WO2007072085A1

WO2007072085A1 - Removal of boundaries in diffuse media

Info

Publication number: WO2007072085A1
Application number: PCT/GR2006/000068
Authority: WO
Inventors: Lorenzo Jorge Ripoll; Vasilios Ntziachristos
Original assignee: Foundation For Research And Technology-Hellas
Priority date: 2005-12-20
Filing date: 2006-12-15
Publication date: 2007-06-28
Also published as: GR1005346B

Abstract

This invention in describes a method which effectively removes the contribution of the boundaries, in essence converting finite diffusive volumes into infinite diffusive volumes of the same absorption and scattering properties. The advantages of such an approach are many, since it opens a new revolutionary way of treating measurements in diffusive media. By making use of such an approach the data can be propagated and backpropagated anywhere in spaces and will enable the use of infinite-case Green functions for solving the inverse problem. This transformation will enable to retain the accuracy of complex numerical methods while boosting the implementation simplicity and computational speed required for obtaining 3D images in diffusive media.

Description

DESCRIPTION

REMOVAL OF BOUNDARIES INDIFFUSE MEDIA

TECHNICAL FIELD

This invention relates to a transformation that can be applied to measurements from diffuse media and removes the contribution of the boundary to the measurements. This invention is strongly linked with accelerating and simplifying tomographic reconstructions of diffuse media and extracting quantitative, three-dimensional structural, functional and molecular information. This is achieved by key theoretical developments taught herein that allow for the use of simple theoretical models in tomography of media with arbitrary geometries. This invention can further enable the use of fast direct inversion and back-propagation algorithms in optical tomography of diffuse media.

BACKGROUND

Imaging of diffusive media has been a topic of interest for several decades in different disciplines due to the fact that the diffusive regime governs propagation in diverse media and instances. Lately, increased attention has been focused on imaging using diffusive light , mainly due to its application in biology and medicine. In this area great advances towards in- vivo imaging have been made which make use of diffuse light propagation with applications ranging from detection and characterization of breast cancer [1, 2], functional activity [3, 4], and drug development [5, 6], to the more recently developed quantification and imaging of molecular function and gene-expression in whole animals [6, 7] (a recent review on this subject may be found in Ref. [8]). Within the last five years we have seen a remarkable increase in the number of measurements used for tomographic imaging in diffusive media, due to the direct use of CCD cameras for high spatial sampling at different projections of photons propagating through diffusive media. While conventional, fiber-based systems utilized at most 10³ surface measurements [1, 2, 7], CCD systems and non-contact approaches can easily yield 10⁶ - 10⁸ measurement data sets [9-12]. Larger data sets yield measurements with greater information content and improve imaging quality; conversely they come with more stringent requirements for theoretical modeling and computational efficiency.

Tomography is based on utilizing these data sets to obtain images or infer the optical properties of the medium under study. Many different numerical approaches have been developed for modeling photon propagation and effectively provide the solutions necessary for tomographic inversion of the data sets collected. This has grown into an active field (see Ref. [13] for a review). In all these numerical methods the objective was to retrieve the distribution of sources inside a heterogeneous or homogeneous medium from a relatively small number of surface measurements (in the order of 10 -10 ). As the number of measurements increased exponentially new numerical methods that could deal with such large data sets were developed so as to reduce the memory and computing time needed to solve for such large systems of equations [14-16]. At present, however, we have reached a point where a compromise is needed when dealing with arbitrary geometries: either accurate methods are used by reducing considerably the size of our data sets [17, 18], or approximate methods are employed to speed up calculations at the expense of reducing the accuracy of the result [15].

SUMMARY

This invention is based on a method which transforms surface measurements to measurements which would be obtained in the absence of the surface, i.e. if the diffusive volume were infinite and homogeneous. This invention has two important consequences. First, it allows the use of infinite Green functions to generate forward solutions for the inverse problem, thus avoiding the use of complex numerical methods which solve for arbitrary geometries. This will have direct impact in significantly accelerating the computation for tomographic images. Second, it allows for propagating the data to virtual detectors located anywhere outside the volume, therefore enabling the use of exact inversion methods such as those proposed in Refs. [16, 19]. Related to this is the fact that once outside the diffusive volume data can be back-propagated inside the diffusive volume, a transformation which is not possible in the presence of arbitrary interfaces. This invention teaches that the method can be used in several instances, presenting two examples which relate to the cases in which we have limited number of detectors on the surface or when the surface can only be partially accessed. Finally examples of the potential of this transformation by propagating and back-propagating measurements to an arbitrary point in space are presented.

The proposed method can be used in combination with several solutions of the diffuse problem, including numerical and analytical solutions of the transport equation or its derivations and approximations such as the Boltzmann Equation or the Diffusion Equation. As such this invention can also be used for resolving absorption, scattering or fluorescence contrast and in combination with contrast agents and molecular probes Additionally the only requisite for this invention to applicable is the fact that the diffusion regime dominates. Hence it is applicable not only to light diffusion but also to electron and neutron diffusion and any other diffusive process.

DESCRIPTION OF THE DRAWINGS

FIGURE 1. Scattering geometry for a diffusive medium of volume V.

FIGURE 2. Effective infinite space geometry once the contribution surface boundary S has been removed by means of the boundary removal equation.

FIGURE 3, Error in retrieving the infinite-case measurements versus the detector area

(dA) for several values of the absorption (μ_a) and scattering coefficients (μ_s ').

In all cases a cylinder of radius R=lcm with index of refraction ni_n=l .333 surrounded by air (n_out⁼l) was considered.

FIGURE 4, Error in retrieving the infinite-case measurements versus the angular coverage of the detectors when these span the top-half (curves to the right) or

the bottom-half (curves to the left). Two cases are presented,

(solid line) and //_α=0.2cm^"' (dotted line). In all cases μ_s -10cm^"1, for the

geometry presented in Fig. 3. FIGURE 5, Propagated value of the boundary-removed data normalized by the direct infinite-case measurement for a cylinder of constant radius R=\cm. Results

for

The error included in each plot corresponds to the error in retrieving the boundary- removed data at the surface. The detector area used for this simulation was dA » 0.05cm .

FIGURE 6, Propagated value of the boundary-removed data normalized by the direct infinite-case measurement for a cylinder of constant radius R=I cm. Results

for μ_a=0.2cm^~ι presented. μ_s —10cm^"1, nj_n=l .333 and n_out=l . The error included in each plot corresponds to the error in retrieving the boundary- removed data at the surface. The detector area used for this simulation was dA » 0.05cm .

FIGURE 7, Propagated value of the boundary-removed data normalized by the direct infinite-case measurement for a cylinder with sinusoidally-varying radius

R = 1 + 0. lcm with period π/6. Results for //_α=0.02cm^"1 presented. In all cases

μ_s - 10cm^{" 1} , nin= 1.333 and n_out= 1. The error included in each plot corresponds

to the error in retrieving the boundary-removed data at the surface. The detector area used for this simulation was dA « 0.05cm . FIGURE 8, Propagated value of the boundary-removed data normalized by the direct infinite-case measurement for a cylinder with sinusoidally-varying radius

R = 1 ± 0.1cm with period π/6. Results for μ^O^cm^"1 presented, hi all cases μ.

_s'=l 0cm^{" ι} , ni_n= 1.333 and n_out= 1. The error included in each plot corresponds to the error in retrieving the boundary-removed data at the surface. The detector area used for this simulation was dA « 0.05cm . FIGURE 9, Backpropagation results for the case presented in Fig. 5 when backpropagating from z_d=\.\cm to zø=0.5cm (see insets for reference). We present results when using two different widths of a Hamming filter, namely,

k_cut=6π cm^'1 (solid lines) and A:_C!,^=8π cm ¹ (dotted lines). The position of the

sources with respect to the backpropagated plane are shown as black bullets on the C=O axis. FIGURE 10, Backpropagation results for the case presented in Fig. 5 when backpropagating from X_d=\.\cm to xo=O.5cm (see insets for reference). We present results when using two different widths of a Hamming filter, namely,

k_cur6π cm^'1 (solid lines) and k_cut ⁼8π cm^'1 (dotted lines). The position of the sources with respect to the backpropagated plane are shown as black bullets on the U=O axis.

DETAILED DESCRIPTION

Let us consider the geometry depicted in Fig. 1 where we have a homogeneous

diffusive volume V with diffusion coefficient D, absorption coefficient μ_a and index

of refraction n_;n. In terms of the reduced scattering coefficient μ_s ' , D is usually

defined as 1 /3μ_s ' (we have neglected the absorption contribution to D since it is not

relevant to this study [20]). We shall denote the surface that encloses volume Fas S. Let us assume that we have a source distribution Φ(r) inside the medium (r <= V )

whose intensity is modulated at a frequency CO . hi this case the average intensity U represents a diffuse photon density wave [21, 22] and obeys the Helmholtz

equation[23], V²t/(r) + κ²U(r) = -Φ(r) ID, r & V with a complex wave-number K given by K = ( -μ_a I D + iωn_in IcD) , being c is the speed of light in vacuum. Due to

the fact that all time-dependent responses may be written in as a sum of contributions with different frequency modulation via a Laplace transform, all cases considered here are general and may be used for time-domain, frequency-domain and continuous illumination sources and detectors. Taking into account rigorously the effect of the interface S, the average intensity U inside volume V is found through Green's theorem as [23-25]:

where n represents the surface normal pointing outwards and,

U°"^c)(r) = Jφ(r')g(Λr|r'-r|)jF (2)

is the average intensity that is obtained in the absence of the surface. In Eqs. (1) and (2) g is Green's function for an infinite homogeneous medium. We shall now introduce in Eq. (1) the boundary condition between the diffusive and non-diffusive medium [26-29]:

U(r) = C_ndJ(r), Vr s S (3)

where the coefficient C_ncι takes into account the refractive index mismatch between both media [26] (C_Λ(/~5 for the example cases presented herein). Introducing Fick's law, J = -DVU and Eq. (3) into Eq. (1) we obtain:

In order to solve the integral Eq. (4) by means of Eq. (3) one usually either solves for the surface flux J_n or for the average intensity U at the boundary. This is typically achieved by using accurate algorithms such as the Boundary Element Method [25, 30] or approximations to it such as the Kirchhoff Approximation [15]. Note that all the Green functions involved are infinite Green's functions.

Interestingly, if we consider an experimental setup which enables us to detect the light that emerges from all points of the surface S, it is possible to experimentally measure the total distribution of emerging flux J_n. In this case J_n does not need to be calculated from Eqs. (3) and (4) but it can be directly substituted by the experimental measurement. Such measurements have become possible by using non-contact approaches recently developed [9-12], by projecting onto the surface the values measured at a CCD detector. These non-contact setups can capture with great accuracy and spatial sampling the distribution of total outward flux on the boundary.

In the case that J_n is known we can directly obtain from Eq. (4)

i.e. the average intensity created by the source distribution in the absence of the interface. This effectively means that volume V has become infinite, filling all space with a diffusive

medium of constant properties D, μ_a and n_in [see Fig. 2]. The measured infinite-case

average intensity U^ at each detector position 17 for a total of N detectors can be

found as:

(5)

where we have represented the surface measurements as J™"" , and dA represents the

detector area. Note that care must be taken for 17=17, in which case the solution depends on the area dA_j. A detailed derivation of these self-induced values in 3D may be found in Ref. [30] for. In order for Eq. (5) to be accurate the following conditions need to be met: a) the (average) optical properties of volume V need to be

known a priori [i.e. the value for K- in Eq. (5)] at least to some degree of

approximation, b) the surface geometry must be known accurately, and c) we must have all values of the surface outward flux measured in order for the series in Eq. (5) to be equivalent to a surface integral. In practice it is possible to obtain accurately the surface geometry [9, 12], and now due to the newly developed non-contact strategies it is experimentally feasible to measure the outward flux in all positions [31]. Finally, even though it is expected that the volumes of interest will not be homogeneous, we have shown that using average optical properties and normalized measurements eliminates for all practical purpouses the contribution of signals due to inhomogeneities (see Ref. [32] for details on this subject). Thus, by making use of non-contact approaches, surface extraction methods and normalized measurements, the proposed transformation will enable 3D imaging while making use of infinite- case Green functions. This approach while retaining the accuracy of complex numerical methods will boost the implementation simplicity and computation speed required to obtain 3D images in diffuse media.

Once the infinite-case measurements have been obtained, a study of the accuracy of Eq. (5) when reducing the number of surface measurements and when the surface S is only partially accessible is presented. To that end we have performed a numerical simulation in which we have a diffusive cylinder of radius lcm and index of refraction of water, «_/„=1.333 surrounded by air (n_OId=l). For simplicity we shall solve the 2D system, without loss of generality. We have placed 4 (line) cw sources (ω=0) inside this volume at locations (x,z) = (-0.1,0.5), (0.1,0.5), (0.5,0.2) cm and (-

0.5,0.5) cm as shown in the inset of Fig. 3. We have solved Eq. (4) by using the Boundary Element Method without any approximation [25] and then retrieved the infinite-case values by using Eq. (5) for a given detector area dA. The error was

measured in % as /l - U^ I U^iinc) ) x 100 , where O denotes average over all

detector positions and where l/^mc) is the direct solution for the source distribution in infinite space, Eq. (2). Results for the error versus different detector sizes are shown in Fig. 3 for several absorption and reduced scattering coefficients within realistic in- vivo values, assuming we have full coverage of surface S by detectors of area dA. As can be inferred from Fig. 3, we see that even for detector areas of ~ I mm we are still below 1% error. Since typical experimental noise values are usually above 1%, limitations due to the finite size of the detectors are not expected.

We shall now present the case when it is not possible to recover the outward flux J_n from all the surface since only a part of it is accessible. In this study we have used the same geometry and source distribution as in Fig 3, looking at two possible configurations: when the detectors span all the upper-half of the cylinder which is closest to the sources, and when the detectors span the lower-half, further from the sources. Results are shown in Fig 4, where both configurations are depicted in the insets. We have studied the error in retrieving the infinite-case values versus the angular coverage: 0 degrees represents full coverage for the upper-half measurements (to the right of Fig. 4) and no coverage for the lower-half, whereas 180 degrees represents full coverage for the lower-half measurements (to the left of Fig. 4) and no coverage for the upper-half. In Fig. 4 we see that if we are near enough the sources (the upper-half spanned by the detectors in this case), we obtain results under 10% error even for a 30 degree coverage. This however is not the case if we choose this coverage in the lower-half where the error is ~80%. Since one in principle does not know where the sources are a priori, Figure 4 evinces the fact that full-angular measurements are needed in order to obtain accurate images [31]. Fortunately, non-contact measurements are capable of providing full-angular coverage and therefore this should not represent a problem experimentally [9]. As expected, in Fig 4 we see that higher absorption values yield lower errors, since the effects of the boundaries dimmish.

Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. Although suitable methods and materials for the practice or testing of the present invention are described below, other methods and materials similar or equivalent to those described herein, which are well known in the art, can also be used. All publications, patent applications, patents, and other references mentioned herein are incorporated by reference in their entirety. In case of conflict, the present specification, including definitions, will control. In addition, the materials, methods, and examples are illustrative only and not intended to be limiting.

This invention in summary describes a method which effectively removes the contribution of the boundaries, in essence converting finite diffusive volumes into infinite diffusive volumes of the same absorption and scattering properties. The advantages of such an approach are many, since it opens a new revolutionary way of treating measurements in diffusive media. A numerical study of the effect that detector size has on the retrieved data has been presented, strongly suggesting that the implementation of this method with experimental data in relevant experimental configurations is feasible. Other features and advantages of the invention will be apparent from the following example were we show examples of how the data can be propagated and backpropagated anywhere in space, stressing the fact that this will allow the application of direct inversion methods in the presence of arbitrary geometries.

EXAMPLES

The invention is further described in the following examples, which do not limit the scope of the invention described in the claims.

Example 1 : Propagation into an arbitrary point in space outside volume V

We shall now show one of the potential applications that the use of Eq. (5) offers. Since we now deal with infinite homogeneous media as depicted in Fig. 2, we may use several approaches to treat our data. We have opted for a simple and visual approach, which can be also applied to direct inversion methods [16, 19], which consists on propagating the infinite-case data to arbitrary points in space outside volume V. To that end we shall use the first Rayleigh-Sommerfeld integral formula which states that for a flat surface A at z—z₀ the field (in this case the average intensity) at Z>ZQ is given by [24]:

where we have written r in cylindrical coordinates (R,z) to stress the fact that this formula holds only for a flat surface. Decomposing surface S into its planar components and assuming that in this case the average intensity U inside the surface integral of Eq. (6) becomes a delta function, we may project the data by adding all the components:

where His the Ηeavyside step function [23] which has been included so that only those points that are above the surface defined by the normal n, are considered for each surface area dAj. This is done to ensure that, at least locally, we do not measure

U_p"_rop inside volume V. In case S is a planar surface, Eq. (7) reduces to the

discretised version of Eq. (6). In all other cases, Eq. (7) is an approximation to the real solution, being termed the Rayleigh Hypothesis in Electromagnetic Theory [24]. Note that Eq. (7) can only be used when S is an interface between two diffusive media, which stresses the importance of the boundary removal formula, Eq. (5).

In order to test the example Eq. (7) we have performed numerical simulations equivalent to those in Figs. 3 and 4, for the same source distribution but two different geometries: a cylinder with constant radius R=I cm and a cylinder with a

sinusoidally- varying radius R = 1 ± 0. lcm with a period of π/6. In all results

presented in Figs. 5 through 8 we have assumed detector areas of dA « 0.05cm . Results for two different absorption coefficients are shown in Figures 5 through 8,

where we plot the ratio of U^ I U^u"^c) using Eqs. (7) and (2). In these figures

several features need to be outlined: first, we can see that near the surface the values are close to unity. This is expected, since the closer we are to the surface the more locally-plane it is and the more Eq. (6) holds. Secondly and most remarkably, even though some of the propagated values may differ 30% from the direct source distribution measurements, this error is quite homogeneous throughout space yielding basically a constant difference. This difference will be considerably diminished and in practice cancelled out when using normalized approaches [32]. Finally, the propagated intensity is always less than the direct source-distribution measurements, except in the convex areas of Figs. 7 and 8 near surface S. In these cases we are adding contributions that actually do go inside volume V, violating Eq.

(6).

Example 2: Back-Propagation into an arbitrary point in space inside volume V

Finally, we would like to show one last example of what can be done once the data has been propagated into r>V. In this case we will backpropagate the data from a plane outside volume V directly onto the sources, in order to prove the fact that all contribution from the surface has vanished. This type of approaches will prove extremely useful particularly in real-time tomography since it will allow the implementation of direct inversion algorithms and make use of extremely large data sets that can be solved in a matter of seconds [16, 19]. We shall consider the case for which forward propagation results are shown in Fig. 5, i.e. a cylinder of i?=lcm,

with μ_a=0.02cm^"' and μ_s'=10cm^"'. We shall backpropagate the data from a distance

of 1.1cm from the center of the cylinder to the sources considering two different measurements, at 0 degrees (shown in Fig. 9 for which we backpropagate from

and at 90 degrees (shown in Fig. 10 for which we backpropagate from X_d=\ .1 cm to xø=0.5cm). In order to backpropagate the data we solved in Fourier space Eq. (6) for U at z=zo, applied a filter, and then performed an inverse Fourier transform (details on this transformation in the context of diffuse waves may be found in Ref. [33]) . In these results we have made use of a Hamming

filter, for which two different widths are presented, k_Cut ⁼6πcm^~! and k_cu^Sπcm^'1. As can be seen in Fig. 9 and Fig. 10 the source positions are retrieved without any additional contribution of the surface values, where we would like to stress the fact that in both cases we have backpropagated inside volume V.

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Claims

CLAIMS REMOVAL OFBOUNDAMESINDIFFUSEMEDIA

1. A method for tomographic or direct imaging of medium comprising:

(a) illuminating a surface that bounds a diffusive medium and detecting an intensity of waves emitted from the medium by using contact or non- contact measurements of intensity outside the medium;

(b) projecting the detected values onto the surface when using non-contact measurements;

(c) fransforrning the surface values to the values that would be obtained in the absence of the surface; and

(d) processing the transformed surfaces values to generate a tomographic image.

2. The method of Claim 1 wherein the medium fills an object of volume V, at least partially bounded by the boundary surface S.

3. The method of Claim 1 wherein step (b) is replaced by contact measurements U, in which case the total flux J_n at the boundary is found as J_n = U/Cnd where Cnd is the boundary coefficient that takes into account the index-mismatch.

4. The method of Claim 1 where the transformed values are projected to anywhere outside the diffusive volume V.

5. The method of Claim 1 where the transformed values are projected into the diffusive volume V.

6. The method of Claim 5 where the transformed values are projected into the diffusive volume V.

7. The method of Claims 1, 3, 4, 5, and 6 where the surface values are transformed using the boundary removal formula: S

where C_n(j is the boundary coefficient that accounts for the index mismatch, D is the diffusion coefficient, dA is the detector area and k is the wave-number for the diffusive wave at the intensity modulation frequency w. g is the Green's function for an infinite homogeneous diffusive medium with a wave number K given by formula : g(Λr l r-r'|) = exp|ϊκ- | r-r^P 0/I>| r-r¹ | .

8. The method of Claim 1 wherein the volume V is of arbitrary geometry.

9. The method of Claim 1 , wherein the volume or object has a fixed geometry whose surface is defined in terms of a continuous function f[z(x,y)] in cartesian, polar or cylindrical coordinates.

10. The method of Claim 9, wherein the object is an animal.

11. The method of Claim 9, wherein the object is a human.

12. The method of Claim 1 further including a step of selecting a tomographic imaging method.

13. The method of claim 1 , wherein the tomographic imaging method is selected from the group consisting of diffuse optical tomography, fluorescence- mediated tomography, near-field optical tomography and thermal tomography.

14. The method of claim 1 , wherein the medium is biological tissue.

15. The method of claim 1, wherein the waves are waves of temperature.

16. The method of claim 1, wherein the waves are light.

17. The method of claim 1 , wherein the wave is continuous wave (CW), time- resolved (TR), intensity modulated (IM) or a combination thereof.

18. The method of claim 1 , wherein the light is near-infrared or infrared light.

19. The method of claim 1 , wherein light is continuous wave (CW), time-resolved (TR) light, intensity modulated (IM) light or any combination thereof.

20. The method of claim 1, wherein contact measurements are made using optical guides, fiber guides, optical matching fluids, lenses or any combination thereof.

21. The method of claim 1 , wherein non-contact measurements are made using a system of lenses, pinholes, apertures or any combination thereof.

22. An apparatus comprising: a machine executable code for a method of tomographic imaging of medium including the steps of:

(a) directing waves into a medium having a boundary S;

(b) detecting an intensity of waves emitted from the medium by using contact or non-contact measurements of waves outside the medium;

(c) projecting the detected intensity (in the case of non-contact measurements) to the surface

(d) transforming the surface data to the data that would be obtained in the absence of the suface. (e) Propagating the transformed data to any point in space outside the diffusive volume V.

(f) Back-propagating the transformed data to any point in space inside the diffusive volume V.