WO2012078819A2 - Appareil, procédé et support accessible par ordinateur pour la détermination non invasive de propriétés électriques de tissus et matériaux - Google Patents

Appareil, procédé et support accessible par ordinateur pour la détermination non invasive de propriétés électriques de tissus et matériaux Download PDF

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WO2012078819A2
WO2012078819A2 PCT/US2011/063844 US2011063844W WO2012078819A2 WO 2012078819 A2 WO2012078819 A2 WO 2012078819A2 US 2011063844 W US2011063844 W US 2011063844W WO 2012078819 A2 WO2012078819 A2 WO 2012078819A2
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field
target
electrical property
quantities
electromagnetic
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WO2012078819A3 (fr
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Daniel K. Sodickson
Yudong Zhu
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New York University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R33/00—Arrangements or instruments for measuring magnetic variables
    • G01R33/20—Arrangements or instruments for measuring magnetic variables involving magnetic resonance
    • G01R33/44—Arrangements or instruments for measuring magnetic variables involving magnetic resonance using nuclear magnetic resonance [NMR]
    • G01R33/443—Assessment of an electric or a magnetic field, e.g. spatial mapping, determination of a B0 drift or dosimetry
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B5/00—Measuring for diagnostic purposes; Identification of persons
    • A61B5/05—Detecting, measuring or recording for diagnosis by means of electric currents or magnetic fields; Measuring using microwaves or radio waves 
    • A61B5/053—Measuring electrical impedance or conductance of a portion of the body
    • A61B5/0536—Impedance imaging, e.g. by tomography
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B5/00—Measuring for diagnostic purposes; Identification of persons
    • A61B5/05—Detecting, measuring or recording for diagnosis by means of electric currents or magnetic fields; Measuring using microwaves or radio waves 
    • A61B5/055—Detecting, measuring or recording for diagnosis by means of electric currents or magnetic fields; Measuring using microwaves or radio waves  involving electronic [EMR] or nuclear [NMR] magnetic resonance, e.g. magnetic resonance imaging
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N24/00—Investigating or analyzing materials by the use of nuclear magnetic resonance, electron paramagnetic resonance or other spin effects
    • G01N24/08—Investigating or analyzing materials by the use of nuclear magnetic resonance, electron paramagnetic resonance or other spin effects by using nuclear magnetic resonance

Definitions

  • the present disclosure relates to mapping electrical properties of tissues or materials, and more specifically to apparatus, method, and computer-accessible medium for a noninvasive mapping of electrical properties of tissues or materials.
  • Certain exemplary embodiments include the mapping of electrical properties of materials using multiple radio frequency ("RF") measurements.
  • RF radio frequency
  • EIT represents the canonical surface-based technique using injected currents.
  • Alternative surface-based techniques which avoid direct application of currents include Microwave Tomography (MWT) and Magnetic Induction Tomography (MIT), as well as less well-known techniques such as noise tomography and Radiofrequency Impedance Mapping (RFIM). All such electrical prospection techniques utilize the solution of ill-posed inverse problems, which carry with them fundamental challenges of robustness, spatial resolution, etc.
  • EPT electrospray senor
  • results with EPT to date have been promising, with early in vivo studies in patient populations just emerging.
  • EPT suffers from a fundamental lack of access to absolute RF phase, as all measurable phases are expressed in relation to some unknown reference phase distribution.
  • This limitation has for many years been considered inescapable ⁇ a basic feature of the elementary processes by which we detect magnetic resonance signals.
  • EPT circumvents this limitation to some extent by using a carefully-chosen coil design (a birdcage) and associated symmetry assumptions dictating field behavior in the body.
  • exemplary architectures, apparatus, methods, and computer-accessible medium for determining at least one electrical property of at least one target.
  • the exemplary embodiments can include an exemplary method.
  • the exemplary method can include applying a plurality of stimulations to the at least one target.
  • the exemplary method can include receiving at least one signal from the at least one target in response to the applied stimulations.
  • the exemplary method can include processing the at least one signal to determine electromagnetic-field-related quantities associated with the stimulations and the target response,
  • the exemplary method can include supplying the eiectromagnetic-fi eld-related quantities to a system of equations relating these quantities to a plurality of electrical property values and residual field-related unknown values of the at least one target.
  • the exemplary method can include determining a solution to the system of equations, including determining at least one electrical property of the at least one target,
  • the at least one target can include at least one of a tissue or a material.
  • the at least one electrical property can include at least one of a conductivity, a permittivity, or a permeability of the at least one target.
  • at least one of the conductivity, permittivity, or the permeability can be at least one of a scalar or a tensor.
  • Certain exemplary embodiments can also include mapping the at least one electrical property of the at least one target.
  • the stimulations can include at least one of an injection of a current or generation of an electromagnetic field.
  • the signal can include information representative of at least one of a current or an electromagnetic field.
  • the stimulations can be created by a plurality of radiofrequency transmitter coils.
  • the signal can be detected in at least one radiofrequency receiver coil.
  • the signal can be a magnetic resonance signal.
  • the residual field-related unknown values can include at least one of an electromagnetic field phase or a magnetization value.
  • electromagnetic-field-related quantities can include a transmit and/or a receive sensitivity distribution.
  • complementary information from transmit and receive sensitivity distributions can be used to resolve ambiguities in electrical property values and residual field-related unknown values.
  • the exemplary embodiments can also include determining local expressions for Maxwell equations relating field curvature to electrical properties of interest, including the at least one electrical property of the at least one target.
  • the exemplary embodiments can also include determining composite expressions by expressing the true laboratory-frame magnetic fields as combinations of measurable quantities and residual unknowns, wherein the measurable quantities include at least one of: directly measurable quantities or quantities derived from the directly measurable quantities.
  • the exemplary embodiments can also include inserting these composite expressions into the local Maxwell equations, and separating terms associated with measurable quantities from those associated with the residual unknowns and local values of the at least one electrical property.
  • the exemplary embodiments can also include grouping the terms to form equations in which known coefficients represent local derivatives of the measurable quantities, and the unknowns include local derivatives of distributions of the residual unknowns , as well as local values of the at least one electrical property.
  • separating terms can include using a product law of differentiation.
  • determining a solution can include solving for electrical conductivity and permittivity separately in two steps.
  • determining a solution can comprise finding and applying one or more linear matrix inverses.
  • determining a solution can comprise applying a nonlinear optimization algorithm.
  • determining a solution can include use of a noise and/or error covariance matrix to control noise/error propagation.
  • the noise/error covariance matrix can include diagonal terms associated with amplitude of field-related quantities.
  • the plurality of stimulations and the at least one signal can be selected so as to maintain good conditioning of the system of equations.
  • the selection of stimulations and signal can be aimed at ensuring sufficient transmit and/or receive field variation in all directions for robust solution of the system of equations.
  • determining a solution can include use of Tikhonov regularization.
  • a transmit- receive array containing at least three elements can be used.
  • more than three coil pairs are used to improve conditioning.
  • the system of equations can be derived by forming local combinations of electromagnetic-field-related quantities which reduce the contributions of some residual unknowns as compared with using uncombined quantities.
  • determining a solution can include use of at least one Savitsky Golay derivative.
  • the system of equations can be derived by forming local combinations of electromagnetic-field-related quantities which improve the robustness of solution as compared with using uncombined quantities.
  • the combinations can be derived from a matched filter or rephased combination.
  • the combinations can be selected to generate slow local variation in the electromagnetic-field-related quantities.
  • the local combinations can be performed on transmit-field-related quantities.
  • the local combinations can be performed on receive-field-related quantities.
  • the local combinations can be selected to produce a tailored phase reference combination at each point of interest.
  • the exemplary embodiment can also include deriving a plurality of estimations of at least one unknown value for the at least one target based on the measured characteristics.
  • the exemplary embodiment can also include determining a consensus of the estimations.
  • the exemplary embodiment can also include determining the at least one property of the at least one target using the consensus.
  • the exemplary embodiment can also include constructing at least one of transmit sensitivity distributions, receive sensitivity distributions, or at least one combination thereof, each having at least one unknown value for the at least one tissue.
  • the exemplary embodiment can also include determining a consensus of the at least one transmit sensitivity distributions, receive sensitivity distributions, or combinations thereof.
  • the exemplary embodiment can include determining the at least one property of the at least one tissue based on the consensus.
  • Additional exemplary embodiments can include a non-transitory computer readable medium including instructions thereon that are accessible by a hardware processing arrangement, wherein, when the processing arrangement executes the instructions, the processing arrangement can be configured to apply a plurality of stimulations to at least one target, receive at least one signal from the at least one target in response to the applied stimulations, process the at least one signal to determine electromagnetic-field-related quantities associated with the stimulations and the target response, supply the electromagnetic-field-related quantities to a system of equations relating these quantities to a plurality of electrical property values and residual field-related unknown values of the at least one target and determine a solution to the system of equations, including determining at least one electrical property of the at least one target.
  • Another exemplary embodiment can include an apparatus for determining at least one property of at least one target.
  • the exemplary apparatus can include a plurality of transmitters which is configured to apply a plurality of stimulations to the at least one target, a plurality of receivers which is configured to receive at least one signal from the at least one target in response to the applied stimulations, and a non-transitory computer readable medium.
  • the exemplary computer readable medium can including instructions thereon that are accessible by a hardware processing arrangement, wherein, when the processing arrangement executes the instructions, the processing arrangement is configured to process the at least one signal to determine electromagnetic-field-related quantities associated with the stimulations and the target response, supply the electromagnetic-field-related quantities to a system of equations relating these quantities to a plurality of electrical property values and residual field-related unknown values of the at least one target, and determine a solution to the system of equations, including determining at least one electrical property of the at least one target.
  • Another exemplary embodiment can include a method for determining a magnetization distribution of at least one target.
  • the exemplary method can apply a plurality of stimulations to the at least one target, receive at least one signal from the at least one target in response to the applied stimulations, process the at least one signal to determine electromagnetic-field-related quantities associated with the stimulations and the target response, supply the electromagnetic-field-related quantities to a system of equations relating these quantities to a plurality of magnetization values and residual field-related unknown values of the at least one target, and determine a solution to the system of equations, including determining at least one magnetization distribution of the at least one target.
  • Another exemplary embodiment can include a method for determining a field-related phase distribution of at least one target.
  • the exemplary method can apply a plurality of stimulations to the at least one target, receive at least one signal from the at least one target in response to the applied stimulations, process the at least one signal to determine electromagnetic-field-related quantities associated with the stimulations and the target response, supply the electromagnetic-field-related quantities to a system of equations relating these quantities to a plurality of field-related phase values and residual field-related unknown values of the at least one target, and determine a solution to the system of equations, including determining at least one field-related phase distribution of the at least one target.
  • an assembly for determining at least one electrical property of an object can be provided.
  • the exemplary apparatus can include at least one transmitter configured to generate a plurality of electromagnetic field distribution patterns directed at an object; and a magnetic resonance imaging ("MRI") apparatus configured to produce at least one image of the object using at least one of a magnitude or a phase modulated by the electromagnetic field distribution patterns.
  • MRI magnetic resonance imaging
  • the assembly can be configured to process data associated with the object to determine the at least one electrical property of the object.
  • the transmitter of the exemplary assembly can include at least one radio frequency ("RF") coil.
  • the RF coil can include: (1) a plurality of RF coils disposed at a plurality of locations; or (2) an RF coil sequentially disposed at a plurality of locations.
  • the RF coil can include at least one RF transmit coil and at least one RF receive coil.
  • the transmitter can include at least one of: (1) a plurality of RF current- inducing leads disposed at a plurality of locations; or (2) an RF current-inducing lead sequentially disposed at a plurality of locations.
  • Transmitter of the exemplary assembly can also include a passive field-altering object sequentially placed at a plurality of locations.
  • a method of mapping at least one electrical property of an object can be provided.
  • the exemplary method can include generating a plurality of electromagnetic field patterns; generating a plurality of MR images with at least one of a magnitude or a phase associated with the generated electromagnetic field patterns; generating a plurality of relationships relating the MR images to the electromagnetic field patterns, and to the at least one electrical property the object; and resolving the relationships for the at least one electrical property.
  • a non-transitory computer readable medium including instructions thereon that are accessible by a hardware processing arrangement can be provided.
  • the processing arrangement executes the instructions, the processing arrangement is configured to generate a plurality of electromagnetic field patterns; generate a plurality of MR images with at least one of a magnitude or a phase associated with the generated electromagnetic field patterns; generate a plurality of relationships relating the MR images to the electromagnetic field patterns, and to the at least one electrical property the object; and resolve the relationships for the at least one electrical property.
  • Figure 1 shows exemplary numerical simulations of conductivity maps, according to an exemplary embodiment of the present disclosure
  • Figure 2 shows an exemplary simulation comparing conductivity maps, according to another exemplary embodiment of the present disclosure
  • Figure 3 shows an alternative exemplary reconstruction of electrical property distributions, according to another exemplary embodiment of the present disclosure
  • Figure 4 is a block diagram of an exemplary embodiment of a system according to the present disclosure.
  • Figure 5 is a flow diagram of an exemplary embodiment of a method according to the present disclosure.
  • Figure 6 is a flow diagram of an exemplary embodiment of a method according to the present disclosure.
  • Figure 7 is an illustration of an exemplary simulation and mapping obtained utilizing an exemplary embodiment of the present disclosure
  • Figure 8 is a flow diagram of an exemplary embodiment of an exemplary procedure according to the present disclosure.
  • Exemplary embodiments of the present disclosure indicate that at least some of the distortions observed in magnetic resonance images can be used to provide a robust solution to the problem of electrical property mapping. It has long been recognized that the distribution of intensities in MR images obtained at high magnetic field strength, and correspondingly high Larmor frequency, reflects the shaping of radiofrequency (RF) fields by tissue, in addition to the underlying distribution of magnetization.
  • the RF transmitter and detector coils used to generate and collect MR signals are known to be perturbed more and more strongly by body structure with increasing frequency, and various characteristic artifacts have been attributed to the interference of applied and induced RF fields in high-field MRI.
  • the exemplary embodiments of the present disclosure demonstrates that certain calibration procedures sometimes used to characterize these artifacts, together with certain experimental and mathematical procedures, can be utilized to obtain all the information required for noncontact mapping of electrical permittivity and conductivity.
  • An exemplary embodiment of the Local Maxwell Tomography (LMT) technique uses measurements of magnetic field curvature in arrays of RF transmitter and detector coils to deduce the underlying distribution of electrical properties via a well-posed local inverse problem free of symmetry assumptions and deductions regarding RF phase that have limited prior methods.
  • the exemplary LMT can solve simultaneously for key functions of the missing RF phase distribution along with unknown electrical properties, e.g., using complementary information from the transmit and receive sensitivity distributions of multiple coils to resolve ambiguities.
  • the exemplary LMT can facilitate an electrical property mapping at arbitrary field strength, with a wide range of coil designs, and free of errors associated with rapid field variation.
  • LMT is a powerfully general approach, from which a wide range of useful special cases may be derived.
  • the use of transmit and receive sensitivities together can be an important feature of the exemplary LMT, since it can be the conjugate relationship of transmit versus receive sensitivities to the missing RF reference phase which enables unique determination of that phase and hence of electrical property distributions.
  • Exemplary embodiments of the LMT procedure can include procedures involving data acquisition as well as image reconstruction.
  • Data acquisition can include recording of signals received in a plurality of detectors from a plurality of excitations in a plurality of transmitters.
  • Image reconstruction can include manipulation of measured data to generate quantities related to transmit and receive fields, followed by construction and then solution of systems of master equations relating the measured/generated field-related quantities to unknowns including the desired electrical properties.
  • the theory underlying construction of exemplary LMT master equations will be described further below. A variety of possible methods of the exemplary solution of such master equations are also described in this disclosure, along with a variety of choices of coil designs and combinations, numerical derivative algorithms, and experimental procedures to improve the robustness of resulting electrical property maps.
  • Exemplary LMT master equations can be constructed using the following general procedure:
  • Maxwell equations e.g., Helmholz equations for time-harmonic fields
  • the tomographic character of the exemplary LMT technique can be provided in the following exemplary characteristics: (1) the use of volumetric tomographic MR data to bypass the ill- posed inverse problem associated with electrical prospection using surface-derived measurements, and (2) the use of complementary data from multiple transmit and receive coils to fix many or all unknown quantities, including those associated with missing phase and magnetization distributions.
  • Exemplary solutions of the exemplary LMT master equations can use any of a number of linear or nonlinear methods known in the art to determine values of the unknown quantities. Exemplary solutions can be provided through the use of a sufficient number of appropriately designed transmitter coils balanced against a sufficient number of suitably designed detector coils
  • Eq. (3) may be insufficient to specify the precise functional form of electromagnetic fields in the absence of information on boundary conditions; however, if sufficient information is available about the structure of the fields, the electrical properties may be deduced even in the absence of boundary condition information.
  • the electrical properties may be deduced even in the absence of boundary condition information.
  • ⁇ ( ⁇ - ⁇ ) ⁇ ⁇ ⁇ ⁇ 0 for piecewise constant electrical properties and/or for property gradients suitably aligned with respect to the local electric field, ( ⁇ ( ⁇ - ⁇ ) ⁇ ⁇ ⁇ ⁇ 0 , and a Helmholtz relation holds for any component B a of the vector magnetic field:
  • the exemplary transmit and receive sensitivities of RF coils in MRI can be known to reflect the distribution of transverse magnetic field components B x ⁇ iB y . If the true amplitude and phase of these field components were known from MRI experiments, it would be possible to compute electrical properties directly from Eq. (4) alone. However, MRI techniques may not provide direct access to pure laboratory-frame field values. Instead, it can measure indirect quantities, depending upon the choice of pulse sequence, transmit and receive coil configuration, and image post-processing algorithm.
  • the absolute phase of transverse RF magnetic field components has until now been considered fundamentally inaccessible to experiment.
  • the phase of the local RF excitation field deviates from the phase of the driving field as a result of varying propagation delays, as well as other perturbations due to the body's particular distribution of electrical properties.
  • the phase of the field produced by a precessing spin suffers various perturbations on its way back to a detector.
  • any measured phase reflects a combination of sources. In other words, there may be no absolute "clock" for MR signal phase. Any measured distribution is referenced to another distribution that cannot be measure. In most applications of MRI, knowledge of relative phase suffices. Nevertheless, accurate solution of Eq. (4) can indicate that phase be specified absolutely, at least up to an overall constant, or else error terms will result from the derivatives on the left hand side operating on the unknown reference phase distribution
  • Exemplary embodiments of the present disclosure can demonstrate that a solution to the problem of unknown reference phase can be derived by exploiting the complementarity of transmit and receive sensitivities.
  • transmit and receive fields can have a conjugate relationship to any particular choice of reference phase ⁇ p 0 (r) , and, in the setting of a sufficient number of coils, this relationship can resolve the phase ambiguity which was previously considered intractable.
  • the function M incorporates all sources of signal variation that are common to all coils and that are not associated with electrodynamics. This can include the spatial distribution of equilibrium magnetization, as well as all non-electrodynamic phase variation ⁇ ⁇ , e.g., due to B 0 inhomogeneity, gradient eddy currents, incomplete RF spoiling, etc. Eq.
  • ⁇ B x l ⁇ iB y uses the field definitions ⁇ B x l ⁇ iB y , relating transmit or receive sensitivity patterns to the complex laboratory-frame transverse magnetic field components B x / and B , generated by a unit current in coil /.
  • the phases ⁇ (+) and ⁇ ( ) define the flip axis of the transmit coil / and the reference phase of the receive coil /' , respectively.
  • the function B ⁇ ) depends upon details of the pulse sequence used for data acquisition. For a simple exemplary gradient-echo sequence with long TR, f ⁇ B ⁇ j « - ⁇ sin [ ⁇ ⁇ ⁇ j / 2 , where / is the gyromagnetic ratio, / is the current applied to the transmit coil, and ⁇ is the pulse duration. In the limit of small flip angle, Eq. (6) then simplifies to,
  • exemplary embodiments can construct various relationships among field quantities of interest. For example, knowing and the
  • LMT master equations can be derived by expressing true field components as products of measurables with unknown quantities, e.g.:
  • Such quantities can be defined with respect to the unknown phase distribution ⁇ 0 of a common reference receive coil / 0 , but any coil combination can also serve as a legitimate phase reference.
  • the exemplary embodiment of the system of matrix equations according to the present disclosure can be composed of concatenated blocks of the form of Eq. (17), with one block resembling the top two rows for each transmit coil /, and one block resembling the bottom two rows for each receive coil /' . Equations involving transmit coils alone or receive coils alone do not have unique solutions, whereas a suitable balance of transmit and receive coils can have a unique solution.
  • the relationships among the elements of the unknown vector x in Eq. (17)deviate from linearity namely:
  • Eq. (17) can thus represent a "nearly" linear system of 2 (Z + L') real equations with 10 real unknowns.
  • the unknowns can include the missing permittivity and conductivity at any chosen local position, as well as the local values of first and second derivatives of the missing reference phase and common magnetization distribution. If the local derivatives in the matrix A can be computed reliably from measured quantities, then the exemplary LMT master questions can also be local, and may be solved voxel by voxel in parallel.
  • Eq. (21) is a purely linear equation, and can be solved in a least-squares sense, for example using a Moore-Penrose inverse of the matrix A (cr) ;
  • ⁇ ( ⁇ ) is an optional noise/eiTor covariance matrix which will be further discussed below, and can provide the weighting and noise decorrelation required for an optimal solution in the presence of noise and error.
  • Standard matrix conditioning strategies can also be used in this inversion.
  • One simple conditioning strategy is to scale the unit elements of ⁇ ( ⁇ ) to match the average value of the other elements, and to scale the corresponding unknown elements of x a) to preserve the product. In practice, this balancing strategy can significantly improve performance. More advanced regularization strategies, such as Tikhonov regularization, can also be employed.
  • Both transmit and receive elements can be used to render ⁇ ( ⁇ ) as defined in Eq. (21) nonsingular.
  • the opposite sign of the coefficient of V ⁇ p Q for transmit as opposed to receive coils can resolve this ambiguity, and uniquely fixes all unknowns. In other words, it is the opposite sign of the reference phase between transmit and receive that can ultimately fix the missing phase which has long plagued electrical property mapping from MR field data. Any deviation from truth in a candidate value of V ⁇ 0 in a transmit equation can result in an oppositely-directed deviation in any of the receive equations, and consistency among equations is spoiled.
  • This exemplary two-stage linear reconstruction strategy can be extremely fast. It also serves to demonstrate requirements for a robust solution of the electrical property mapping problem. As one example, a total of at least five field maps, taken from at least one transmit and at least one receive coil, is required to fix the values of the five unknowns in Eq. (21). The same five coil dataset suffices to solve Eq. (24). Receive coils can also share the same structure as transmit coils, if the array is operated in transmit-receive mode. Thus, a 3- element transmit-receive array is sufficient to determine both ⁇ and ⁇ .
  • the conditioning of the problem generally improves as the number of independent coils increases, and also as the number of transmit coils is balanced against the number of receive coils. Otherwise, the condition number of the matrix ⁇ ( ⁇ ) depends upon the balance of the relevant gradients of measured field amplitudes and phases, which in turn depends upon the design of the transmit and receive coil array/s.
  • Eq. (26) resembles Eq. (21), with the addition of three new unknowns and corresponding matrix coefficients for receive equations on the left-hand side, and with a single nonlinear receive term on the right hand side.
  • This can be solved using nonlinear optimization to minimize a quantity such as ( ⁇ , ⁇ ) - ( ⁇ , ⁇ ) ⁇ . ( ⁇ , ⁇ )
  • the right hand side vector b is composed of second derivatives or products of first derivatives, whereas the elements of A involve only first derivatives. Since higher-order derivatives are generally significantly more error-prone than lower-order derivatives (a fact verified experimentally for MR-derived field maps), and since the condition number of the encoding matrix A is generally smaller than the relative scaling between first and second derivatives, approximate expressions for noise and error propagation may be derived by neglecting errors in A:
  • x is the reconstructed estimate of the unknown vector x
  • quantities labeled with a subscript 0 represent the true values of those quantities
  • quantities preceded by a ⁇ represent errors (random and/or systematic) in those quantities
  • a " ' represents the computed reconstruction matrix which inverts the effects of the encoding matrix A.
  • a signal-to- noise ratio (SNR) optimizing reconstruction may take the form of Eq. (23), e.g.,
  • An exemplary noise covariance matrix for two-stage reconstruction according to Eqs. (21)-(24) can be formulated using partial derivatives with respect to each measured quantity according to standard error-propagation approaches. Since there are several nonlinear steps involved both in the mapping of and
  • a choice of ⁇ can be made based on the observation that, in regions of low transmit or receive sensitivity, the determination of field amplitudes or
  • Another way of improving the robustness of LMT reconstruction can be to choose RF coils or coil combinations such that the condition number of the encoding matrix A is kept as low as possible so as to yield the highest possible SNR via Eq. (29).
  • the exemplary encoding matrix in the exemplary LMT can consist of various directional derivatives, it can be important to ensure that there is suitable field variation along all directions in regions of interest.
  • the use of an encircling loop or strip elements with a large extent along the Bo field direction z can lead to singularity of the encoding matrix for centrally- located voxels, since the z gradients of field amplitudes and phases then tend to vanish for all coil elements.
  • the condition number of the exemplary encoding matrix increases beyond a certain limit, errors in A may no longer be neglected as in Eq. (28), and alternative approaches may be taken to estimation of error propagation and to optimized reconstruction.
  • EPT as a special case of LMT: EPT is based on the fundamental assumption that, for a birdcage coil operated with a traditional quadrature hybrid, the reversal of effective circular polarization between transmission and reception results in very similar transmit and receive field phase distr butions: ⁇ . » . To the extent that this assumption holds true, the phase of the MR signal in Eq. (6) becomes 2 ⁇ (+) + ⁇ ⁇ , and correction for or minimization of any background phase ⁇ ⁇ can result in full knowledge of the transmit field phase ⁇ ( ) , perhaps up to an overall constant phase. Thus, it is possible to set ⁇ 0 « constant , yielding V ⁇ 3 ⁇ 4 « 0 and ' v ' o ⁇ 0 ⁇ In this case, Eq. (15) simplifies to
  • EPT represents just one special case of the more general LMT formalism.
  • the central EPT approximation ⁇ (+) « ⁇ is reliable at low field strength or in selected situations of suitable symmetry in
  • the exemplary LMT can eliminate the need for such symmetries and opens up a broader range of allowed coil designs and field strengths, at the cost of increased requirements for data acquisition and reconstruction.
  • the exemplary local shimming approach can be viewed as a kind of physical back-substitution procedure which, though in principle may be no better conditioned than general LMT, may have certain advantages of stability and simplicity.
  • phase sum ⁇ , t) + #> ⁇ .. ) is just the phase of the MR signal formed by combining signals from different transmitters according to the local transmit shim, and signals from different receivers according to the local receive shim.
  • the net result can be a single composite signal which, when background phase ⁇ ⁇ is removed, behaves like an ideal
  • EPT birdcage signal in the sense that the curvature of transmit and receive phases are each directly proportional to the conductivity (e.g., even if these phases themselves are not necessarily equal), and there therefore may be no need to separate them. It may be that only when rapidly varying magnetization interferes with the receive field shim will the locally- adjusted EPT condition fail to hold rigorously.
  • Choice of exemplary derivative algorithm Any suitable numerical derivative procedure can be selected to compute the matrix and vector elements in the LMT master equations.
  • the formalism does not assume any particular choice of quadrature for derivative estimation, thus embodiments of the exemplary LMT can be free to use whichever algorithm works most robustly for a case or even a voxel of interest.
  • Exemplary embodiments can use Savitsky-Golay (SG) derivatives, in which the function to be differentiated can be fitted to a low-order polynomial over a small kernel region around each voxel, and analytic derivatives of the best-fit polynomial can be computed.
  • SG Savitsky-Golay
  • Exemplary phase unwrapping Field phases ⁇ p(x,y, z) are typically more slowly varying than complex exponentials exp(/ ?(x,_y, z)) , and can therefore be better suited to SG derivative estimation.
  • exemplary phase wrapping can cause artificially high derivative values in the vicinity of 2 ⁇ discontinuities.
  • Exemplary embodiments according to the present disclosure can use a simple unwrapping procedure taking advantage of the fact that only phase derivatives and not the phase values themselves are of interest.
  • the source complex exponentials defining measured phases can be incremented by a fixed phase in multiple steps ranging from 0 to 2 ⁇ , thereby shifting the location of phase discontinuities, and the phase functions and their derivatives can be recomputed for each increment.
  • the median derivative value can then be selected as the true derivative, since only a small minority of increments may result in a phase discontinuity within the small computation kernel and in a corresponding anomalous derivative value.
  • Exemplary local coil recombination The concept of region-by-region shimming introduced in the description of a simplified LMT solution above can also be valuable in improving the accuracy of the exemplary LMT matrix and its inverse. As was mentioned earlier, SG derivative estimates may be particularly error-prone in regions of low transmit or receive sensitivity, and these errors may propagate unfavorably into the exemplary LMT reconstruction. In an attempt to maximize signal and minimize dynamic range, exemplary embodiments can also use all-but-one transmit coil combinations. These combinations, though useful over much of the field of view, typically result in destructive interference in certain regions, often yielding even more rapid field variation and more pathological derivative errors than single-coil transmission.
  • exemplary embodiments can use a local matched filter field recombination approach in which known phase relations between transmit and receive fields at the center of each SG kernel can be used to ensure constructive interference and to create synthetic coil sets with comparatively slow field variation over the kernel.
  • exemplary embodiments can use the amplitudes and relative phases ⁇ ⁇ to form the following set of composite field combinations :
  • Such exemplary approach entails forming a matched filter combination by rephasing each coil by its relative phase at the kernel center (x Q , y 0 , z 0 ) , multiplying by its sensitivity at that point, and summing, then subtracting each rephased coil in turn from the matched filter baseline combination to yield a set of complex fields whose amplitude and phase can then be used for LMT.
  • the availability of full field amplitude and relative phase information at every spatial location can facilitate an exemplary local post-acquisition retuning of the all-but-one approach which can avoid destructive interference and corresponding rapid field variation and/or signal nulls which would otherwise be unavoidable with spatially invariant coil combinations.
  • the exemplary reference coil combination used to define the missing phase ⁇ 0 also need not remain constant from region to region, as long as it does not change within each region.
  • a local matched filter combination can also be used to minimize destructive interference which might contribute to phase uncertainty: ⁇ ,
  • Exemplary consensus solutions using particular choices of exemplary derivative discretization can involve choosing a particular discretization for local derivatives in advance and rewriting the equations in discrete form. For example, assuming a regular grid of voxels defined by coordinates ( ,, ⁇ , ⁇ ), the terms of the Laplacian operator can be approximated with simple finite differences on that grid:
  • Eq. (43) can be rewritten separating known and unknown quantities: b ⁇ P ⁇ + 'Pi bWpW +b IMA pW + b ⁇ WpW +b 'P ⁇ i k ) * c (44) where, for the case of Eq. (11), B,
  • the balanced cost function in Eq. (50) can simultaneously penalize variances among transmit coil property estimates (term 2), variances among receive coil estimates (term 3), and differences between transmit and receive estimates (term 1).
  • the exemplary relative weight of each term may be controlled by appropriate weighting parameters , , L, .
  • Figure 1 shows numerical simulations of conductivity maps comparing EPT 100 (e.g., simulated birdcage coil) with LMT 101 (e.g., birdcage rungs used as 16 individual elements) in a simplified cylindrical body model with heart, lung, spinal cord, kidney, and muscle compartments assigned literature values of electrical properties at 300 MHz (corresponding to the proton Larmor frequency for 7 Tesla field strength).
  • Elements 105 and 107 show artifacts due to derivative errors 105 and EPT phase assumption errors 107, which are removed by LMT.
  • Figure 2 shows results of experiments at 7 Tesla field strength comparing conductivity maps for EPT 200 (e.g., birdcage) with LMT 201 (e.g., encircling loop array with 8 transmit and 16 receive elements) in a single-compartrnent cylindrical phantom. Ring artifacts 207 in EPT due to a null in birdcage sensitivity and corresponding derivative errors are removed by LMT, which yields correct electrical property values (validated by dielectric probe) as well as a correct spatial distribution.
  • EPT 200 e.g., birdcage
  • LMT 201 e.g., encircling loop array with 8 transmit and 16 receive elements
  • FIG. 1 The simulations shown in Figure 1 use Finite Difference Time Domain software executed by a computer processor to compute magnetic fields resulting from a time-harmonic stimulus in the selected coils. Measurable quantities were formed from computed fields, and these quantities were used for electrical property map reconstruction without reference to true fields or electrical properties.
  • Reconstruction times for custom-designed Matlab code were approximately 0.1 sec per voxel, for our particular choice of numerical derivative algorithm. Substantial increases in reconstruction speed may be expected both from use of compiled code and from trivial voxel-wise parallelization, e.g., using the parallel Matlab toolbox or other multicore/GPU implementations.
  • Nonlinear reconstruction including variable local magnetization density was found to increase reconstruction times by a factor of 2-3. The choice of whether to include magnetization gradients as unknowns can be dictated by prior knowledge about the imaged object, or by a desire for full generality.
  • Figure 3 illustrates an alternative reconstruction of electrical property distributions (including both permittivity and conductivity) obtained using an exemplary pre-discretized consensus reconstruction procedure on simulated data with noise added (average SNR of 200 for MR signal, 20 for B !+ maps) in the same simple body model used for Figure 1.
  • Figure 4 shows an exemplary block diagram of an exemplary embodiment of a system according to the present disclosure. For example, exemplary procedures in accordance with the present disclosure described herein can be performed by transmitters 122 (TX 1 through TX N), receivers 124 (RX 1 through RX N), and a processing arrangement and/or a computing arrangement 102.
  • Such processing/computing arrangement 102 can be, e.g., entirely or a part of, or include, but not limited to, a computer/processor 104 that can include, e.g., one or more microprocessors, and use instructions stored on a computer-accessible medium (e.g., RAM, ROM, hard drive, or other storage device).
  • a computer/processor 104 can include, e.g., one or more microprocessors, and use instructions stored on a computer-accessible medium (e.g., RAM, ROM, hard drive, or other storage device).
  • a computer-accessible medium 106 e.g., as described herein above, a storage device such as a hard disk, floppy disk, memory stick, CD-ROM, RAM, ROM, etc., or a collection thereof
  • the computer-accessible medium 106 can contain executable instructions 108 thereon.
  • a storage arrangement 1 10 can be provided separately from the computer-accessible medium 106, which can provide the instructions to the processing arrangement 102 so as to configure the processing arrangement to execute certain exemplary procedures, processes and methods, as described herein above, for example.
  • the exemplary processing arrangement 102 can be provided with or include an input/output arrangement 1 14, which can include, e.g., a wired network, a wireless network, the internet, an intranet, a data collection probe, a sensor, etc.
  • the exemplary processing arrangement 102 can be in communication with an exemplary display arrangement 1 12, which, according to certain exemplary embodiments of the present disclosure, can be a touch-screen configured for inputting information to the processing arrangement in addition to outputting information from the processing arrangement, for example.
  • the exemplary display 1 12 and/or a storage arrangement 110 can be used to display and/or store data in a user-accessible format and/or user-readable format.
  • the exemplary method can implement a magnetic resonance imaging scanner equipped with at least one transmit channel and at least one receive channel to measure several variables.
  • the exemplary method can measure a magnetic resonance signal resulting from excitation of the imaged body via each of a plurality of transmit coils and reception in each of a plurality of detector coils.
  • Each transmit and receive coil may be connected to a distinct transmit or receive channel in the MR scanner, or else distinct coils may be interfaced sequentially to a smaller number of channels, or else a smaller number of coils may be moved sequentially into distinct positions to accomplish equivalent measurements.
  • the exemplary method can determine a transmit field amplitude map in the imaged body associated with each of the plurality of transmit coils. Additionally, the exemplary method can determine a phase map of the magnetization distribution in the imaged body.
  • the exemplary method can generate, from such
  • the exemplary method can provide these quantities, and/or any appropriate derivatives thereof, to suitably derived LMT master equations relating measured field-related quantities to functions of unknown phases, magnetizations, and electrical conductivity and permittivity values.
  • the exemplary method can solve a system of exemplary equations using any of the techniques outlined in this disclosure or other appropriate techniques to determine values of electrical properties and other unknowns.
  • the exemplary embodiment can generate maps or images of the resulting electrical properties throughout the imaged body.
  • Figure 6 illustrates an exemplary embodiment for deriving the system of equations, e.g., as discussed in Figure 5, element 530.
  • the exemplary procedure can determine local expressions for Maxwell equations relating field curvature to electrical properties of interest.
  • the exemplary procedure can determine composite expressions by expressing the true laboratory-frame magnetic fields as combinations of measurable quantities and residual unknowns.
  • the measurable quantities can include those quantities that can be directly measured, as well as quantities derived from the directly measurable quantities or a combination thereof.
  • the exemplary procedure can insert these composite expressions into the local Maxwell equations.
  • the exemplary procedure can use the product law of differentiation to separate terms associated with measurable quantities from those associated with the residual unknowns.
  • the exemplary procedure can group the terms to form equations in which known coefficients represent local derivatives of the measurable quantities, and the unknowns include local derivatives of distributions of the residual unknowns , as well as local values of the at least one electrical property.
  • RF radio frequency
  • ⁇ ) B (X) V B 1 (x) + higher order terms (52)
  • x can be the spatial coordinate vector
  • * can denote complex conjugate
  • B and B 1 can be phasor representations of, respectively, the time-harmonic radio-frequency transmit and receive fields
  • ⁇ ( ⁇ ) can represent ( ⁇ ( ⁇ )+ a composite quantity that can be composed of angular frequency ( ⁇ ) as well as such electrical properties as conductivity ( ⁇ ), permittivity ( ⁇ ) and permeability ( ⁇ ).
  • the higher order terms can involve spatial derivatives of electrical properties, and tend to vanish when the properties vary slowly in space. Typically, the higher the frequency at which an MR experiment is conducted, the stronger the impact the electrical property distributions tend to exert on the spatial variations of RF fields.
  • Eqn. 51 and 52 typically does not rely on assumptions beyond Maxwell equations. Further, when the higher order terms are negligible, the two equations can involve RF transmit and receive field quantities. Because of relatively robust MR-based techniques that map RF transmit field using spin flip angles, non-invasive mapping of RF transmit field quantities can be more manageable than that of other electromagnetic field quantities. Eqn. 51 or Eqn. 52 can point to a more accurate / practical method for electrical property mapping. The Laplace's differential operator can be local and the higher order terms can be negligible in regions of slowly varying electrical property. This can give rise to local calculation of electrical properties:
  • c(x) complex conjugate( V 2 B 1 ' (x) / B 1 (x) ) (54)
  • a discretized approximation of the Laplacian can be used as follows and can give electrical properties:
  • the subscripts can be voxel indices
  • the b's can denote phasor representations of true Bi + values
  • the bracketed expression can be an example finite difference
  • the true phase distribution of Bi + can be evasive and can be an obstacle for applying Eqn. 55-based property mapping in practice.
  • a Bi + map acquired by an existing MR ⁇ based Bj + mapping schemes can have its phase corrupted by an unknown phase distribution.
  • the acquired B i + map can be an unknown phase offset away from the true ⁇ :
  • B;(x) B
  • Exemplary embodiments of the present disclosure can include a technique and/or a procedure that can employ multiple (albeit corrupted) measurements of RF transmit / receive fields to constrain and resolve the electrical property distributions.
  • parallel RF transmitters / receivers can be used to obtain the measurements.
  • each of a plurality of true Bi + maps preferably satisfies Eqn. 51.
  • a common unknown phase distribution due to, e.g., the receive coil sensitivity's phase distribution and/or B 0 inhomogeneity
  • Eqn. 59 indicates that the use of a parallel transmit system and the
  • Eqn 60 can represent a set of N equations in 7 unknowns ( ⁇ and the z's), and each of the z's can be additionally constrained to be of unit modulus.
  • This formulation can resemble a hyperplane fitting problem, and a singular value decomposition can offer a solution for
  • An exemplary result obtained by employing an embodiment of the present disclosure in an FDTD simulation can be shown, for example, in Figure 7. More involved calculations leveraging the unit modulus constraints can be applied. This, for example, can take the form of a constrained least squares.
  • ⁇ ( ⁇ ) can represent ( ⁇ ( ⁇ )- ( - -1 ⁇ ( ⁇ )) V-1 ⁇ ( ⁇ )
  • can capture conductivity and permittivity variations within the scanned object. This can provide noninvasive detection / characterization of pathology of the scanned object.
  • the concept of pooling equations that can constrain and resolve electrical property distributions can be integrated with other data acquisition schemes.
  • an expanded hardware setup where M parallel receive coils as well as N parallel transmit coils are available for use in MR scans can be used.
  • a single transmit coil (which can be one of the N transmit coils or one that is synthesized by combining several of the N transmit coils) and the M receive coils can be utilized to additionally acquire M number of MR images (e.g., one from each of the receive coil) that can differ in individual receive coil sensitivity profiles (B ⁇ (m) ):
  • S (m) can be a common complex- valued scaling factor away from the true B ⁇ (m) ,
  • Eqn. 63 can represent a set of M equations in 7 unknowns (e.g., ⁇ and the y's).
  • the constraints represented by Eqn. 63 can augment those represented by Eqn. 60, allowing further determination of both the real and imaginary components of ⁇ ⁇ > ⁇ and, subsequently, the conductivity and permittivity maps (see Eqns. 66 and 67).
  • Eqn. 60 is capable of determining
  • Eqn. 60 by itself typically cannot resolve the phase of ⁇ ⁇ ⁇ +6/ ⁇ , This limitation can be addressed with the incorporation of Eqn. 63.
  • Exemplary setups can include a plurality of RF current-inducing leads which can be positioned at a set of locations, an RF coil or RF current-inducing lead which can be sequentially placed at set of locations, and a passive, field-altering object which can be sequentially placed at a set of locations.
  • Figure 7 shows an illustration of an exemplary simulation utilizing data that simulated MR-based RF transmit field measurements and utilized an exemplary embodiment of the present disclosure to create an electrical property map.
  • the exemplary object 700 contained four materials of different electrical properties.
  • the exemplary hardware included 32 parallel RF transmit / receive coils operating at 298MHz (7T MRI).
  • the exemplary ⁇ ⁇ +6/ ⁇ 2 ⁇ map 701 over an exemplary slice showing the four materials and an electrical property-based contrast.
  • Figure 8 illustrates an exemplary procedure according to an exemplary embodiment of the present disclosure.
  • the exemplary procedure of Figure 8 can generate a plurality of electromagnetic field patters at 810, e.g., using at least one transmit coil. These can be from a plurality of generators (e.g., transmitting coil) or one generator that is repositioned (e.g., over time).
  • the exemplary procedure can generate a plurality of MR images, which can include at least a magnitude or a phase associated with the generated electromagnetic field patterns.
  • the exemplary procedure can generate a plurality of relationships relating the M images to the electromagnetic field patterns and to at least one electrical property of a target object.
  • the exemplary procedure can resolve the relationship for the electrical property.

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Abstract

L'invention concerne un appareil, un procédé et un support accessible par ordinateur pour une cartographie non invasive de propriétés électriques de tissus ou matériaux. Par exemple, il est possible d'appliquer une pluralité de stimulations à une cible. Il est possible de recevoir au moins un signal de la cible en réponse aux stimulations appliquées. En outre, il est possible de traiter le au moins un signal pour déterminer des quantités relatives à un champ électromagnétique associé aux stimulations et à la réponse cible. De même, il est possible de fournir les quantités relatives à un champ électromagnétique à un système d'équations reliant ces quantités à une pluralité de valeurs de propriétés électriques et des valeurs inconnues relatives à un champ résiduel de la ou des cibles. Il est également possible de déterminer une solution au système d'équations, y compris la détermination d'au moins une propriété électrique de la ou des cibles.
PCT/US2011/063844 2010-12-07 2011-12-07 Appareil, procédé et support accessible par ordinateur pour la détermination non invasive de propriétés électriques de tissus et matériaux WO2012078819A2 (fr)

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US20060125475A1 (en) * 2002-09-17 2006-06-15 Sodickson Daniel K Radio frequency impedance mapping
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US7795870B2 (en) * 2006-02-21 2010-09-14 Beth Israel Deaconess Medical Center, Inc. Magnetic resonance imaging and radio frequency impedance mapping methods and apparatus

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US20060125475A1 (en) * 2002-09-17 2006-06-15 Sodickson Daniel K Radio frequency impedance mapping
US7795870B2 (en) * 2006-02-21 2010-09-14 Beth Israel Deaconess Medical Center, Inc. Magnetic resonance imaging and radio frequency impedance mapping methods and apparatus
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