exp(-ikxXs)exp[i(kxxˆs-γzˆs)·r]·- dkxγexp(-ikxXd)exp[-γL+i(kxxˆd+γzˆd)·r],
ψ˜s(kd, ks)=--ψs(Xd, Xs)×exp[i(kdXd+ksXs)]dXddXs.
ψ˜s(kd, ks)=-iωIs4γsγdexp(-γdL)δβ(r)×exp[i(ksxˆs-γszˆs+kdxˆd+γdzˆd)·r]d2r,
δβ˜(k)=δβ(r)exp(ik·r)d2r.
δβ˜(ksxˆs-γszˆs+kdxˆd+γdzˆd)
=4iγsγdωIsψ˜s(kd, ks)exp(γdL).
F˜(kd, ks)=4iγsγdωIsψ˜s(kd, ks)exp(γdL).
F˜(kd, ks)=0Lz0Lxδβ(x, z)×exp[i(kd+ks)x+i(γd-γs)z]dxdz.
kxkd+ks,
kzγd-γs=k02-kd2-k02-ks2,
ksk0 sin(p+q+iv),
kdk0 sin(p-q+iv),
kx=2k0 sin(p+iv)cos q,
kz=2k0 sin(p+iv)sin q.
kx=2ωβ0(1+i)sin(p+iv)cos q,
kz=2ωβ0(1+i)sin(p+iv)sin q.
Im{(1+i)sin(p+iv)}
=sin p cosh v+cos p sinh v=0,
tan p=-tanh v.
kx=2ωβ0 sin 2pcos 2pcos q,
kz=2ωβ0 sin 2pcos 2psin q.
F˜(kd, ks)=0Lz0Lxδβ(x, z)×exp[i(kxx+kzz)]dxdz.
δβ(x, z)=1(2π)2--F˜(kd, ks)×exp[-i(kxx+kzz)]dkxdkz.
J(p)=2ωβ01+cos2 2pcos2 2psin 2p.
δβˆ(x, z)=1(2π)20π/4dpJ(p)A(p)02πdqF˜(kd, ks)×exp-i2ωβ0 sin 2pcos 2p×(x cos q+z sin q),
A(p)1forp<pmax0forp>pmax
A(p)exp-α sin 2pcos 2p,
F˜(kd, ks)=4iγsγdωIsexp(γdL)0Ls0Ldψs(Xd, Xs)×exp[i(kdXd+ksXs)]dXddXs,
δβˆ(x, z)=0Ls0LdK(x, z|Xd, Xs)×ψs(Xd, Xs)dXddXs,
K(x, z|Xd, Xs)
iωIsπ20π/4dpJ(p)A(p)×02πdqγsγd exp[γdL+i(kdXd+ksXs)]×exp-i2ωβ0 sin 2pcos 2p(x cos q+z sin q),
ks=ωβ02 cos 2p [sin(2p+q)+i sin q]
kd=ωβ02 cos 2p [sin(2p-q)-i sin q],
γsγd=12k02[cos(2p+2iv)+cos(2q)].
02π exp[inq+i(P cos q+Q sin q)]dq
=2πJn(P2+Q2)exp[in tan-1(P/Q)],
δβ(x, z)=c1δ(x-x1)δ(z-z1)+c2δ(x-x2)δ(z-z2),
F˜(kd, ks)=c1 exp[i(kd+ks)x1+i(γd-γs)z1]+c2 exp[i(kd+ks)x2+i(γd-γs)z2]+˜(kd, ks),
˜(kd, ks)=0Ls0Ld(Xd, Xs)×exp[-i(kdXd+ksXs)]dXddXs.
(Xd, Xs)=1(2π)2--˜(kd, ks)×exp[i(kdXd+ksXs)]dkddks.
˜(kd, ks)=1(2π)2--˜(kd, ks)×exp[i(kd-kd)Ld]-1i(kd-kd)×exp[i(ks-ks)Ls]-1i(ks-ks)dkddks.
kx2+kz2|max=2ωβ0 sin 2pmaxcos 2pmax=2δsin 2pmaxcos 2pmax

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