Systems and methods of constrained reconstruction of images with white noise

A magnetic resonance imaging (MRI) system can include a processor and a memory. The processor can receive an acquired magnetic resonance (MR) dataset having a first signal-to-noise ratio (SNR). The processor can extract, from the acquired MR dataset, a first set of values corresponding to a first variable having a second SNR and a second set of values corresponding to a second variable. The processor can apply a constraint function that includes a function of the first variable and the second variable. The processor can minimize a cost function according to the constraint function to generate a cost function solution. The processor can input the first variable and the second variable into the cost function solution to generate a modified first variable having a third SNR, the third SNR being greater than the second SNR.

BACKGROUND

Magnetic resonance imaging (MRI) is an accepted modality for imaging soft tissue in the human body as well as in animals and some materials. Mill is intrinsically inefficient in that at room temperatures the available magnetization is limited. The more spins that can be polarized, the better the signal-to-noise ratio (SNR) of the imaging will be. For this reason, the MM field has pushed the limits of high field human imaging from 3 T to 7 T and more recently to almost 12 T. Reasons to increase the magnetic field can include better SNR, better resolution, faster imaging, novel contrast mechanisms, and better spectral resolution. However, problems with imaging at a high magnetic field can include high costs, increased power deposition, and decreased susceptibility effects for a given echo time. This can lead to the need for greater gradient strengths to compensate for bad local fields. Under these conditions, the SNR can be expected only to increase with the square root of the main field strength. One of the main goals in MRI research today is to find ways to increase the SNR through signal processing means and bypass the need for higher field strengths and their associated higher expense.

SUMMARY

Systems and methods for constrained reconstruction of images with white noise (CROWN) can offer a powerful solution to improve SNR without modifying or blurring the image structures as is the case in most other methods that purport to improve SNR. CROWN can lead to improved SNR for proton spin density (PD) estimates, images for any arbitrary flip angles, and images for any other MRI pulse sequences which can be derived from knowledge of T1, PD, and T2*. CROWN can operate in conjunction with strategically acquired gradient echo (STAGE) imaging, any other multi-flip angle approach, or any method that generates both a spin density map and T1 map. CROWN can be used in low field strength applications where the SNR is inherently limited. CROWN can also be applied in any system where two variables are related.

At least one aspect of the present disclosure is directed to a magnetic resonance imaging (Mill) system. The Mill system can include a processor and a memory. The processor can receive an acquired magnetic resonance (MR) dataset having a first signal-to-noise ratio (SNR). The processor can extract, from the acquired MR dataset, a first set of values corresponding to a first variable having a second SNR and a second set of values corresponding to a second variable. The processor can apply a constraint function that includes a function of the first variable and the second variable. The processor can minimize a cost function according to the constraint function to generate a cost function solution. The processor can input the first variable and the second variable into the cost function solution to generate a modified first variable having a third SNR, the third SNR being greater than the second SNR.

Another aspect of the present disclosure is directed to a method of magnetic resonance imaging. The method can include receiving, by at least one processor, an acquired MR dataset having a first SNR. The method can include extracting, from the acquired MR dataset by the at least one processor, a first set of values corresponding to a first variable having a second SNR and a second set of values corresponding to a second variable. The method can include using, by the at least one processor, a constraint function that includes a function of the first variable and the second variable. The method can include minimizing, by the at least one processor, a cost function according to the constraint function to generate a cost function solution. The method can include inputting, by the at least one processor, the first variable into the cost function solution to generate a modified first variable having a third SNR, the third SNR being greater than the second SNR.

DETAILED DESCRIPTION

Following below are more detailed descriptions of various concepts related to, and implementations of, methods, apparatuses, and systems for the constrained reconstruction of images with white noise. The various concepts introduced above and discussed in greater detail below may be implemented in any of a number of ways, as the described concepts are not limited to any particular manner of implementation. Examples of specific implementations and applications are provided primarily for illustrative purposes.

One way to improve the SNR is to average the data. However, acquiring the data N times can take N times longer but only give an improvement in the SNR of √{square root over (N)}. Therefore, fewer patients can be scanned clinically because the Mill scans take longer. Another way to improve the SNR is to filter the data. For example, a Hanning filter can remove Gibbs ringing and improve the SNR. However, this approach leads to blurring of the image (e.g., loss of resolution). Usual (e.g., conventional) approaches to improving the SNR and, hence, image quality without destroying the resolution in the process can include using an edge preserving filter. This can be accomplished in a variety of ways to keep the high spatial frequency components in the image. Suppressing high k-space signals, as is done with the Hanning filter, may not be desired. An anisotropic diffusion filter (ADF) can be used to improve the SNR. Although the ADF can preserve the edges when the SNR is high, the ADF still blurs the images to some degree. Some techniques use artificial intelligence to reduce noise by training different models. However, these models may be trained using a single type of contrast and only work for that contrast. These models may need to be trained over hundreds or thousands of cases every time contrast is changed. These algorithms can lead to some degradation of the image which is recognizable as a remnant blurring relative to the original data.

In the present disclosure, systems and methods for constrained reconstruction of images with white noise (e.g., CROWN, CROWN processing, etc.) are described. CROWN can allow for enhanced (e.g., improved) SNR without loss of detail by using constraints. CROWN can be used with strategically acquired gradient echo (STAGE) imaging, which is described below. However, CROWN can be used for any approach which can generate spin density (e.g., water content) and T1 maps. More generally, CROWN can be used for any system and/or method that collects two or more images that are functionally related to each other through two or more variables. CROWN can be used to improve SNR without loss of image resolution.

STAGE imaging can include a rapid, multi-contrast, multi-echo, gradient echo imaging method. It can use at least two flip angles to estimate proton spin density (PD) and T1 (e.g., longitudinal relaxation time) maps. Generally, in MRI, water content (e.g., PD) drives T1 and T2*. Therefore, a relationship (e.g., linear relationship) between water content and T1 or water content and T2* can be determined (e.g., written, specified, formed, etc.). The relationship can be used to reduce the noise level in the PD image and/or generate higher quality simulated images for any flip angle for a given echo time. CROWN can be applied to data with low SNR and can improve data collected with high parallel imaging acceleration factors, high resolution, or from radiofrequency receive coils (e.g., RF coils) with a small number of channels. In addition, CROWN processing can be extremely fast compared to conventional methods and can reduce the white noise level without degrading the edges or image details in general.

The relationship between variables can be used to constrain the data (e.g., MM data) in a way that reduces noise. For example, this can include a relationship between a spin density,

=1β,
and longitudinal relaxation time,

T⁢1=1R⁢1.
The relationship between spin density and T1 can be used to constrain the data in a way that reduces noise.

Consider, for example, the following linear relationship described by Equation 1:
R1=αβ+b(1)

STAGE can collect two datasets. Each of the datasets can have noise. This in turn can lead to noise being generated in both the spin density map and the T1 map. Often the spin density maps can be quite noisy, especially after correcting for T2*(1/R2*) using a separate T2* map that also has noise. A map (e.g., spin density map, T1 map, etc.) can include a set of values of the variable (e.g., spin density, T1, etc.) at points throughout the image. For example, a spin density map can indicate the value of the spin density at each pixel.

The SNR in the spin density image can be enhanced using CROWN processing. To accomplish this, the following cost function described in Equation 2 can be determined to calculate the estimated point (R1est, βest) from the measured data (R1′, β′):
C(β,R1)=min((β−β′)2+(R1−R1′)2)  (2)

Taking the derivative of C(β, R1) with respect to and setting it equal to zero yields Equation 4 and Equation 5:

These predicted values (e.g., βestand R1est) can include projections of the values (R1′, β′) onto (R1, β) which fall on the line represented by Equation 1. These new values of (R1est, βest) represent the new CROWN values. The relationship between R1 (1/T1) and μ (1/ρ) can be modified for different TE. As TE changes, the “effective” spin density, ρeff, described in Equation 6 changes accordingly.
ρeff(TE)=ρ·e−TE R2*  (6)

The effect of the minimization for one point is graphically/geometrically shown inFIG. 1.FIG. 1illustrates a plot of the minimization of the cost function for one point. A graphical representation of the minimization of the cost function (e.g., Equation 2) is shown. Once the linear relationship between β and R1 has been determined, any measurement point with noise (β′, R1′) that lies outside the line can be mapped back to its estimate (βest, R1est) using the proposed solution (e.g., cost function solution) to the cost function of Equation 2.

The effect of the minimization for multiple points is graphically/geometrically shown inFIG. 2.FIG. 2illustrates a plot of the minimization of the cost function for multiple points. A graphical representation of the linear relationship between R1 and β representing Equation 1 and the projection of noisy measurements (crosses) to their estimates (circles) to the line after CROWN processing is shown.

To apply CROWN, the linear coefficients a and b from Equation 1 can be determined. Assumptions about the tissue parameters of T1, spin density, and T2* (e.g., if effective spin density is desired) in the brain's white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF) can be made. The values used for 3T field strength are shown in Table 1.

Table 1 illustrates assumed tissue parameter values at 3T to determine CROWN coefficients.

These values can provide three sample points on the (β, R1) axes which can be used to perform a linear regression and extract the slope a and intercept b. In this case, the line can be fit by forcing the regression line to pass through the point representing CSF. With TE=0 ms (e.g., no T2* effects) the coefficients a and b can be 2.03/sec and −1.81/sec respectively. To take T2* effects into account, values for spin density can be adjusted by the factor E2*=e(−TE/T2*), and linear regression can be re-performed using the new coefficients for different TE provided in Table 2.

Table 2 illustrates the relationship between R1 and β for different echo times. The linear relationship between R1 and β can be chosen based on the echo time for multi-echo STAGE images.

An issue that can be addressed before applying CROWN is that the spin density map values generated from STAGE are scaled arbitrarily while the CROWN coefficients can be determined by representing spin density as a unitless percentage on a scale of 0 to 1. Thus, the spin density maps can be normalized to the CSF, which can be assumed to have a spin density of unity (e.g., the maximum spin density is assumed to be water and is normalized to 1.0). Another issue that can be addressed before applying CROWN is that CSF values in both the T1 and spin density maps from STAGE can be noisy. This can be a by-product of only using two flip angles (e.g., one flip angle roughly equal to and the other flip angle greater than the Ernst angle of CSF) to generate these maps. The following procedure can address these issues.

The CSF can be used to normalize the spin density to a unitless measure of water on a scale of 0 to 1 (since CSF is 100% water). To determine the required scale value and normalize the entire PD map, the location of CSF pixels in the ventricles can be determined using a T1 map. This can be achieved through the following approaches. First, a T1 threshold of CSF within a middle slab of the field of view (FOV) can be used to do this. Using a slab rather than the entire 3D dataset can help avoid any global variations of spin density across the data. The slab can consist of enough slices to effectively guarantee the ventricles will reside within it. For the typical STAGE brain scan FOV of 256 mm×192 mm×128 mm, assuming the subject is properly centered, the central 20 mm of the FOV can capture the ventricles. To reduce global variations in the effective spin density and ensure CSF spin density is unity throughout the brain, radiofrequency (RF) penetration effects and RF receive coil effects can be removed using STAGE. As an alternative approach, to ensure that the region that is used to normalize the spin density has the best possible signal-to-noise, the maximum value of the sum of correlating a 3D object with dimensions a×b×c (e.g., where a×b could be 20 mm×20 mm and c could be 10 mm) where each pixel has a value of unity with the T1 map either in the vertical direction above and below the center slice or throughout the entire object can be determined. This process can choose the region-of-interest (ROI) that has the most pixels with CSF (e.g., an ROI centered around on the ventricles.) This approach can ensure that only the ventricles are captured and not any of the cortical CSF. Mathematically, this can be described by Equation 7:
rcsf=argmaxr{W(r)*T1(r)}  (7)
where W(r) represents the a×b×c 3D rectangular box, with a value of 1 inside and a value of 0 outside, T1(r) is the T1 map, * is the convolution operator, and argmaxrreturns the value of r that maximizes the value inside the brackets. Thus rcsfcan be the location of the box that provides the highest value of the convolution, which is expected to represent predominantly the ventricles, assuming that the box is large enough to contain them all.

SCSF-MIDcan be the set of pixel locations, ri,j,k, inside the slab or inside the chosen ROI that also satisfy the condition {(4300 ms-ΔT1CSF)<T1(ri,j,k)<(4300 ms+ΔT1*CSF)}, where ΔT1*CSFcan be adjusted to change the width of the CSF T1 window. ΔT1*CSFcan be 500 ms. Most of these pixels in SCSF-MIDcan be inside the ventricles. The average PD value over the set of pixels SCSF-MIDcan be taken to represent the PD scale value described by Equation 8:

P⁢Dscale=1N⁢∑ri,j,k∈SCSF-MIDP⁢D⁡(ri,j,k)(8)
where N is the number of pixel locations in SCSF-MID. The normalized PD map can be calculated from the expression PDnorm=PD/PDscale.

The poorly behaved CSF values can be identified and replaced in both the T1 and PD maps. A T1 map threshold can again be used to do this. However, rather than only identifying the bad pixels, it can be easier to capture as many CSF pixels as possible and replace them with the known CSF values plus some noise. A low pass filter of the T1 map can also be used to help capture some of the pixel locations within the ventricles which have extremely low T1 values due to the noise. SCSFcan be the set of pixel locations, ri,j,k, that satisfy the condition {T1(ri,j,k)>(4300 ms-|offset|) or LP{T1}(ri,j,k)>(4300 ms-|offset|)}, where LP represents a low-pass averaging filter, and “offset” can be adjusted to capture more of the CSF at the risk of including other tissues. The final T1 and PD maps can be described by Equation 9 and Equation 10:

Improved ρ and R2* can be generated using CROWN. Improved p maps (e.g., with no R2* weighting) and R2* maps can be generated in a variety of different ways. The conventional STAGE method of generating these maps and three different methods to generate improved ρ and R2* maps with CROWN are described.

In some approaches, STAGE can utilize data from multiple flip angles and echoes to generate maps of PD, T1, and R2*. For M flip angles and N echoes, the total number of datasets used can be M×N. For each of the M flip angles, a least squares fit can be performed over the N echoes to generate a map of R2*. This can result in M R2* maps which can then be averaged together for a final STAGE R2* map. Following that, for each echo, a least squares fit can be performed over the set of M flip angles to generate N T1 maps and effective PD maps (T2* weighted), ρeff. The N T1 maps can be averaged together for a final STAGE T1 map. The STAGE R1 map can be the inverse of this averaged T1 map. Next, each of the N effective PD maps can be corrected using the STAGE R2* map to generate N PD maps (with TE=0 ms). The N PD maps can be averaged together for a final STAGE PD map (ρAVG). These steps are presented as a data flow diagram inFIGS. 3A-3C.FIG. 3Aillustrates a flow chart for calculating R2* using STAGE.FIG. 3Billustrates a flow chart for calculating ρ using STAGE.FIG. 3Cillustrates a flow chart for calculating T1 using STAGE.

Constrained reconstruction of images with white noise (CROWN) can include a magnetic resonance imaging system. The MRI system can include at least one processor. The MRI system can include a memory, with computer code instructions stored thereon. The computer code instructions, when executed by the at least one processor, can cause the at least one processor to receive an acquired MR dataset. The acquired MR dataset can have a first SNR. The first SNR can include the SNR of the acquired MR dataset.

The processor can extract, from the acquired MR dataset, a first set of values corresponding to a first variable. The first variable can have a second SNR. The second SNR can include the SNR of the first variable. The first variable can include (e.g., correspond to), for example, spin density, β, susceptibility, or another variable. The first variable can correspond to an inverse of spin density (1/β). The first variable can correspond to susceptibility. The acquired data can include values that correspond to the first variable. The acquired data can be processed to produce the first variable.

The processor can extract, from the acquired MR dataset, a second set of values corresponding to a second variable. The second variable can include (e.g., correspond to), for example, T1, R1, T2*, or another variable. The second variable can correspond to an inverse of T1. The second variable can correspond to R2*. The second variable can be related (e.g., functionally related) to the first variable. For example, the inverse of T1 can be related to the inverse of spin density via a relationship (e.g., linear relationship). R2* can be related to susceptibility via a relationship (e.g., linear relationship). R2* can be related to the inverse of spin density via a relationship (e.g., linear relationship). The relationship between the two or more variables can be used to derive a constraint function. The acquired data can include values that correspond to the second variable. The acquired data can be processed to produce the second variable.

The processor can apply (e.g., use, establish, identify, implement, execute, etc.) a constraint function (e.g., relationship, linear relationship, relation, etc.). The constraint function can include a function of the first variable and the second variable. The constraint function can be described by Equation 1. The constraint function can include a component that accounts for a presence of iron in a tissue. For example, the relation between two or more variables can account for and be used to correct for the presence of iron. The constraint function can include a component that accounts for each echo time (e.g., TE1, TE2, TE3, etc.).

The computer code instructions can cause the at least one processor to minimize a cost function (e.g., cost) according to the constraint function to generate a cost function solution. The cost function can be described by Equation 2. The cost function solution can include predicted values described by Equation 4 and Equation 5.

The processor can input the first variable and the second variable into the cost function solution to generate a modified first variable having a third SNR. The third SNR can be greater than the second SNR. The third SNR can include the SNR of the modified first variable. The processor can generate a simulated dataset for an arbitrary flip angle (e.g., 0°, 3°, 6°, 9°, 12°, 15°, 18°, 21°, 24°, etc.). For example, the processor can generate, using the modified first variable and the second variable, a simulated dataset for an arbitrary flip angle. One or more images can be created for arbitrary flip angles. Also, one or more images can also be created for any MRI pulse sequence.

The processor can extract, from the acquired MR dataset, a third set of values corresponding to a third variable. The third variable can include, for example, R1* or T2*. The processor can apply (e.g., use) the constraint function. The constraint function can be a function of the first variable, the second variable, and the third variable. The MRI dataset can include more than three variables (e.g., multiple variables). The constraint function can be a function of the multiple variables or a subset of the multiple variables. For the case of a linear relationship between two variables, the constraint function can be described by the equation: y=mx+b. For the case of a linear relationship between multiple variables, the constraint function can be described by the equation: y=Σimixi+b.

Constrained reconstruction of images with white noise (CROWN) can include a variety of approaches. For example, in a first approach (e.g., Approach 1, CROWN Approach 1, etc.), CROWN can be used to improve each effective spin density map, giving ρeff-CROWNat each TE. Once CROWN has been used to improve each effective spin density map, the expression ln ρeff-CROWN=ln ρCROWN−TE R2*CROWNand R2*CROWNcan be fitted on a pixel-by-pixel basis. This formula can give the slope of the signal as a function of TE as −R2* and the intercept as ln ρCROWNfrom which the value of ρCROWNand R2*CROWNcan be obtained. These new images ρCROWNand R2*CROWNcan have higher SNR compared to the SNR of images ρ and R2*. Multiple echo CROWN data can be used to generate ρCROWNand R2*CROWN. The first approach can include the following steps: (1) generate ρeff-CROWNfor each echo, (2) perform a least squares fit across the logarithm of these data, (3) use the intercept to determine ρCROWN, and (4) use the slope to determine R2*CROWN.FIG. 4illustrates a flow chart for calculating ρCROWNand R2*CROWNusing CROWN processing.

The acquired MR dataset can include data corresponding to a first flip angle and a second flip angle. For example, the first flip angle can include a low flip angle (e.g., an angle less than the Ernst angle). The Ernst angle, (θE), is defined via cos(θE)=exp(−TR/T1). The Ernst angle can include the flip angle (e.g., tip angle, nutation angle, etc.) for excitation of a particular spin that gives the maximal signal intensity for any flip angle for a given TR and T1. The second flip angle can include a high flip angle (e.g., an angle greater than the Ernst angle). The acquired MR dataset can be acquired by imaging an anatomical region using at least one echo time (e.g., TE1, TE2, TE3, etc.).

In a second approach (e.g., Approach 2, CROWN Approach 2, etc.), R2* can be determined from fitting the original multi-echo STAGE data or from Approach 1 using CROWN processing. Once R2* is determined, the effective spin density from each echo can be corrected to produce a p map with no R2* weighting, by rearranging Equation 6, ρ=ρeff/e−TE R2*. With the use of CROWN R2* , each of these ρ maps can be quite noisy but with CROWN processing the SNR can be dramatically improved. One implementation of this is to use the coefficients of 2.03/sec and −1.81/sec for a and b, respectively, and then averaging the resulting images for a final CROWN ρ map, ρAVG-CROWN. This can provide an alternative to Approach 1 to generate ρCROWN.

To further generate an improved R2* map, the log of the ratio of the original effective spin density maps from STAGE with the averaged zero echo time spin density can be taken to give ln [ρeff/ρAVG-CROWN]=−TE R2*. Since TE is known, an R2* map can be calculated from each echo and then, again, averaged for a final R2*CROWNmap.

The second approach can include the following steps: (1) generate ρefffor each echo using the conventional STAGE approach, (2) generate STAGE R2* using the conventional STAGE approach, (3) use Equation 6 with ρeffand STAGE R2* to determine p for each echo (e.g., ρTE1, ρTE2, etc.), (4) apply CROWN to each ρ to obtain ρCROWNfor each echo (e.g., ρTE1-CROWN, ρTE2-CROWN, etc.), (5) average these together to obtain AVG-CROWN, (6) use Equation 6 with ρeffand ρAVG-CROWNto determine a new R2* for each echo, (7) average these together to obtain a final R2*CROWN.FIG. 5illustrates a flow chart for calculating ρAVG-CROWNusing CROWN processing.FIG. 6illustrates a flow chart for calculating R2*AVG-CROWNusing CROWN processing.

In the second approach, the acquired MR dataset can include data corresponding to a first flip angle and a second flip angle. The acquired MR dataset can be acquired by imaging an anatomical region using at least one echo time (e.g., TE1, TE2, TE3, etc.). The processor can generate a first calculated value of a third variable. The third variable can correspond to R2*. The third variable can have a fourth SNR. The processor can generate, using the modified first variable, a second calculated value of a modified third variable. The modified third variable can include R2*CROWN. The modified third variable can have a fifth SNR. The fifth SNR can be greater than the fourth SNR.

In some embodiments, the acquired MR dataset can include data corresponding to a first flip angle and a second flip angle for a plurality of echo times. The processor can generate a weighted average value for at least one of the first variable and the modified first variable over the plurality of echo times. For example, the processor can generate a weighted average of spin density (e.g., ρAVG-CROWN) or R2* (e.g., R2*CROWN).

A third approach (e.g., Approach 3, CROWN Approach 3, etc.) can be similar to the second approach, except for starting with ρeff-CROWNrather than ρeffto improve PD and R2* (e.g., using ρeff-CROWN/e−TE R2*to obtain ρAVG-CROWNand then ln[ρeff-CROWN/ρAVG-CROWN] to obtain R2*CROWN). The third approach can include the following steps: (1) generate ρeff-CROWNfor each echo, (2) generate STAGE R2* using the conventional STAGE approach, (3) use Equation 6 with ρeff-CROWNand STAGE R2* to determine a ρCROWNfor each echo (e.g., ρTE1|CROWN, ρTE2|CROWN, etc.), (4) apply CROWN again to each ρCROWNto obtain ρCROWN2for each echo (e.g., ρTE1-CROWN2, ρTE2-CROWN2, etc.), (5) average these together to obtain ρAVG-CROWN2, (6) use Equation 6 with ρeff-CROWNand ρAVG-CROWN2to determine a new R2* for each echo, (7) average these together to obtain a final R2*CROWN2.FIG. 7illustrates a flow chart for calculating ρAVG-CROWN2using CROWN processing.FIG. 8illustrates a flow chart for calculating R2*AVG-CROWN2using CROWN processing.

In some embodiments of the third approach, the acquired MR dataset can include data corresponding to a first flip angle and a second flip angle. The acquired MR dataset can be acquired by imaging an anatomical region using at least one echo time (e.g., TE1, TE2, TE3, etc.). The modified first variable can include a first modified first variable. The first modified first variable can include effective spin density (e.g., ρeff). The processor can generate a first calculated value of a third variable. The third variable can correspond to R2*. The third variable can have a fourth SNR. The processor can use the first modified first variable at each of the at least two echo times and the third variable to generate a second modified first variable (e.g., ρTE1|CROWN, ρTE2|CROWN, ρTE3|CROWN, etc.). The processor can use (e.g., apply) the second modified first variable at each of the at least two echo times and the second variable as inputs into the cost function solution to generate a third modified first variable (e.g., ρTE1-CROWN2, ρTE2-CROWN2, ρTE3-CROWN2, etc.) having a sixth SNR. The sixth SNR can be greater than the third SNR.

In some embodiments of the third approach, the processor can generate a second calculated value of the third variable (e.g., R2*TE1-CROWN2, R2*TE2-CROwN2, R2*TE3-CROWN2, etc.). The third variable can correspond to R2*. The third variable can have a seventh SNR. The seventh SNR can be greater than the fourth SNR.

In some embodiments, the acquired MR dataset can include data corresponding to a first flip angle and a second flip angle for a plurality of echo times. The processor can generate a weighted average value for at least one of the first variable and the modified first variable over the plurality of echo times. For example, the processor can generate a weighted average of spin density (e.g., ρAVG-CROWN2) or R2* (e.g., R2*AVG-CROWN2). In some embodiments, the processor can generate a spin density image (e.g., TE=0, true spin density image, etc.).

Some slightly different notation can be used for some of the intermediate steps in Approach 3 compared to Approach 2. Since in Approach 3, CROWN is applied prior to correcting for R2* (to generate ρeff-CROWN), these ρ maps after R2* correction can be called ρTEi|CROWN. Then, after CROWN is applied a second time, the notationCROWN2can be used. A table of terms with descriptions is shown in Table 3.

ApproachesApproachTermExplanationUsedGeneratedρeff(TEi)The effective (R2* weighted) PD map generated by1, 2STAGEconventional STAGE for a single echo time, TEi.Defined in Equation 6.ρeff-CROWN(TEi)The result of applying CROWN to ρeff(TEi)1, 31ρThe general term for a “true” PD map with no R2*N/AN/Aweighting. Can also be thought of as ρeff(TE = 0)ρTEiThe result of correcting ρeff(TEi) for R2*, specifically2STAGEat TEi. Can also be viewed as ρeff(TE = 0)ρAVGThe ρ map generated by averaging ρTEiover allN/ASTAGEpossible echoes. This is the conventional STAGEway to generate a “true” PD map.ρCROWNThe intercept result of performing a least squares fitN/A1across all available ρeff-CROWN(TEi)ρTEi-CROWNThe result of applying CROWN to ρTEi22ρTEi|CROWNThe result of correcting ρeff-CROWN(TEi) for R2*,33specifically at TEiρAVG-CROWNThe result of averaging together all available ρTEi-CROWNN/A2ρTEi-CROWN2The result of applying CROWN to ρTEi|CROWN33ρAVG-CROWN2The result averaging together all available ρTEi-CROWN2N/A3R2*CROWNThe slope result of performing a least squares fitN/A1across all available ρeff-CROWN(TEi)R2*TEi-CROWNThe R2* map generated from ρeff(TEi) and ρAVG-CROWN22R2*TEi-CROWN2The R2* map generated from ρeff-CROWN(TEi) and33ρAVG-CROWN2R2*AVG-CROWNThe result of averaging together all available R2*TEi-CROWNN/A2R2*AVG-CROWN2The result of averaging together all available R2*TEi-CROWN2N/A3STAGE R2*The average of all R2* maps generated for each flip2, 3STAGEangle collected for STAGE (by fitting over echotime)STAGE T1The average of all T1 maps generated for each echo1, 2, 3STAGEtime collected for STAGE (by fitting over flip angle)

Table 3 illustrates a description of the different terms used inFIGS. 3A-3CandFIGS. 4-8described in the present disclosure. Table 3 illustrates which approach the terms are used in and/or generated from. For example, ρAVGcan be generated in the conventional STAGE approach, but not used in any of Approaches 1-3. On the other hand, ρTE1-CROWNcan be generated in Approach 2, and used further within it to generate the final ρAVG-CROWNand R2* AVG-CROWN.

The processor can generate a simulated MR dataset having an eighth SNR. The eighth SNR can be greater than the first SNR. The processor can generate a simulated MR dataset by using the modified first variable. The simulated MR dataset can have a greater SNR than the SNR of the acquired dataset.

In some embodiments, the acquired MR dataset includes a first set of echo times. The processor can generate a simulated MR dataset. The simulated MR dataset can include a second set of echo times. The second set of echo times can be different from the first set of echo times. The processor can generate the simulated MR dataset by using the modified first variable and the second variable and a modified third variable.

FIGS. 9A-9Eillustrate ρeffmaps and ρ maps depicted inFIGS. 3A-3CandFIGS. 4-8. The ρeffmaps and ρ maps can correspond to a single echo with an echo time of 7.5 ms.FIG. 9Adepicts ρeff(TE1) before any CROWN is performed and corresponds toFIG. 3B.FIG. 9Cdepicts ρTE1before any CROWN is performed and corresponds toFIG. 5.FIG. 9Bdepicts ρeff-CROWN(TE1) and how it is used can be seen inFIGS. 4, 7, and 8.FIG. 9Ddepicts ρTE1-CROWN, which can be the result of CROWN being performed on ρTE1ofFIG. 9C. The contrast of the substantia nigra (SN) and red nucleus (RN) can appear to increase after CROWN, which could be due to not correcting the R1 map for iron.FIG. 9Edepicts ρTE1-CROWN2fromFIG. 7and is the result of correcting ρeff-CROWN(TE1) for R2* and then applying CROWN a second time.

New images can be generated from the CROWN PD and T1 maps. The CROWN images can also be used to simulate images with the original flip angles and create higher quality images than were originally collected. For given tissue parameters T1, ρ, and T2*, together with imaging parameters TR, TE, and flip angle θ, the predicted signal from a spoiled GRE scan, S, is given as Equation 11:

These CROWN processed maps can be input into Equation 11 to regenerate the spoiled GRE datasets which were used to create them. The SNR of the results can be compared with the original images. Further, any scan that has well mathematically understood signal behavior as a function of T1, PD, and T2* can be simulated with any imaging parameters. This can provide better SNR than what would have been acquired using the actual sequence, making CROWN processing a powerful tool to improve the SNR even for the originally collected data.

FIG. 10illustrates a test image used in the simulations. Five different tissue types can be used in the simulations. Each annular region can represent one tissue type. A background square with no intensity can be added as the outer region. The test image can be built out of a set of embedded squares with each annular-like region representing one tissue type. First, the original signal intensity can be calculated using Equation 11. Five types of tissues with spin density values ranging from 0.6, 0.7, 0.8, 0.9 and 1 can be used in the simulations and the corresponding T1 values can be calculated using R1=2.03/sec*β−1.81/sec. A TR of 25 ms and TE of 0 ms can be used to generate both 6° and 24° data. The outer border region can be set to have zero signal intensity. Final values for 6° and 24° magnitude images are given in Table 4.

Table 4 illustrates the initial signals used in the simulation of the MRI magnitude for regions1to5to generate the conventional and CROWN PD and T1 maps.

As a demonstrative case, Gaussian noise can then be added to the original signal. The amount of noise can be set to be 10% of the signal from region1at the 6° magnitude image. Second, the PD and T1 maps can be generated based on the linear transformation of Equation 11. Finally, CROWN can be applied on the PD and T1 maps to remove the noise. 6° and 240 magnitude images can be reproduced to compare the results with the original data.FIG. 11illustrates simulation results for CROWN processing. To simulate a more realistic situation, 10% Gaussian noise (e.g., relative to the signal of the center square for the 6° magnitude image) can be added to each image. PD and T1 maps can be generated using 6° and 24° magnitude images, then the CROWN process can be applied on both maps. CROWN processed PD maps and original T1 maps can be used to reproduce 6° and 24° magnitude images.

For an imaging approach, two sets of STAGE data can be collected from two different individuals on two different MRI scanners. The first set of STAGE data can be collected on a 3T Siemens Prisma scanner with a 64-channel head/neck coil using the following imaging parameters: a resolution of 0.67 mm×1 mm×1.34 mm; FOV=256 mm×192 mm (final matrix 384×288); TE1=7.5 ms, TE2=15 ms, and TE3=22.5 ms; TR=29 ms; bandwidth=180 Hz/pixel; FA=6° and 27°; PFP=7/8; using elliptical sampling; axial imaging plane; and a parallel acceleration factor=3. The total imaging time can be 6 minutes 42 seconds.

The second set of STAGE data can be collected on a 3T Siemens Skyra with a 16-channel head/neck coil using the following imaging parameters: a resolution of 0.67 mm×1 mm×1.34 mm; FOV=256 mm×192 mm (final matrix 384×288); TE1=7.5 ms, TE2=12.5 ms, and TE3=17.5 ms; TR=25 ms; bandwidth=220 Hz/pixel; FA=6° and 24°; partial Fourier factor=7/8; using elliptical sampling; and a parallel acceleration factor=2. The total imaging time can be 8 minutes 30 seconds. After acquisition, the STAGE data can be processed to generate T1 and effective spin density maps, both corrected for B1 transmit/receive inhomogeneity.

FIG. 12illustrates an example of CROWN processing performed on multiple STAGE PD maps. The STAGE PD maps can be created from the 3T Siemens Prisma STAGE data. The echo time of zero (TE=0 ms) for the STAGE row (e.g., top row) can be determined from the intercept of a linear fit across the three effective STAGE PD maps (e.g., similar to Approach 1 except using ρeffrather than ρeff-CROWN). The CROWN coefficients used can be from Table 2. The CROWN row (e.g., bottom row) is shown. The most dramatic effect of noise reduction can be seen at an echo time of zero (TE=0 ms) where the initial estimate is the noisiest. The TE=60 ms PD map can be simulated using both the R2* map and the ρAVGmap (e.g., both CROWN and STAGE versions) to show how long echo times can be generated to create unique T2* contrast.

FIG. 13illustrates a scatter plot of R1 values versus β values for different combinations of STAGE and CROWN processed maps using the pixels from the same slice shown inFIG. 12(at TE=0 ms). The effect of CROWN processing can be seen in steps, from the noisy distribution of points before CROWN (STAGE R1 vs STAGE β), to the tighter distribution seen from applying CROWN to just the R values first using Equation 4 (STAGE R1 vs CROWN β), and finally to the perfectly linear distribution seen from re-calculating R1 from the CROWN R values using Equation 5 (CROWN R1 vs CROWN β).

FIGS. 14A-14Hillustrate the efficacy of CROWN processing for forward simulating data using 3T Siemens Skyra STAGE data. The CROWN simulated images can be generated using STAGE T1, ρAVG-CROWN, and R2*AVG-CROWNas inputs into Equation 11. The original data from the Skyra can be seen inFIGS. 14A, 14C, 14E, and 14Gwith their corresponding forward simulated data from CROWN inFIGS. 14B, 14D, 14F, and 14H, respectively.FIGS. 14A, 14B, 14C, and 14Dshow the 6° STAGE data andFIGS. 14E, 14F, 14G, and 14Hshow the 24° STAGE data.FIGS. 14A, 14B, 14E, and 14Fis from TE=7.5 ms andFIGS. 14C, 14D, 14G, and 14His from TE=22.5 ms. The low flip angle (FA=6°) images can be quite noisy and can suffer from receive field inhomogeneity, as shown inFIGS. 14A and 14Cwhile the CROWN/STAGE simulated FA=6° image has significantly improved SNR and uniformity, as shown inFIGS. 14B and 14D.

FIG. 14Aillustrates a spin density weighted image with a short echo time (TE) before (e.g., prior to) CROWN processing.FIG. 14Billustrates a spin density weighted image with a short echo time after (e.g., post) CROWN processing of the data used to generateFIG. 14A.FIGS. 14A and 14Bdepict a region (e.g., same region) encompassed by a dashed circle.FIG. 14Bdepicts a CROWN simulated image showing an inhomogeneity (e.g., tumor) that is distinguishable (e.g., visually distinguishable, distinct) from the surrounding tissue. In this image, the inhomogeneity can be distinguishable from the surrounding tissue because the CROWN processed image has a higher signal-to-noise ratio than the image before CROWN processing (e.g., STAGE image, pre-processed image).FIG. 14Adepicts the corresponding region encompassed by the dashed circle. In this image, the inhomogeneity may not be clearly distinguishable from the surrounding tissue because the image has a lower signal-to-noise ratio than that ofFIG. 14B.

FIG. 14Cillustrates a spin density weighted image with a long echo time (e.g., long compared to the echo time of the image shown inFIGS. 14A and 14B) before CROWN processing.FIG. 14Dillustrates a spin density weighted image with a long echo time after CROWN processing of the data used to generateFIG. 14C.FIG. 14Eillustrates a T1 weighted image with a short echo time before CROWN processing.FIG. 14Fillustrates a T1 weighted image with a short echo time after CROWN processing of the data used to generateFIG. 14E.FIG. 14Gillustrates a T1 weighted image with a long echo time (e.g., long compared to the echo time of the image shown inFIGS. 14E and 14F) before CROWN processing.FIG. 14Hillustrates a T1 weighted image with a long echo time after CROWN processing of the data used to generateFIG. 14G.

The MRI system can include a third set of values corresponding to the acquired MR dataset at a set of positions. The set of positions can correspond to a map (e.g., spin density map). A subset of the third set of values can satisfy a predetermined threshold. For example, the subset of the third set of values can be greater than a predetermined threshold or less than a predetermined threshold. The subset of the third set of values can include a subset of the set of positions. The subset of the set of positions can correspond to a location of blood vessels. The MRI system can include a simulated MR dataset. The simulated MR dataset can include a fourth set of values at the subset of the set of positions. The fourth set of values can include values at the location of blood vessels. The subset of the set of positions can include the location of blood vessels. The processor can replace the fourth set of values at the set of positions with the subset of the third set of values at the set of positions. For example, the processor can overlay the image of blood vessels from an original spin density map onto an improved spin density map. The improved spin density map can have a greater SNR than the SNR of the original spin density map.

FIG. 15illustrates examples of different methods of generating ρ and R2* maps from the Prisma dataset. Using the notations depicted in Table 3 andFIGS. 3A-3CandFIGS. 4-8, from left to right, the top row depicts STAGE R2*, R2*CROWN, R2*AVG-CROWN, and R2*AVG-CROWN2. The bottom row depicts ρAVG, ρCROWN, ρAVG-CROWN, and ρAVG-CROWN2. Approaches 2 and 3 can work well for p. Approach 3 can work well for R2*.

The efficacy of CROWN processing can be demonstrated inFIGS. 10 and 11for the simulation results andFIGS. 12, 13, 14A-14H, and 15for the in vivo results where the original noisy spin density maps, R2* maps, and low flip angle images can be seen to have significantly improved.

The values of (β, R1) from region1in the simulation can be plotted to compare the change of β and R1 before and after CROWN, as shown inFIG. 2. The range of R can be reduced from [1.25, 2.2] to [1.5, 1.82], which can imply that the noise in the spin density map should decrease, and the SNR of the spin density map, therefore, should increase. However, the range of R1 can increase from [1.4, 1.8] to [1.25, 2.0], which can imply that the noise in the T1 map does not decrease, and the SNR of the T1 map may not improve after the CROWN processing. This same effect can be seen on the in vivo data, as shown inFIG. 13. These predictions can be verified in the simulations shown in Table 5, where it can be seen that the SNR of the spin density map for each region can increase after the CROWN processing. However, the SNR of the T1 map for regions1,2, and3can decrease after CROWN. With this effect in mind, the original T1 map can be used instead of the CROWN processed T1 map to reproduce the new CROWN 6° (FIG. 14D) and 24° magnitude images.

Table 5 illustrates SNR values for the simulated data before and after CROWN processing. Gaussian noise of 10% of the signal from region1for the 6° magnitude image can be used for all regions in the 6° and 240 generated magnitude images.

As shown inFIG. 11, the noise can be significantly reduced after CROWN processing for the spin density map. The SNR can be significantly improved for all regions but with dramatic improvements in regions4and5of Table 5. The noise can be significantly reduced for the 6° and 240 magnitude images after CROWN, especially for regions1to4of Table 5. The SNR for region5that represents CSF may not improve significantly. This may be a result of the low SNR for region5from the original T1 map and the fact that 6° is not below the Ernst angle for CSF. Had a flip angle dataset lower than 6°, such as 2° or some other angle, also been collected, a similar improvement in SNR effect would have been seen. The boxes shown inFIG. 11can have a different tissue property (e.g., spin density values, T1 values, etc.). The tissue properties can be related to each other.

CROWN can offer a powerful tool to improve SNR without modifying or blurring the image structures as is the case in most other methods that purport to improve SNR. CROWN processing can lead to improved SNR not only in the original PD estimates but also for the original low and high flip angle images and any other images for any arbitrary flip angles or images from any other MRI pulse sequence. CROWN processing can work in conjunction with STAGE imaging, any other multi-flip angle approach, or any method that generates both a spin density map and T1 map. CROWN processing can be “scanner agnostic” by being successfully applied to two separate in vivo datasets from two separate scanners. However, for other field strengths, new coefficients can be determined since the T1 values for tissues can change for different field strengths. CROWN processing can be used in low field strength applications where the SNR is inherently worse.

The effects of iron content on T1 can be corrected. The relationship of R1 versus R=1/p can be reasonably well behaved when there is no iron present in the tissue. However, iron can change the R1 value of tissue. The R1 values of different tissues used to determine the linear coefficients a and b for CROWN processing can be under the assumption that no iron is present. If an incorrect relationship is forced onto the tissue, the expected outcome can be modified, correct tissue properties can be lost, and contrast in the images can be lost. The iron content of the different tissues can be used to correct the measured values of R1 before applying CROWN. A set of data from a healthy adult volunteer can be acquired on a 3T scanner in the example shown inFIGS. 16A-16Bto calculate the R1 value as a function of water content for the different deep GM. The susceptibility (e.g., quantitative susceptibility), T1, and proton density can be measured for each of the different deep GM structures. The R1 value can be corrected to remove the T1 reducing nature of iron, which can act like a paramagnetic contrast agent in the tissue, and to find the pristine relationship between R1 and water content. The constraint function can include a component that accounts for a presence of iron in a tissue. For example, the relation between two or more variables can account for and be used to correct for the presence of iron.

To correct the effect of iron on T1 shortening, the relationship between R1 and susceptibility can be evaluated. As shown inFIG. 16A, the relationship between R1 and susceptibility at 3T for deep GM can be R1=1.37/sec/ppm*χ(ppm)+0.86/sec. From this linear relation, the change in R1 from tissue to tissue can be related to the change in susceptibility via ΔR1=1.37/sec/ppm*Ax (ppm). The corrected R1 can be calculated by subtracting ΔR1 from the measured R1. The presence of iron can decrease the T1 values, but the proton density is not affected. Therefore, the relation between R1 and susceptibility can be used to correct the T1 reducing effect of iron.

FIG. 16Bshows that after iron correction, the slope of R1 vs. β decreased from 2.42/sec to 2.01/sec and the intercept from 2.19/sec to 1.78/sec. The result from fitting to Table 1 and including WM as well which leads to the earlier formula that R1=2.03/sec*β−1.81/sec which can give a better fit for the T1 of cortical GM. When using CROWN, the formula for R1 that includes susceptibility effects becomes R1*ironfree=R1meas−ΔR1=2.03/sec*β−1.81/sec.

An example of R1 before and after the iron correction, along with the resulting PD maps produced from CROWN is shown inFIGS. 17A-17E. A comparison between using an R1 map, uncorrected for iron (FIG. 17A), to generate a CROWN PD map (FIG. 17D) versus using an R1 map, corrected for iron (FIG. 17C), to generate a CROWN PD map (FIG. 17E). Approach 2 depicted inFIG. 5can be used to generate these PD maps. The R1 map can be the inverse of the STAGE T1 map depicted inFIG. 3C. The correction map, ΔR=1.37/sec/ppm*Δχ(ppm) can be seen inFIG. 17Busing a threshold of 20 ppb. The 3T Siemens Prisma dataset can be used.

A threshold can be set below which the susceptibility is assumed to be zero. This can have a very small effect when the threshold is 25 ppb, for example, leading to only a 0.035/sec change in R1. This threshold can be chosen based on the inherent SNR in the QSM dataset. For a given noise level (e.g., as determined in the white matter or thalamus regions, for example), choosing the threshold to be 2 times the noise level can be acceptable while 3 times would be a more conservative approach to have less of an effect of noise on estimating the T1 change from iron content. The initial measurements of susceptibility, T1, and β, as well as the R1 and T1 values after correcting for the T1 reducing effect of iron are shown in Table 6.

Table 6 illustrates the initial measurements of susceptibility, T1, and β, as well as the R1 and T1 values after correcting for the T1 reducing effect of iron.

FIG. 18illustrates a method of magnetic resonance imaging (MRI). In brief summary, the method1800can include receiving an acquired magnetic resonance (MR) dataset (BLOCK1805). The method1800can include extracting a first set of values and a second set of values (BLOCK1810). The method1800can include applying a constraint function (BLOCK1815). The method1800can include minimizing a cost function (BLOCK1820). The method1800can include inputting the first variable into the cost function (BLOCK1825).

The method1800can include receiving an acquired MR dataset (BLOCK1805). The acquired MR dataset can have a first SNR. The acquired MR dataset can include data corresponding to a first flip angle and a second flip angle. The acquired MR dataset can be acquired by imaging an anatomical region using at least one echo time

The method1800can include extracting values (e.g., a first set of values and a second set of values) (BLOCK1810). The first set of values can correspond to a first variable. The first variable can have a second SNR. The second set of values can correspond to a second variable. The method1800can include extracting, from the acquired MR dataset, a third set of values. The third set of values can correspond to a third variable. The first variable can correspond to the inverse of spin density and the second variable can correspond to the inverse of T1. The first variable can correspond to susceptibility and the second variable can correspond to R2*. The first variable can correspond to an inverse of spin density and the second variable can correspond to R2*.

The method1800can include applying a constraint function (BLOCK1815). The constraint function can be a function of the first variable and the second variable. The constraint function can be a function of the first variable, the second variable, and the third variable. The constraint function can be a function of multiple variables. The constraint function can include a component that accounts for a presence of iron in a tissue. The constraint function can include a component that accounts for each echo time. The constraint function can include the constraint function described by Equation 1.

The method1800can include minimizing a cost function (BLOCK1820). The cost function can be minimized according to the constraint function to generate a cost function solution. The cost function can include the cost function described by Equation 2.

The method1800can include inputting a variable (e.g., the first variable) into the cost function (BLOCK1825). The first variable can be input into the cost function solution to generate a modified first variable having a third SNR. The third SNR can be greater than the second SNR. The method1800can include generating, using the modified first variable and the second variable, a simulated dataset for an arbitrary flip angle.

In some embodiments, the method1800can include generating a first calculated value of a third variable corresponding to R2*. The third variable can have a fourth SNR. The method1800can include generating, using the modified first variable, a second calculated value of a modified third variable. The modified third variable can have a fifth SNR. The fifth SNR can be greater than the fourth SNR.

In some embodiments, the method1800can include generating a first calculated value of a third variable corresponding to R2*. The third variable can have a fourth SNR. The method1800can include using the first modified first variable at each of the at least two echo times and the third variable to generate a second modified first variable. The method1800can include using (e.g., apply) the second modified first variable at each of the at least two echo times and the second variable as inputs into the cost function solution to generate a third modified first variable having a sixth SNR. The sixth SNR can be greater than the third SNR.

In some embodiments, the method1800can include generating a second calculated value of the third variable corresponding to R2*. The third variable can have a seventh SNR. The seventh SNR can be greater than the fourth SNR. In some embodiments, the method1800can include generating, using the modified first variable, a simulated MR dataset having an eighth SNR. The eighth SNR can be greater than the first SNR.

In some embodiments, the acquired MR dataset includes a first set of echo times. The method1800can include generating, using the modified first variable and the second variable and a modified third variable, a simulated MR dataset. The simulated MR dataset can include a second set of echo times. The second set of echo times can be different from the first set of echo times.

In some embodiments, the acquired MR dataset includes data corresponding to a first flip angle and a second flip angle for a plurality of echo times. The method1800can include generating a weighted average value for at least one of the first variable and the modified first variable over the plurality of echo times. In some embodiments, the method1800can include generating a spin density image.

The implementations described herein can be implemented in any of numerous ways including, for example, using hardware, software or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers.

A computer employed to implement at least a portion of the functionality described herein may comprise a memory, one or more processing units (also referred to herein simply as “processors”), one or more communication interfaces, one or more display units, and one or more user input devices. The memory may comprise any computer-readable media, and may store computer instructions (also referred to herein as “processor-executable instructions”) for implementing the various functionalities described herein. The processing unit(s) may be used to execute the instructions. The communication interface(s) may be coupled to a wired or wireless network, bus, or other communication means and may therefore allow the computer to transmit communications to or receive communications from other devices. The display unit(s) may be provided, for example, to allow a user to view various information in connection with execution of the instructions. The user input device(s) may be provided, for example, to allow the user to make manual adjustments, make selections, enter data or various other information, or interact in any of a variety of manners with the processor during execution of the instructions.

The terms “program” or “software” are used herein to refer to any type of computer code or set of computer-executable instructions that can be employed to program a computer or other processor to implement various aspects of embodiments as discussed above. One or more computer programs that when executed perform methods of the present solution need not reside on a single computer or processor, but may be distributed in a modular fashion amongst a number of different computers or processors to implement various aspects of the present solution.

Computer-executable instructions may be in many forms, such as program modules, executed by one or more computers or other devices. Program modules can include routines, programs, objects, components, data structures, or other components that perform particular tasks or implement particular abstract data types. The functionality of the program modules can be combined or distributed as desired in various embodiments.

Any references to implementations or elements or acts of the systems and methods herein referred to in the singular can include implementations including a plurality of these elements, and any references in plural to any implementation or element or act herein can include implementations including only a single element. References in the singular or plural form are not intended to limit the presently disclosed systems or methods, their components, acts, or elements to single or plural configurations. References to any act or element being based on any information, act or element may include implementations where the act or element is based at least in part on any information, act, or element.

The systems and methods described herein may be embodied in other specific forms without departing from the characteristics thereof. The foregoing implementations are illustrative rather than limiting of the described systems and methods.