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  • Editor: Xi-Cheng Zhang
  • Vol. 39, Iss. 4 — Feb. 15, 2014
  • pp: 826–829
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Closed-form solution of the steady-state photon diffusion equation in the presence of absorbing inclusions

Rosario Esposito, Fabrizio Martelli, and Sergio De Nicola  »View Author Affiliations


Optics Letters, Vol. 39, Issue 4, pp. 826-829 (2014)
http://dx.doi.org/10.1364/OL.39.000826


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Abstract

We have developed a theoretical model for photon migration through scattering media in the presence of an absorbing inhomogeneity. A closed-form solution for the average diffuse intensity has been obtained through an iterative approximation scheme of the steady-state diffusion equation. The model describes absorbing defects in a wide range of values. Comparisons with the results of Monte Carlo simulations show that the error of the model is lower than 3% for size inclusion lower than 4 mm and absorption contrast up to the threshold value of the “black defect.” The proposed model provides a tractable mathematical basis for diffuse optical and photoacoustic tomographic reconstruction techniques.

© 2014 Optical Society of America

Many approaches have attempted to provide forward solvers describing the effects of absorbing inclusions using a perturbation scheme applied to the DE [6

6. M. R. Ostermeyer and S. L. Jacques, J. Opt. Soc. Am. A 14, 255 (1997). [CrossRef]

12

12. A. Sassaroli, F. Martelli, and S. Fantini, J. Opt. Soc. Am. A 27, 1723 (2010). [CrossRef]

] or a random walk scheme [13

13. A. G. Gandjbakhche, R. F. Bonner, R. Nossal, and G. H. Weiss, Appl. Opt. 35, 1767 (1996). [CrossRef]

,14

14. V. Chernomordik, D. Hattery, A. H. Gandjbakhche, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, and R. Cubeddu, Opt. Lett. 25, 951 (2000). [CrossRef]

]. In particular, the Born approximation has been widely used to describe absorbing defects for imaging applications to biological tissues, although the accuracy of the method is limited to weak perturbations of a few percent. To overcome this limitation, several iterative schemes have been developed to improve the accuracy of the calculated diffuse intensity. These approaches have been based on the calculation of high-order terms in the expansion series of the intensity [10

10. D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, Phys. Rev. E 76, 061908 (2007). [CrossRef]

12

12. A. Sassaroli, F. Martelli, and S. Fantini, J. Opt. Soc. Am. A 27, 1723 (2010). [CrossRef]

]. However, in many cases, the use of these approximations has been severely limited by the long computation time [10

10. D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, Phys. Rev. E 76, 061908 (2007). [CrossRef]

]. On the other hand, computationally faster methods using a heuristic perturbative approach based on higher-order perturbation theory and the Padé approximants were developed [11

11. A. Sassaroli, F. Martelli, and S. Fantini, Appl. Opt. 48, D62 (2009). [CrossRef]

,12

12. A. Sassaroli, F. Martelli, and S. Fantini, J. Opt. Soc. Am. A 27, 1723 (2010). [CrossRef]

]. These forward solvers show a negligible computation time for the calculation of the average intensity, although the complexity of the theory behind such solvers may discourage their use. Moreover, all these theories show deficiencies when used with high or totally absorbing (“black”) defects where a saturation effect of contrast is achieved. Situations like this can be usually found in biological tissues such as in the case of vessels, which are structures with a strong localized absorption and produce a marked relative contrast in optical and photoacoustic tomography. For this evident argument, it will be desired to provide improved forward solvers that are able to overcome the limitations of the existing ones.

The second-order term is
φ2(r,r0)=Δμa2V24πG¯(rc)G(rcr0)G¯(r),
(7)
where the product of two average values of the Green’s function appears. Proceeding in this way, we find that the general expression of the nth order term is
φn(r,r0)=(ΔμaVG¯(rc))n4πG¯(rc)G(rcr0)G¯(r).
(8)

According to Eq. (8), the sum of the geometrical series at the righthand side of the Eq. (2) can be easily performed, and we obtain
Δφ(r,r0;Δμa)=(ΔμaV+Δμa2V2G¯(rc)1+ΔμaVG¯(rc))G(rcr0)G¯(r)4π,
(9)
where we have defined the perturbation of the average intensity as Δφ(r,r0;Δμa)=n=1φn(r,r0).

Equation (9) gives a simple closed-form expression for the correction to the unperturbed intensity φ0(r,r0) due to an absorptive inclusion, and it is written as the sum of two terms. The first term is proportional to the product ΔμaV of the absorption contrast and its volume, and it is the leading contribution of the first-order perturbation approximation, while the second term is the contribution of high-order terms in the perturbation expansion. Equation (9) also shows that, in case of a “black defect,” i.e., when Δμa, the limit value of the correction to the perturbed intensity reduces to
Δφ(r,r0;Δμaμa)=G(rcr0)G¯(r)4πG¯(rc).
(10)
If the linear size of the volume V of the inclusion is smaller than the distance between the center rc of the defect and the measurement point r, as is often the case, we can further approximate the average value G¯(r) in Eq. (9) with G(rrc). In the following, we specialize the developed general formalism to the case of an absorptive spherical inclusion of radius R embedded in an infinitely extended slab of thickness d, as illustrated in Fig. 1. The source, for the case of a pencil light beam impinging the medium at (x0,y0,0), can be placed at r0=(x0,y0,z0=1/(μs+μa)) [17

17. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).

], and the inclusion is located at rc=(xc,yc,zc). The transmitted flux is calculated at rm=(xm,ym,zm=d), while in a reflectance configuration, we have rm=(xm,ym,zm=0). The Green’s function G(rr0) of the homogeneous absorbing slab of Fig. 1 can be expressed as an infinite sum of dipoles located along the z axis that represents pairs of negative and positive point-like sources, namely
G(rr0)=14πDn=exp(μaDρ+)ρ+exp(μaDρ)ρ.
(11)
We have defined the quantities ρ±2=ρ2+(zz±)2, z+=2nde+z0, and z=2nde2zez0 where the extrapolated length is de=d+2ze. The square of the radial distance from the z axis is ρ2=(xx0)2+(yy0)2. The average intensity given by Eq. (10) vanishes at extrapolated flat surfaces outside the slab at a distance ze=2AD from the boundaries, where the coefficient A accounts for the refractive index mismatch between the slab and the external [17

17. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).

]. Taking into account the extrapolated boundary conditions, the outward photon flux measured at the position rm on the surface of the slab is given by I(rm,r0)=(2π/A)φ(rm,r0) [17

17. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).

,18

18. R. Esposito, S. De Nicola, M. Lepore, and P. L. Indovina, J. Opt. Soc. Am. A 23, 1937 (2006). [CrossRef]

].

Fig. 1. Schematic of the heterogeneous slab with the absorptive inclusion.

The calculation of the perturbed intensity φ(r,r0;Δμa) through the slab requires the calculation of the average value G¯(r) of the Green’s function. According to the definition given by Eq. (6), the average value is obtained by integrating the Green’s function over the volume V of the spherical region of the defect centered at rc=(xc,yc,zc). Although it is possible to formulate a general expression for G¯(r) which takes into account the contributions of both the positive and negative sources, it turns out that the average value at the center of the defect is essentially determined by the positive zero-order term, corresponding to the solution for a homogeneous infinite medium geometry, because the higher-order terms in Eq. (10) give a negligible contribution for R|rcrm|. Accordingly, we have the following simple expression for the average value at the center of the defect
G¯(rc)=1Vμa[1eRα(1+Rα)],
(12)
where α=μa/D.

The accuracy of the model for the slab geometry is investigated by means of comparisons with the results of Monte Carlo (MC) simulations [17

17. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).

] for the relative contrast |ΔI/I0|=|(II0)/I0|, where I0 is the unperturbed transmittance. In this regard, Fig. 2 shows the dependence of |ΔI/I0| for the transmittance through a slab versus the absorption contrast Δμa of the defect for three values of the volume V: 100, 300, and 500mm3. The perturbation of the flux ΔI in the slab geometry is calculated as ΔI(rm,r0)=(2π/A)Δφ(rm,r0), where the perturbed fluence Δφ is obtained with Eq. (9). The inclusion is assumed to be located midway between the source and the detector at zc=d/2 in coaxial configuration (ρ=0) in panel (a), and at radial distance ρ=5mm in panels (b) and (c). The thickness of the slab d and the optical properties are reported in the inset of the figure. The discrepancies between the presented solver and MC simulations are within 11% for the all situations considered in the figure. Remarkably, the theory also accounts for the saturation of the relative intensity for increasing values of the absorption contrast when the inclusion behaves like a black defect, and this is observed for a wide range of optical and geometrical properties that cover tissue optics applications. Figure 3 shows the relative contrast |ΔI/I0| in the case of a black defect versus the inclusion volume: the red dotted curve is the model and the solid black curve is the MC simulation. The results are obtained for the same slab and the same optical parameters used for case (a) of Fig. 2. As can be noticed, the theoretical predictions are in good agreement with MC simulations for volumes of the black defect ranging from 50 to 500mm3 (R5mm). In particular, excellent agreement between the theory and the MC results curve is obtained for volumes lower than 300mm3. Indeed, the discrepancies (solid blue curve) are lower than 3% for volumes increasing from 50 to 300mm3. This is an expected result because the theoretical model is derived within the small volume approximation. Conversely, the theory overestimates the MC simulations when V300mm3, with discrepancies that tend to increase up to a value within 11% in the considered range. In this case, the accuracy is limited essentially by the larger values of the volume of the inclusion.

Fig. 2. Relative contrast |ΔI/I0| for the transmittance through a slab plotted against the absorption contrast Δμa between a spherical absorptive inclusion and the background medium.
Fig. 3. Relative contrast as a function of the inclusion volume V of the black defect. The red dotted curve is the model prediction and the solid black curve is the MC simulation. The solid blue curve shows the relative discrepancies. The slab geometry and the optical parameters are the same as in Fig. 2(a).

In conclusion, we have developed a simple theoretical approach for photon migration through scattering media in the presence of an absorptive inhomogeneity. A closed-form expression for the average diffuse intensity is derived in the framework of an iterative approximation scheme of the steady-state DE. This method addresses the case of a single inclusion but it should be remarked that this limitation, which is also present in current analytical and perturbation schemes, can be released somewhat in the presence of multiple defects whose relative distances are large enough to make negligible their interactions. The slab geometry was used to test the closed-form solution given by Eq. (9) in transmission configuration. Comparable accuracy was also obtained in the case of reflectance geometry. It is worth pointing out that this solution can be applied to both homogeneous or inhomogeneous geometries. Indeed, there exists a variety of available analytical solutions of the DE such as, for instance, the parallelepiped, the sphere, the cylinder, and several layered geometries [17

17. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).

]. Furthermore, Eq. (9) is applicable also in case no explicit or analytical expressions of the Green’s function are available.

References

1.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, Rep. Prog. Phys. 73, 076701 (2010). [CrossRef]

2.

B. Cox, S. Arridge, K. Kostli, and P. Beard, Appl. Opt. 45, 1866 (2006).

3.

A. Q. Bauer, R. E. Nothdurft, T. N. Erpelding, L. V. Wang, and J. P. Culver, J. Biomed. Opt. 16, 096016 (2011). [CrossRef]

4.

T. Correia, N. Ducros, C. D’Andrea, M. Schweiger, and S. Arridge, Opt. Lett. 38, 1903 (2013). [CrossRef]

5.

V. Ntziachristos and R. Weissleder, Opt. Lett. 26, 893 (2001). [CrossRef]

6.

M. R. Ostermeyer and S. L. Jacques, J. Opt. Soc. Am. A 14, 255 (1997). [CrossRef]

7.

S. Carraresi, T. S. M. Shatir, F. Martelli, and G. Zaccanti, Appl. Opt. 40, 4622 (2001). [CrossRef]

8.

S. De Nicola, R. Esposito, and M. Lepore, Phys. Rev. E 68, 21901 (2003). [CrossRef]

9.

S. Feng, F. A. Zeng, and B. Chance, Appl. Opt. 34, 3826 (1995). [CrossRef]

10.

D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, Phys. Rev. E 76, 061908 (2007). [CrossRef]

11.

A. Sassaroli, F. Martelli, and S. Fantini, Appl. Opt. 48, D62 (2009). [CrossRef]

12.

A. Sassaroli, F. Martelli, and S. Fantini, J. Opt. Soc. Am. A 27, 1723 (2010). [CrossRef]

13.

A. G. Gandjbakhche, R. F. Bonner, R. Nossal, and G. H. Weiss, Appl. Opt. 35, 1767 (1996). [CrossRef]

14.

V. Chernomordik, D. Hattery, A. H. Gandjbakhche, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, and R. Cubeddu, Opt. Lett. 25, 951 (2000). [CrossRef]

15.

G. Started, The Language of Technical Computing (Prentice Hall, 1997), Vol. L, p. 186.

16.

G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Elsevier, 1971), Vol. 39.

17.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).

18.

R. Esposito, S. De Nicola, M. Lepore, and P. L. Indovina, J. Opt. Soc. Am. A 23, 1937 (2006). [CrossRef]

OCIS Codes
(110.7050) Imaging systems : Turbid media
(170.3010) Medical optics and biotechnology : Image reconstruction techniques
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.3880) Medical optics and biotechnology : Medical and biological imaging
(170.5280) Medical optics and biotechnology : Photon migration
(170.6510) Medical optics and biotechnology : Spectroscopy, tissue diagnostics

ToC Category:
Medical Optics and Biotechnology

History
Original Manuscript: October 18, 2013
Revised Manuscript: December 11, 2013
Manuscript Accepted: December 26, 2013
Published: February 6, 2014

Virtual Issues
Vol. 9, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Rosario Esposito, Fabrizio Martelli, and Sergio De Nicola, "Closed-form solution of the steady-state photon diffusion equation in the presence of absorbing inclusions," Opt. Lett. 39, 826-829 (2014)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-39-4-826


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References

  1. T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, Rep. Prog. Phys. 73, 076701 (2010). [CrossRef]
  2. B. Cox, S. Arridge, K. Kostli, and P. Beard, Appl. Opt. 45, 1866 (2006).
  3. A. Q. Bauer, R. E. Nothdurft, T. N. Erpelding, L. V. Wang, and J. P. Culver, J. Biomed. Opt. 16, 096016 (2011). [CrossRef]
  4. T. Correia, N. Ducros, C. D’Andrea, M. Schweiger, and S. Arridge, Opt. Lett. 38, 1903 (2013). [CrossRef]
  5. V. Ntziachristos and R. Weissleder, Opt. Lett. 26, 893 (2001). [CrossRef]
  6. M. R. Ostermeyer and S. L. Jacques, J. Opt. Soc. Am. A 14, 255 (1997). [CrossRef]
  7. S. Carraresi, T. S. M. Shatir, F. Martelli, and G. Zaccanti, Appl. Opt. 40, 4622 (2001). [CrossRef]
  8. S. De Nicola, R. Esposito, and M. Lepore, Phys. Rev. E 68, 21901 (2003). [CrossRef]
  9. S. Feng, F. A. Zeng, and B. Chance, Appl. Opt. 34, 3826 (1995). [CrossRef]
  10. D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, Phys. Rev. E 76, 061908 (2007). [CrossRef]
  11. A. Sassaroli, F. Martelli, and S. Fantini, Appl. Opt. 48, D62 (2009). [CrossRef]
  12. A. Sassaroli, F. Martelli, and S. Fantini, J. Opt. Soc. Am. A 27, 1723 (2010). [CrossRef]
  13. A. G. Gandjbakhche, R. F. Bonner, R. Nossal, and G. H. Weiss, Appl. Opt. 35, 1767 (1996). [CrossRef]
  14. V. Chernomordik, D. Hattery, A. H. Gandjbakhche, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, and R. Cubeddu, Opt. Lett. 25, 951 (2000). [CrossRef]
  15. G. Started, The Language of Technical Computing (Prentice Hall, 1997), Vol. L, p. 186.
  16. G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Elsevier, 1971), Vol. 39.
  17. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2010).
  18. R. Esposito, S. De Nicola, M. Lepore, and P. L. Indovina, J. Opt. Soc. Am. A 23, 1937 (2006). [CrossRef]

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