Download Heuristic Green`s function of the time dependent

Heuristic Green’s function of the time
dependent radiative transfer equation
for a semi-infinite medium
Fabrizio Martelli,1 Angelo Sassaroli,2 Antonio Pifferi,3 Alessandro
Torricelli,3 Lorenzo Spinelli,3 and Giovanni Zaccanti 1
1 Dipartimento
di Fisica dell’Università degli Studi di Firenze,
Via G. Sansone 1, 50019 Sesto Fiorentino, Firenze, Italy
2 Tufts University, Department of Biomedical Engineering, Bioengineering Center,
4 Colby Street, Medford, MA 02155, USA
3 IIT, ULTRAS-INFM-CNR and IFN-CNR, Politecnico di Milano-Dipartimento di Fisica,
Piazza Leonardo da Vinci 32, I-20133 Milano, Italy
[email protected]
Abstract: The Green’s function of the time dependent radiative transfer
equation for the semi-infinite medium is derived for the first time by a
heuristic approach based on the extrapolated boundary condition and on
an almost exact solution for the infinite medium. Monte Carlo simulations
performed both in the simple case of isotropic scattering and of an isotropic
point-like source, and in the more realistic case of anisotropic scattering
and pencil beam source, are used to validate the heuristic Green’s function.
Except for the very early times, the proposed solution has an excellent
accuracy (> 98 % for the isotropic case, and > 97 % for the anisotropic
case) significantly better than the diffusion equation. The use of this solution
could be extremely useful in the biomedical optics field where it can be
directly employed in conditions where the use of the diffusion equation is
limited, e.g. small volume samples, high absorption and/or low scattering
media, short source-receiver distances and early times. Also it represents a
first step to derive tools for other geometries (e.g. slab and slab with inhomogeneities inside) of practical interest for noninvasive spectroscopy and
diffuse optical imaging. Moreover the proposed solution can be useful to
several research fields where the study of a transport process is fundamental.
© 2007 Optical Society of America
OCIS codes: (170.5280) Photon migration; (170.7050) Turbid media; (170.3660) Light propagation in tissues; 170.6510 Spectroscopy, tissue diagnostics.
References and links
1. J. J. Duderstadt and W. R. Martin, Transport Theory (John Wiley&Sons, New York, 1979).
2. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
3. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.
50, R1-R43 (2000).
4. T. Feng, P. Edström, and M. Gulliksson, “Levenberg–Marquardt methods for parameter estimation problems in
the radiative transfer equation,” Inverse Probl. 23, 879-891 (2007).
5. L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, “Time-dependent photon migration using
path integrals,” Phys. Rev. E 51, 6134-6141 (1995).
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Received 31 Oct 2007; revised 7 Dec 2007; accepted 17 Dec 2007; published 19 Dec 2007
24 December 2007 / Vol. 15, No. 26 / OPTICS EXPRESS 18168
6. F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe
photon migration through an infinite medium: Numerical and experimental investigation,” Phys. Med. Biol. 45,
1359-1373 (2000).
7. For recent results: Special issue on recent development in biomedical optics, Phys. Med. Biol. 49, N. 7 (2004).
8. A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, “Time-resolved
reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging,”
Phys. Rev. Lett. 95, 078101 (2005).
9. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion
calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285-1302
(1998).
10. H. Dehghani, S. R. Arridge, and M. Schweiger, “Optical tomography in the presence of void regions,” J. Opt.
Soc. Am. A 17,1659-1670 (2000).
11. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of
whole-body photonic imaging ,” Nat. Biotechnol. 23, 313-320 (2005).
12. J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E 56, 1135-1141 (1997).
13. D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on
diffusion approximation. I. Theory,” Appl. Opt. 36, 4587-4599 (1997).
14. E. Zauderer, Partial Differential Equations of Applied Mathematics, (John Wiley&Sons, New York, 1989) Sec.
7.5, p. 484.
15. M. H. Lee, “Fick’s Law, Green-Kubo Formula, and Heisenberg’s Equation of Motion,” Phys. Rev. Lett. 85,
2422-2425 (2000).
16. F. Martelli, D. Contini, A. Taddeucci, and G. Zaccanti, “Photon migration through a turbid slab described by a
model based on diffusion approximation. II. Comparison with Monte Carlo results,” Appl. Opt. 30, 4600-4612
(1997).
17. G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of
scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897-905 (1994).
18. F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation model for light propagation through diffusive layered
media,” Phys. Med. Biol. 50, 2159-2166 (2005).
1. Introduction
The radiative transfer equation (RTE) provides a full description of light propagation in diffusive media like biological tissue in the red and near infrared spectral range. Unfortunately the
RTE is an integro-differential equation and the retrieval of solutions is an extremely computationally expensive process [1]. Solutions of the RTE are usually based on numerical methods
like the finite element method [1, 2, 3] or other methods like the discrete ordinates method
[1, 4], the spherical harmonics method [1], the integral transport method [1] and the path integral formalism [5]. Among the numerical procedures there are also statistical methods like
the Monte Carlo (MC) that is largely used to reconstruct solutions of the RTE [1, 6]. To our
knowledge, no analytical (closed-form) solutions of the RTE are available [1] and simpler approximated models are usually sought [2, 3]. The most widely and successfully used approach
makes use of the diffusion equation (DE) to yield a variety of solutions capable of providing
a modelling of light propagation in homogeneous, layered and perturbed diffusive media, both
in the steady state and in the time-domain. Solutions of the DE are available for the slab and
for the semi-infinite medium that represent typical geometrical schematics for most of the experiments carried out in biomedical optics applications. In the last decade it has been possible
to develop noninvasive tissue spectroscopy and imaging for diagnostic and therapeutic applications (e.g. optical mammography, brain imaging, photodynamic therapy) due to the availability
of these DE based easy-to-use tools for light dosimetry and for the assessment of tissue optical
properties [7].
However, the investigations based on the DE are subjected to the intrinsic approximations
of this theory. In particular for continuous wave instrumentation the basic configuration needs
a source-detector separation ρ , typically in the range 20-40 mm, because photons received at
shorter distances are not well described by the DE [6]. Moreover, in the time-domain early
received photons must be disregarded because the validity of the DE is limited to photons
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Received 31 Oct 2007; revised 7 Dec 2007; accepted 17 Dec 2007; published 19 Dec 2007
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that, in their migration, undergo many scattering events [8]. Conversely, by exploiting an RTE
solution it can be foreseen to overcome the limits of the DE and to immediately obtain a twofold
improvement: (i) the error determined by approximations of the theory would be drastically
reduced, therefore modelling by simulations would be more accurate; (ii) measurements carried
out in conditions where the use of the DE is strongly limited (like small volume samples, high
absorption and/or low scattering media, short source-receiver distances and early times) could
be more easily interpreted. For these reasons the availability of analytical solutions of the RTE
would have generally a strong impact on many practical applications in the biomedical optics
[9, 10]. A first example is the optical molecular imaging of small animals where the use of
quantitative methods to estimate tumor growth or the effect of drugs is hampered by the rough
approximation of the DE [11]. Further, the possibility of performing measurements at null or
small source-receiver distance has been proposed as a new imaging modality offering improved
contrast, sensitivity and spatial resolution and as a novel approach to local spectroscopy of
tissue [8]. Finally, weakly diffusive media of interest can be found in the biomedical field (e.g.
blood, due to the high absorption) and in the industrial field (e.g. vegetable oil, due to the weak
scattering). In general the applications of RTE solutions are addressed to several research fields
where a transport process is involved.
Ten years ago Paasschens [12] proposed an almost exact solution of the time dependent RTE
in the infinite medium geometry and for isotropic scattering. Paasschens, using a path-integral
method, gave explicit time domain solutions of the RTE for two and four dimensions, while
in three dimensions the solution was given in terms of its Fourier transform. The time domain
solution in three dimensions was obtained by an interpolation between the two exact solutions
found for two and four dimensions. It is not surprising that Paasschens’ paper received scarce
attention within the biomedical optics community. The solution proposed by Paasschens is too
far from being a workable solution. The possibility of modelling a sample as an infinite medium
is in fact rarely encountered in real applications.
In this paper, starting from the work of Paasschens [12] and following a heuristic approach,
an analytical solution of the time dependent RTE for the semi-infinite medium is derived for
the first time. The solution is obtained in the simple case of isotropic scattering and point-like
isotropic source. The accuracy of the proposed solution is validated by a comparison with results of MC simulations. Moreover, for the more realistic case of anisotropic scattering and
pencil beam source, the derived solution still shows an excellent behavior. The proposed solution can be directly used in applications where a reflectance approach is used to noninvasively
characterize a sample. Also it represents a first step to derive analytical solutions of the RTE
for other geometries (e.g. slab).
2. Theory
Our derivation starts from the interpolating analytical formula for the time-dependent fluence
rate [12], Φi (r, t), emitted by a pulsed point-like isotropic source (i.e. source = δ (r) δ (t)) in an
infinite non-absorbing medium
1
8
3 v 1 − (r/vt)2
v e−μs vt
2 4
e−μs vt Θ(r − vt), (1)
δ
(r
−
vt)
+
G
μ
vt
1
−
(r/vt)
Φi (r, t) s
3
2
4π r
2
4π vt/3μs
where r is the distance from the source, Θ(x) is the step function (zero for x <0 and 1 for x >0),
μs , μs and v are respectively the scattering coefficient, the reduced scattering coefficient and
the speed of light in the medium, while the expression for the function G is provided as the
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following
G(x) = 8(3x)−3/2
∞
√
Γ( 34 N + 32 ) xN
∑ Γ( 3 N) N! ex 1 + 2.026x.
N=1
4
(2)
At early times (small values of x) the first expression of Eq. (2) has to be preferred to the second
one because it is more accurate, while for late times they tend to give the same values. The first
term of Eq. (1) represents the ballistic peak, while the second term is the scattered component.
The accuracy of Eq. (1) is better than 2% [12]. In the second term of Eq. (1), differently from
the paper by Paasschens [12], we introduced μ s instead of μs . Since μs = μs (1 − g), where
the symbol, g, denotes the anisotropy factor, for isotropic scattering (g =0) the two formalisms
are identical. Furthermore, our approach would be useful to treat anisotropic scattering (g =0)
for which Eq. (2) can still be heuristically used. The effect of absorption can be introduced,
according to the RTE, by multiplying the above time-domain solution obtained for the case of
zero absorption by exp(−μ a vt).
When the semi-infinite geometry is considered the main problem to be addressed for obtaining the Green’s function is the modeling of the boundary condition with the external non
scattering region. The boundary condition for the RTE and for an isotropic source embedded
in the medium at a depth z 0 would involve the radiance, I(r, ŝ, t), emerging from the medium
along any direction ŝ. If the refractive indexes of both media are the same (as it is assumed in
our derivation), the boundary condition results: I(r, ŝ, t) = 0, for every point r ≡ (x, y, z) at the
external boundary and for all directions ŝ inwardly directed. Although rigorous, this approach
cannot be implemented to solve the RTE with an analytical method. In order to overcome
this problem the same boundary condition used for the DE [13], usually named extrapolated
boundary condition (EBC), is here heuristically used for the RTE. More in details, we derived
the Green’s function by using the method of images [14]. The locations, r + and r− , of the real
source (positive) and of the image source (negative) are shown in Fig. 1. In this way the fluence
rate will vanish at z = −ze , which identifies the extrapolated boundary, and it is composed by
two terms of fluence from an infinite medium, Φ i : one positive from the source at r + and the
other negative from the source at r − . This heuristic approach leads to the following expression
for the fluence rate in a semi-infinite medium, Φ si (x, y, z, z0 , t), with refractive index matched
at the boundary
Φsi (x, y, z, z0 , t) = Φi (|r − r+|, t) − Φi (|r − r− |, t) ,
1/2
1/2
(3)
|r − r+ | = x2 + y2 + (z − z0 )2
, |r − r−| = x2 + y2 + (z + z0 + 2ze )2
.
According to Ref. [13], we assume an extrapolated distance z e = 2/3μs . Note that it differs in
about 6 % from the extrapolated end point, d e , of the Milne problem [1] (d e = 0.7104 /μ s ) that is
accounted as the distance from the interface at which the asymptotic component of the density
of energy extrapolates to zero. Although this fact does not represent a ground for the heuristic
boundary condition employed, it anyway suggests that for a source deeply sinked inside the
medium the EBC tends within few per cents toward the boundary conditions of the classical
Milne problem of photon transport. It can be also noticed that the differences between these two
solutions become negligible for a source far from the boundary and for the cases analyzed in
this work. For instance, for z 0 = 1/μs the differences observed are within 2% and the accuracy
of these two boundary conditions is thus similar. The time-resolved reflectance emerging from
the medium is derived by Fick’s law [15]
R(ρ ,
t) = D ∂ Φsi (x, y, z = 0, z0 , t)/∂ z ,
(4)
ρ = x2 + y2 , D = 1/(3μs ) .
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Fick’s law introduces an additional approximation at early times and the effects of this approx
imation decrease to zero for t 1/(μ s v).
Fig. 1. Schematic of the medium, symbols used and positions of real and image sources,
r+ and, r− , respectively.
We remark a feature of Eq.(4) that needs to be known in its use. This feature is the tail of the
ballistic peak due to radiation which has undergone a single forward scattering event, already
mentioned by Paasschens
[12] for Eq.(1). This tail determines two logarithmic singularities in
Eq.(4), one at t 1 = (ρ 2 + z20 )/v and the other at t 2 = (ρ 2 + (z0 + 2ze )2 )/v. Moreover, at t 1
and t2 we have the singularities due to the ballistic peak of the two sources. These effects arise
from the delta source considered in the solution and are confined to very early times (e.g. t 1 =
4.7 ps and t2 = 8.7 ps for ρ = 1 mm, μ s = 1 mm−1 and v = 0.3 mm/ps). We want to stress that
Eq. (4) is zero for t < t 1 and, with the exception of the singularities at t 1 and t2 , it is a continuous
function.
3. Results
The proposed solution has been obtained with a heuristic hypothesis that was validated by comparing the results of Eq. (4) with the results of MC simulations, here used as a gold standard.
Details on the MC code used can be found in Ref. [16]. The correctness of the routines used to
determine the scattering points was checked comparing some statistical parameters with analytical formulae [17]. The scattering functions used for the MC simulations were based on the
Henyey-Greenstein (HG) model.
The comparison between the results of MC and RTE has been done for the time-resolved
reflectance in a semi-infinite medium with μ s = 1 mm −1 . Since both Eq. (4) (denoted RTE
in the figures) and the MC use the same dependence on μ a the comparison is done for a non
absorbing medium without any loss of generality. The results obtained with the DE theory
(Eq. (36) of Ref. [13] where only the first term of the series is retained) are also shown. The
simulations have been firstly carried out for an isotropic source and for isotropic scattering
as it is assumed in the derivation of Eq. (1). The position of the isotropic point-like source is
considered at z0 = 1/μs and v is assumed to be 0.3 mm/ps in the medium. In Fig. 2 (a) and (b)
the reflectance from a semi-infinite medium is shown at ρ = 1 mm and 5 mm, respectively. In
Fig. 2 (c) and (d) the plots show the ratio of the MC results to those of RTE and DE. In Fig. 2 (a)
and (b) the error bars on the MC data are omitted since they have similar size as the symbols. In
Fig. 2 (c) and (d) the error bars are obtained from the statistical standard deviations on the MC
data. Similar results have been obtained for ρ in the range 0.1-20 mm and for z 0 > 0.2 mm.
Both for RTE and for MC response the contribution of ballistic photons has not been included.
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Fig. 2. Comparison between the results of MC, RTE and DE for the time-resolved re
flectance from a non absorbing semi-infinite medium with μs = 1 mm−1 , ρ = 1 and 5 mm,
where an isotropic point-like source at depth z0 = 1/μs and isotropic scattering is assumed.
Figures (a) and (b) show the reflectance and figures (c) and (d) show the ratio of the MC
results to those of RTE and DE.
Figures 1(c) and 1(d) show that, except for the very first temporal windows (4.7 ≤ t ≤ 12
ps for ρ = 1 mm or 16.9 ≤ t ≤ 26 ps for ρ = 5 mm), the RTE solution is within 2% in
agreement with the results of the MC simulations. Approximately, for ρ = 0.1, 1, 5, 10 and 20
mm the error is less than 2% just 8 ps after t 1 (that corresponds to 2.4 mean free paths). This
accuracy is similar to that of Eq. (1). Thus even at very short distances the accuracy of Eq. (4) is
similar to that of Eq. (1), assuring that the introduction of the heuristic boundary condition does
not diminish the accuracy of the Paasschens’ solution. These results also show for the DE an
accuracy significantly worse than the proposed heuristic solution. Moreover the solution from
the DE is nonzero also for t < t 1 , due to the non-causality hypothesis assumed in this approach.
The good agreement obtained at early times suggests that the CW reflectance obtained from an
integration of Eq. (4) works well even at short distances and for high values of μ a .
The validation of the presented heuristic solution has been done for the case of an isotropic
source and of isotropic scattering. However, when light propagates through biological tissues
or real media, this exact situation is almost never found. In order to test the applicability of Eq.
(4) in more realistic conditions, with the MC code we simulated the case of a pencil light beam
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impiging a semi-infinite medium where a scattering function with anisotropy factor g = 0.9,
typical of biological tissue, was assumed. In Fig. 3 (a) and (b) the reflectance is shown at ρ =1
mm and 5 mm respectively. The ratio of the MC results to those of RTE and DE is plotted in
figures (c) and (d). Equation (4) and the DE are evaluated with the source at z 0 = 1/μs . The
distance ρ = 1 mm could be an actual value for measurements at null source-detector distance.
These results show an agreement between the heuristic solution and the MC results similar to
that of Fig. 2 and we still can see clear advantages in using this solution instead of the DE. These
results support the use of Eq. (4) for analyzing measurements on tissue at short source-receiver
distances and at early times.
Fig. 3. Comparison between the results of MC, RTE and DE for the time-resolved re
flectance from a non absorbing semi-infinite medium with μs = 1 mm−1 , ρ = 1 and 5 mm.
The MC simulations assume a pencil beam impinging the medium at z = 0 and anisotropic
scattering with g = 0.9. Equation (4) for the RTE and Eq. (36) of Ref. [13] for the DE have
been used with z0 = 1/μs . Figures (a) and (b) show the reflectance and figures (c) and (d)
show the ratio of the MC results to those of RTE and DE.
4. Discussion and conclusions
The theory we have presented has been validated for refractive index matched at the boundary
with the external medium. This situation has been chosen because it significantly simplifies
the boundary condition and thus a better accuracy of the solution is expected. All the excel#89251 - $15.00 USD
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lent agreement observed in the comparisons we showed refer to matched refractive index at
the boundary. The agreement becomes worse for mismatched refractive index. However, we
point out that for many applications the use of a matching liquid makes possible to carry out
measurements with the refractive index matched at the boundary.
One of the main drawbacks of using RTE solutions is the long computation time that is
usually required when numerical methods are used to solve integro-differential equations. The
RTE solution proposed in this paper has the advantage to be an analytical closed-form solution
with a negligible computation time. This fact is particularly important for inverse problems
where Eq. (4) can be used to retrieve the optical properties of the medium in short time with a
procedure that does not require to handle complex mathematics.
In conclusion, we have presented for the first time a heuristic analytical solution of the time
dependent RTE for a semi-infinite medium based on an almost exact solution for the infinite
medium where isotropic scattering is assumed. The comparisons with MC simulations for a
point-like isotropic source and isotropic scattering (Fig. 2) have shown excellent performances:
Except for the first temporal windows (i.e. approximately t 1 <t ≤t1 + 8 ps for v = 0.3 mm/ps
and μs = 1 mm−1 ) the proposed solution shows an error less than 2% even in regions where
the DE shows errors larger than 10%. Also the comparisons with MC results for a medium
with anisotropic scattering (g = 0.9) illuminated by a pencil light beam (Fig. 3) showed an
excellent agreement. This is surprising, since the proposed solution is based on the response
for the infinite medium that is exact only for isotropic scatterers. These surprising good results
are probably due to a compensation effect between the different approximations introduced
to obtain the heuristic solution (boundary conditions, pencil beam modelled with a point-like
isotropic source, isotropic scattering). Anyway, the excellent agreement shown by the comparisons in Fig. 3 for the temporal response even at short distances and at early times, suggests
that also the CW reflectance obtained integrating Eq. (4) gives accurate results even at short
distances and for high values of absorption for diffusive media with anisotropic scattering.
Thus, the use of Eq. (4) can significantly improve the analysis of measurements from tissue
at short source-receiver distances and at early times with respect to solutions based on the DE.
The use of this new formula could improve the retrieval of the optical properties from CW
measurements at short distances since the DE has a poor accuracy in modelling light close to
the source. These kind of measurements can be particularly important for small volume samples
of biological tissues where the use of short interfiber distances is mandatory. We also stress that,
following a similar procedure, it is possible to derive a similar expression to Eq. (4) for the slab
geometry. Finally, we point out that the perturbation theory can also be implemented for the
solution here described following the lines of Ref. [18].
The importance of the proposed solution is not only restricted to the biomedical optics field
but it is in general addressed to a wide variety of physical phenomena where transport processes
are involved.
Acknowledgements
This work was supported by MIUR under the project PRIN2005 (prot. 2005025333).
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