System and method for the tomography of the primary electric current of the brain and of the heart

US 20030093004A1
Filed: 07/26/2002
Published: 05/15/2003
Est. Priority Date: 02/18/2000
Status: Active Grant

First Claim

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1. Method for the Tomography of the Primary Electric Current (TPEC) of the brain (TPECc) and of the heart (TPECk) characterized as comprising the following operations:

Pre-processing of the signals recorded from the subject for the elimination of artifacts and of non-physiological frequency components of, Conversion of the coordinates of the sensors to the reference system of the head or torso by means of the application of a visco-elastic deformation, in the most general case nonlinear, that puts into correspondence the coordinates of the external anatomical markers of the subject with those of the structural image stored in the CU;

Calculation of the electric Kernel and magnetic Kernel linear operators that predict the o(t) that would be produced in the subject by the presence of the PEC j(t) in any part of the brain or heart;

Calculation of the TPEC by means of Bayesian Hierarchical Estimation Procedure that determines the primary electric current j(t) of the brain and heart using the structural a priori information obtained from an anatomical atlas, the metabolic a priori information obtained from functional images and the assumption that the solution belongs to a space of Besov space of given smoothness or determined by a norm in a Megadictionary;

Calculation of descriptive parameters either on the basis of the observations o(t) directly, or of the j(t);

Calculation of the probability that all or some of the DP are similar to those of a given test group; and

The obtained TPECc is coded by means of a pseudocolor scale and is overlaid on the anatomical atlas, the scale is fitted according to the probability previously calculated in such a way that only statistically significant anatomical sites are highlighted.

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Abstract

A three-dimensional map of the probability of brain or heart functional states is obtained based on electric and or magnetic signals measured in the surface of body. From said signals, the statistical descriptive parameters are obtained and a map of its distribution is calculated. The map is the inverse solution of the problem based upon: a) the restriction of the solution to structures with high probability of generating electric activity using for this restriction an Anatomical Atlas and b) imposing that the solution belong to a pre-specified functional space. The probability is determined that this map belongs to a test group. The spatial and temporal correlations of the map are modeled as well as their dependence on experimental covariables. The resulting probabilities are coded in a pseudocolor scale and they are superimposed on a Anatomical Atlas for their interactive three-dimensional visualization.

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20 Claims

1. Method for the Tomography of the Primary Electric Current (TPEC) of the brain (TPECc) and of the heart (TPECk) characterized as comprising the following operations:
- Pre-processing of the signals recorded from the subject for the elimination of artifacts and of non-physiological frequency components of, Conversion of the coordinates of the sensors to the reference system of the head or torso by means of the application of a visco-elastic deformation, in the most general case nonlinear, that puts into correspondence the coordinates of the external anatomical markers of the subject with those of the structural image stored in the CU;
  
  Calculation of the electric Kernel and magnetic Kernel linear operators that predict the o(t) that would be produced in the subject by the presence of the PEC j(t) in any part of the brain or heart;
  
  Calculation of the TPEC by means of Bayesian Hierarchical Estimation Procedure that determines the primary electric current j(t) of the brain and heart using the structural a priori information obtained from an anatomical atlas, the metabolic a priori information obtained from functional images and the assumption that the solution belongs to a space of Besov space of given smoothness or determined by a norm in a Megadictionary;
  
  Calculation of descriptive parameters either on the basis of the observations o(t) directly, or of the j(t);
  
  Calculation of the probability that all or some of the DP are similar to those of a given test group; and
  
  The obtained TPECc is coded by means of a pseudocolor scale and is overlaid on the anatomical atlas, the scale is fitted according to the probability previously calculated in such a way that only statistically significant anatomical sites are highlighted.
- View Dependent Claims (2, 3, 4, 5, 20)
- - 2. The method of claim 1, wherein vector of descriptive parameters DP is calculated that are sufficient statistics of models of stochastic processes for o(t) or j(t). Particulars, but not exhaustive instances, of DP are:
    - Subsets of o(t) or j(t), selected in a suitable manner;
      
      Subsets of the o(t) or J(t) or μ
      
      (t), or γ
      
      (t) selected in a suitable manner, Subsets of averages, variances and covariances or higher order o(t) or J(t) or μ
      
      (t), or γ
      
      (t) or of their representation in the basis of a Megadictionary;
      
      Subsets of cross polyspectra of o(t) or J(t) or μ
      
      (t), or γ
      
      (t) of their representation in the basis of a Megadictionary;
      
      Subsets of parametric multivariate Autoregressive functions (linear or of any order) fitted to o(t) or J(t) or μ
      
      (t), or γ
      
      (t) or their representation in the basis of a Megadictionary;
      
      Subsets of non parametric multivariate Autoregressive functions fitted to o(t) or J(t) or μ
      
      (t), or γ
      
      (t) of their representation in the basis of a Megadictionary;
      
      Correlation measures, mutual information, Granger causality among components of o(t) or J(t) or μ
      
      (t), or γ
      
      (t) of their representation in the basis of a Megadictionary; and
      
      Regression coefficients of any one of the previously described types of DP with the stimuli emitted by the system or with the subject'"'"'s responses.
  - 3. The method of claim 1, wherein the probability is calculated that all or part of the TPEC belongs to a test group by means of a theoretical correction or by means of statistical resampling methods that compensate for the effect of the multiple spatial and temporal correlations, that are due both to the large number of measurements, as well as a consequence of either the estimation process or the physiology of the brain or heart, the effect of said correlations being to increase the type error I in the identification of brain or heart states, eliminating additionally the effects of the dependence of the estimated maps with experimental covariables of interest.
  - 4. The method of claim 1, wherein test groups are constituted to evaluate intra or interindividual changes of brain or heart function, examples of test groups being:
    - Groups of subjects defined as normal by criteria external to their TPEC, which will be specified implicitly by means of a sample of DP of TPEC of normal subjects contained in a database the objective of statistical analysis being to determine the probability that the DP of the obtained TPEC is obtained from a healthy subject. Groups of subjects defined as having pathologies predefined by criteria external to the TPEC, which will be specified implicitly by means of samples of DP of TPEC of patient with the pathologies of interest defined in a database, the objective of statistical analysis being to determine the probability that the TPEC is obtained from a subject with the specified illnesses. Groups determined by methods of Unsupervised Pattern Recognition on the basis of the DP of the TPEC of a sample having in common physiological characteristics of interest, the objective of statistical analysis is to determine the probability that the TPEC is obtained from a subject with said physiological characteristics. Groups consisting of a sample of DP of TPEC obtained in previous moments from the subject under study, where all of the DPs for each analysis moment constitutes a vector time series and the objective of the statistical analysis of this vector time series has two purpose;
      
      to determine changes with time in the functioning of predetermined parts of the brain or heart as reflected in changes of the DP, as well as to carry out predictions about the probability that a vector of DP will attain certain values (reflecting the functional state of the brain or heart) in a moment posterior to the current examination.
  - 5. The method of claim 1, wherein the following statistical operations are carried out:
    - Calculation of the mean μ
      
      _x(v), and variance, σ
      
      _x(v), of each Test Group for each one of the parameters and each point of the lattice of the generating volume, said means and variances are be expressed by means of the non parametric regression as a function of control variables;
      
      Transformation towards gaussianity x=T(DP), using a non-linear function to obtain the vector x;
      
      Calculation of the z transform z(x_i)=(x_I−
      
      μ
      
      _x(v))/σ
      
      _x(v);
      
      Calculation of the probabilities that z_maxand z_min, the maximum and minimum value of z in a region of interest C respectively, exceeds a threshold U;
      
      Calculation of the following transformation;
      
      ${\begin{matrix} z_{w} = F^{- 1} (P_{f, \max} (z)) & si z \geq U_{\max} \\ z_{w} = 0 & si U_{\min} < z < U_{\max} \\ z_{w} = F^{- 1} (P_{f, \min} (z)) & si z \leq U_{\min} \end{matrix},$
      
      in which the complete image is adopted as the region of interest, F^−
      
      1is the inverse gaussian function. U_max, U_minare thresholds chosen by the user;
      
      z_xor z_ware plotted as a one-dimensional graph where the point of the lattice is fixed, said graph being denominated a z curve or, in the case of the frequency domain, z spectrum of the TPEC; and
      
      z_xor z_ware plotted as two or three-dimensional image where t is fixed, said image being denominated z image of the TPEC the image being rendered according to a color code that indicates the degree of deviation of the values from the normative, said image being superimposed an anatomical atlas, which constitutes the background of the image.
  - 20. The method defined in claim 5, wherein the calculation of the element I_B→
    - A/C that quantifies the directed influence of brain or heart region B on region A that is not due to the activity coming from region C comprises the steps of;
      
      a) Defining regions A, B, C as sets of arbitrary voxels r_iε
      
      _gchosen from the generating volume b) Defining the vector valued time series;
      
      a(t) as the values of γ
      
      _m(t) for all voxels belonging to the set A;
      
      b(t) as the values of γ
      
      _m(t) for all voxels belonging to the set B; and
      
      c(t) as the values of γ
      
      _m(t) for all voxels belonging to the set C. c) Defining the matrices;
      
      A_t=[a(t),a(t−
      
      1), . . . ,a(t−
      
      k), . . . ,a(t−
      
      p)]B_t=[b(t),b(t−
      
      1), . . . ,b(t−
      
      k), . . . ,b(t−
      
      p)]C_t=[c(t),c(t−
      
      1), . . . ,c(t−
      
      k), . . . ,c(t−
      
      p)]d) Defining the matrices;
      
      $Γ_{t} (A_{t} / {} \cdot C_{t}) = [\begin{matrix} A_{t} \\ C_{t} \end{matrix}] and Γ_{t} (A_{t} / B_{t} \cdot C_{t}) = [\begin{matrix} A_{t} \\ B_{t} \\ C_{t} \end{matrix}],$ a) Calculating $\sum γ (Γ_{t} (A_{t}  {} \cdot C_{t})) = \frac{1}{N_{t}} \sum_{t = 1}^{N_{t}} ɛ_{t} (Γ_{t} (A_{t}  {} \cdot C_{t})) \circ ɛ_{t} (Γ_{t} (A_{t}  {} \cdot C_{t})),$
      
      where $ɛ_{t} (Γ_{t} (A_{t}  {} \cdot C_{t})) = a (t) - \sum_{k = 1}^{p} α_{k} a (t - k) - \sum_{k = 1}^{s} χ_{k} c (t - k);$
      
      α
      
      _kand χ
      
      _k, for any A and C, are estimated by minimizing;
      
      $E_{A  {} \cdot C} = \sum_{t = 1}^{N_{t}} (ɛ_{t} (Γ (A  {} \cdot C)) \circ ɛ_{t} (Γ (A  {} \cdot C))) K_{h} ( Γ (A  {} \cdot C) - (Γ_{t} (A_{t}  {} \cdot C_{t})) );$ $K_{h} (x) = \exp (- \frac{1}{2} { \frac{x}{h} }^{2});$
      
      p, s, and h being selected by cross-validation;
      
      b) Calculating $\sum γ (Γ_{t} (A_{t}  B_{t} \cdot C_{t})) = \frac{1}{N_{t}} \sum_{t = 1}^{N_{t}} ɛ_{t} (Γ_{t} (A_{t}  B_{t} \cdot C_{t})) \circ ɛ_{t} (Γ_{t} (A_{t}  B_{t} \cdot C_{t})),$
      
      where $ɛ_{t} (Γ_{t} (A_{t} | B_{t} \cdot C_{t})) = a (t) - \sum_{k = 1}^{p} α_{k} a (t - k) - \sum_{k = 1}^{q} β_{k} b (t - k) - \sum_{k = 1}^{s} χ_{k} c (t - k);$
      
      α
      
      _k, β
      
      _k, and χ
      
      _k, for any A, B, and C, are estimated by minimizing;
      
      $E_{A | B \cdot C} = \sum_{t = 1}^{N_{t}} (ɛ_{t} (Γ (A | B \cdot C)) \circ ɛ_{t} (Γ (A | B \cdot C))) K_{h} ( Γ (A | B \cdot C) - (Γ_{t} (A_{t} | B_{t} \cdot C_{t})) );$ $K_{h} (x) = \exp (- \frac{1}{2} { \frac{x}{h} }^{2});$
      
      p, s, and h being selected by cross-validation; and
      
      p1 c) Calculating the influence measure $I_{B -> A / C} = \ln (\frac{\langle \sum γ (Γ_{t}, (A_{t}  {} \cdot C_{t})) \rangle}{\langle \sum γ (Γ_{t}, (A_{t}  B_{t} \cdot C_{t})) \rangle}) .$

6. A Method of obtaining a Tomography of the Primary Electric Current (PEC) from the signals originated either from the brain or the heart using .an array of external electrical and biomagnetic sensors and a set of anatomical and medical images, the method comprising the steps of:
- a) Measuring the electrical and/or magnetic signals from the sensor array, defining observations for a common time t, designating the EEG/EKG as o₁(t) and the MEG/MKG as o₂(t) respectively, obtaining as a result a vector time series defined at time instants t_I,I=1, . . . ,N_t^Iwhich specify the lattice of time instants for electrophysiological signals ℑ
  
  ₁;
  
  b) Specifying lattices of sensor positions as the set of Cartesian coordinates that define the position and orientation of each sensor in a predefined common body reference system, said lattices of sensor positions being designated ₁and ₂for the EEG/EKG and MEG/MKG respectively;
  
  c) Selecting an anatomical image and specifying the Cartesian coordinates, in the common body reference system, of its constituent voxels as the lattice of the volume conductor _vhaving points r_v;
  
  d) Specifying for each point r_vof _v, the probability p(s,r_v) of that voxel belonging to a given tissue type s thus defining an Anatomical Atlas;
  
  e) Labeling each point r_vof the lattice of the volume conductor with the tissue type that has the highest probability, that is s_v=arg max_v(p(s,r_v));
  
  f) Specifying for each point r_v, of the lattice of the volume conductor _vthe conductivity value corresponding to the tissue label s_v, said values being taken from a predefined set of conductivities, designated as the conductivity profile σ
  
  ;
  
  g) Selecting the label b that indicates excitable tissue in order to define;
  
  a probability p(b,r_v) of excitable tissue for each points r_v, a set _gof N_gpoints r_gfor which said probability is non-zero, and a probability function p_g=p(b,r_g) defined on _g;
  
  h) Calculating the electric K₁and magnetic K₂lead field matrices by a vector boundary element method such that, given σ
  
  , ₁, ₂, _g, and _v, the following equations hold;
  
  o₁(t)=K₁f₁(t) and o₂(t)=K₂f₂(t), where f₁(t) and f₂(t) are used interchangeably to denote the PEC;
  
  i) Selecting one or more of images selected from the group consisting of the following functional modalities;
  
  Functional magnetic resonance (fMRI), Positron Emission Tomography (PET), and Single Photon Emission Computed Tomography (SPECT), and defining o₃(t), o₄(t), and o₅(t) observations, respectively;
  
  j) Specifying the lattices of voxel positions as the set of Cartesian coordinates, in the common body reference system, for each selected functional image, said lattices being designated ₃, ₄, and ₅for fMRI, PET, and SPECT respectively;
  
  k) Specifying the lattices of time instants t_I,I=1, . . . ,N_t^mfor which each selected functional image m has been sampled, in a common time scale with ℑ
  
  ₁, said lattices being designated ℑ
  
  ₃, ℑ
  
  ₄, and ℑ
  
  ₅for fMRI, PET, and SPECT respectively;
  
  l) Calculating the aggregation operator K_mthat expresses our knowledge that the ideal functional indicator f_m(t) associated to the m-th image modality (m=3,4,5) defined on _vand ℑ
  
  _mis modified by the image formation process according to the transformation o_m(t)=(K_m*f_m)(t) which;
  
  Reduces the spatial resolution of f_m(t) from that of _vto that of _m, and Reduces the temporal resolution of f_m(t) from that of ℑ
  
  ₁, to that of ℑ
  
  _m;
  
  m) Calculating the physiological operator H_mthat expresses our knowledge that the neural or cardiac tissue activation a(t) defined on _gand ℑ
  
  _mproduces physiological processes associated to the m-th image modality according to the transformation f_m(t)=(H_m•
  
  a)(t);
  
  n) Calculating estimates of the PEC by joint estimation of all the f_m(t) and of a(t) for all observed m, said estimation comprising steps of;
  
  i. Assigning arbitrary initial values to the f_m(t) and a(t), ii. Iteratively modifying the values of f_m(t) and a(t), iii. Calculating a probability measure p which increases when values of f_m(t) and a(t) simultaneously Reconstruct the o_m(t) with small error as quantified by a risk function, and Have a small norm in a given Besov space, and iv. Continuing step (ii) until the probability measure p does not increase;
  
  o) Defining γ
  
  _m(t)=f_m(t), m=1, 2, 3, 4, 5, and γ
  
  ₆(t)=a(t);
  
  p) Calculating from γ
  
  _m(t) the following quantities;
  
  a vector s^m={s_j^m} that quantifies the magnitude of γ
  
  _m(t) for r_iε
  
  _g, and a value I_B→
  
  A/Cthat quantifies the directed influence of brain or heart region B on region A that is not due to the activity coming from region C;
  
  q) Calculating the probabilities p_G(s_i^m) and p_G(s_i→
  
  j^m) that the magnitudes and influences obtained for vector s^m={s_j^m} and I_B→
  
  A/Care typical of a given reference group G; and
  
  r) Coding the values of p_G(s_i^m) and p_G(s_i→
  
  j^m) by means of a color scale and displaying said codes overlaid on the selected anatomical image highlighting statistically significant regions by thresholding to zero those values beneath a chosen significance level to obtain a statistical parametric map for the Tomography of PEC.
- View Dependent Claims (7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19)
- - 7. The method defined in claim 6, wherein the common reference system is the Talairach coordinate system.
  - 8. The method defined in claim 6, wherein the anatomical atlas is selected from the group consisting of following variants:
    - a) As an Individual Anatomical Atlas by the extraction of tissue probabilities of anatomical images obtained for the specific subject under study being these any one of the following types;
      
      Computed Axial Tomography (CAT), Magnetic Resonance Images (MRI), and post-mortem sections of the head, and b) As a Probabilistic Anatomical Atlas by a rigid or elastic transformation of the subject'"'"'s volume conductor lattice in order to ensure the best correspondence possible to a canonical volume conductor lattice for which a tissue probability distribution has obtained from a sample of normal or pathological anatomical images.
  - 9. The method defined in claim 6, wherein the computation of the Electric and Magnetic Lead Fields matrices by a vector boundary element method that comprises the steps of:
    - a) Specifying the positions of the sensors for the lattice of EEG/EKG and MEG/MKG modality ₁and ₂;
      
      b) Specifying the positions r_gand r_vin the lattice of the generating volume _g, and of the volume conductor _vrespectively;
      
      c) Specifying the conductivity profile using an approximation such that each set of tissue labels constitutes a collection of N embedded regions for which the conductivity value is constant, σ
      
      ={σ
      
      ₁, . . . , σ
      
      _N};
      
      d) Calculating numerically the ohmic current densities j_kV=j_k(r_v) and j_kV^ω=j_k(r_v,ω
      
      ) on each r_vof the lattice of volume conductor _vthat belongs to the surfaces limiting the regions by means of the following Linear Algebraic Systems of Equations;
      
      Dc=c_∞−
      
      Γ
      
      c;
      
      wherein c=(c₁;
      
      . . . ;
      
      c_N);
      
      said c_k={j_vK,V=1, . . . ,N_k,k+1} for the EEG/EKG or c_k={j_kV^ω,V=1, . . . ,N_k,k+1} for the MEG/MKG;
      
      N_k,k+1is the number point r_vthat belong to the surface separating the regions k and k+1, the matrices Γ and
      
      D being defined as;
      
      $Γ = \frac{1}{4 π} (\begin{matrix} \frac{(σ_{1} - σ_{2})}{σ_{1}} Γ_{11} & \frac{(σ_{2} - σ_{3})}{σ_{2}} Γ_{12} & \dots & Γ_{1 N} \\ \frac{(σ_{1} - σ_{2})}{σ_{1}} Γ_{21} & \frac{(σ_{2} - σ_{3})}{σ_{2}} Γ_{22} & \dots & Γ_{2 N} \\ ⋮ & ⋮ & ⋰ & ⋮ \\ \frac{(σ_{1} - σ_{2})}{σ_{1}} Γ_{N1} & \frac{(σ_{2} - σ_{3})}{σ_{2}} Γ_{N2} & \dots & Γ_{NN} \end{matrix})$ $D = (\begin{matrix} α_{1} I_{N_{1, 2}} & 0 & \dots & 0 \\ 0 & 0 & ⋮ \\ ⋮ & ⋰ & 0 \\ 0 & \dots & 0 & α_{N} I_{N_{N, N}} \end{matrix})$ $where α_{j} = \frac{(2 σ_{j} + σ_{j + 1})}{3 σ_{j}};$ $Γ_{jk} = (\begin{matrix} Γ_{1}^{k} (r_{1}^{j}) & \dots & Γ_{N_{k, k + 1}^{k}} (r_{1}^{j}) \\ ⋮ & ⋰ & ⋮ \\ Γ_{1}^{k} (r_{N_{j, j + 1}}^{j}) & \dots & Γ_{N_{k, k + 1}}^{k} (r_{N_{j, j + 1}}^{j}) \end{matrix});$ r_i^jlabels the i-th point of _vthat belong to j-th surface;
      
      c_∞=(c_∞¹;
      
      . . . ;
      
      c_∞^N) with c_∞^k={j_kV^∞,V=1, . . . ,N_k,k+1} for the EEG/EKG and c_∞^k={j_kV^ω
      
      ∞,V=1, . . . ,N_k,k+1} for the MEG/MKG, wherein the magnitudes j_kV^∞=j_k∞(r_v) and j_kV^ω
      
      ∞=j_k∞(r_v,ω
      
      ) are evaluated from the expressions;
      
      $j_{k \infty} (r) = - \frac{I_{θ r}}{4 π} \nabla \cdot (g (r, r_{θ}) - g (r, r_{r})) j_{k \infty} (r, ω) = \frac{ ω}{4 π} \sum_{k = 1}^{N} (σ_{k} - σ_{k + 1}) \oint_{S_{k, k + 1}} g (r, r^{'}) \times b_{c} (r^{'}, ω)) \partial r^{′2}$ wherein b_c(r′
      
      ,ω
      
      ) is the magnetic field generated by the low frequency ac current passing through a single coil in a infinite and homogeneous volume conductor, and the function $Γ_{n}^{k} (r) = \int_{Δ_{n}^{k}} \nabla^{'} g (r, r^{'}) \cdot \partial r^{′2} - \int_{Δ_{n}^{k}} \nabla^{'} g (r, r^{'}) \circ \partial r^{′2}$ where g(r_g,r′
      
      ) is the Green Function of the infinite domain. e) Calculating the Electric and Magnetic Lead Field in each compartment of j.(r) an IPHC by using the reciprocity theorems;
      
      $k_{eg} = - \frac{j_{j} (r_{g})}{σ_{j} I_{cr}}$
      
      at the lead electrode “
      
      er”
      
      , and $k_{eg} = - \frac{j_{j} (r_{g}, ω)}{i ω σ_{j} I_{c} (ω) Δ S_{c}}$
      
      at the coil “
      
      co”
      
      , respectively, the ohmic current densities being evaluated at any point r_gof the lattice of the generating volume _gby the use of the following expressions;
      
      Electric Lead Field;
      
      $j_{j} (r_{g}) = j_{k \infty} (r_{g}) - \frac{1}{4 π} \sum_{k = 1}^{N} \frac{(σ_{k} - σ_{k + 1})}{σ_{k}} \sum_{V = 1}^{N_{k, k + 1}} Γ_{V}^{k} (r_{g}) j_{k V}$ Magnetic Lead Field;
      
      $j_{j} (r_{g}, ω) = j_{k \infty} (r_{g}, ω) - \frac{1}{4 π} \sum_{k = 1}^{N} \frac{(σ_{k} - σ_{k + 1})}{σ_{k}} \sum_{V = 1}^{N_{k, k + 1}} Γ_{V}^{k} (r_{g}) j_{k V}^{ω}; and$ f) Calculating the Electric K₁={k_θ
      
      g^t} and Magnetic K₂={k_θ
      
      g^t} Lead Fields matrices specifying a row of said matrices for each sensor (EEG/EKG e=1, . . . ,N_eand MEG/MKG c=1, . . . ,N_c) and a column for each point of the lattice of the generating volume r_gε
      
      _g.
  - 10. The method defined in claim 6, wherein the calculation of the aggregation operator K_massociated to the m-th image modality (i=3,4,5) is defined by the following operation
  - 11. The method defined in claim 6, wherein the estimation of the PEC f₁(t) for a given subject comprises steps of:
    - a) Assigning initial values to the following sets of variables;
      
      1) Tissue activation a(t)=[a(r_g,t)]_r_g_ε_gfor each point r_gof _g2) Functional indicators f_m(t)=[f_m(r_g,t)]_r_g_ε_gfor each point r_gof _gwhere f(t)=[f_m(t)]_{1≦
      
      m≦
      
      M}denotes the totality of measurement modalities obtained for the subject, 3) Hyperparameter of the activation Θ
      
      _a, 4) Hyperparameter of the intrinsic noise of the functional indicators Θ
      
      _β=[Θ
      
      _β_m]_{1≦
      
      m≦
      
      M}, and 5) Hyperparameter of the instrumental noise Θ
      
      _θ=[Θ
      
      _θ_m]_{1≦
      
      m≦
      
      M};
      
      b) Computing the a posteriori probability;
      
      $P (a (t), Θ_{a}, f (t), Θ_{β}, Θ_{e}  o (t)) \propto \prod_{m = 1}^{M} I_{o_{m}} (o_{m} (t) - (K_{m} * f_{m}) (t)  Θ_{e_{m}}) \times \prod_{m = 1}^{M} π_{f_{m}} (f_{m} (t) - (H_{m} \cdot a) (t)  Θ_{β_{m}}) \times \prod_{m = 1}^{M} {π_{Θ_{a_{m}}} (Θ_{e_{m}}) \cdot π_{Θ_{β_{m}}} (Θ_{β_{m}})} \times π_{a} (a (t)  Θ_{a}) \times π_{Θ_{a}} (Θ_{a})$
      
      that comprises the following multiplicative terms;
      
      1) Likelihood terms I_o_mfor each modality mI_o_m(o_m(t)−
      
      (K_m*f_m)(t)|Θ
      
      _β_m,2) The a priori probabilities for the functional indicatorsπ
      
      _f_m(f_m(t)−
      
      (H_m•
      
      a)(t)|Θ
      
      _β_m,3) The a priori probability for the activation π
      
      _a(a(t)|Θ
      
      _a), 4) The a priori probability for the hyperparameters of the activation π
      
      _Θ_a(Θ
      
      _a), 5) The a priori probabilities for the hyperparameters of the intrinsic noise of the functional indicators π
      
      _Θ_β
      
      m(Θ
      
      _β_m), and 6) The a priori probabities for the hyperparameters of the instrumental noise π
      
      _Θ_θ
      
      m(Θ
      
      _θ_m);
      
      c) Iteratively modifying the parameters a(t), Θ
      
      _a, f(t), Θ
      
      _β, Θ
      
      _euntil a maximum value of the a posteriori probability is achieved, said modification being carried out according to a scheme selected from the group consisting of;
      
      1. Successive Maximization of P(a(t),Θ
      
      _a,f(t),Θ
      
      _β,Θ
      
      _θ|o(t))for subsets of the parameters, while maintaining fixed all the others, “
      
      Iterated Conditional Maximization”
      
      (ICM), 2. Successive Maximization of P(a(t),Θ
      
      _a, f(t),Θ
      
      _β,Θ
      
      _θ|o(t)) for subsets of the parameters, fixing the other parameters to their expected value, “
      
      Expectation Maximization”
      
      (EM), and 3. Choosing the mode of the distribution of P(a(t),Θ
      
      _a,f(t),Θ
      
      _β,Θ
      
      _θ|o(t)) obtained by means of Monte Carlo Markov Chain methods (MCMC).
  - 12. The method in claim 11, wherein the electrophysiological observations o_m(t) m=1,2 for the estimation of f₁(t) is selected from the group consisting of:
    - EEG alone, MEG alone, and EEG and MEG together.
  - 13. The method in claim 11, wherein the electrophysiological observations o_m(t) m=3,4,5 for the estimation of f₁(t) are selected from the group consisting of:
    - fMRI alone, PET alone, SPECT alone, fMRI and SPECT, fMRI and SPECT, and PET and SPECT.
  - 14. The method in claim 11, wherein the calculation of the likelihoods I_o_m(o_m(t)−
    - (K_m*f_m)(t)|Θ
      
      _θ_mcomprise the following steps;
      
      a) Selecting the use of either e insensitive polynomial risk or a generalized polynomial risk R_x[x|Θ
      
      _x] for the specific purpose of evaluating non gaussian likelihoods;
      
      b) Calculating the discrepancy vector Δ
      
      =o_m(t)−
      
      (K_m*f_m)(t);
      
      c) Reseating a(t) as a function of the hyperparameters Θ
      
      _θ_m;
      
      d) Calculating R_x^C[Δ
      
      |Θ
      
      _x], the risk of Δ
      
      ; and
      
      e) Calculating I_o_m(Δ
      
      |Θ
      
      _θ_m)=exp(−
      
      BR_x^C[Δ
      
      |Θ
      
      _x]) where B is a preselected constant.
  - 15. The method in claim 11, wherein the calculation of the a priori probability π
    - _f_m(f_m(t)−
      
      (H_m•
      
      a)(t)|Θ
      
      _β_mfor the functional indicators comprises the following steps;
      
      a) Selecting the indices indices m,n,s of the Besov space B_n,s^m, with the purpose of defining the smoothness of the estimated functional indicator, a particular but not exclusive choice being m=1,n=1,s=1 which defines the “
      
      algebra of bumps;
      
      ”
      
      b) Calculating the discrepancy vector Δ
      
      =f_m(t)−
      
      (H_m•
      
      a)(t);
      
      c) Rescaling the Δ
      
      as a function of the hyperparameters Θ
      
      _β_m;
      
      d) Calculating ∥
      
      Δ
      
      ∥
      
      _B_n,s^m, the norm of the Δ
      
      in B_n,s^m, said calculation being carried out by first expanding Δ
      
      as $Δ = \sum_{k} F_{k} \cdot Ψ_{k}$ where the Ψ
      
      _kbelong to a Dictionary of Atoms and then calculating ${ Δ }_{B_{n, s}^{m}} \approx \sum_{k} a_{k} \cdot { F_{k} }^{m};$ and e) Calculating π
      
      _f_m(Δ
      
      |Θ
      
      _B_m)=exp(−
      
      C∥
      
      Δ
      
      ∥
      
      _B_n,s^m) where C is a preselected constant.
  - 16. The method in claim 11 wherein the calculation of the a priori probability of the activation π
    - _a(a(t)|Θ
      
      _a) comprises the following steps;
      
      a) Calculating a non decreasing function h(•
      
      ) of the probabilistic atlas h(p_g);
      
      b) Calculating the norm ∥
      
      a(t)∥
      
      _B_n,s^m;
      
      and c) Calculating π
      
      _a(a(t)|Θ
      
      _a)=exp(−
      
      Dh(p(s,r_g))−
      
      E∥
      
      a(t)∥
      
      _B_n,s^m), where D and E are prespecified constants.
  - 17. The method in claim 15, wherein the of the a priori probability for the PEC takes the following specific form:
    - π
      
      _f₁(f₁(t)−
      
      (H₁•
      
      a)(t)|Θ
      
      _β₁)=π
      
      ₁(f₁(t)−
      
      Ma(t)|Θ
      
      ₁π
      
      ₂(μ
      
      |Θ
      
      ₂)where π
      
      ₁and π
      
      ₂are a priori probabilities specified as in claim 15,
  - 18. The method in claim 17, wherein the a priori probability π
    - ₂(μ
      
      |Θ
      
      ₂) defined in claim 17 is calculated by an expression selected from the group of expressions consisting of;
      
      π
      
      ₂(μ
      
      |Θ
      
      ₂)=exp(−
      
      ∥
      
      Λ
      
      _s·
      
      μ
      
      ·
      
      Λ
      
      _m∥
      
      _B_n,s^m) where Λ
      
      _sis a diagonal matrix that specifies the degree of smoothness that will be imposed at each point of _gand Λ
      
      _m=W·
      
      P, where W is a prespecified weighting matrix and P is a diagonal matrix containing p_gfor all r_g, and by means of successive evaluation of the marginal distributions of π
      
      ₂(μ
      
      _g|Θ
      
      ₂)=p_g·
      
      N(0,σ
      
      _A²)+(1−
      
      p_g)·
      
      N(0,σ
      
      _B²) with N(0,σ
      
      ²) the univariate gaussian distribution with mean zero and standard deviation σ
      
      , σ
      
      _A², and σ
      
      _B²being constants selected to be large and small respectively with respect to the expected variation of the orientations.
  - 19. The method in claim 11, wherein the calculations are carried out for o₁(t) and/or o₂(t) in the frequency domain by specifying:
    - That the electrical signals have been subjected to the Fourier transform, t now denoting frequency. I_o_m(o_m(t)−
      
      (K_m*f_m)(t)|Θ
      
      _θ_m)=N_R^2,p(0,Σ
      
      _θ), where N_R^2,p(μ
      
      _x,Σ
      
      _x) denotes the multivariate real gaussian distribution with mean and variance (μ
      
      _x,Σ
      
      _x) π
      
      _f_m(f_m(t)−
      
      (H_m•
      
      a)(t)|Θ
      
      _β_m)=N_R^2,p(0,0), H_m=I, the identity matrix π
      
      _a(a(t)|Θ
      
      _a)=N_R^2,p(0,Σ
      
      _a) $π_{Θ_{a}} (Θ_{a}) = { \sum_{a} }^{- \frac{a}{2}} \exp (- \frac{1}{2} Tr (Q \sum_{a}^{- 1}))$
      
      a natural conjugate a priori distribution for complex Wishart matrices, where α and
      
      Q are prespecified in accordance with prior knowledge, π
      
      _Θ_β
      
      m(Θ
      
      _β_m) and π
      
      _Θ_θ
      
      m(Θ
      
      _θ_m) being constants defining improper priors.

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System and method for the tomography of the primary electric current of the brain and of the heart

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System and method for the tomography of the primary electric current of the brain and of the heart

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