US 20050283071 A1 Abstract A method for tomographic imaging of diffuse medium includes directing waves into a diffusive medium, solving a surface-bounded inversion problem by forward field calculations through decomposition of contributions from the multiple reflections from an arbitrary surface within the diffusive medium or outside the diffusive medium into a sum of different orders of reflection up to an arbitrary order, and using contact or non-contact measurements of waves outside said diffusive medium to generate a tomographic image.
Claims(38) 1. A method for tomographic imaging of medium comprising:
(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; and (c) processing the detected intensity to generate a tomographic image. 2. The method of 3. The method of 4. The method of 5. The method of 6. The method of _{DRBM} ^{(N) }according to: where G
^{(N)} _{DRBM}(r_{p}) is the N-order Green function in the DRBM approximation, D is the diffusion coefficient inside the diffusive medium, n is a unity vector normal to boundary surface S and pointing into the non-diffusive medium, κ is a diffusive wave number κ=√{square root over (−μ_{a}/D+iω/c)}, for a modulation frequency ω, c is a speed of light in the medium, Sa is an absorption coefficient, τ is an evolution step, and r_{s}, and r_{p }are source and detector positions respectively, and wherein the detector is located at the surface, and where g is the Green's function for an infinite homogeneous diffusive medium with a wave number κ. given by formula g(κ|r−r′|)=exp[iκ|r−r′|]/D|r−r′|, and N is an arbitrary integer not smaller than 1, and where C
_{nd }is a total reflectivity of the boundary surface S, integrated over all angles, and expressed through the Fresnel reflection coefficients r as: where
R _{J} ^{1→0}=∫_{0} ^{1}[1−|r _{10}(μ)|^{2}]μ^{2} dμ R _{J} ^{0→1}=∫_{0} ^{1}[1−|r _{01}(μ)|^{2} ]μdμ R _{U} ^{0→1}=∫_{0} ^{1}[1−|r _{01}(μ)|^{2}]μ^{2} dμ and where μ=cos θ for an incidence angle θ, r
_{01 }and r_{10 }represent the reflection coefficients when the incident wave comes from the inner, diffusive medium having an index of refraction n_{in}, or outer, non-diffusive medium having an index of refraction n_{out}, respectively, and are defined in terms of the parallel and perpendicular polarization components as: and where cos θ
_{t }is the cosine of the transmitted angle, which is found from Snell's law n_{out }sin θ_{t}=n_{in }sin θ_{i}. 7. The method of _{DRBM} ^{(N) }and U_{DRBM} ^{(N) }according to: where G
^{(N)} _{DRBM}(r_{p}) is the N-order Green function in the DRBM approximation, D is the diffusion coefficient inside the diffusive medium, n is a unity vector normal to boundary surface S and pointing into the non-diffusive medium, κ is a diffusive wave number κ=√{square root over (−μ_{a}/D+iω/c)}, for a modulation frequency ω, c is a speed of light in the medium, μ_{a }is an absorption coefficient, τ is an evolution step, and r_{s}, and r_{p }are source and detector positions respectively, and wherein the detector is located at the surface, and where g is the Green's function for an infinite homogeneous diffusive medium with a wave number κ given by formula g(κ|r−r′|)=exp[iκ|r−r′|]/D|r−r′|, N is an arbitrary integer not smaller than 1, U
_{DRBM} ^{(N)}(r_{d}) is a wave intensity at point r_{d}, and S(r′) is the strength of the light source at position r′ expressed in units of energy density, and where C
_{nd }is a total reflectivity of the boundary surface S, integrated over all angles, and expressed through the Fresnel reflection coefficients r as: where
R _{J} ^{1→0}=∫_{0} ^{1}[1−|r _{10}(μ)|^{2}]μ^{2} dμ R _{J} ^{0→1}=∫_{0} ^{1}[1−|r _{01}(μ)|^{2} ]μdμ R _{U} ^{0→1}=∫_{0} ^{1}[1−|r _{01}(μ)|^{2}]μ^{2} dμ and where μ=cos θ for an incidence angle θ, r
_{01 }and r_{10 }represent the reflection coefficients when the incident wave comes from the inner, diffusive medium having an index of refraction n_{in}, or outer, non-diffusive medium having an index of refraction n_{out}, respectively, and are defined in terms of the parallel and perpendicular polarization components as: and where cos θ
_{t }is the cosine of the transmitted angle, which is found from Snell's law n_{out }sin θ_{t}=n_{in }sin θ_{i}. 8. The method of monitoring a gradient of the boundary surface to detect complex boundaries; and automatically increasing a density of local surface discretization and the number N of terms in a series, if the boundary is complex. 9. The method of monitoring relative change in a value of the calculated intensity added by each term of the series; and truncating the series by selecting a finite number N of terms in a series, when the relative change in a value of the calculated intensity meets convergence criteria. 10. The method of monitoring the gradient of a surface boundary to detect complex boundaries; automatically increasing a density of local surface discretization and the number N of terms in a series, if the boundary is complex; and optimizing an evolution step r by assigning a value, of about τ=2imag{κ}/√{square root over (W)}+1, wherein W is a mean diameter of the diffusive medium. 11. The method of 12. The method of 13. The method of 14. The method of 15. The method of 16. The method of 17. The method of 18. The method of 19. The method of 20. The method of 21. The method of 22. The method of 23. The method of 24. The method of 25. A method of obtaining a tomographic image of a target region within an object, the method comprising:
(a) directing light waves from multiple points into an object; (b) detecting light waves emitted from multiple points from the object, wherein the light is emitted from an intrinsic absorber, fluorochrome, or scatterer; (c) processing the detected light by representing the contribution of each wave into the detected intensity as a sum of an arbitrary number N of terms in a series and wherein each term in the series is an intensity of a wave reflected from an arbitrary surface within or outside a diffusive medium; and (d) forming a tomographic image that corresponds to a three-dimensional target region within said object and to a quantity of intrinsic absorber, fluorochrome, or scatterer in the target region. 26. The method of 27. A method of obtaining a tomographic image of a target region within an object, the method comprising:
(a) administering to an object a fluorescent imaging probe; (b) directing light waves from multiple points into the object; (c) detecting fluorescent light emitted from multiple points from the object; (d) processing the detected light by representing the contribution of each wave into the detected intensity as a sum of an arbitrary number N of terms in a series and wherein each term in the series is an intensity of a wave reflected from an arbitrary surface within or outside a diffusive medium; and (e) forming a tomographic image that corresponds to a three-dimensional target region within the object and to a quantity of fluorescent imaging probe in the target region. 28. The method of 29. The method of 30. The method of 31. The method of 32. The method of 33. The method of 34. A tomographic imaging system comprising:
(a) a wave source block to direct waves into an object; (b) a wave detector block to detect the intensity of waves emitted from the object and to convert the intensity of the waves into a digital signal representing waves emitted from the object; (c) a processor to control the detector block and, optionally, the source block and to process the digital signal representing waves emitted from the object into a tomographic image on an output device, wherein the processor is programmed to process the digital signal by representing the contribution of each wave into the detected intensity as a sum of an arbitrary integer number N of terms in a series and wherein each term in the series is an intensity of a wave reflected from an arbitrary surface within or outside a medium. 35. The system of _{DRBM} ^{(N) }according to: where G
^{(N)} _{DRBM}(r_{p}) is the N-order Green function in the DRBM approximation, D is the diffusion coefficient inside the diffusive medium, n is a unity vector normal to boundary surface S and pointing into the non-diffusive medium, κ is a diffusive wave number κ=√{square root over (−μ_{a}/D+iω/c)}, for a modulation frequency ω, c is a speed of light in the medium, μ_{a }is an absorption coefficient, τ is an evolution step, and r_{s}, and r_{p }are source and detector positions respectively, and wherein the detector is located at the surface, and where g is the Green's function for an infinite homogeneous diffusive medium with a wave number κ. given by formula g(κ|r−r′|)=exp[iκ|r−r′|]/D|r−r′|, and N is an arbitrary integer not smaller than 1, and
where C
_{nd }is a total reflectivity of the boundary surface S, integrated over all angles, and expressed through the Fresnel reflection coefficients r as: where
R _{J} ^{1→0}=∫_{0} ^{1}[1−|r _{10}(μ)|^{2}]μ^{2} dμ R _{J} ^{0→1}=∫_{0} ^{1}[1−|r _{01}(μ)|^{2} ]μdμ R _{U} ^{0→1}=∫_{0} ^{1}[1−|r _{01}(μ)|^{2}]μ^{2} dμ and where μ=cos θ for an incidence angle θ, r
_{01 }and r_{10 }represent the reflection coefficients when the incident wave comes from the inner, diffusive medium having an index of refraction n_{in}, or outer, non-diffusive medium having an index of refraction n_{out}, respectively, and are defined in terms of the parallel and perpendicular polarization components as: and where cos θ
_{t }is the cosine of the transmitted angle, which is found from Snell's law n_{out }sin θ_{t}=n_{in }sin θ_{i}. 36. The system of _{DRBM} ^{(N) }and U_{DRBM} ^{(N) }according to: wherein where G
^{(N)} _{DRBM}(r_{p}) is the N-order Green function in the DRBM approximation, D is the diffusion coefficient inside the diffusive medium, n is a unity vector normal to boundary surface S and pointing into the non-diffusive medium, κ is a diffusive wave number κ=√{square root over (−μ_{a}/D+iω/c)}, for a modulation frequency ω, c is a speed of light in the medium, μ_{a }is an absorption coefficient, τ is an evolution step, and r_{s}, and r_{p }are source and detector positions respectively, and wherein the detector is located at the surface, and where g is the Green's function for an infinite homogeneous diffusive medium with a wave number κ given by formula g(κ|r−r′|)=exp[iκ|r−r′|]/D|r−r′|, N is an arbitrary integer not smaller than 1, U_{DRBM} ^{(N)}(r_{d}) is a wave intensity at point r_{d}, and S(r′) is the strength of the light source at position r′ expressed in units of energy density, and
_{nd }is a total reflectivity of the boundary surface S, integrated over all angles, and expressed through the Fresnel reflection coefficients r as: where
R _{J} ^{1→0}=∫_{0} ^{1}[1−|r _{10}(μ)|^{2}]μ^{2} dμ R _{J} ^{0→1}=∫_{0} ^{1}[1−|r _{01}(μ)|^{2} ]μdμ R _{U} ^{0→1}=∫_{0} ^{1}[1−|r _{01}(μ)|^{2}]μ^{2} dμ _{01 }and r_{10 }represent the reflection coefficients when the incident wave comes from the inner, diffusive medium having an index of refraction n_{in}, or outer, non-diffusive medium having an index of refraction n_{out}, respectively, and are defined in terms of the parallel and perpendicular polarization components as: _{t }is the cosine of the transmitted angle, which is found from Snell's law n_{out }sin θ_{t}=n_{in }sin θ_{i}. 37. The system of 38. 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) processing the detected intensity to generate a tomographic image by representing the contribution of each wave into the detected intensity as a sum of an arbitrary integer number N of terms in a series and wherein each term in the series is an intensity of a wave reflected from an arbitrary surface within or outside the medium. Description This application is a continuation of International Application No. PCT/US03/17558, which designated the United States and was filed on Jun. 4, 2003, published in English, which claims the benefit of U.S. Provisional Application No. 60/385,931, filed on Jun. 4, 2002. The entire teachings of the above applications are incorporated herein by reference. Optical imaging is an evolving clinical imaging modality that uses penetrating lights rays to create images of both intrinsic and extrinsic biological scatterers. Light offers unique contrast mechanisms that can be based on absorption, e.g., probing of hemoglobin concentration or blood saturation, and/or fluorescence, e.g., probing for weak auto-fluorescence, or exogenously administered fluorescent probes (Neri et al., Diffuse optical tomography (DOT) is one type of optical tomography that has been used for quantitative, three-dimensional imaging of biological tissue, based on intrinsic absorption and scattering. (Ntziachristos et al., Currently, optical tomography utilizes either numerical or analytical methods for solving the equations governing propagation of light through biological tissue. Numerical methods, such as the finite element method (FEM), finite differences (FD) or the boundary element method (BEM), are used for complex geometries of air/tissue boundaries and are extremely computationally costly and therefore are currently non-viable in a real-time three-dimensional research and clinical setting. (For example, reported numerical-based reconstruction times for a simulated typical 3D breast-imaging problem on state of the art single processor computers range from 6 to 36 -hours.) Analytical methods are much faster (for example, an analytical-based 3D reconstruction case of the breast ranges between 2-15 minutes), but are available only for simple geometries of air/tissue boundary such as a slab, a cylinder or a sphere, and often lack adequate accuracy for imaging complex objects such as a human breast. An alternative method to perform tomography of diffuse or diffuse-like medium with complex geometries is by means of an analytical approach called the Kirchhoff Approximation (KA). This method uses the angular spectrum representation of the propagating average intensity and employs the reflection coefficients for light waves to calculate the light intensity inside any arbitrary geometry. Although in contact imaging applications the KA can achieve relatively good computational efficiency (Ripoll et al., There is a need for fast computational methods for real-time optical tomographic diagnostics and other research and clinical uses that can deal with arbitrary sizes, shapes and boundaries that (1) attain the computational simplicity and efficiency of analytical methods while (2) retaining the accuracy and capacity of numerical methods to model arbitrary shapes and boundaries, and (3) can be applied both to contact and non-contact measurements of diffuse and diffuse-like medium. The invention is directed to an imaging system and a method for imaging volumes or subjects containing interfaces such as diffuse and non-diffuse tissue interfaces or other interfaces using analytical methods. In one embodiment, the present invention is a method for tomographic imaging of diffuse medium comprising directing waves into a medium having a boundary S, detecting an intensity of waves emitted from the medium by using contact or non-contact measurements of waves outside the medium, and processing the detected intensity to generate a tomographic image. In another embodiment, the present invention is a method of obtaining a tomographic image of a target region within an object, the method comprising directing light waves from multiple points into an object, detecting light waves emitted from multiple points from the object, wherein the light is emitted from an intrinsic absorber, fluorochrome, or scatterer, processing the detected light by representing the contribution of each wave into the detected intensity as a sum of an arbitrary number N of terms in a series and wherein each term in the series is an intensity of a wave reflected from an arbitrary surface within or outside a diffusive medium, and forming a tomographic image that corresponds to a three-dimensional target region within said object and to a quantity of intrinsic absorber, fluorochrome, or scatterer in the target region. In another embodiment, the present invention is a method of obtaining a tomographic image of a target region within an object comprising administering to an object a fluorescent imaging probe, directing light waves from multiple points into the object, detecting fluorescent light emitted from multiple points from the object, processing the detected light by representing the contribution of each wave into the detected intensity as a sum of an arbitrary number N of terms in a series and wherein each term in the series is an intensity of a wave reflected from an arbitrary surface within or outside a diffusive medium, and forming a tomographic image that corresponds to a three-dimensional target region within the object and to a quantity of fluorescent imaging probe in the target region. In another embodiment, the present invention is a tomographic imaging system comprising a wave source block to direct waves into an object, a wave detector block to detect the intensity of waves emitted from the object and to convert the intensity of the waves into a digital signal representing waves emitted from the object, a processor to control the detector block and, optionally, the source block and to process the digital signal representing waves emitted from the object into a tomographic image on an output device, wherein the processor is programmed to process the digital signal by representing the contribution of each wave into the detected intensity as a sum of an arbitrary integer number N of terms in a series and wherein each term in the series is an intensity of a wave reflected from an arbitrary surface within or outside a medium. In another embodiment, the present invention is an apparatus comprising machine executable code for a method of tomographic imaging of medium including directing waves into a medium having a boundary S, detecting an intensity of waves emitted from the medium by using contact or non-contact measurements of waves outside the medium, processing the detected intensity to generate a tomographic image by representing the contribution of each wave into the detected intensity as a sum of an arbitrary integer number N of terms in a series and wherein each term in the series is an intensity of a wave reflected from an arbitrary surface within or outside the medium. The approach described herein provide several advantages over the existing art, including (1) the ability to image volumes with 2D or 3D geometries of arbitrary size, shape and boundaries using fast and accurate analytical methods, and (2) the application of these new methods for both contact and non-contact tomography of diffuse and diffuse-like medium. These new imaging methods can have broad applications in a wide variety of areas in research and clinical imaging. Importantly, these methods significantly improve existing tomographic imaging techniques and make possible the use of these imaging techniques in real-time animal and human subject imaging and clinical medical diagnostics by allowing the implementation of practical systems with unprecedented capacity for data collection. Furthermore, the approach described herein can be applied both to contact and non-contact measurements, as well as for any diffuse and non-diffuse interfaces as related to tomography, including optical tomography, fluorescence-mediated tomography, near-field optical tomography, tomography with thermal waves and generally any surface-bounded inversion problems of tissues and other diffuse or diffuse-like medium. The foregoing and other objects, features and advantages of the invention will be apparent from the following more particular description of preferred embodiments of the invention, as illustrated in the accompanying drawings. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention. The purpose of optical tomography is to recover the optical properties, location, and shape of objects buried deep inside a specific volume by solving equations that describe light propagation through diffuse medium, such as biological tissue. As used herein, the terms “diffuse medium” or “diffusive medium” are used interchangeably and are defined to mean media where waves suffer multiple scattering events with small particles (the scatterers) within an otherwise homogeneous medium, randomizing their phase; in this case it is the average wave intensity that is studied. The average wave intensity will follow the diffusion equation, behaving in itself as a “diffuse wave” and interacting with surfaces and boundaries. The terms “non-diffuse medium” or “non-diffusive medium” are used interchangeably and are defined to mean media where waves do not suffer multiple scattering events with scatterers within the medium and maintain their phase; within these media, waves will interact and suffer multiple scattering events with surfaces and boundaries. Analytical solutions are available only for a small number of simple geometries of the air/tissue boundaries, such as cylinders, spheres, and slabs. Due to the lack of an appropriate theoretical method for more complex boundaries, numerical methods need to be used. Numerical methods offer practical simplicity but also significant computational burden, especially when large three-dimensional reconstructions are involved. Such methods include the finite element method (FEM), finite differences (FD), and the extinction theorem (ET) or Boundary Element Method (BEM). Generally, optical tomographic analysis is divided into two steps. The first step is solving the “forward problem,” in which a solution of the wave transport or “diffusion” equation, is used to describe the wave propagation through a medium with assumed optical or other wave-transmitting properties, e.g., tissue, and is used to predict the intensity of light detected emerging from this medium. In a preferred embodiment, the “diffusion equation” is
There are several ways to solve the forward problem (by obtaining analytical and numerical solutions of the diffusion equation) and inverse problem (direct inversion, χ An embodiment of the invention is a tomographic imaging system In another embodiment, the present invention is an apparatus comprising machine executable code for a method of tomographic imaging of medium including the steps of directing waves into a medium having a boundary S; detecting an intensity of waves emitted from the medium by using contact or non-contact measurements of waves outside the medium; and processing the detected intensity to generate a tomographic image by representing the contribution of each wave into the detected intensity as a sum of an arbitrary integer number N of terms in a series and wherein each term in the series is an intensity of a wave reflected from an arbitrary surface within or outside the medium. Without being limited to any particular theory, it is believed that the physical foundation of the method of the present invention is as set forth below. Green's Function In an infinite geometry (no air/time boundary), the so-called homogenous Green's function g(r) is obtained by solving Eq. (3):
Considering the geometry of the problem, a complete Green's function G(r In infinite space the equation is G(r For a homogeneous diffusive volume limited by surface S, i.e., a volume of spatially invariant absorption and scattering coefficients inside S, the solution to Eq. (5) for arbitrary geometries of S can be expressed in terms of a surface integral by means of Green's Theorem (Ripoll et al, When many solutions to a forward problem need to be generated, such as in iterative reconstruction schemes, a first-order approximation to Eq. (8), applicable to arbitrary geometries in 3D is needed, both for the sake of computing time and memory. One such approximation applicable to arbitrary geometries is the Kirchhoff Approximation (KA), sometimes also known as the physical-optics or the tangent-plane method (Ogilvy, London, IOP publishing, 1991; Nieto-Vesperinas, Pergamon, New York, 1996 the entire teachings of which are incorporated herein by reference), so long as the curvature of the surface under study is larger than the wavelength. The KA therefore only considers first order reflections at an interface decomposed into planar components and in the case of a planar surface yields exact solutions. For diffusive point sources, the Kirchhoff Approximation (KA) assumes that the surface is replaced at each point by its tangent plane. Referring to Taking into consideration the different propagation directions of the incident and reflected wave with respect to the local plane, the gradient of the Green's function is:
In most practical cases a more accurate solution of Eq. (8) than the one given by the KA is needed. Thus, this invention describes and teaches the use of an iterative approach that uses an arbitrary number of multiple reflections from the boundary. This approach, called the N-order Diffuse Reflection Boundary Method (DRBM), solves the exact integral Eq. (8) by iteratively decomposing each reflecting event into a sum of the series. Each term in a series is referred herein as an order of reflection. The series can be truncated when the sum meets convergence criteria defined below. Thus, the DRBM method finds the solution of the exact surface integral equation Eq. (8) by representing the solution as a sum of different orders of reflection. In this manner, one can obtain fast solutions to any degree of accuracy by adding orders of reflection. This approach takes into account the shadowing effect due to the higher reflection orders (i.e., the effect caused at those areas where the source is not directly visible due to the presence of the interface), and the high reflectivities at the interfaces (i.e., high values of the reflection coefficient Eq. (10) that introduce multiple scattering interactions at the boundary) can be modeled up to any degree of accuracy. The set of N iterations can be performed without steps of matrix inversion or solving a system of equations. Hence, the computation time of the DRBM is extremely low when compared to a rigorous numerical method such as the ET or BEM, whereas the accuracy is greatly enhanced compared to the KA. In practice, the number of DRBM orders needed may not exceed two, due to the fact that DPDWs are highly damped (i.e., suffer from strong absorption on propagation) so that multiple scattering of DPDWs along the boundary rarely takes place. About one or two first orders of the DRBM are crucial for Eq. (8) in the case of non-convex surfaces that create shadowing. This is because the 1 To develop the DRBM method, Eq. (8) is written in an iterative manner by using the Euler method with an evolution step τ (C. Macaskill and B. J. Kachoyan, Applied Optics 32, 2839-2847, 1993), and the assumption that the detectors are located at each of the surface points:
These values are well known and are dependent on the medium and surface discretization area (Yaghjian, Once the expression at the boundary for the N-th approximation is obtained through Eq. (DRBM.1), the intensity U It is worth noting that a direct relation between computing times for different orders can easily be found by:
In the following paragraphs, preferred embodiments of the invention are described that further accelerate the computation of U Adaptive-DRBM In a preferred embodiment of the invention, the DRBM can adaptively adjust the number of N-order reflections. In one embodiment, the adaptive adjustment can be achieved by detecting complex boundaries, i.e., surfaces with high spatial variation, by monitoring the gradient of the boundary and automatically increasing the density of local surface discretization i.e., the number of discretization areas that define surface S at that local point, and the number of orders of reflections to include in the DRBM calculations. (It is worth noting that in the case of a plane interface, the 0 In another embodiment, the adaptive adjustment can be achieved by monitoring the relative change in value of the calculated intensity added by each iteration step and stopping the number of iterations based on a convergence criterion. Typical criteria include limits on the relative change in a value of a function after each iterated step and limits on new contributions of the new iteration step to the overall value of the intensity. For example, the iterative process can be stopped when the relative change in the value of intensity falls below about 0.1% to about 10%, preferably below 1%. In another example, the iterative process can be stopped when a new contribution falls under a certain threshold value ξ. The threshold value can be selected, for example, to be about twice the value of noise. Additionally, since the DRBM is based on a surface integral equation (DRBM.1) for waves in an absorbing medium, not all surface points contribute equally to the intensity at a certain detector point since the distances between each surface point and the detector vary. Therefore, contribution from surface points further away from the detector will suffer higher attenuation on propagation than points nearer to the detector. Therefore, in another embodiment of the invention, a threshold value that determines which surface points and their corresponding surface values G In another embodiment of the invention, a convenient approximation to the solution of Eq. (8) may be found by the method of images. The method of images is applied by taking into account the boundary condition at a diffusive/non-diffusive interface given by Eq (7), which can be approximated to:
With this expression, the 1 The method of the instant invention can be applied both to contact and non-contact measurements, as well as for any diffuse and non-diffuse interfaces as related to forming a tomographic image using waves and including. As used herein, the term “contact measurement” means measurement that takes place with the detector in contact with the surface or at a distance of less than about 1 mm from the surface. The term “non-contact measurement” refers to measurements that take place with the detector at a distance of greater than 1 mm from the surface. The contemplated waves include light waves, infra-red waves, waves of temperature, acoustic waves and the like. Contemplated modes of tomography include optical tomography, fluorescence-mediated tomography, near-field optical tomography, tomography with temperature waves and generally any surface-bounded inversion problems of tissues and other diffuse or diffuse-like or highly scattering medium. The application of DRBM to performing non-contact measurements will now be described. Referring to the geometrical scheme described in The expression for F in Eq. (21) is given by:
Upon discretization of surfaces called for by the DRBM, Eq. (21) can be rewritten as:
In order to find the average intensity U at the detector, we will approximate it by U(r)=C The expression Eq. (24) for non-contact detectors is independent of the sources. In order to obtain an expression for non-contact sources, the same formulation can be used, due to the source-detector invariance. It is to be understood that while the invention has been described in conjunction with the detailed description thereof, the foregoing description is intended to illustrate and not limit the scope of the invention, which is defined by the scope of the appended claims. In a preferred embodiment, the present invention is a method for tomographic imaging of medium comprising directing waves into a medium having a boundary S; detecting an intensity of waves emitted from the medium by using contact or non-contact measurements of waves outside the medium; and processing the detected intensity to generate a tomographic image. The medium is diffusive or non-diffusive. Preferably, the medium is a diffusive medium. In one preferred embodiment, the medium fills an object of volume V, at least partially bounded by the boundary surface S. The waves are directed into the medium by a source. The waves directed into the medium to be imaged can be waves of temperature, waves of light or acoustic waves. Light is preferred. Near-infrared light is most preferred. The light source of the imaging system of the instant invention can be one or more sources specific to different chromophores, fluorophores or fluorescent imaging probes. Laser diodes can be used as light sources since they produce adequate power, are within the FDA class I and class II limits, and are stable, wavelength-specific and economical. Alternatively, filtered light can also be used. The processing step includes representing the contribution of each wave into the detected intensity as a sum of an arbitrary integer number N of terms in a series. Each term in the series is an intensity of a wave reflected from an arbitrary surface within or outside the medium. In a preferred embodiment, the processing step includes solving the equation (DRBM.1) for an unknown function G In another preferred embodiment, the processing step includes solving the equation (DRBM.2) for an unknown function G The processing step can further include monitoring a gradient of the boundary surface to detect complex boundaries and automatically increasing a density of local surface discretization and the number N of terms in a series, if the boundary is complex. Alternatively, the processing step includes monitoring relative change in a value of the calculated intensity added by each term of the series and truncating the series by selecting a finite number N of terms in a series, when the relative change in a value of the calculated intensity meets convergence criteria. In a preferred embodiment, the processing step comprises monitoring the gradient of a surface boundary to detect complex boundaries, automatically increasing a density of local surface discretization and the number N of terms in a series, if the boundary is complex, and optimizing an evolution step τ by assigning a value, of about τ=2imag{κ}/√{square root over (W)}+1, wherein W is a mean diameter of the diffusive medium. Preferably, the medium being imaged is filling a volume V of arbitrary geometry. Alternatively, 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. Arbitrary geometry is preferred. The object can be a sample of a biological tissue or an animal, including a human. The object may also be non-mammalian, i.e., Three different source-detection technologies exist. Any combination of them can be used for tomography applications as described herein. The simplest is continuous wave (CW) imaging. This technique uses light of constant intensity and measures either (1) the signal due to a distribution of excited fluorophores or (2) the attenuation of light (due to tissue absorption and scattering) employing multiple source-detector pairs. The technique is technically relatively simple and usually offers the best signal-to-noise (SNR) characteristics. However, it is not best suited for imaging of intrinsic tissue contrast since it usually introduces significant cross-talk between the calculations and imaging of absorption and scattering coefficients. On the other hand, if the background optical properties are known, the method is well-suited for imaging fluorophore concentration in the steady-state. A more elaborate approach is to use intensity modulated (IM) light at a single or at multiple frequencies. With this method, modulated light attenuation and phase shifts, relative to the incident light, can be measured for multiple source-detector pairs. Compared to a CW measurement, which yields intensity attenuation, the IM technique offers two pieces of information, i.e., intensity attenuation and phase shift per source-detector pair. Amplitude and phase are usually uncorrelated measurements and can more efficiently resolve the absorption and scattering coefficient of intrinsic contrast. In the fluorescence mode, the technique can image two sets of information, fluorophore concentration and fluorescence lifetime. The third approach, the time-resolved (TR) technique uses short pulses of light injected into the tissue. The technique resolves the distribution of times that the detected photons travel into the medium for multiple source-detector pairs. Time-resolved methods contain the highest information content per source-detector pair, comparable only to the IM method performed simultaneously at multiple frequencies. This can be easily explained when one considers that the Fourier transform of the time-resolved data yields information at multiple frequencies up to 1 GHz, including the continuous wave components (f=0 MHz) used by the previous two methods. Therefore, the time-resolved method offers a CW component for direct comparison with the CW system, but also intensity attenuation and phase-shift measurements at multiple-frequencies (via the Fourier transform) that can image intrinsic absorption and scattering, and also fluorophore concentration and fluorescence lifetime. The step of detection can be accomplished by either contact or non-contact measurements of emitted wave intensity. In one embodiment, contact measurements are made using optical guides, fiber guides, optical matching fluids, lenses or any combination thereof. In another embodiment non-contact measurements are made using a system of lenses, pinholes, apertures or any combination thereof. Non-contact measurements are preferred. The method of the present invention can further include selecting a tomographic imaging method. The tomographic imaging method can be selected from the group consisting of diffuse optical tomography, fluorescence-mediated tomography, near-field optical tomography and thermal tomography. The preferred intrinsic absorbers, fluorochrome, or scatterer is selected from the group comprising hemoglobin, water, lipid, myoglobin, tissue chromophores and organelles. These steps can also be repeated at predetermined intervals thereby allowing for the evaluation of emitted light in an object over time. Preferred fluorescent imaging probes that can be used with the present invention include, but are not limited to (1) probes that become activated after target interaction (Weissleder, et al., The methods of the invention can be used to determine a number of indicia, including tracking the localization of the fluorescent imaging probe in an object over time or assessing changes or alterations in the metabolism and/or excretion of the fluorescent imaging probe in the subject over time. The methods can also be used to follow therapy for such diseases by imaging molecular events and biological pathways modulated by such therapy, including but not limited to determining efficacy, optimal timing, optimal dosing levels (including for individual patients or test subjects), and synergistic effects of combinations of therapy. The invention can be used to help a physician or surgeon to identify and characterize areas of disease, such as arthritis, cancers and specifically colon polyps, or vulnerable plaque, to distinguish diseased and normal tissue, such as detecting tumor margins that are difficult to detect using an ordinary operating microscope, e.g., in brain surgery, help dictate a therapeutic or surgical intervention, e.g., by determining whether a lesion is cancerous and should be removed or non-cancerous and left alone, or in surgically staging a disease, e.g., intraoperative lymph node staging. The methods of the invention can also be used in the detection, characterization and/or determination of the localization of a disease, especially early disease, the severity of a disease or a disease-associated condition, the staging of a disease, and monitoring and guiding various therapeutic interventions, such as surgical procedures, and monitoring drug therapy, including cell based therapies. Examples of such disease or disease conditions include inflammation (e.g., inflammation caused by arthritis, for example, rheumatoid arthritis), all types of cancer (e.g., detection, assessing treatment efficacy, prognosis, characterization), cardiovascular disease (e.g., atherosclerosis and inflammatory conditions of blood vessels, ischemia, stroke, thrombosis), dermatologic disease (e.g., Kaposi's Sarcoma, psoriasis), ophthalmic disease (e.g., macular degeneration, diabetic retinopathy), infectious disease (e.g., bacterial, viral, fungal and parasitic infections), immunologic disease (e.g., Acquired Immunodeficiency Syndrome, lymphoma, multiple sclerosis, rheumatoid arthritis, diabetes mellitus), central nervous system disease (e.g., Parkinson's disease, Alzheimer's disease), and bone-related disease (e.g., osteoporosis, primary and metastatic bone tumors, osteoarthritis). The methods of the invention can therefore be used, for example, to determine the presence of tumor cells and localization of tumor cells, the presence and localization of inflammation, including the presence of activated macrophages, for instance in athlerosclerosis or arthritis, the presence and localization of vascular disease including areas at risk for acute occlusion (i.e., vulnerable plaques) in coronary and peripheral arteries, regions of expanding aneurysms, unstable plaque in carotid arteries, and ischemic areas. The methods and compositions of the invention can also be used in identification and evaluation of apoptosis, necrosis, hypoxia and angiogenesis. Importantly, the methods of the present invention may be used in combination with other imaging compositions and methods. For example, the methods of the present invention may be used in combination with other traditional imaging modalities such as X-ray, CT, PET, SPECT, and MRI. For instance, the methods of the present invention may be used in combination with CT and MRI to obtain both anatomical and molecular information simultaneously, for instance by co-registration of a tomographic image with an image generated by another imaging modality. In particular, the combination with MRI or CT is preferable given the high spatial resolution of these imaging techniques. DOT imaging (absorption only) has already been combined with MRI imaging (Ntziachristos et al., The invention is further described in the following examples, which do not limit the scope of the invention described in the claims. Referring to By way of a practical example, the number of discretization points for a sphere of radius 2 cm needed in order to maintain a one transport mean free path (ltr=D/3) distance between points (N˜5000) was used to compare the speed of the KA and ET methods. In this case, it takes the KA methods 70 seconds and the ET method 50 minutes to solve the problem, indicating that the KA as approximately 40 times faster than the ET method. A more realistic surface such as the adult head, would imply an equivalent radius of at least 4 cm, and thus N˜20000. In this case, the KA takes on the order of 90 seconds, whereas the ET takes on the order of 45 hours for only one forward solution. In this more realistic case, the KA is 1800 times faster. Similar conclusions can be reached for the FD method. Also, due to its linearity, large numbers of surface points are possible with the KA, namely N˜10E+6, whereas with the ET, such large matrices are impossible to solve at the present time. These results indicate that utilizing KA allows fast, real-time application of optical tomographic techniques. So-called shadowing effect appears when certain surface areas are blocked from the source by the geometry of the interface. Since the KA only considers first order reflections at the interface, errors appear in the proximities of the shadow regions of a source. Furthermore, since these shadow areas are not taken into account in the KA, it predicts higher values of the intensity for these areas. Referring to FIGS. In general, it had been demonstrated that the KA calculates the average intensity with errors that are less than 5% inside the volume (see A comparison of the intensity generated by a point source in a cylindrical geometry using the KA versus the exact solution, (ET) with different several radius and optical properties is shown in The smaller the dimensions of the geometry and the smaller the absorption coefficient the larger the error committed by the KA method. As seen in this figure, the error increases as the volume of the medium or the absorption decreases. From Computation times of the first four orders of the DRBM are shown in Computation times using equation (DRBM.4) for the 1 To demonstrate the efficiency of the DRBM to image small volumes with complex boundaries, simulations were performed using an exact forward solver to evaluate its performance when a) the boundary is not exactly known and b) when an approximation to the forward model is used, i.e., when one approximates the actual arbitrary boundary with a generic boundary such as an ellipse or a slab. The DRBM method was also compared to an exact solution (ET method) and to the KA method. Data Collection and Processing Referring to In step In step In step In step In step In step In step In step In step In step In step In step In step In step When all N It will be appreciated by one skilled in the art that the computational methods of the present invention can be applied both to contact and non-contact measurements of any diffuse or diffuse-like medium from many different optical tomographic imaging system configurations. While this invention has been particularly shown and described with references to preferred embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the invention encompassed by the appended claims. It will be apparent to those of ordinary skill in the art that methods disclosed herein may be embodied in a computer program product that includes a computer usable medium. For example, such a computer usable medium can include a readable memory device, such as a hard drive device, a CD-ROM, a DVD-ROM, or a computer diskette, having computer readable program code segments stored thereon. The computer readable medium can also include a communications or transmission medium, such as a bus or a communications link, either optical, wired, or wireless, having program code segments carried thereon as digital or analog data signals. Referenced by
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