| WO/2024/026004 | METHOD AND SYSTEM FOR AUTOMATIC HER2 SCORING |
| JP6745400 | Data acquisition device for assistance device |
| WO/2021/039889 | PATHOLOGY IMAGE MANAGEMENT SYSTEM |
CRAFTS EVAN (US)
| CLAIMS What is claimed is: 1. A method of performing a diagnostic scan of a subject, comprising: defining a set of data acquisition parameter sequences; defining a set of upper bounds and a set of lower bounds for each of the set of data acquisition parameter sequences; defining a set of representative tissue parameters for a tissue of the subject; selecting a set of basis functions; calculating values for a set of basis function coefficients to yield a piecewise polynomial representation of each of the set of data acquisition parameter sequences within the sets of upper and lower bounds based on the set of desired tissue parameters; and performing a diagnostic scan of the subject using the calculated data acquisition parameter sequences. 2. The method of claim 1, wherein the diagnostic scan is a magnetic resonance imaging (MRI) scan. 3. The method of claim 2, wherein the diagnostic scan is a magnetic resonance fingerprinting scan. 4. The method of claim 1, wherein the set of data acquisition parameter sequences are selected from flip angles, radiofrequency phases, repetition times, echo times, time-bandwidth products, or k-space sampling locations. 5. The method of claim 1, wherein the set of basis functions is selected from B-splines, wavelets, radial basis functions, or Fourier basis functions. 6. The method of claim 5, wherein the set of basis functions is a B-spline basis function. 7. The method of claim 6, further comprising the step of selecting a degree of the B-spline basis functions to use in the step of calculating the piecewise polynomial. 8. The method of claim 7, wherein the degree of the B-spline basis function is selected using a machine learning algorithm. 9. The method of claim 1, further comprising the steps of: generating a set of possible values for each of the set of basis function coefficients; iterating through each combination of the possible values in the set of possible values to generate a set of piecewise polynomials; calculating a result of a simulated diagnostic scan for each piecewise polynomial in the set of piecewise polynomials to create a multidimensional set of results; and calculating a maximum value among the multidimensional set of results using an algorithm to yield the piecewise polynomial representation; wherein the step of calculating the values of the set of basis function coefficients to yield the piecewise polynomial comprises executing an iterative algorithm. 10. The method of claim 9, wherein the iterative algorithm is selected from a sequential quadratic programming algorithm, an interior point algorithm, a genetic algorithm, and a simulated annealing algorithm. 11. The method of claim 10, wherein the algorithm is a sequential quadratic programming algorithm. 12. The method of claim 9, wherein the algorithm comprises setting a tolerance threshold and terminating the algorithm when the norm of a gradient of the multidimensional set of results is less than the tolerance threshold. 13. The method of claim 9, wherein the simulated diagnostic scan is a Bloch simulation. 14. The method of claim 1, further comprising selecting a set of time offsets for the set of basis functions to calculate the piecewise polynomial. 15. The method of claim 1, wherein the tissue of the subject is selected from liver, spleen, kidney medulla, kidney cortex, kidney, skeletal muscle, fat, myocardium, abdomen, lungs, stomach, intestines, brain, or blood. 16. The method of claim 1, wherein the step of defining the set of upper and lower bounds comprises calculating at least one of the upper and lower bound based on a desired total acquisition time or a specific absorption rate. 17. The method of claim 1, wherein the step of calculating values for each of the set of basis function coefficients to yield a piecewise polynomial representation of each of the set of data acquisition parameters comprises constructing the piecewise polynomials using the Cox-de Boor recursion formula. |
for n = 1, ⋯ , N . In matrix-vector form, Eq. [8] and Eq. [9] can be rewritten as denotes the B-spline coefficient matrix for FA with denotes the B-spline coefficient matrix for TR with and g ∈ contain the B-spline coefficients for the FA and TR sequences, respectively. [0070] Note that Equation 10 and Equation 11 enforce the subspace constraints for the acquisition parameter sequences, and that the two subspaces spanned by the columns of B α and B TR are controlled by the degrees and the knot sequences of the corresponding B-spline basis functions. According to the Schoenberg-Whitney condition (Höllig & Hörner, Approximation and Modeling with B-Splines. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2013), the dimensions of the two subspaces are L α and L TR , which in practice can be chosen to be much smaller than N to represent the desired acquisition sequences. In some embodiments, the dimensions used for the acquisition parameter sequences disclosed herein may be 5, 10, 20, 25, 50, 100, or any other suitable dimension. Integrating the subspace constraints with the upper and lower bounds on the FAs and TRs, the proposed constraint set can be written as Equation 12 [0071] With the low-dimensional B-spline subspace constraints, an provide a much smaller search space than n Equation 3, while respecting the underlying physical constraints. This method of calculating the acquisition sequences greatly improves the computational efficiency of solving the OED problem using numerical algorithms. [0072] In various embodiments, different degrees of B-spline basis functions may be chosen to construct piecewise polynomial representations of time series functions. In some embodiments, the degree may be chosen manually, for example based on prior knowledge of existing time series functions and the typical shapes those functions take – for example, if a particular time series was a binary sequence of square waves oscillating between two values, a degree of 0 may be chosen. In another example, if a particular time series is typically a sawtooth wave with linear progressions between two or more values, a degree of 1 may be used, and so on. In some embodiments, the ideal degree for a given parameter may be algorithmically chosen, for example using a machine learning algorithm or via an algorithmic function of a number of inputs, including but not limited to parameters related to the subject(s) being imaged, the devices being used in the imaging, and/or the target tissue(s) of the imaging. [0073] Certain embodiments disclosed herein include the calculation of improved time series functions, for example stimulus patterns over a fixed time sequence, using a set of basis functions and accompanying coefficients, for example B-spline basis functions with magnitude/damping/gain coefficients and knot/phase coefficients. It is understood that in other contemplated embodiments, other basis functions may be used to calculate improved time series, including but not limited to wavelets, radial basis, Fourier basis, or any other set of basis functions. [0074] Combining the equations disclosed above, the OED problem can be formulated as follows:
[0075] Solutions to Equation 13 can be obtained using a number of algorithms, such as nonlinear programming (Nocedal & Wright, Numerical Optimization. New York: Springer, 2nd edition, 2006) or stochastic programming methods (Spall JC, Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control. New York, NY: John Wiley & Sons, 2003). As in (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861), a state-of-the-art nonlinear programming algorithm is applied, which in some embodiments is sequential quadratic programming (SQP), to solve this problem in Equation 13. The SQP algorithm is a gradient- based iterative algorithm which, at each iteration, solves a constrained quadratic programming problem in which the objective function is a quadratic approximation of the original objective function and the constraint set is the same as Equation 13. Other exemplary algorithms which may be used to solve the equation in Equation 13 include, but are not limited to, other nonlinear programming algorithms (e.g., interior point algorithms) and stochastic programming algorithms (e.g., genetic algorithms and simulated annealing algorithms). [0076] The dominant computational cost of the SQP algorithm for solving Equation 13 is the gradient evaluation, which involves the simulation of a Bloch equation based dynamical system. While there are several methods to calculate the gradient of an objective function (Nocedal & Wright, Numerical Optimization. New York: Springer, 2nd edition, 2006), e.g., finite differencing and automatic differentiation, an OED problem with a reduced search space generally enables more efficient gradient calculation. For example, the computational complexity of gradient calculation via finite differences is linear with respect to the dimension of the search space in Equation 13, which, in theory, would provide a factor of improvement in computational efficiency, as compared to the OED without the subspace constraints. EXPERIMENTAL EXAMPLES [0077] The invention is further described in detail by reference to the following experimental examples. These examples are provided for purposes of illustration only, and are not intended to be limiting unless otherwise specified. Thus, the invention should in no way be construed as being limited to the following examples, but rather, should be construed to encompass any and all variations which become evident as a result of the teaching provided herein. [0078] Without further description, it is believed that one of ordinary skill in the art can, using the preceding description and the following illustrative examples, make and utilize the system and method of the present invention. The following working examples therefore, specifically point out the exemplary embodiments of the present invention, and are not to be construed as limiting in any way the remainder of the disclosure. Example #1 – Solution Algorithm Methods [0079] First, the performance of the disclosed approach in optimizing a representative MR Fingerprinting pulse sequence was evaluated, i.e., the inversion recovery fast imaging with steady state precession (IR-FISP) sequence (Jiang et al., Magn Reson Med 2015;74:1621–1631), for neuroimaging. The disclosed approach was compared with the state-of-the-art methods in (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861) using numerical simulations, phantom experiments, and in vivo experiments. The related details are provided in the following subsections. Implementation Details [0080] The implementation of the disclosed approach is described below. First, the tissue parameters were specified for the design metric in Equation 13. Three representative tissues relevant to neuroimaging applications were selected for the experimental design, i.e., [1100 ms, 102 ms, 0.6] T . Furthermore, tissue relaxation times (i.e., T 1 and T 2 ) were assumed to be of primary interest, and so [ω 1 , ω 2 , ω 3 ] = [3.45,1,1.6] were manually chosen for the weighting matrix W (l) with diagonal entries to ensure that the optimized pulse sequence achieved good performance for different tissue components of interest. [0081] The bound constraints for the design acquisition sequences were then specified. For the FA sequence, the lower bounds: upper bounds were imposed. Here, the upper bound on the first flip angle allowed the use of a 180 ∘ inversion pulse, which is often advantageous to the T 1 estimation. Moreover, the upper bounds and lower bounds for the TR sequence were set as 15 ms for 1 ≤ n ≤ N . Note that the above bound constraints are consistent with the range of FAs and TRs in conventional MR Fingerprinting experiments (Jiang et al., Magn Reson Med 2015;74:1621– 1631). [0082] The B-spline subspace constraints were also specified. These constraints, which enforce the desired structure of the FA and TR sequences, are controlled by the degrees and knot sequences of the B-spline basis functions. To exploit the special structure of the optimized FA and TR sequences (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861), in the present disclosure, the respective degrees of the B-spline basis functions were set as d α = 3 and d TR = 0. Moreover, uniformly spaced knots in the interval [1, N ] were used for the FA and TR sequences such that the dimensions of the two subspaces are L α = L TR = N/20. Here repeated knots were used at the boundary of the interval, which ensured that the B-spline basis functions formed a complete basis for the subspaces. Additionally, the first flip angle was excluded from the spline representation to allow a potential discontinuity from an inversion pulse to the subsequent RF pulse. As described in the Results section below, the above B-spline parameters consistently yielded good empirical results for the OED problem. Further discussions on the parameter selection are included below in the Discussion section. [0083] The SQP algorithm was applied to solve the OED problem in Equation 13. The SQP algorithm requires the gradient evaluation at each iteration, for which a finite difference approximation was used. This requires a large number of objective function evaluations at each iteration, which involves extensive Bloch simulations. In some embodiments, the number of evaluations required is on the order of 2N, where N is the acquisition length, i.e., the number of RF pulses applied. In general, the cost is on the order of N*D, where D is the number of optimizable acquisition parameter sequences. [0084] Bloch simulations were performed using the summation-of-isochromats approach (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861; Maidens et al., Parallel dynamic programming for optimal experiment design in nonlinear systems. In: Proc. IEEE Conf. Decis. Control, 2016. pp.2894–2899; Malik et al., Equivalence of EPG and isochromat-based simulation of MR signals. In: Proc. Int. Symp. Magn. Reson. Med., 2016. p.3196) with a total number of 400 isochromats. The SQP algorithm was initialized by performing a least-squares fit of the B-spline coefficients to the FA and TR sequences in the conventional MR Fingerprinting experiments (Jiang et al., Magn Reson Med 2015;74:1621–1631), and the algorithm was terminated when the norm of the gradient was less than the pre-specified tolerance ∈ = 10 -6 or the maximum number of iterations N Iter = 4 × 10 4 was reached. In some embodiments, the maximum iteration is chosen as a number large enough to comfortably reach the selected pre- specified tolerance, while also restricting the algorithmic runtime to a reasonable duration in case the algorithm fails to converge. Finally, the algorithm was implemented on a Linux server with dual Intel Xeon Gold 6248R processors (each with 3.00 GHz and 24 cores) and 1.50 TB RAM running MATLAB ® R2020a. Parallel computing was used for the gradient evaluation across all 48 cores. Method Comparison [0085] The disclosed approach was compared with the two state-of-the-art methods described in (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861). The first method, referred to herein as Optimized-I, formulates the OED problem as follows: [0086] Note that this method optimizes the same objective function as the disclosed approach, except it only imposes the bound constraints on the FAs and TRs. [0087] It has been demonstrated that constraining the flip angle variation in the experimental design is often beneficial for parameter decoding from highly-undersampled MR Fingerprinting experiments (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861). This leads to the following OED formulation: Equation 16 where the maximum FA variation is set as as in (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861). This approach is referred to herein as Optimized-II. It is clear that Optimized-II also uses the same objective function as the disclosed approach, but with a different constraint set, i.e., n Equation 3. Here, by constraining the flip angle variation, Optimized-II reduces the highly oscillatory behavior of the magnetization evolutions, which has been shown to improve the accuracy of parameter estimation (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861). As disclosed in the Results section below, the disclosed approach provides similar benefits as Optimized-II, but uses completely different constraints, i.e., the B-spline based subspace constraints, which results in a much smaller search space for the OED problem. [0088] The disclosed approach was compared with Optimized-I and Optimized-II in terms of computational efficiency and estimation accuracy. The same algorithm was used (i.e., SQP) as the disclosed method to solve Optimized-I and Optimized-II with an identical algorithmic setup (e.g., initialization and stopping criteria). The three approaches were evaluated on the same computing platform as described above. Numerical Simulations [0089] Numerical simulations were conducted to evaluate the disclosed approach. Specifically, a numerical brain phantom was constructed using T 1 , T 2 , and spin density maps from the Brainweb database (Collins et al., Design and construction of a realistic digital brain phantom. IEEE Trans Med Imaging 1998;17:463–468), which models a single-channel MR Fingerprinting experiment. The field-of-view (FOV) was set as 300 × 300 mm 2 and the matrix size as 256 × 256. he IR- FISP sequence was simulated to generate contrast-weighted images with the acquisition parameters from conventional experiments (Jiang et al., Magn Reson Med 2015;74:1621–1631), Optimized-I, Optimized-II, and the disclosed approach at two acquisition lengths: N = 400 and 600. [0090] The undersampled k-space data was synthesized from the contrast-weighted images using a nonuniform Fourier transform (Fessler & Sutton, Nonuniform fast Fourier transforms using min-max interpolation. IEEE Trans Signal Process 2003;51:560–574). As in (Jiang et al., Magn Reson Med 2015;74:1621–1631), one spiral interleaf was acquired (out of the full set of 48 interleaves) per each TR in the simulation. Complex-valued white Gaussian noise was added to the measured k-space data with a pre-specified signal-to-noise ratio level. Here the signal-to- noise ratio was defined as SNR = 20log 10 ( s/σ) , where s denotes the average value of spin density in a region of white matter and σ denotes the noise standard deviation. Specifically, in the disclosed experiment, the numerical simulations were performed at SNR = 33 dB. Although various examples presented herein specify an SNR at a given value, for example 33 dB, it is understood that noise variance appears in the algorithm as a global scaling constant, and so does not in some embodiments impact the performance of the experimental design. [0091] The tissue parameter maps were reconstructed from the undersampled k-space data using a maximum likelihood (ML) reconstruction (Zhao et al., Maximum likelihood reconstruction for magnetic resonance fingerprinting. IEEE Trans Med Imaging 2016;35:1812–1823), in which a dictionary for the magnetization evolutions was generated with T 1 and T 2 values relevant to neuroimaging. Specifically, the value of T 1 was in the range [20,3000] ms, with an increment of 10 ms for [20,1500] ms and an increment of 30 ms for [1501,3000] ms. The value of T 2 was in the range [30,500] ms, with an increment of 1 ms for [30,200] ms and an increment of 5 ms for [201,500] ms. [0092] The reconstruction accuracy was evaluated using the following two metrics: (a) the overall relative error, i.e., where respectively denote the ground truth and reconstructed tissue parameter maps, and (b) the voxelwise relative error, i.e., where are the values of the ground truth and reconstructed tissue parameter maps at the vth voxel, respectively. The bias and variance of the reconstructed tissue parameter maps were evaluated using Monte Carlo (MC) simulations. Specifically, MC simulations were constructed with 100 trials and evaluating the normalized bias: Equation 17 and the normalized standard deviation: Equation 18 where denotes the sample mean over the MC trials. Phantom Experiments [0093] Phantom experiments were performed to evaluate the disclosed approach. Specifically, a physical phantom was used that consisted of 12 tubes filled with solutions of gadolinium and agar at different concentrations, creating different combinations of T 1 and T 2 values relevant to neuroimaging. The experiments were conducted on a 3T Siemens Magnetom Prisma scanner (Siemens Healthineers, Erlangen, Germany) equipped with a 20-channel head receiver coil. Here the acquisition parameters from the conventional experiment were used (Jiang et al., Magn Reson Med 2015;74:1621–1631), Optimized-I, Optimized-II, and the disclosed approach at acquisition length N = 400. Other relevant imaging parameters included FOV = 300 × 300 mm 2 , matrix size = 256 × 256, and slice thickness = 5 mm. [0094] The MR Fingerprinting experiments were carried out using the same set of spiral trajectories as in (Jiang et al., Magn Reson Med 2015;74:1621–1631), along with a pilot scan using the vendor-provided gradient echo sequence for the coil sensitivity map estimation. The tissue parameter maps were reconstructed using ML reconstruction (Zhao et al., Maximum likelihood reconstruction for magnetic resonance fingerprinting. IEEE Trans Med Imaging 2016;35:1812–1823) with a slice-profile corrected dictionary (Ma et al., Slice profile and B_1 corrections in 2D magnetic resonance fingerprinting. Magn Reson Med 2017;78:1781–1789) containing the same T 1 and T 2 resolutions as in the numerical simulations. To assess the estimation performance associated with different experimental designs, the imaging experiments were repeated 15 times for each acquisition scheme and the voxelwise normalized standard deviation of the reconstructed tissue parameters was evaluated. Note that the above procedure is often employed to evaluate the variance of tissue parameters in experimental studies for quantitative MRI (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861; Nataraj et al., Optimizing MR scan design for model-based T1, T2 estimation from steady-state sequences. IEEE Trans Med Imaging 2017;36:467–477). In Vivo Experiments [0095] In vivo experiments were conducted to further evaluate the disclosed approach. Two healthy subjects were recruited with the approval of the local Institutional Review Board and informed consent obtained from both subjects. The imaging experiments were conducted on the same MR scanner and with the same set of imaging parameters as the phantom experiments. The conventional, Optimized-I, Optimized-II, and disclosed approaches were evaluated at acquisition length N = 400, and each experiment was repeated 15 times. The ML reconstruction was performed and the voxelwise normalized standard deviation of the reconstructed parameter maps were calculated as in the phantom experiments. Results [0096] Representative results are shown to illustrate the performance of the disclosed approach. First, the superior computational efficiency of the disclosed approach was demonstrated. Specifically, the MR Fingerprinting experiments were calculated using Optimized-I, Optimized- II, and the disclosed approach at acquisition lengths N = 300,400,500, and 600, and the algorithm runtimes recorded below in Table 1. Note that the above approaches optimize the same objective function but with different constraint sets. As can be seen, with a reduced search space enabled by the subspace constraints, the disclosed approach provides a two-order-of-magnitude improvement of the computational efficiency over Optimized-I and Optimized-II. Remarkably, it solves the OED problem in approximately one minute, which significantly enhances the practical utility of the disclosed approach. The times below are recorded in minutes and the acquisition lengths are the number of RF pulses applied during the imaging experiment. [0097] Fig.3A, Fig.3B, Fig.3C, and Fig.3D show the optimized FA and TR sequences using the above approaches at N = 400, with the corresponding magnetization evolutions shown in Fig.4A, Fig.4B, Fig.4C, and Fig.4D. For all the acquisition schemes, the first RF pulses were 180 ∘ , which exceeded the scales of the vertical axes. The total acquisition times for the conventional, Optimized-I, Optimized-II, and disclosed approaches are 5.30 sec, 5.30 sec, 5.25 sec, and 5.22 sec, respectively. As can be seen, with the subspace constraints that enforce the desired structure, the disclosed approach produces very similar FA and TR sequences as Optimized-II. As in the early work (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861), Optimized-II yields highly structured FA and TR sequences, and by constraining the FA variation, it overcomes the oscillatory behavior of the magnetization evolutions in Optimized-I. With the spline subspace constraints that enforce the desired structure, the disclosed approach results in similar FA and TR sequences and magnetization evolutions as Optimized-II but with significantly better computational efficiency. Similar observations have been obtained at other acquisition lengths. [0098] In another example, an acquisition length of N=600 was used. Fig.5A, Fig.5B, Fig.5C, and Fig.5D show the FA and TR sequences for the conventional (Fig.5A), Optimized-I (Fig. 5B), Optimized-II (Fig.5C), and disclosed (Fig.5D) approaches with N = 600. As in the N = 400 case (see Fig.3A, Fig.3B, Fig.3C, and Fig.3D in the main text), the disclosed approach yields optimized acquisition parameters that are similar to those from Optimized-II. Fig.6A and Fig.6B show the reconstructed tissue parameter maps from the numerical simulations using the acquisition parameter sequences at N = 600 and SNR = 33 dB. Along with the reconstructions, the underlying ground truth, associated error maps, and overall relative errors are also shown. As can be seen, the disclosed approach provides improved estimation performance compared to the conventional and Optimized-I approaches and the performance of the disclosed approach is similar to that of Optimized-II. [0099] Next, competitive estimation performance of the MR Fingerprinting experiments improved by the disclosed approach is discussed. Fig.7A and Fig.7B show the reconstructed T 1 and T 2 maps and the corresponding error maps at N = 400 and SNR = 33 dB in the simulations. Consistent with the early observation in (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861), Optimized-II improves the reconstruction accuracy over Optimized-I by controlling the flip angle variation. With the subspace constraints, the disclosed approach provides similar improvement over Optimized-I. Note that the overall relative errors are displayed at the lower right corner of each error map. As can be seen, the disclosed approach provides comparable estimation performance as Optimized-II, while enabling a substantial improvement of the T 2 accuracy and slightly improving the T 1 accuracy over the conventional method. Also note that with smooth magnetization evolutions, both the disclosed approach and Optimized-II offer improvements over Optimized-I. [0100] Fig.8A and Fig.8B show the bias-variance analysis of the reconstructed MR tissue parameter maps using the acquisition parameters from the conventional experiment, Optimized-I, Optimized-II, and the disclosed approach at N = 400 and SNR = 33 dB. The normalized biases and standard deviations are shown for the T 1 reconstruction (Fig.8A) and the T 2 reconstruction (Fig.8B). The regions associated with the background, skull, scalp, and CSF are set to zero. It is clear that, compared to the conventional experiment (Jiang et al., Magn Reson Med 2015;74:1621–1631), the disclosed approach reduces the normalized standard deviation of T 2 by a factor of two, while slightly improving that of T 1 . With the subspace constraint, it also improves the estimation performance over the unconstrained case, i.e., Optimized-I. Note that the improvement is comparable to that from Optimized-II. [0101] Fig.9A, Fig.9B, Fig.10A, and Fig.10B show the T 1 and T 2 reconstruction results, respectively, from the phantom experiments at N = 400, using the acquisition parameters from the conventional experiment, Optimized-I, Optimized-II, and the disclosed approach.. Fig.9A shows the reconstructed maps and associated normalized standard deviation maps for T1, and Fig.9B shoes the normalized standard deviation averaged over each tube. Each of the 12 tubes shows a group of four measurements which are, left to right, the Conventional, Optimized-I, Optimized-II, and disclosed methods. Fig.10A corresponds to Fig.9A and Fig.10B corresponds to Fig.9B, but Fig.10A and Fig.10B show data for T2. [0102] Fig.11A, Fig.11B, Fig.12A, and Fig.12B show the corresponding results for the in vivo experiments for T1 (Fig.11A and Fig.11B) and T2 (Fig.12A and Fig.12B) using the acquisition parameters from the conventional experiment, Optimized-I, Optimized-II, and proposed method. The measurement bars in Fig.11B and Fig.12B correspond to the measurement bars in Fig.9B and Fig.10B. Fig.11A and Fig.12A show reconstructed maps and associated normalized standard deviation maps. Fig.11B and Fig.12B show normalized standard deviation averaged over the ROIs in the gray matter and white matter, marked as black squares in 11A.The improvement from the disclosed approach is consistent with the numerical simulations, which further demonstrates that the disclosed approach enables competitive estimation performance while significantly improving the computational efficiency. Discussion [0103] The disclosed experimental examples demonstrate that the disclosed approach with the B-spline subspace constraints significantly improves the computational efficiency of solving the OED problem. The use of the B-spline subspace constraints was originally motivated from the structure observed in solving the OED problem (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861), and has been demonstrated to be effective in enabling competitive estimation performance in numerical, phantom, and in vivo studies. [0104] Note that the observed structure pertains to the formulations in (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861) and this disclosure - specifically, their design metrics and design constraints. It is worth emphasizing that if the design metric and/or constraints are changed, there is no guarantee that the same structure will appear. For example, the flip angle constraint could be replaced by a direct SAR constraint, which is linearly proportional to the square of the FAs and inversely proportional to the TRs (Prost et al., SAR reduced pulse sequences. Magn Reson Imaging 1988;6:125–130; Brown et al., Magnetic Resonance Imaging: Physical Principles and Sequence Design. John Wiley & Sons, 2014). In this case, the resulting formulation would include a highly nonlinear constraint, and it would be interesting to examine the structure of the solution and the further utility of using B-spline subspace constraints for improved computational efficiency. [0105] Additionally, it is important to gain an in-depth understanding of the structure arising in solving the OED problem from different perspectives using different approaches. While some spin physics perspectives (e.g., spin-coherence arguments in (Assländer et al., Cloos MA, Optimized quantification of spin relaxation times in the hybrid state. Magn Reson Med 2019;82:1385–1397; Cloos et al., Multiparametric imaging with heterogeneous radiofrequency fields. Nat Commun 2016;7:12445–12445)) are intuitive and may help understand the structure, the problem can be approached from an optimal control viewpoint. Specifically, the OED problem for MR Fingerprinting can be formulated as an optimal control problem with which the binary structure of the TR sequence could be characterized using bang-bang control (Lewis et al., Optimal Control. New York, NY: John Wiley & Sons, 2012). However, understanding the structure of the FA sequence remains an open problem that requires further theoretical study. [0106] The present experimental example shows that the class of B-spline basis functions has good representation power and flexibility to represent the desired structure in the OED. In particular, by controlling the underlying knot sequences and the degree of the spline functions, the B-spline representations represent different structures associated with the flip angle and repetition time sequences. In general, the function classes chosen to represent the data acquisition parameter sequences depend on the structure enforced in the OED problem. Besides B-spline basis functions, it is understood that there are alternative function classes (e.g., wavelets, radial basis, Fourier basis) that may in some embodiments be well suited for the OED problem. [0107] Note that the performance of the disclosed approach depends on the spline subspaces, which are governed by the knot sequences and degrees of the B-spline basis functions. In general, the selection of the B-spline parameters balances the dimensionality and representation power of the subspaces, and also allows for incorporating prior knowledge on the desired structure of the acquisition sequences. In the disclosed methods and systems, these parameters were manually selected to ensure a good trade-off between computational efficiency and estimation accuracy, while incorporating the observed optimal structure (e.g., the binary structure of the TR sequence (Zhao et al., IEEE Trans Med Imaging 2019;38:844–861)) into the experimental design. More details regarding the impact of B-spline parameters on the performance of the disclosed approach are included in the second Experimental Example below. [0108] The disclosed approach was initialized using the acquisition parameters from conventional MR Fingerprinting (Jiang et al., Magn Reson Med 2015;74:1621–1631), which resulted in good performance as illustrated in the Results section. Note that the disclosed approach involves solving a nonconvex optimization problem, and its performance generally depends on the initialization of the algorithm. In Experimental Example #3 below, the performance of the disclosed approach was evaluated with different initialization schemes. It was observed that the initialization of the flip angle sequence plays a dominant role in shaping the optimized acquisition parameters, and that among the schemes evaluated, the initialization from conventional MR Fingerprinting leads to the best performance. In some embodiments, for solving a nonconvex optimization problem in the disclosed approach, a multi-start strategy is used. [0109] The SQP algorithm was applied to solve an OED problem whose computational bottleneck, i.e., the gradient evaluation with finite differences, significantly benefits from the reduced search space enabled by the disclosed subspace constraints. It is worth noting that the benefits of the disclosed constraints go beyond this specific gradient evaluation method. In one embodiment, the improvement of the computational efficiency for the SQP algorithm is demonstrated with another widely used gradient calculation method, i.e., automatic differentiation (Lee et al., Buonincontri G, Hargreaves BA, Flexible and efficient optimization of quantitative sequences using automatic differentiation of Bloch simulations. Magn Reson Med 2019;82:1438–1451; Nocedal & Wright, Numerical Optimization. New York: Springer, 2nd edition, 2006). [0110] As an alternative approach to numerical differentiation, automatic differentiation can be used in some embodiments to calculate the gradient of an objective function in optimization (J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. New York: Springer, 2006; A. G. Baydin, et al., “Automatic differentiation in machine learning: a survey,” J. Mach. Learn. Res., vol.18, 2018.) and was previously employed for the optimization of acquisition parameter sequences in MR Fingerprinting (P. K. Lee, et al., “Flexible and efficient optimization of quantitative sequences using automatic differentiation of Bloch simulations,” Magn. Reson. Med., vol.82, pp.1438–1451, 2019.) Here it was demonstrated that the algorithm, i.e., sequential quadratic programming, with automatic differentiation also benefits from the proposed subspace constraints. [0111] Specifically, the OED problem in Equation 13 was solved using the sequential quadratic programming (SQP) algorithm for which the gradient was calculated using the reverse-mode automatic differentiation implementation from the Deep Learning Toolbox in MATLAB ® . The algorithm was run on the computational platform described above, and, for simplicity, only a single CPU core of the server was used, as in P. K. Lee, et al., Magn. Reson. Med., vol.82, pp. 1438–1451, 2019. Table 2 below shows the runtimes (in minutes) of solving the OED problems associated with Optimized-I, Optimized-II, and the disclosed approach with the sequential quadratic programming (SQP) algorithm with automatic differentiation for different acquisition lengths. As can be seen, the disclosed approach provides a significant improvement in computational efficiency over Optimized-I and Optimized-II for all the acquisition lengths. Finally, as expected, both automatic differentiation and numerical differentiation yield the same optimized acquisition parameter sequences. [0112] Also note that the spline-based representations could be advantageous over other OED problems apart from the CRB-based experimental design. For example, Hu et al. presented an OED approach for MR Fingerprinting (Hu et al., Sequence design for fast and robust MR fingerprinting scans using quantum optimization. In: Proc. Int. Symp. Magn. Reson. Med., 2021. p.169) that employed a spline-based representation for a supervised learning formulation with a stochastic optimization algorithm. Although a reduced search space generally brings computational benefits, the improvement is certainly problem and/or algorithm specific. [0113] The present disclosure uses ML reconstruction to illustrate the estimation performance enabled by the CRB-based OED. However, the benefits of the CRB-based experimental design are not tied to any specific reconstruction algorithm because the CRB, as an estimation-theoretic bound, is independent of the reconstruction process. This is generally different from the emerging supervised learning based OED approaches (Heesterbeek et al., Sequence optimisation for multi-component analysis in magnetic resonance fingerprinting. In: Proc. Int. Symp. Magn. Reson. Med., 2021. p.1561; Hu et al., Sequence design for fast and robust MR fingerprinting scans using quantum optimization. In: Proc. Int. Symp. Magn. Reson. Med., 2021. p.169; Loktyushin et al., MRzero - Automated discovery of MRI sequences using supervised learning. Magn Reson Med 2021;86:709–724), which often incorporate a specific reconstruction algorithm into an objective function that performs empirical risk minimization. In practice, it was observed that a variety of reconstruction algorithms, including both model-based (Zhao B, Model-based iterative reconstruction for magnetic resonance fingerprinting. In: Proc. Int. Conf. Image Process., 2015. pp.3392–3396; Doneva et al., Matrix completion-based reconstruction for undersampled magnetic resonance fingerprinting data. Magn Reson Med 2017;41:41–52; Zhao et al., Improved magnetic resonance fingerprinting reconstruction with low-rank and subspace modeling. Magn Reson Med 2018;79:933–942; Mazor et al., Low-rank magnetic resonance fingerprinting. Med Phys 2018;45:4066–4084) and data-driven methods (Cohen et al., MR fingerprinting Deep RecOnstruction NEtwork (DRONE). Magn Reson Med 2018;80:885–894; Fang et al., Deep learning for fast and spatially constrained tissue quantification from highly accelerated data in magnetic resonance fingerprinting. IEEE Trans Med Imaging 2019;38:2364– 2374), can benefit from the CRB-based experimental design, given the SNR benefit such design provides. [0114] The CRB-based OED focuses on minimizing the variance of the MR tissue parameter estimates, although the overall estimation performance also depends on the bias of the tissue parameters. In numerical simulations, it was demonstrated that the disclosed approach yielded a similar level of bias as the conventional experiment and the state-of-the-art approaches. However, assessing the bias for experimental studies, and, in particular, in vivo studies, can be substantially more complex, since a variety of potential model mismatches (e.g., tissue microstructure, acquisition imperfections, etc.) in real imaging experiments could complicate the bias evaluation, and in many cases, finding a viable ground truth for the bias evaluation can be very challenging (Stikov et al., On the accuracy of T1 mapping: Searching for common ground. Magn Reson Med 2015;73:514–522). A thorough investigation of the bias for experimental studies remains an open problem for MR Fingerprinting. [0115] It is also worth pointing out several future extensions of the disclosed approach. First, this work has demonstrated the efficacy of the disclosed approach for optimizing the FAs and TRs for a representative MR Fingerprinting sequence, i.e., the IR-FISP sequence. The disclosed approach can also be applied or extended to include other acquisition parameters (e.g., TE) and design other MR Fingerprinting pulse sequences (Ma et al., Magnetic resonance fingerprinting. Nature 2013;495:187–192; Cloos et al., Multiparametric imaging with heterogeneous radiofrequency fields. Nat Commun 2016;7:12445–12445; Jiang et al., MR fingerprinting using the quick echo splitting NMR imaging technique. Magn Reson Med 2017;77:979–988; Cui et al., A multi-inversion-recovery magnetic resonance fingerprinting for multi-compartment water mapping. Magn Reson Imaging 2021;81:82–87). In particular, an interesting extension is to design MR Fingerprinting sequences with multiple inversion pulses (e.g., (Cui et al., A multi- inversion-recovery magnetic resonance fingerprinting for multi-compartment water mapping. Magn Reson Imaging 2021;81:82–87)). In some such embodiments, the disclosed approach could be generalized from utilizing a single subspace to a union of subspaces. Here one additional subspace could be introduced to represent isolated inversion pulses, with the canonical basis as the basis functions. A sparsity constraint on the basis coefficients could further be imposed to control the number of inversion pulses. In some embodiments, this allows for the design of MR Fingerprinting sequences that have piecewise smooth FAs together with multiple isolated inversion pulses. [0116] Second, although the effectiveness of employing a small number of representative tissues for the OED problem is demonstrated, the improved computational efficiency with the disclosed approach facilitates the inclusion of a more extensive range of tissue parameters into the OED problem or even a prior distribution of tissue parameters as in the Bayesian OED paradigm (Chaloner & Verdinelli, Bayesian experimental design: A review. Stat Sci 1995;10:273–304). In some embodiments, the disclosed approach can also be extended to optimize MR Fingerprinting experiments with more complex tissue biophysical models, such as those used in myelin water imaging (Piredda et al., Probing myelin content of the human brain with MRI: A review. Magn Reson Med 2021;85:627–652) and quantitative magnetization transfer imaging (Hilbert et al., Magnetization transfer in magnetic resonance fingerprinting. Magn Reson Med 2020;84:128– 141). Conclusions [0117] The present experimental example and disclosure have introduced an efficient approach to solving the OED problem for MR Fingerprinting. Some embodiments of the disclosed approach employ B-spline based subspace constraints to incorporate the desired structure of data acquisition parameters and reduce the search space of the OED problem. Some embodiments of the disclosed approach enable a two-order-of-magnitude improvement of the computational efficiency over the state-of-the-art approaches, while providing a comparable SNR benefit. Example #2 – Impact of B-Spline Parameters on the Performance of the Disclosed Approach [0118] The proposed subspace constraints are governed by the degrees and knots sequences of the B-spline basis functions. In the first Experimental Example above the effectiveness of the disclosed approach using one set of B-spline parameters is shown. Here the impact of the B- spline parameters on the performance of the proposed approach is examined. [0119] First, the impact of the choice of B-spline degree was evaluated. Specifically, different combinations of FA and TR B-spline degrees were used with the same knot sequence as the one used in the main text. Fig.13A, Fig.13B, and Fig.13C show the resulting optimized FA and TR sequences and associated representative magnetization dynamics from the disclosed approach at N = 400. Specifically, Fig.13A shows FA at degrees of 1 and 2, and TR at degree 0. Fig.13B shows FA at degree 2 and 3, and TR at degree 1 and 0, and Fig.13C shows FA at degree 3 and TR at degree 1. As can be seen, the optimized acquisition parameter sequences and magnetization evolutions have similar structure for all choices of degrees considered. In Fig. 14A and Fig.14B, the corresponding reconstructed T 1 and T 2 maps from the numerical simulations with N = 400 and SNR = 33dB are shown. Here various combinations of B-spline degrees were used to represent the flip angle and repetition time sequences in the experimental design. As can be seen, in all cases, the performance is comparable to that of the disclosed approach with the choice of B-spline degree used in the first Experimental Example, which demonstrates the relative robustness with respect to this B-spline parameter. [0120] Second, the impact of the choice of B-spline knot sequence was evaluated. Here the same B-spline degrees (i.e., d α = 3 and d TR = 0) were used as in the first Experimental Example but different knot sequences were used. Specifically, five different uniformly spaced knot sequences were evaluated at acquisition length N = 400 with corresponding B-spline subspaces of dimensions L α = L TR = 10,20,40,100, and 200. Fig.15A (10 and 20), Fig.15B (40 and 100), and Fig.15C (200) show the resulting acquisition parameters and associated magnetization dynamics. As can be seen, the choice of knot sequence plays a significant role in shaping the structure of the acquisition parameter sequences and magnetization evolutions. With a proper selection of the knot sequence, the desired magnetization evolutions can be obtained that lead to better estimation performance. The corresponding reconstructed T 1 and T 2 maps are shown in Fig.16A and Fig.16B. Specifically, Knot Seq.1, 2, 3, 4, and 5 are uniformly spaced knot vectors with associated B-spline subspaces of dimension L α = L TR = 10,20,40,100, and 200, respectively. As can be seen, the knot sequence plays an important role in determining the structure of the B-spline subspaces and the resulting estimation performance. [0121] Furthermore, as illustrated in Fig.15A, Fig.15B, and Fig.15C, as the dimension of the B-spline subspaces approaches the acquisition length, the optimized acquisition parameters approach those from Optimized-I. When the dimension of the B-spline subspace is equal to the acquisition length, the solution space of the proposed approach is equal to that of Optimized-I, and with equivalent initializations, it can be shown that the solution of the proposed approach is the same as the one for Optimized-I. Example #3 - Impact of Initialization on the Experimental Design [0122] Next, the impact of the initialization of the algorithm on the performance of the disclosed approach was evaluated. Specifically, the SQP algorithm was initialized by performing a least- squares fit of the B-spline coefficients to the following four acquisition parameter sequences: [0123] (1) FA sequence and TR sequence both from conventional MR Fingerprinting (D. Ma, et al., “Magnetic resonance fingerprinting,” Nature, vol.495, pp.187–192, 2013.) (i.e., the one used in the first Experimental Example) [0124] (2) FA sequence from conventional MR Fingerprinting and TR sequence with each TRs from the uniform distribution on [11ms, 15ms] [0125] (3) FA sequence with FAs from the uniform distribution on the interval [10 ∘ , 60 ∘ ] and TR sequence from conventional MR Fingerprinting [0126] (4) FA sequence with FAs from the uniform distribution on the interval [10 ∘ , 60 ∘ ] and TR sequence with TRs from the uniform distribution on [11ms, 15ms]. [0127] Fig.17 shows the optimized FA and TR sequences using the above initializations as well as the cost function values associated with these solutions. As can be seen, the solution of the algorithm depends on the initialization. In particular, it was observed that the initialization of the FA sequence plays a dominant role in shaping the optimized acquisition parameters. Note that the cost function values for the solution using the four initializations are 3.75, 3.75, 3.88, and 3.88, respectively Compared with other initializations, the two initialization schemes using the FA sequence from the conventional MR Fingerprinting (Initialization-I and Initialization-II) result in the best cost function value. Fig.18A and Fig.18B show the reconstructed T 1 and T 2 maps using the acquisition sequences in Fig.17 at acquisition length N=300 and SNR=33dB, which illustrates the impact of these optimized acquisition parameters on the reconstructed tissue parameter maps. Example #4 – Impact of the Acceleration Factor [0128] The performance of the OED was evaluated using various acceleration factors for different choices of spline subspace dimension. Specifically, spline subspaces of dimensions L α = L TR = 20,40, and 200 were considered for MR Fingerprinting experiments with acceleration factors (AF) = 1,16,24, and 48. Note that the case with L α = L TR = 20 and AF = 48 (i.e., one spiral interleave per TR) corresponds to the one shown in the first Experimental Example. Fig.19 shows the resulting overall relative reconstruction errors for T 1 and T 2 . using the OED with B-spline subspaces of dimension L α = L TR = 20,40, and 200 and acceleration factors AF = 1, 16, 24, and 48 at N = 400 and SNR = 33 dB. As can be seen, in a highly- undersampled regime (e.g., AF = 48 or 24), enforcing the smoothness constraint in the OED is beneficial for improving the reconstruction performance. However, such benefit reduces as the acceleration factor decreases. Note that this is consistent with the data model used in the OED, which corresponds to a fully-sampled imaging experiment. Nonetheless, for an MR Fingerprinting experiment with a lower acceleration factor (e.g., AF = 16) or even the fully- sampled case (i.e., AF = 1), imposing the smoothness constraint in the OED does not appear to compromise the reconstruction performance much, which is highly desirable in practice. Note that the observations for the fully-sampled spiral MRF experiments are consistent with early observations for the fully-sampled Cartesian imaging experiment (see B. Zhao, et al., IEEE Trans. Med Imaging, vol.38, no.3, pp.844–861, 2019.) [0129] Despite this insightful illustration in numerical simulations, it is worth noting that MR Fingerprinting is a transient-state imaging technique in which there is no steady-state for magnetizations. As a result, having a lower acceleration factor than 48 (equivalently, acquiring multiple spiral interleaves per TR) requires a substantial wait time (e.g., about 10 seconds) between different repetitions of acquisitions to ensure that the magnetization returns to the thermal equilibrium (a more detailed description of fully-sampled MR Fingerprinting experiments can be found in the early work B. Zhao, et al., IEEE Trans. Med Imaging, vol.38, no.3, pp.844–861, 2019). In practice, this waiting time can be even longer than the actual acquisition time (e.g., about 5 seconds for N = 400), which makes MR Fingerprinting experiments with a low AF less practically appealing. Thus, only the highly-undersampled MR Fingerprinting experiment with AF = 48 was considered in the first Experimental Example. Example #5 – Different Solution Algorithms [0130] In the first Experimental Example, the SQP algorithm was used to solve the proposed formulation in Equation 13, which is a constrained nonlinear optimization problem with both bound constraints and linear inequality constraints. Here the results of solving this optimization problem using two alternative solution algorithms are shown. The first alternative algorithm was another deterministic nonlinear programming algorithm, i.e., an interior-point algorithm (J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. New York: Springer, 2006; A. Wächter and L. Biegler, “On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,” Math. Program., vol.106, no.1, pp.25–57, 2006.) which has the same local convergence guarantee as the disclosed approach. Second, a stochastic nonlinear programming algorithm, i.e., a genetic algorithm, was applied which has completely different convergence behavior. More specifically, the same initialization was used for the interior-point algorithm as the disclosed approach, which is used to confirm the convergence of the disclosed approach. For the genetic algorithm, a uniform random initialization was used. Fig. 20 shows the optimized acquisition parameters from the SQP algorithm, the interior-point algorithm, and the genetic algorithm. The cost function values associated with the three solutions are 3.75, 3.77, and 3.77, respectively. As can be seen, three dramatically different algorithms produce highly similar FA and TR sequences, which is remarkable. Although the cost function values associated with the optimal solutions from the three algorithms are very similar, the SQP algorithm used in the present disclosure produces the best cost function value. [0131] The disclosures of each and every patent, patent application, and publication cited herein are hereby incorporated herein by reference in their entirety. While this invention has been disclosed with reference to specific embodiments, it is apparent that other embodiments and variations of this invention may be devised by others skilled in the art without departing from the true spirit and scope of the invention. The appended claims are intended to be construed to include all such embodiments and equivalent variations. REFERENCES [0132] The following publications are incorporated herein by reference in their entirety: [0133] Ma D, Gulani V, Seiberlich N, Liu K, Sunshine JL, Duerk JL, Griswold MA, Magnetic resonance fingerprinting. Nature 2013;495:187–192. [0134] Chen Y, Jiang Y, Pahwa S, Ma D, Lu L, Twieg MD, Wright KL, Seiberlich N, Griswold MA, Gulani V, MR fingerprinting for rapid quantitative abdominal imaging. Radiology 2016;279:278–286. [0135] Badve C, Yu A, Dastmalchian S, Rogers M, Ma D, Jiang Y, Margevicius S, Pahwa S, Lu Z, Schluchter M, Sunshine J, Griswold MA, Sloan A, Gulani V, MR fingerprinting of adult brain tumors: Initial experience. Am J Neuroradiol 2017;38:492–499. [0136] Yu A, Badve C, Ponsky LE, Pahwa S, Dastmalchian S, Rogers M, Jiang Y, Margevicius S, Schluchter M, Tabayoyong W, Abouassaly R, McGivney D, Griswold MA, Gulani V, Development of a combined MR fingerprinting and diffusion examination for prostate cancer. Radiology 2017;283:729–738. [0137] Liu Y, Hamilton J, Rajagopalan S, Seiberlich N, Cardiac magnetic resonance fingerprinting: Technical overview and initial results. JACC: Cardiovasc Imaging 2018;11:1837– 1853. [0138] Chen Y, Panda A, Pahwa S, Hamilton JI, Dastmalchian S, McGivney DF, Ma D, Batesole J, Seiberlich N, Griswold MA, Plecha D, Gulani V, Three-dimensional MR fingerprinting for quantitative breast imaging. Radiology 2019;290:33–40. [0139] Cloos MA, Assländer J, Abbas B, Fishbaugh J, Babb JS, Gerig G, Lattanzi R, Rapid radial T1 and T2 mapping of the hip articular cartilage with magnetic resonance fingerprinting. J Magn Reson Imaging 2019;50:810–815. [0140] Kiselev VG, Körzdörfer G, Gall P, Toward quantification: Microstructure and magnetic resonance fingerprinting. Investig Radiol 2021;56:1–9. [0141] Jaubert O, Cruz G, Bustin A, Hajhosseiny R, Nazir S, Schneider T, Koken P, Doneva M, Rueckert D, Masci PG, Botnar RM, Prieto C, T1, T2, and fat fraction cardiac MR fingerprinting: Preliminary clinical evaluation. J Magn Reson Imaging 2021;53:1253–1265. [0142] Zhao B, Haldar JP, Setsompop K, Wald LL, Optimal experiment design for magnetic resonance fingerprinting. In: Proc. IEEE Eng. Med. Bio. Conf., 2016. pp.453–456. [0143] Zhao B, Haldar JP, Setsompop K, Wald LL, Towards optimized experiment design for magnetic resonance fingerprinting. In: Proc. Int. Symp. Magn. Reson. Med., 2016. p.2835. [0144] Zhao B, Haldar JP, Liao C, Ma D, Jiang Y, Griswold MA, Setsompop K, Wald LL, Optimal experiment design for magnetic resonance fingerprinting: Cramér-Rao bound meets spin dynamics. IEEE Trans Med Imaging 2019;38:844–861. [0145] Jones J, Hodgkinson P, Barker A, Hore P, Optimal sampling strategies for the measurement of spin–spin relaxation times. J Magn Reson 1996;113:25–34. [0146] Deoni SCL, Peters TM, Rutt BK, Determination of optimal angles for variable nutation proton magnetic spin-lattice, T1, and spin-spin, T2, relaxation times measurement. Magn Reson Med 2004;51:194–199. [0147] Fleysher L, Fleysher R, Liu S, Zaaraoui W, Gonen O, Optimizing the precision-per-unit- time of quantitative MR metrics: examples for T1, T2, and DTI. Magn Reson Med 2007;57:380– 387. [0148] Akçakaya M, Weingärtner S, Roujol S, Nezafat R, On the selection of sampling points for myocardial T1 mapping. Magn Reson Med 2015;73:1741–1753. [0149] Whitaker ST, Nataraj G, Nielsen J, Fessler JA, Myelin water fraction estimation using small-tip fast recovery MRI. Magn Reson Med 2020;84:1977–1990. [0150] Maidens J, Packard A, Arcak M, Parallel dynamic programming for optimal experiment design in nonlinear systems. In: Proc. IEEE Conf. Decis. Control, 2016. pp.2894–2899. [0151] Lee PK, Watkins LE, Anderson TI, Buonincontri G, Hargreaves BA, Flexible and efficient optimization of quantitative sequences using automatic differentiation of Bloch simulations. Magn Reson Med 2019;82:1438–1451. [0152] Assländer J, Lattanzi R, Sodickson DK, Cloos MA, Optimized quantification of spin relaxation times in the hybrid state. Magn Reson Med 2019;82:1385–1397. [0153] Lahiri A, Fessler JA, Hernandez-Garcia L, Optimizing MRF-ASL scan design for precise quantification of brain hemodynamics using neural network regression. Magn Reson Med 2020;83:1979–1991. [0154] Heesterbeek D, Vos F, Gijzen M, Nagtegaal M, Sequence optimisation for multi- component analysis in magnetic resonance fingerprinting. In: Proc. Int. Symp. Magn. Reson. Med., 2021. p.1561. [0155] Scope Crafts E, Zhao B, An efficient approach to optimal design of MR fingerprinting experiments with B-splines. In: Proc. Int. Symp. Magn. Reson. Med., 2021. p.1566. [0156] Jiang Y, Ma D, Seiberlich N, Gulani V, Griswold MA, MR fingerprinting using fast imaging with steady state precession (FISP) with spiral readout. Magn Reson Med 2015;74:1621–1631. [0157] Höllig K, Hörner J, Approximation and Modeling with B-Splines. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2013. [0158] Schumaker L, Spline Functions: Basic Theory. Cambridge: Cambridge University Press, 2007. [0159] Heath MT, Scientific Computing: An Introductory Survey. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2018. [0160] Hastie T, Tibshirani R, Friedman J, The Elements of Statistical Learning: Data Mining, Inference, and Prediction. New York: Springer, 2009. [0161] Fey M, Lenssen JE, Weichert F, Müller H, SplineCNN: Fast geometric deep learning with continuous B-spline kernels. In: Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., 2018. pp.869–877. [0162] Unser M, Aldroub A, Eden M, B-spline signal processing: Part I–theory. IEEE Trans Signal Process 1993;41:821–833. [0163] Unser M, Aldroub A, Eden M, B-spline signal processing: Part II–efficient design and applications. IEEE Trans Signal Process 1993;41:834–848. [0164] Unser M, Splines: A perfect fit for signal and image processing. IEEE Signal Process Mag 1999;16:22–38. [0165] Rueckert D, Sonoda L, Hayes C, Hill D, Leach M, Hawkes D, Nonrigid registration using free-form deformations: Application to breast MR images. IEEE Trans Med Imaging 1999;18:712–721. [0166] Hao S, Fessler JA, Noll DC, Nielsen JF, Joint design of excitation k-space trajectory and RF pulse for small-tip 3D tailored excitation in MRI. IEEE Trans Med Imaging 2016;35:468– 479. [0167] de Boor C, On calculating with B-splines. J Approx Theory 1972;6:50–62. [0168] Nocedal J, Wright SJ, Numerical Optimization. New York: Springer, 2nd edition, 2006. [0169] Spall JC, Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control. New York, NY: John Wiley & Sons, 2003. [0170] Shkarin P, Spencer RG, Time domain simulation of Fourier imaging by summation of isochromats. Int J Imaging Syst Technol 1997;8:419–426. [0171] Malik SJ, Sbrizzi A, Hoogduin H, Hajnal JV, Equivalence of EPG and isochromat-based simulation of MR signals. In: Proc. Int. Symp. Magn. Reson. Med., 2016. p.3196. [0172] Collins DL, Zijdenbos AP, Kollokian V, Sled JG, Kabani NJ, Holmes CJ, Evans AC, Design and construction of a realistic digital brain phantom. IEEE Trans Med Imaging 1998;17:463–468. [0173] Fessler JA, Sutton BP, Nonuniform fast Fourier transforms using min-max interpolation. IEEE Trans Signal Process 2003;51:560–574. [0174] Zhao B, Setsompop K, Ye H, Cauley SF, Wald LL, Maximum likelihood reconstruction for magnetic resonance fingerprinting. IEEE Trans Med Imaging 2016;35:1812–1823. [0175] Ma D, Coppo S, Chen Y, McGivney DF, Jiang Y, Pahwa S, Gulani V, Griswold MA, Slice profile and B_1 corrections in 2D magnetic resonance fingerprinting. Magn Reson Med 2017;78:1781–1789. [0176] Nataraj G, Nielsen JF, Fessler JA, Optimizing MR scan design for model-based T1, T2 estimation from steady-state sequences. IEEE Trans Med Imaging 2017;36:467–477. [0177] Prost JEH, Wehrli FW, Drayer B, Froelich J, Hearshen D, Plewes D, SAR reduced pulse sequences. Magn Reson Imaging 1988;6:125–130. [0178] Brown RW, Cheng YCN, Haacke EM, Thompson MR, Venkatesan R, Magnetic Resonance Imaging: Physical Principles and Sequence Design. John Wiley & Sons, 2014. [0179] Cloos MA, Knoll F, Zhao T, Block KT, Bruno M, Wiggins GC, Sodickson DK, Multiparametric imaging with heterogeneous radiofrequency fields. Nat Commun 2016;7:12445–12445. [0180] Lewis FL, Vrabie D, Syrmos V, Optimal Control. New York, NY: John Wiley & Sons, 2012. [0181] Hu S, Rozada I, Boyacioglu R, Jordan S, Huang S, Troyer M, Griswold MA, McGivney D, Ma D, Sequence design for fast and robust MR fingerprinting scans using quantum optimization. In: Proc. Int. Symp. Magn. Reson. Med., 2021. p.169. [0182] Loktyushin A, Herz K, Dang N, Glang F, Deshmane A, Weinmüller S, Doerfler A, Schölkopf B, Scheffler K, Zaiss M, MRzero - Automated discovery of MRI sequences using supervised learning. Magn Reson Med 2021;86:709–724. [0183] Zhao B, Model-based iterative reconstruction for magnetic resonance fingerprinting. In: Proc. Int. Conf. Image Process., 2015. pp.3392–3396. [0184] Doneva M, Amthor T, Koken P, Sommer K, Börnert P, Matrix completion-based reconstruction for undersampled magnetic resonance fingerprinting data. Magn Reson Med 2017;41:41–52. [0185] Zhao B, Setsompop K, Adalsteinsson E, Gagoski B, Ye H, Ma D, Jiang Y, Grant PE, Griswold MA, Wald LL, Improved magnetic resonance fingerprinting reconstruction with low- rank and subspace modeling. Magn Reson Med 2018;79:933–942. [0186] Mazor G, Weizman L, Tal A, Eldar YC, Low-rank magnetic resonance fingerprinting. Med Phys 2018;45:4066–4084. [0187] Cohen O, Zhu B, Rosen MS, MR fingerprinting Deep RecOnstruction NEtwork (DRONE). Magn Reson Med 2018;80:885–894. [0188] Fang Z, Chen Y, Liu M, Xiang L, Zhang Q, Wang Q, Lin W, Shen D, Deep learning for fast and spatially constrained tissue quantification from highly accelerated data in magnetic resonance fingerprinting. IEEE Trans Med Imaging 2019;38:2364–2374. [0189] Stikov N, Boudreau M, Levesque IR, Tardif CL, Barral JK, Pike GB, On the accuracy of T1 mapping: Searching for common ground. Magn Reson Med 2015;73:514–522. [0190] Jiang Y, Ma D, Jerecic R, Duerk J, Seiberlich N, Gulani V, Griswold MA, MR fingerprinting using the quick echo splitting NMR imaging technique. Magn Reson Med 2017;77:979–988. [0191] Cui D, Hui ES, Cao P, A multi-inversion-recovery magnetic resonance fingerprinting for multi-compartment water mapping. Magn Reson Imaging 2021;81:82–87. [0192] Chaloner K, Verdinelli I, Bayesian experimental design: A review. Stat Sci 1995;10:273– 304. [0193] Piredda GF, Hilbert T, Thiran JP, Kober T, Probing myelin content of the human brain with MRI: A review. Magn Reson Med 2021;85:627–652. [0194] Hilbert T, Xia D, Block KT, Yu Z, Lattanzi R, Sodickson DK, Kober T, Cloos MA, Magnetization transfer in magnetic resonance fingerprinting. Magn Reson Med 2020;84:128– 141.