WO2015051256A1 - Analyse du flux sanguin myocardique par résolution de voxels - Google Patents

Analyse du flux sanguin myocardique par résolution de voxels Download PDF

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WO2015051256A1
WO2015051256A1 PCT/US2014/059062 US2014059062W WO2015051256A1 WO 2015051256 A1 WO2015051256 A1 WO 2015051256A1 US 2014059062 W US2014059062 W US 2014059062W WO 2015051256 A1 WO2015051256 A1 WO 2015051256A1
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model
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processing device
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John M.M. ANDERSON
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Howard University
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    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B6/00—Apparatus for radiation diagnosis, e.g. combined with radiation therapy equipment
    • A61B6/50—Clinical applications
    • A61B6/503—Clinical applications involving diagnosis of heart
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B5/00—Measuring for diagnostic purposes; Identification of persons
    • A61B5/02—Detecting, measuring or recording pulse, heart rate, blood pressure or blood flow; Combined pulse/heart-rate/blood pressure determination; Evaluating a cardiovascular condition not otherwise provided for, e.g. using combinations of techniques provided for in this group with electrocardiography or electroauscultation; Heart catheters for measuring blood pressure
    • A61B5/026—Measuring blood flow
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B6/00—Apparatus for radiation diagnosis, e.g. combined with radiation therapy equipment
    • A61B6/02—Devices for diagnosis sequentially in different planes; Stereoscopic radiation diagnosis
    • A61B6/03—Computerised tomographs
    • A61B6/037—Emission tomography
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B6/00—Apparatus for radiation diagnosis, e.g. combined with radiation therapy equipment
    • A61B6/50—Clinical applications
    • A61B6/507—Clinical applications involving determination of haemodynamic parameters, e.g. perfusion CT
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B6/00—Apparatus for radiation diagnosis, e.g. combined with radiation therapy equipment
    • A61B6/52—Devices using data or image processing specially adapted for radiation diagnosis
    • A61B6/5211—Devices using data or image processing specially adapted for radiation diagnosis involving processing of medical diagnostic data
    • A61B6/5217—Devices using data or image processing specially adapted for radiation diagnosis involving processing of medical diagnostic data extracting a diagnostic or physiological parameter from medical diagnostic data
    • G—PHYSICS
    • G16—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS
    • G16H—HEALTHCARE INFORMATICS, i.e. INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR THE HANDLING OR PROCESSING OF MEDICAL OR HEALTHCARE DATA
    • G16H50/00—ICT specially adapted for medical diagnosis, medical simulation or medical data mining; ICT specially adapted for detecting, monitoring or modelling epidemics or pandemics
    • G16H50/50—ICT specially adapted for medical diagnosis, medical simulation or medical data mining; ICT specially adapted for detecting, monitoring or modelling epidemics or pandemics for simulation or modelling of medical disorders

Definitions

  • This invention relates generally to estimating myocardial blood flow (MBF) and more specifically to a positron emission tomography (PET) based method for estimating myocardial blood flo (MBF).
  • MAF myocardial blood flow
  • PET positron emission tomography
  • Imaging of blood flow through the heart and associated veins can improve diagnosis and treatment of cardiac diseases.
  • estimation of myocardial blood flow or blood flow through heart muscular tissue can be useful as descr bed below.
  • nuclear based medicine can be used to produce useful medical images.
  • radioactive elements are introduced into the bloodstream such that when the radioactive elements experience a radioactive decay, the byproducts of that decay (often the reaction with particles called positrons) can be sensed to produce an image of the area where the radioactive elements are placed.
  • PET positron emission tomography
  • positron emitting tracers are available for these studies with the most common being * 2 Rb and 1 N-Ammonia.
  • Coronary flow reserve can be defined as a ratio of a maximum hyperemic flow to baseline flow, m normal patients this ratio can typically range between 3-5, which is a essentially a measure of the function of coronary circulation and is particularly useful in the detection of early abnormalities due to coronary artery disease. Because the coronary flow reserve determination is a ratio, it is unaffected by a uniform reduction in both baseline and maximal flow.
  • coronary flow reserve does not reflect true vasodilation capacity.
  • a reduction in coronar flow reserve could be caused either by increased flow in the baseline state or by reduced maximum hyperemia flow.
  • Factors that increase myocardial oxygen demand for example hypertension, increased left ventricular wall stress, increases in inotropic state, and tachycardia, can lead to an increased basal flow. Differentiating between this case and the reduced maximal hyperemic flow due to significant coronary stenosis is difficult without absolute myocardial blood flo measurements. Measurements of hyperemic blood flow in absolute units provide a more direct estimate of vasodilation capacity.
  • the primary factor curves for this application are the right ventricular blood pool, the left ventricular blood pool, and the myocardial tissue cisrve.
  • the mathematical task is to find both the factors and coefficients so that the linear combination of factor curves for every pixel in the image matches the measured curve as close as possible. This problem is constrained by requiring that the tissue curves and the linear coefficients are all positive.
  • Ammonia or Rubidium uptake is generally analyzed with a two or three compartment model.
  • the models all have a blood compartment in contact with an extracellular free distribution compartment, which is in turn in contact with a metabolically trapped compartment.
  • These models are much easier to calculate if it is assumed that the clearance from the metabolically trapped compartment is zero (or near zero) over the duration of the experiment.
  • several authors recommend collecting and analyzing only two minutes of data. When using smooth data generated by a veraging all pixels within a large range of interest, this is a reasonable approach.
  • MVF myocardial blood flow
  • K-FADS pharmacological kinetics based factor analysis of dynamic structures
  • V-MBF Voxel-Resolution myocardial blood flow
  • a myocardial blood flow analysis includes a processing device applying a pharmacological kinetic model to a data set stored in a storage device.
  • the data set may be compiled from a PET scan or other imaging approach that can monitor fluid flow in a voxel set.
  • the data set may be derived from an imaging technique based on monitoring fluid based tracers in a left ventricle, a right ventricle, and myocardium of a patient or animal subject.
  • the pharmacological kmetic model includes incorporating a model of changing concentrations of bound fluid based iracers, unbound fluid based tracers, and blood plasma fluid based iracers into a standard factor analysis of dynamic structures model combined with a model of fluid based tracer activity in the left ventricle as a time shifted and dispersed function of blood flow from the right ventricle.
  • the tracer activity is modeled without assumption that a right ventricle tissue curve and a left ventricle tissue curve obey a particular mathematical relationship.
  • the processing device is configured to output a processed data set based on the application of the pharmacological kinetic model to the data set for providing a representation of blood flow in the myocardium.
  • the processed data set may be usable to create a visual representation, an audio representation, a textural description of the myocardial blood flow using known methods for conveying such information.
  • the processing device optionally estimates parameters of the standard factor analysis of dynamic structures model.
  • the estimating may be done by estimating maximum values of fluid based tracer activity in one or both of the right ventricle or the left ventricle and modifying a corresponding signal vector value for the one of the right ventricle or the left ventricle using the estimated maximum values of fluid based tracer activity.
  • the estimating may be done by estimating a left ventricle time activity curve, a right ventricle time activity curve, and a time activity curve, wherein the left ventricle time activity curve is assumed to be approximately equal to a response of an mth-order, all-pole filter applied to the right ventricle time activity curve and determining a set of parameters that produce a smallest least-squares error for the pharmacological kinetic model.
  • This estimation may include for a given initial estimate right ventricle time activity curve and a given initial estimated left ventricle time activity curve, determining initial estimates for parameters of the pharmacological kinetic model.
  • a least squares objective function can be applied to obtain estimates for parameters of the pharmacological kinetic model.
  • the least squares objective function is minimized by applying a maj rize-minimize optimization technique to iteratively estimate the right ventricle tissue curve and the left ventricle tissue curve.
  • initial estimates for an initial estimated right ventricle time activity curve and an initial estimated left ventricle time activity curve are estimated and use the initial estimates for the initial estimate right ventricle time activity curve and the initial estimated left ventricle time activity curve to determine initial parameters of the pharmacological kinetic model.
  • the processing device smoothing estimates for blood flow for a given voxel by applying a limiting factor or penalty factor on data from voxels located within a given distance from the given voxel .
  • the pharmacological kinetic model includes a semi -parametric model configured to drive time activity curves for the right ventricle imaging activity from the fluid based tracers and the left ventricle imaging activity from the fluid based tracers to zero over time.
  • Figure 1 illustrates a generalization of a 3-compartment model used to model Rubidium or Ammonia kinetics in the heart
  • Figure 2 illustrates results from a V-MBF algorithm applied to synthetic data: (a) true, initial, and estimated RV factor, (b) comparison of true TAG (computed analytically) and estimated TAG generated using the discrete-time K-FADS model and estimated parameters, (c) same as (b) except for a different TAG, and (4) least-squares objective function as a function of iteration number.
  • FIG. 3 illustrates example results for plane 9 using the IV-MBF algorithm
  • FIG. 3A illustrates MBF estimates displayed as an image
  • FTG illustrates a mask used to define myocardium voxels
  • FIG. 3C illustrates the image obtained by averaging the PET images for the last six sub-scans
  • FIG. 3D illustrates a histogram of the MBF estimates.
  • FIG. 4 illustrates example results for plane 16 using the IV-MBF algorithm, where FIG. 4A illustrates MBF estimates displayed as an image, FTG. 413 illustrates a mask used to define myocardium voxels, FIG. 4C illustrates the image obtained by averaging the PET images for the last six sub-scans, and FIG. 4D illustrates a histogram of the MBF estimates.
  • FIG. 5 illustrates example results for a specific myocardium voxel
  • FIG. 5A illustrates a plot of the measured interpolated TAG (solid) and corresponding estimated TAC using the IV-MBF algorithm (dashed-dotted)
  • FIG 5B illustrates signal components that combine to form estimated TAC: sampled RV and LV tissue curves, and curves due to activity in the free and bound states are represented by the dashed, dotted, solid, and dashed- dotted lines, respectively
  • 5C illustrates the LS objective function.
  • Figure 6 illustrates an embodiment of a illustrative system that can incorporate the present embodiments.
  • a method and apparatus for estimating myocardial blood flow (MBF) in each voxel in the myocardium is described.
  • the algorithm is based on a factor analysis of dynamic structures (FADS) model that has been enhanced to constrain the factor analysis curves to be physiologically appropriate.
  • FADS factor analysis of dynamic structures
  • Figure 1 illustrates a generalization of a 3-compartment model used to model Rubidium or Ammonia kinetics in the heart. Tracer in the second compartment is freely diffusible, while the tracer in the third compartment is trapped.
  • the unknown, parameters ki, k2, and ks are first order rate constants that describe the tracer movement between the compartments.
  • the parameter ki (ml/min/g) is identified as myocardial blood flow.
  • C P is the ammonia concentration in blood plasma
  • Cu is the concentration of the free (i.e., unbound) ammonia
  • Cb is the concentration of the trapped (i.e., bound) ammonia.
  • a model is applied that incorporates a pharmacological kinetic model with the standard FADS model, which is a model where each time activity curve is assumed to be a linear combination of factor curves.
  • the resulting model which can be called the pharmacological kinetics based factor analysis of dynamic structures (K-FADS) model, provides a means for estimating factor curves in the myocardium that are
  • V-MBF Voxel-Resolution myocardial blood flow
  • the method disclosed herein is feasible for determining physiologically meaningful estimates of absolute MBF.
  • the initial model is a continuous-time K-FADS model from which a discrete-time K-FADS model can be obtained by applying a bilinear transform to the continuous-time K- FADS model. It should be noted that there is a simple relationship between the discrete-time and continuous-time K-FADS parameters. Using a systems theory framework, the problem of estimating the discrete-time K-FADS parameters is related to the problem of identifying a discrete-time system from given input and output data (i.e., input-output system
  • the V-MBF algorithm iteratively estimates the MBF for each voxel in the myocardium.
  • the V-MBF algorithm can be initialized using an input-output system identification method, such as one described by Steiglitz-McBride, which is applicable to a class of discrete-time systems that includes the discrete K-FADS model.
  • Steiglitz-McBride an input-output system identification method, such as one described by Steiglitz-McBride, which is applicable to a class of discrete-time systems that includes the discrete K-FADS model.
  • the RV factor, LV factor (i.e.,Cp), Cu , and Cb can be denoted by the continuous-time functions fi(t), f2(t), fi,3(t), and f (t), respectively, where i is the voxel index.
  • fi and f?. represent the activity concentration of the ammonia in the right ventricle and left ventricle, respectively.
  • fijrt), and f (t) represent the activity concentration of free and trapped ammonia in my ocardial tissue, respectively.
  • the RV (right ventrical) and LV (left ventrical) factors are spatially constant (i.e., RV and LV factor are voxel independent).
  • the LV factor is modeled as a time shifted and dispersed function of the RV factor
  • ⁇ , ⁇ , and ⁇ (tau) are the unknown gain, time constant, and delay of the LV, respectively.
  • ⁇ (tau) accounts for the fact that the ammonia activity first appears in the right ventricle and then, after a period of time, appears in the LV.
  • the function u is the unit step function, and the notation * denotes the continuous-time convolution operator.
  • the model for the LV factor in (4) can be motivated by observations of "isolated" RV and LV factors obtained from dynamic PET sequences and, in part, by the need for mathematical tractability.
  • ki is the preferred MBF parameters.
  • the activity for the ith pixel can be expressed as
  • Oi(i) ⁇ 3 ⁇ 4, ⁇ / ⁇ ( ⁇ ) + ⁇ 3 ⁇ 4,2/2 ( ⁇ ) + ⁇ 3 ⁇ 4, 3 / ⁇ , 3 ( ⁇ ) + ⁇ 3 ⁇ 4 ⁇ 4 / ⁇ ,4 ( ⁇ ) ⁇ (7)
  • the first term in (7) can be identified as the amount of spillover from the right ventricle, and the second term can be a combination of the ammonia activity in blood vessels within the myocardium and spillover from the left ventricle.
  • the constant Ci,i accounts for the amount of the measured radioactivity in voxel in the case of a PET scan that is due to the blood plasma in the RV.
  • the constant a,? accounts for the amount of the measured radioactivity in voxel i that is due to blood plasma in the LV (i.e., LV spill over) and blood plasma in the blood vessels of the myocardium.
  • c . :;; 0.05 it is assumed that c . :;; 0.05.
  • the third and fourth terms in (7) are the activity of the free and trapped ammonia in the myocardial tissue, respectively.
  • ⁇ 3 ⁇ 4(*) ⁇ 3 ⁇ 4/( «) + f ⁇ t) * [3 ⁇ 4,i + 1 ⁇ 4,2 exp(-/?(i - r)) + 3 ⁇ 4 i3 exp(-fci (t - r)) ⁇ u(t - r) (1 1)
  • the next aspect of this approach is to address the problem of estimating the parameters ⁇ , ⁇ ki ⁇ , ⁇ a ⁇ , ⁇ by ⁇ , ⁇ bi.2 ⁇ , and ⁇ by ⁇ from discrete-time Time Activity Curve (TAG) data because continuous-time TAC data is not available in practice.
  • TAG Time Activity Curve
  • the bilinear transformation is a way to transform a linear time-invariant continuous- time system into a linear time-invariant discrete-time system.
  • a limitation of the bilinear transform is that a delay in a continuous -time system must be an int eger multiple of the sampling interval. Taking the Laplace transform of (1 1), we get the following relationship
  • ⁇ dT for some integer d
  • ri 1
  • ri (2/T + ⁇ )-'(2/ ⁇ - ⁇ )
  • r, ,3 (2/T +ki) "1 (2/T - ki)
  • b'u bi,i(2/T)- !
  • b'i, 2 bi, (2/T + ⁇ )->
  • b' b (2/T + ki)->.
  • the delay term (i.e., z "d term) in equation (21) follows from the ⁇ second delay in the continuous-time K-FADS model, which is the delay between activity appearing in the right and left ventricles.
  • the bilinear transformation has the property that it maps a stable continuous-time system into a stable discrete-time system. Moreover, the bilinear transformation avoids the problem of aliasing by mapping the jO axis into the unit circle of the complex plane. However, "frequency warping" can occur as a result of mapping the entire jO axis into the unit circle. Note, the frequency warping problem can be ameliorated by choosing a sufficiently high sampling rate fs.
  • ⁇ ⁇ 3 [ri,3,r 2 ,3, ⁇ . ⁇ ,T7, 3 ]
  • c [ci,C 2 , ... ,Ci] .
  • Additionall is a ffeeaassiibbllee sseett ooff tthhee p totalraammeetteerrs; ⁇ and
  • Step 2.2 advantage is taken of the fact that the objective function ⁇ is de-coupled in terms of the parameters ri ' 3 ' ⁇ 2 > 3 ' ' ' ' ' r I,3
  • a 1-D line search algorithm such as the golden section method can be used to solve the ID minimization problems in Steps 2.1 and 2.2.
  • Ci > 2 ⁇ ⁇ and ⁇ 3 ⁇ 4,3— 0.8 f or ⁇ ⁇ i
  • initial estimates for the RV factor x, r 2 , and 6 r needed.
  • One method for obtaining an initial estimate for the RV factor x ⁇ °> is for a user selected TAG of a voxel to be used that can essentially remain in the right ventricle throughout the duration of the scan.
  • the known Steiglitz-McBride algorithm may be used.
  • first observations can be used from eqation (22) that for some 3rd-order polynomial Q : ⁇ : ⁇ ' ⁇ and 2nd-order polynomial P'i(z), the z-transform of the ith TAG, is given by
  • equation (40) can be written as
  • the Steiglitz-McBride algorithm Given an input-output pair for a linear, time-invariant system modeled as an ARMA model, the Steiglitz-McBride algorithm can provide estimates for ARMA parameters. Thus, given a TAC, yi[n], and initial RV factor x (0) , the Steiglitz-McBride algorithm can be used, which is an iterative algorithm, to estimate Pi(z) and Q;(z), The Steiglitz-McBride algorithm can be summarized below
  • V pa,Pi,P2,Pa,Pd,Pd+i,Pd+2,Pd+3] (47)
  • i ⁇ >, ⁇ i ⁇ p i , and 9 « can be similarly defined, and
  • the objective function in (42) can be equivalent to a linear least-squares objective function.
  • the Steigiitz-McBride algorithm entails a filtering step (i.e., initial RV factor x (0) [n] and ith TAC, y;[n], are filtered by l/Q " v ( ⁇ and minimization of a linear least- squares objective function.
  • (0) (o) _ estimates for the parameters n and ry can be T i ⁇ av gi z i,2 > 3 ⁇ 4,2, ⁇ , 3 ⁇ 4,2j anc f 3 ⁇ 4 — 3 ⁇ 4, 3 ; respectively.
  • V-MBF algorithm does not require that the parameters ci, ki,2, and ku be voxel independent. The values for these parameters were simply chosen for exemplary purposes.
  • the simulated TACs were integrated using a time- varying window.
  • the resulting integrated data simulated the activity data that would actually be available in practice.
  • a typical protocol used at an exemplary PET Center could leads to the following specification for the simulated integrated TACs (I-TACs)
  • T2 12 seconds
  • T3 30 seconds
  • the V-MBF algorithm can be based on standard TAG data (i.e., ai(nT)).
  • the I-TAC data is preferrably pre-processed.
  • the I-TAC data is assumed to be nearly piece-wise linear. It follows using a known method from Kuhle that the standard TAC data at the midpoints of the windows is approximately:
  • a preferred sampling interval is T 0.05 sec. It is noted that the approach described above for obtaining standard sampled TAC data would also be used to generate an initial RV factor from I-TAC data located in the RV.
  • V-MBF algorithm described above was applied for 5000 iterations where one sub-iteration was used to update the estimate for the RV curve. In this simulation, the maximum error in the MBF estimates was 1.5 percent.
  • a typical result of a V-MBF algorithm is summarized in the Figure 2, where (a) shows a true, initial, and estimated RV factor, (b) compares a true TAC (computed analytically) and with one generated using the discrete-time K-FADS model and estimated parameters, (c) is the same as (b) except for a different voxel, and (d) is a plot of the least-squares objective function as a function of iteration number. In the exemplary study, the V-MBF algorithm was stable and
  • the V-MBF algorithm can be based on a model that accounts for the fact that the shape of TACs due to ischemic and normal tissue are different.
  • the model can allow for the factors that represent free and trapped ammonia to be voxel dependent and physiologically appropriate.
  • TACs in ischemic and normal tissue can be modeled as a linear combination of the same three factors.
  • the present methods and systems represent a significant improvement in the art as a more appropriate model to provide more accurate MBF estimates than available methods.
  • the problem of estimating the weights and signal vectors of the above described models is a blind inverse problem.
  • the data are nonnegative J x 1 vectors
  • a 3 ⁇ 4 and ey are the jth components of a; and ei, respectively, and fjk is the jth value of the kth signal vector.
  • An example where this model is used is cardiac imaging using dynamic positron emission tomography (PET), where ajj is a measure of the radiopharmaceutical concentration in the ith voxel at the jth time point.
  • PET dynamic positron emission tomography
  • Another example is multispectral imaging, where a;; represent the value of the ith pixel in the jth spectral plane.
  • the problem Given the data ij , the problem is to estimate the weights ⁇ 3 ⁇ 4 ⁇ and signal vector values ⁇ fjk) .
  • the least- squares estimates of the weights and signal vector values are obtained by minimizing the least-squares objective function L subject to a non-negativity constraint,
  • the vectors c and f contain the signal weights and signal vector values, respectively:
  • the least squares estimation problem is ill-posed, so the results are highly dependent on the initial estimates.
  • Steps 1 ' and 2 " imply that the resulting algorithm monotonically decreases the least- squares objective function
  • MM algorithms are a viable approach provided a majorizing function can be found that is easier to minimize than the original objective function. Assuming that g is a suitable majorizing function for f, the corresponding MM algorithm is
  • Dc and Df are the set of non-negative vectors of dimension IK x 1 and JK x 1 , respectively.
  • the accuracy of the MBF estimates would greatly depend on the performance of the standard least-squares algorithm, which, as mentioned previously, is highly dependent on the initial estimates. Therefore, we develop extensions of the least- squares method that greatly reduce the parameter space by incorporating a priori information.
  • the proposed algorithms are expected to be more stable and produce more accurate estimates of the factor curves and weights than the standard feast-squares algorithm. Therefore, improved MBF estimation is anticipated when the proposed algorithms are used instead of the standard least-squares algorithm.
  • g is a real J x 1 vector. It can be seen that ⁇ (g) is the energy of g after the user chosen time frame jo. [00112]
  • PLS penalized least-squares
  • the PLS framework can incorporate other known penalty functions.
  • ⁇ and , 112 represent the unknown maximum values of the right and l eft ventricles, respectively, and ji and j2 denote the locations of the maximum values of the right and left ventricles.
  • / (n+1) arg min r(f, / ⁇ »>, c ⁇ n >) subject to / > 0, ⁇ f kl - fa ⁇ e, ⁇ f h2 - flz ⁇ e, (50) where e is a tolerance parameter chosen by the user, and fii and ⁇ are estimates of the maximum values ⁇ and ya.
  • the bilinear transformation is a popular way to transform a linear time-invariant continuous-time system into a linear time-invariant discrete- time system.
  • ⁇ dT s for some integer d, g ⁇ a(2/T s + b) ⁇ t m+ 1) , and p— (2/T s + _j) ⁇ 1 (2/r s — b) (note: h c (t) h c (t - ⁇ )).
  • equation (71) is substituted into equation (63) to get a majorizing function that satisfies equations (66) and (67) and can be easily minimized with respect to ft
  • equation (76) is set to zero to get the desired update for the right ventricle tissue curve, which is ,J (77)
  • n 0, 1 ,2, . . .
  • the desired parameter estimates are the estimates from the CLA algorithm that produce the smallest least-squares error. It should be noted that the objective function Le decreases monotonically with increasing iterations.
  • IC Step 2 Using a ID line search, such as the golden section method [17], get p new
  • Step 3 Set p old — p ew and repeat the above two steps, for a set number of iterations or until some desired stopping criterion is met.
  • H (z)f ⁇ °' (z) where f ⁇ and f ' denote the initial right and left ventricle tissue curves, and H(z) is given by equation (58). Therefore,
  • Initial g, p - Step 1 Using an input-output system identification method, such as the Steiglitz-McBride algorithm, find the parameters bo, ai, a 2 , ... , a m that provide the best least-squares fit for the ing model:
  • the set of parameters that produce the smallest least-squares error are the desired initial estimates for the parameters g, p, m, and d, which we denote respectively as g*°>, ⁇ ⁇ 0 ', m ⁇ °>, and d (0) .
  • the initial estimates for the parameters m and d are not explicitly used in the CLA algorithm. However, they could be used to reduce the range of values considered for the parameters m and d.
  • the "for loop" for the delay in the CLA algorithm could be instead: for a— d 0 ⁇ — ⁇ : d ' 0) + ⁇ , where A> 0, would be an integer chosen by the user.
  • ⁇ fy ⁇ are the values of the factor curves and the coefficients ⁇ m ⁇ are the factor weights.
  • the primary factor curves for conventional MBF estimation applications are the right ventricle (RV), left ventricle (LV), and myocardial tissue curves, which model the ammonia concentration as a function of time in the RV, LV, and myocardium, respectively.
  • the mathematical task is to find both the factor curves and weights so that the linear combination of factor curves for every voxel in the myocardium matches the corresponding measured time activity curve (TAG) as close as possible. This problem is constrained by requiring that the factor curves and weights all be nonnegative.
  • K-FADS-n MODEL [00145]
  • K-FADS-n MODEL we first present a model that combines Hutchins' pharmacological kinetic model with the standard factor analysis model (see equation (3)). The resulting model is an improvement over an earlier version discussed above, so we call it the second pharmacological kinetics based FADS model (K-FADS-II).
  • K-FADS-II the second pharmacological kinetics based FADS model
  • K-FADS-II the second pharmacological kinetics based FADS model
  • discrete-time K-FADS-II model is obtained by applying the bilinear transform to the continuous-time K-FADS-II model. It should be noted that there is a simple relationship between the discrete-time and continuous-time K-FADS-II parameters.
  • the first term in (6) is identified as the amount of spillover from the RV, and the second term is a combination of the ammonia activity in blood vessels within the myocardium and spillover from the LV. More specifically, the constant a,i accounts for the amount of the ammonia activity in voxel that is due to the blood plasma in the RV. Further, the constant a,2 accounts for the amount of the ammonia activity in voxel i that is due to the blood plasma in the LV (i.e., LV spill over) and blood plasma in the blood vessels of the myocardium.
  • the third and fourth terms in (6) model the free and trapped ammonia activity in the myocardium, respectively.
  • the images are separated into three sets of voxels that lie in the RV, LV, and myocardium such as that described by V. Appia, B.
  • uniform samples of the TACs can be obtained from non-uniform samples of the TACs via a suitable interpolation using known methods.
  • H(s) the system function of a continuous-time system
  • H z the system function of a discrete-time system
  • Lei x[n] be an arbitrary discrete-time sequence.
  • the bilinear transformation has the property that it maps a stable continuous-time system into a stable discrete-time system. Moreover, the bilinear transformation avoids the problem of aliasing by mapping the / ⁇ axis into the unit circle of the complex plane.
  • frequency warping occurs as a result of mapping the entire fil axis into the unit circle. Note, the frequency warping problem can be ameliorated by choosing a sufficiently high sampling rate ⁇ .
  • the parameters k lt are the desired MBF parameters.
  • the parameters r, I, c, ki, and ki can be viewed as nuisance parameters because they must be accounted for in the analysis e ven though they are not of direct interest,
  • IV-MBF Improved Voxel Resolution MFB
  • the IV-MBF algorithm is based on a model that accounts for the fact that the shape of TACs due to ischemic and normal tissue are different. In fact, the model allows for tissue curves that represent free and trapped ammonia to be voxel dependent and physiologically appropriate.
  • TACs in ischemic and normal tissue can be modeled as a linear combination of the same three factors. We believe the use of a more appropriate model offers the possibility of more accurate MBF estimates than available methods.
  • the IV-MBF algorithm performed well in a limited study using real patient data.
  • the resu lts of the study suggest that the IV-MBF algorithm is robust and w ould perform well in practice, where MBF values due to ischemic and normal tissue can vary over a wide range.
  • MM algorithms are a viable approach provided a majorizing function can be found that is easier to minimize than the original objective function. Assuming that g is a suitable majorizing function for/, the corresponding MM algorithm is
  • Step 1 A simpler alternative to Step 1 is to use the MM technique described above to obtain iteraies a n( j - ) t
  • iat decrease the least-squares objective L in the sense that, for all m 0, 1 , 2, i(r (mM) , c ⁇ m >, (m) , k 2 (m) , k 3 ⁇ m) ) ( 33)
  • Step 1 a Find iterates &' L ' ) and / (,K+i) such that
  • Step la can be obtained by solving the following optimization problem:
  • Equation (45) has not consistently resulted in accurate estimates for the sampled RV tissue cm've. We believe the inaccuracy may be due to the fact that the myocardium TACs contains limited iniormaiion about the RV tissue curve. Therefore, although the update in equation (45) is theoretically meaningful, it may be advisable to use the estimate for the RV tissue curve obtained from the algorithm described below based on a semi-parametric model.
  • Step 2 The minimization in Step 2 is equivalent to the following / minimization problems: For z ⁇ 1, 2, ... , / , c( ⁇ ) r (r i+ l) [3 ⁇ 4] + c (m) ( ?Ti + i) [ ]
  • Step 3 The minimization in Step 3 is equivalent to the followmg / one-dimensional minimization problems: For i 1,2,...,/,
  • equation (51) can be conveniently expressed as:
  • 0 j (m+1) argmin ⁇ [di[ ] - (c i( iS ⁇ +1) [n] + c i>2 s ⁇ +l [n] + k itl sQ +l [n] arq mm
  • Step 3 requires the solution to equation (56) for all i - ----- ⁇ , 2, .. ., I.
  • Step 3a we use a majorizing function for a certain class of linear LS objective functions that was put forth by De Pierro in 'On the relation between the ISRA and the EM algorithms for positron emission tomography," IEEE Transactions on Medical Imaging, vol. 12, pp. 328-333, 1993, which is incorporated by reference. Given that 0 £ , d it and the matrix A are nonnegative, De Pierro' s result can be used to obtain th )
  • N;- is the intersection of the set of eight nearest voxels to the i ih voxel and the set of voxels that lie in the myocardium. Note, penalty functions could also be used to enforce smoothness on the other parameters c, la, and ks.
  • n o l and ⁇ are unknown constants.
  • this model is a semi-parametric model because only the sampled LV tissue curve values for n > no,! are described by a parametric model.
  • r(0) initial RV tissue curve estimate
  • the parameter n Q r is unknown.
  • the parameter ⁇ must satisfy the constraint 0 ⁇ ⁇ 1.
  • Step la an alternative approach to Step la are the following two steps:
  • Model Based Step la Find an iterate ' lj such that:
  • N x 1 vectors s°" and s? ld denote an estimate of the semi-parametric RV and L V tissue curves, respectively: s? ld [s? w [0; 3 ⁇ 4f ., ⁇ 01 ⁇ ], s? ld [l; 3 ⁇ 4f ., ⁇ ° ⁇ ] s? ld [N - l;3 ⁇ 4 , ? oid ] (75) r ( ⁇ ) [0], r (0) [1], ...
  • v follows from equation (44) and is equal to dq L ⁇ (ig, ⁇ >, cTM, k ( k >, > s r m) , sf m) )
  • d avg ,r[n] and davgjfnjto be the average of the RV and LV TACs, respectively.
  • n r ,max and ni.max equal the time points where dav S ,r[n] and d avg [nj equal their maximum values, respectively.
  • «r,*e//represenf the time point after n r ,mox where d avg ,r[n] is halfway down from its maximum value (note: nika!f is defined similarly).
  • nika!f is defined similarly.
  • Another approach would be to first compute the K-FADS-II parameters for each (n 0 r , n o i ) pair that comes from a set of ( ⁇ , 3 ⁇ 4, , , ) pairs, such as the set
  • RV and LV TACs Due to the motion of the heart and the finite resolution of PET imaging, the RV and LV TACs are corrupted by activity in the myocardium so they do not decay to zero. Keeping in mind the reasoning behind the semi-parametric model for the R V and LV tissue curves, the initial RV and LV tissue curves are chosen to be
  • n n Q r + 1 , ? ⁇ ⁇ + 2, . , ,, N- 1 do
  • the dynamic PET data set comes from scanning an unhealthy patient at rest using a standard protocol consisting of 20 scans of duration 3 sec, followed by 5 scans of duration 12 see, and ending with 6 scans of duration 30 sec (i.e., 31 sub-scans in toial).
  • the scanner used in the study has 22 planes and the reconstructed cardiac images are of size 42 x 30.
  • the data set consists of 31 images of size 42 x 30 per plane.
  • the measured myocardium data ⁇ 3 ⁇ 4/ ⁇
  • a popular approach is to first assume the measured myocardium TAC data is nearly piece-wise linear. Under this assumption and as discussed in W. G. Kuhle, G. Porenta, S. C. Huang, D. Buxton, S. S. Gambhir, H. Hansen, M. E. Phelps, and H. R. Schelbert, "Qitantification of regional myocardial blood flow using 13N-ammonia and reoriented dynamic positron emission tomographic imaging," Circulation, vol. 86, pp. 1004-17, 1992, which is incorporated by reference, it follows that the values of ai(t) at the midpoints of the sub-scan windows are approxim tely equal to
  • the regularly sampled myocardium TAC data can be estimated from the measured myocardium TAC data using interpolation.
  • T 3 /2) ⁇ . and linear interpolation to obtain estimates for the regularlv sampled myocardium TACs. Note, in our experiment we used T s 0.5 sec for the sampling interval. It should be mentioned that the approach described above for generating regularly sampled myocardium TACs from the measured myocardium TAC data would also be used to generate the regularly sampled RV and LV TACs,
  • Figs. 3 and 4 Some results for the IV-MBF algorithm are shown in Figs. 3 and 4.
  • Fig. 3 the MBF estimates for the 9 th plane are displayed as an image in Fig. 3 A while the corresponding myocardium mask, which specifies the myocardium voxels, is shown in Fig. 3B.
  • the PET image that results from averaging the images for the last six sub-scans and the histogram of the MBF estimates are shown in Figs. 3C and 3D, respectively.
  • the same information for plane 16 is provided in Fig. 4. Because the data is from a patient the true MBF values are not known. However, Dr. Votaw reviewed the results and considered them to be promising and noted that the range of the MBF estimates was consistent with his expectation and human physiology.
  • Fig. 5 A is a plot of the measured interpolated TAG (solid) and the corresponding estimated TAG using the TV-MBF algorithm (dashed- dotted). Also, the signal components due to activity in the RV (dashed) and LV (dotted), and activity in the free (solid) and trapped ( dashed- dotted) states are shown in Fig. 5B. Lastly, a plot of the LS objective function, which decreases monotonically as expected, is shown in Fig. 5C.
  • zi[ti], and w[n] be positive, casual sequences of length K.
  • the term (zi[ti] + Z2'[n] * w[n]) 2 can be expressed as
  • FIG. 6 a system 400 is illustrated that may be u sed for any such implementations.
  • One or more components of the system 400 may be used for implementing any system or device mentioned above, such as for example even a handheld device.
  • the use of the system 400 or any portion thereof is not necessarily required.
  • the system 400 may include, but is not required to include, a central processing unit (CPU) 410, a random access memory (RAM) 420, and a mass storage unit 430, such as a disk drive.
  • the system 400 may be coupled to, or integrated with, any of the other components described herein, such as an input device 450, 460 and other input device 470.
  • the system 400 comprises an example of a processor based system.
  • the CPU 410 may be used to execute or assist in executing the s teps of the methods and techniques described herein.
  • the system 400 may further comprise a graphics processing unit to execute or assist in executing the steps of the methods and techniques described herein.
  • the input device 450 may comprise a first touch sensitive panel and the input device 460 may comprise a second touch sensitive panel.
  • the system 400 comprises another input device 460 that may comprise other user input means such as buttons, keyboard, mouse, joy stick, and the like.
  • other input device 460 may further comprise output means, such as displays, sound emitters, light emitters, and the like configured to provide feedback or output to a user.
  • one or more of the input device 450, input device 460 and other input device 470 comprise display functionality.
  • various program content, images, shadows, lighting, and the like may be rendered on one or more of the input device 450, 460 and other input device 470.
  • the mass storage unit 430 may include or comprise any type of computer readable storage or recording medium or media.
  • the computer readable storage or recording medium or media may be fixed in the mass storage unit 430, or the mass storage unit 430 may optionally include external memory 470, such as a digital video disk (DVD), Blu-ray disc, compact disk (CD), USB storage device, floppy disk, or other media.
  • DVD digital video disk
  • CD compact disk
  • USB storage device floppy disk
  • the mass storage unit 430 may comprise a disk drive, a hard disk drive, flash memory device, USB storage device, Blu-ray disc drive, DVD drive, CD drive, floppy disk drive, and the like.
  • the mass storage unit 430 or external memory 470 may be used for storing program code or macros that implement the methods and techniques described herein.
  • external memory 470 may optionally be used with the mass storage unit 430, which may be used for storing program code that implements the methods and techniques described herein.
  • any of the storage devices such as the RAM 420 or mass storage unit 430, may be used for storing such program code.
  • any of such storage devices may serve as a tangible computer readable storage medium for storing or embodying a computer program for causing a console, system, computer, or other processor based system to execute or perform the steps of any of the methods, code, and/or techniques described herein.
  • any of the storage devices, such as the RAM 420 or mass storage unit 430 may be used for storing any needed database(s), gestures, lists, macros, etc.
  • one or more of the embodiments, methods, approaches, and/or techniques described above may be implemented in a computer program executable by a processor based system.
  • processor based system may comprise the processor based system 400, or a computer, console, graphics workstation, and the like.
  • Such computer program may be used for executing various steps and/or features of the above-described methods and/or techniques. That is, the computer program may be adapted to cause or configure a processor based system to execute and achieve the functions described above.
  • such computer program may be used for implementing any embodiment of the above-described steps or techniques for performing a task at the handheld device.
  • such computer program may be used for implementing any type of tool or similar utility that uses any one or more of the above described embodiments, methods, approaches, and/or techniques.
  • the computer program may comprise a computer simulation, or system software such as an operating system, BIOS, macro, or other utility.
  • program code macros, modules, loops, subroutines, etc., within the computer program may be used for executing various steps and/or features of the above- described methods and/or techniques.
  • the computer program may be stored or embodied on a computer readable storage or recording medium or media, such as any of the computer readable storage or recording medium or media described herein.
  • a computer program product comprising a non- transitory medium for embodying a compuier program for input to a compuier and a computer program embodied in the medium for causmg the computer to perform or execute steps comp rising any o ne or more of the steps involved in any one or more of the embodiments, methods, approaches, and/or techniques described herein.

Abstract

L'invention concerne une analyse de flux sanguin myocardique par scanner comprenant l'incorporation d'un modèle cinétique pharmacologique au modèle d'analyse de coefficients standard, chaque courbe d'activité dans le temps étant présumée être une combinaison linéaire de courbes de coefficients. Un modèle K-FADS - utilisant une analyse de coefficients des structures dynamiques à base de cinétique pharmacologique - peut être appliqué, l'évaluation des courbes de coefficients dans le myocarde pouvant être physiologiquement riche d'enseignement. Des aspects facultatifs supplémentaires comportent la réalisation d'une discrétisation pour transformer un modèle K-FADS-II continu dans le temps en un modèle K-FADS-II discret dans le temps, et l'application d'un algorithme de flux sanguin myocardique par résolution de voxels (V-MBF) itératif amélioré. Là où le modèle est appliqué sans présomption qu'une courbe de tissu du ventricule droit et qu'une courbe de tissu du ventricule gauche n'obéissent à une relation mathématique particulière, au moins une fonction objective des carrés peut être appliquée pour obtenir des estimées de paramètres du modèle cinétique pharmacologique par l'application d'une technique d'optimisation majoration-minoration pour l'estimation itérative des courbes.
PCT/US2014/059062 2011-03-29 2014-10-03 Analyse du flux sanguin myocardique par résolution de voxels WO2015051256A1 (fr)

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