Initial in vivo experience with steady-state subzone-based MR elastography of the human breast

Authors


Abstract

Purpose

To describe initial in vivo experiences with a subzone-based, steady-state MR elastography (MRE) method. This sparse collection of in vivo results is intended to shed light on some of the strengths and weaknesses of existing clinical MRE approaches and to indicate important areas of future research.

Materials and Methods

Elastic property reconstruction results are compared with data compiled from the limited existing body of published studies in breast elasticity. Mechanical parameter distributions are also investigated in terms of their implications for the nature of biological soft tissue. Additionally, a derivation of the statistical variance of the elastic parameter reconstruction is given and the resulting confidence intervals (CIs) for different parameter solutions are examined.

Results

By comparison with existing estimates of the elastic properties of breast tissue, the subzone-based, steady-state MRE method is seen to produce reasonable estimates for the mechanical properties of in vivo tissue.

Conclusion

MRE shows potential as an effective way to determine the elastic properties of breast tissue, and may be of significant clinical interest. J. Magn. Reson. Imaging 2003;17:72–85. © 2002 Wiley-Liss, Inc.

THE FIELD of MR elastography (MRE) has reached a point where clinical experimentation is emerging with many of the available methods for noninvasive, elastic property imaging. While this is certainly an exciting juncture in the development of what promises to be a fertile clinical imaging tool, it is also a period of some uncertainty, as early results are yet to be validated in terms of the accuracy with which they portray the mechanical properties of tissue. It is certainly difficult to accurately describe the true mechanical behavior of biological tissue under loading, and one can assume that accurate determinations of the mechanical characteristics of a tissue from its behavior under load through use of a mathematical model are equally difficult to achieve and prone to error. All existing MRE methods make use of some type of model to generate quantitative elasticity estimates; hence, they all experience errors that arise from the inevitable shortcomings in their respective model descriptions. Nonetheless, the limited body of clinical MRE experience in the literature to date indicates that valuable results can be achieved with elastographic imaging of in vivo tissue, despite obvious inaccuracies in the models on which the estimated values are based. This provides hope that by carefully accounting for the dominant mechanical effects, an MRE model should be able to produce clinically meaningful results.

Outside of the field of elastography, in vivo measurements of the elastic properties of biological tissues are hard to find in the literature. Saravazyan et al. (1) have generated and collected data on the mechanical properties of soft tissues from available sources. While the elasticity values of particular tissues vary widely in this collection of results, there is strong evidence of an increase in stiffness between healthy and cancerous tissues of up to several orders of magnitude. Krouskop et al. (2) developed a protocol for testing the stiffness of biological tissue samples at low frequencies using a servo-hydraulic Instron device (Instron, Inc., Canton, MA). These data also suggest an order of magnitude increase in stiffness between healthy and malignant tissues.

Within the MRE community a handful of research groups have reported in vivo results for breast tissue. Plewes et al. (3) presented quasi-static strain images from the breast of a healthy volunteer. A basic study of shear wave propagation in excised tissue by Bishop et al. (4) showed that wave speed exhibits relatively small changes with frequency in a variety of tissues. Attenuation effects were found to be mostly constant as a function of frequency, except in the case of fat, which exhibited a noticeable increase in wave speed with frequency. Kruse et al. (5) described a variety of early tissue characterization results based on the use of local frequency estimation under plane wave conditions and an assumption of linear elastic wave propagation. These data show frequency and temperature dependence in modulus measurements in in vitro kidney and liver samples, as well as the possible effects of anisotropy on varying measurements in highly ordered skeletal muscle. Additionally, an example of an in vivo breast elasticity image was given that showed a localized area roughly two to three times stiffer than the surrounding fibrous tissues corresponding to a biopsy-proven cancerous tumor. Sinkus et al. (6, 7) presented a method for measuring the tensor elasticity of tissue based on a regularized algebraic inversion of the equations of linear elasticity with an additional attenuation factor. While their method is not a continuum approach to anisotropic materials per se, this group has shown results from their tensor-valued Young's modulus reconstructions indicating that while malignant and benign tumors can exhibit similar isotropic stiffnesses, they may be differentiable based on their level of anisotropy. In vivo data obtained with the method have revealed increases in stiffness of roughly two to three times between background tissue and lesions.

In this work a collection of initial in vivo results in healthy volunteers from a three-dimensional, steady-state MRE method are presented and discussed. The approach is based on a statistically formulated nonlinear error minimization between the gradient-echo, phase-contrast, MR-generated harmonic displacement data and a fully three-dimensional linear elastic model of the steady-state mechanical wave propagation (8–11). As the computational load associated with performing this type of parameter reconstruction at resolutions approaching that of the MR is beyond the capability of most modern computers, the MRE mechanical property estimate is developed on a set of overlapping subzones that support the global imaging region. This subzone method has been shown to be effective in estimating the elasticity distributions within gelatin, and more recently has been applied in the clinical setting (12–14). Through use of the statistical basis for the reconstruction approach, a standard covariance measure of the reconstructed parameter distribution can be generated that allows the magnitude of the 95% confidence interval for the estimated modulus to be calculated, as outlined in Appendix A.

Materials and Methods

Application of the subzone elastography reconstruction method to displacement data generated by phase-contrast MR techniques has been described previously, and encouraging results have been obtained in three-dimensional, asymmetric gel phantoms (14, 15). Clinical imaging (as in the gel phantom case) requires that steady-state harmonic motion be generated within the tissue during the phase-contrast imaging procedure. To this end a piezocrystal-based clinical breast actuator (Fig. 1) has been developed, which allows the application of precision mechanical excitation to the breast during imaging without the creation of electromagnetic artifacts within the MR phase data. Figure 1A shows the clinical breast actuator with its form-fit padding removed. Both breasts hang pendant through the large oval opening while mild compression is applied to the tissue by a movable plate and the vibrating pillow-block/actuator mechanism seen in the lower right corner. Figure 1B shows a closeup of this actuating mechanism and the six piezoactuators set in two serially arranged groups of three. Total displacement of the device has a magnitude of roughly 30 μm.

Figure 1.

A clinical breast actuator for MRE exams. Piezocrystal-based mechanical excitation is applied to the breast through the ceramic pillow-block construction seen in the lower right corner. The low-field piezocrystal actuator itself is detailed in image B.

While gradient-echo phase-contrast methods for detecting motion fields have been available for some time, their application to the imaging of steady-state motion fields is nontrivial. In order to maintain a steady motion pattern within the tissue, a separate signal generator is fixed to the gradient frequency of the MR sequence and used to drive the piezocrystal actuator throughout the imaging procedure. A receive-only 5-inch circular coil is used for imaging and data acquisition. Phase data is collected for a variety of phase offsets between the motion-encoding gradients (MEGs) and the applied motion from which a complex valued description of the steady-state motion field is constructed (14).

The computational process for the estimation of mechanical properties from displacement data has been described in detail elsewhere (8, 10, 11). The basic premise of the algorithm is to match the observed displacement behavior measured in the tissue to the displacements generated within a discretized model through mechanical property parameter optimization. The discretized model used for motion calculation based on the most current property estimate is driven by displacement boundary conditions taken from the measured data. This ensures accurate representation of the actuator-tissue coupling as any effects of this coupling will be expressed in the motion patterns at the tissue boundary. Using the squared error between the measured and calculated displacement patterns, a nonlinear minimization problem is developed which is recast as a series of smaller subzone solution spaces to facilitate computation. While this nonlinear optimization approach to parameter reconstruction is computationally intensive, the total run time of the calculations can be significantly reduced by processing the subzone minimization problems in parallel. One of the direct benefits of using such a least-squared-error approach is the ability to directly generate a covariance matrix for the reconstructed parameter set. These statistical variance values can be used to generate a confidence interval for the reconstructed parameter results, giving a quantifiable measure of how well the calculated model matches the behavior observed within the imaging body. The model used for the results shown here is the standard three-dimensional, isotropic equation of linear elasticity, written in partial differential equation (PDE) form as

equation image(1)

where u is the three-dimensional displacement vector within the material, μ and λ are the two Lamé moduli for the material, and ρ is the material density. Two other common material property descriptors are Young's modulus, E, and Poisson's ratio, ν, which describes the displacement in one direction relative to the induced displacement in an orthogonal direction. These parameters may be written in terms of Lamé's constants through the following relationships (16):

equation image(2)

The subzone reconstruction process is based on the model description shown in Eq. [1] and generates spatial distributions of μ and λ, along with their variances, σmath image and σmath image, and covariance, σμλ, as its solution. The relations shown in Eq. [2] can then be used to create the alternative elastic parameter distributions E and ν, while the variances for these parameters can be estimated by error propagation, where

equation image

and

equation image(3)

The magnitude of the 95% confidence interval for a particular parameter value can be calculated from its variance through the following relation:

equation image(4)

For the in vivo human subject study presented here, five volunteers were selected on the basis of their having no abnormalities or lesions detected during standard clinical screening exams taken at the time of the MRE imaging procedure.

Results

Here a subset of reconstructed elastic parameter images from several subjects are selected to highlight characteristic features in the elastograms recovered from this initial series of exams. None of the participants in the group were diagnosed with benign or malignant lesions. Analysis of the elastic property results based on general segmentation of internal tissue types in the corresponding T2*-weighted MR magnitude images for all five subjects is also described. This approach is used to reach preliminary estimates of the bulk in vivo properties for breast tissue constituents (specifically, fat and glandular tissues) and examine their variability within the breast of an individual as well as across the subject pool contained in this report.

Reconstructed Images

Overall, the three-dimensional, full-volume reconstructed images of Young's modulus (E) or related (via Eq. [2]) shear modulus (μ) appear to be promising. They have exhibited a level of anatomical correlation with their T2*-weighted MR counterparts that is physically intuitive in cases in which the breast has well-demarcated structural characteristics, while also portraying a distinct contrast signature. The 3D volume recoveries have not been plagued by artifacts associated with edges or termination of the image volume, provided the MR displacement data is robust at these boundaries.

For example, Figure 2 shows a montage of T2*-weighted MR magnitude images (subject 1003) relative to the corresponding views of the E distribution. The central conduit of glandular tissue extending distally from the chest wall toward the nipple that dominates the mid-series of MR images in the set appears as a stiffer medium in the elastogram, surrounded on both sides by a softer material corresponding to an outer layer of fat, as expected. Figure 3 illustrates another example, this time presented in terms of μ. In these images (subject 1004), the glandular structure is more centrally isolated throughout much of the T2* image volume, but again appears as a stiffer area in the elastogram surrounded by a zone of softer fatty composition. The correspondence between T2*-weighted MR and the elasticity images is not always obvious, however. Figure 4 shows Young's and shear modulus images from a single slice (subject 1005) in which structural correlations with stiffness properties are less evident.

Figure 2.

Slice montages of the MR image and corresponding estimated Young's modulus (E) distribution for subject 1003. Image A shows a slice montage of the T2*-weighted MR magnitude images. Image B shows the reconstructed Young's modulus solution in units of [kPa].

Figure 3.

Slice montages of the MR image and corresponding estimated shear modulus (μ) distribution for subject 1004. Image A is a slice montage of the T2*-weighted MR magnitude series. Image B shows the reconstructed shear modulus solution in units of [kPa].

Figure 4.

Single slices of the MR image and the corresponding estimated elastic parameter distribution for subject 1005. Image A shows a slice of the T2*-weighted MR magnitude image, while images B and C show the shear modulus (μ) reconstruction and the Young's modulus (E) solution, respectively, within the same slice, both in units of [kPa].

In addition to Young's and shear moduli, the algorithm recovers a second elastic property: either Poisson's ratio (ν) or Lamé's λ modulus. Figure 5 contains these parameter images for the cases highlighted in Figures 2–4. They are less distinctive and more difficult to interpret. The Poisson ratio images are relatively uniform, with most values occurring between 0.4–0.5, suggesting that the breast is modestly compressible. The images do contain some speckle, but it is not yet clear whether this is due to noise sensitivity in the parameter estimate or to variations that reflect changes in the physical characteristics of the tissue. The λ modulus images are even more challenging to understand because this parameter does not have a physically intuitive meaning. Again, the reconstructed values are relatively uniform but do exhibit variations, especially over the more central areas of the breast.

Figure 5.

Slice montages of the estimated Poisson's ratio (ν) distribution for subject 1003 and the estimated λ modulus distribution for subject 1004. Image A shows the reconstructed Poisson's ratio solution [unitless]. Image B shows the reconstructed λ modulus solution in units of [kPa].

As described in detail in Appendix A, it is possible to determine the 95% confidence interval (CI) on each reconstructed elastic property estimate, allowing the construction of a CI image. The CI maps for Young's modulus and Poisson ratio corresponding to the property estimates in Figures 2 and 5 (subject 1003) are reported in Figure 6. The CI on Young's modulus is typically an order of magnitude below the estimated pixel/voxel value. Generally, the CI is in the range of 1–3 kPa relative to 10–30 kPa for the elastic property itself. Some widening of the CI does occur in isolated locations, with values reaching as high as 10 kPa. Based on the CI map, the Poisson ratio estimates appear to be less robust. Pixel CIs are generally below 0.15, but do range over the full scale of the reconstructed property in the image. Figure 7 shows the CI maps for the elasticity images in Figures 3 and 5 (subject 1004). Here the robustness of the shear images is very high, with the CIs nearly two orders of magnitude smaller than the estimate in the majority of locations. The accompanying λ modulus images exhibit less confidence, similar to the Poisson's ratio, although they appear to be even more sensitive to noise. The CIs are only a factor of 2 or less than the estimated modulus, suggesting that these images may be the least reliable in general.

Figure 6.

Slice montages of the 95% confidence intervals of the estimated property distributions for subject 1003. The grayscale images show how the size of the 95% confidence interval for the particular parameter reconstruction varies throughout the imaging region. Image A shows the magnitude of the 95% confidence interval for the reconstructed Young's modulus distribution in units of [kPa]. Image B shows the magnitude of the 95% confidence interval for the reconstructed Poisson's ratio distribution [unitless].

Figure 7.

Slice montages of the 95% confidence intervals of the estimated property distributions for subject 1004. The grayscale images show how the size of the 95% confidence interval for the particular parameter reconstruction varies throughout the imaging region. Image A shows the magnitude of the 95% confidence interval for the reconstructed shear modulus distribution in units of [kPa]. Image B shows the magnitude of the 95% confidence interval for the reconstructed λ modulus distribution [kPa].

Tissue-Type Property Analysis

Manual segmentation was performed on the volumetric mechanical property data sets in order to investigate differences in estimated mechanical properties between fat and fibroglandular (FBG) tissue. Segmentation was performed using the T2*-weighted MR images acquired in the same geometry as the reconstructed mechanical property images. The results presented here are from one particular segmentation; however, multiple segmentations were performed, leading to marginal changes in numerical property values, and no alterations in the general trends were noted. The segmentations were purposely coarse and perhaps are better termed as region-of-interest (ROI) identifications, as the goal was to analyze reconstructed mechanical properties in zones of the breast that clearly correspond to fat or FBG in the anatomical MR scans rather than to separate the fine FBG structure from the fat through detailed image analysis. Examples of the manual segmentation based on the T2*-weighted MR images are provided in Figure 8. The white areas in the segmentation images indicate tissue classified as adipose, while the black areas represent regions classified as FBG. The gray zones were left unclassified, and property values within these areas were not compiled into the segmentation results.

Figure 8.

An example of the manual segmentation used for differentiating adipose and fibroglandular tissues. Image A shows a representative slice of the T2*-weighted MR for subject 1001. Image B shows the manual thresholding used for this slice, with white indicating regions classified as fat, black indicating regions defined as fibroglandular, and gray indicating unspecified tissue. Images C and D show a corresponding slice used in the manual segmentation for subject 1002.

Figure 9 shows the results from full-volume segmentations performed on the images acquired during exams from all five subjects. In general, fat is less stiff than fibroglandular tissue based on comparisons of Young's modulus. A perhaps less intuitive result is the noticeably lower Poisson's ratio within fat relative to FBG tissue in all subjects except subject 1004, in whom Poisson's ratio was found to be slightly higher in the adipose compartment. Tabular representation of the data illustrated in Figure 9 is given below.

Figure 9.

Errorbar plots showing the segmented tissue property results for all five subjects. White bars show average property values within regions of adipose tissue, while gray bars show average property values for regions of FBG tissue. Error bars represent 1 SD in magnitude. Image A shows results for Young's modulus estimates, and image B shows results for Poisson's ratio estimates.

Tissue Property Analysis Data in Tabular Format

Tables 1 and 2 show the numerical data illustrated in Fig. 9. In addition to the mean and standard deviation (SD) values for each subject and each mechanical property, the statistically calculated differences between property values for fat and FBG tissue, with 95% confidence, are indicated. These give an indication of the magnitude of change between mechanical properties in the two tissue types.

Table 1. Average Values for μ and λ From Manually Segmented Areas of Each Imaging Volume
Subject ID numberShear modulus (Pa)λ modulus (Pa)
DifferenceaFat meanFat stdDifferenceaFat meanFat std
(FBG mean)(FBG std)(FBG mean)(FBG sd)
  • a

    Range of statistical difference, for P ≤ 0.05, between parameter values for the two tissues types.

Sub. 1001967.7–1003.18487.51404.313886.1–14603.039366.024546.1
  (9473.0)(1527.3) (53610.6)(36320.0)
Sub. 10021918.5–1957.57082.71127.617375.6–18442.827273.320290.3
  (9020.7)(1069.5) (45182.5)(32023.8)
Sub. 10032647.0–2675.06339.5980.729051.1–29779.720534.515797.9
  (9000.5)(827.6) (49949.9)(30453.0)
Sub. 10042588.2–2615.48209.21183.27977.2–8673.548536.726107.4
  (10811.0)(1183.0) (56862.0)(33263.6)
Sub. 1005955.8–1004.37705.51104.611142.6–12278.532689.417967.4
  (8685.6)(1050.3) (44399.9)(26529.3)
Table 2. Average Values for E and ν From Manually Segmented Areas of Each Imaging Volume
Subject ID numberYoung's modulus (Pa)Poisson's ratio
DifferenceaFat meanFat stdDifferenceaFat meanFat std
(FBG mean)(FBG std)(FBG mean)(FBG std)
  • a

    Range of statistical difference, for P ≤ 0.05, between parameter values for the two tissues types.

Sub. 10013028.8–3131.523536.74034.70.017–0.0180.3860.060
  (26616.8)(4487.4) (0.404)(0.047)
Sub. 10025723.1–5846.119348.93657.20.030–0.0320.3610.075
  (25133.5)(3331.1) (0.392)(0.053)
Sub. 10038131.0–8217.017106.73028.30.057–0.0590.3470.074
  (25280.7)(2521.2) (0.404)(0.045)
Sub. 10047123.7–7203.023136.13516.80.007–0.0060.4090.052
  (30299.5)(3400.0) (0.402)(0.045)
Sub. 10052897.9–3040.421318.03221.00.014–0.0160.3830.059
  (24287.1)(3093.9) (0.398)(0.049)

Discussion

Measurements of elastic moduli for breast tissue are relatively rare within the literature. Saravazyan et al. (1) compiled elastic modulus values for different breast tissues from several sources, showing a broad range of values for normal breast and a consistent trend of significantly increased stiffness within detectable lesions. Krouskop et al. (2) presented elastic moduli measurements for breast tissue based on a specialized mechanical testing technique. For breast tissue specimens with 5% precompression, their method found the Young's moduli for normal fat and normal glandular tissues to be on the order of tens of kPa, while the Young's moduli for fibrous tissues and invasive ductal carcinomas were on the order of hundreds of kPa. The moduli for glandular and fibrous tissues, and invasive ductal carcinomas were found to increase significantly with additional precompression. Table 3 summarizes the combined results from these two reports on the mechanical properties of the breast.

Table 3. A Summary of the Results From Mechanical Testing on Biological Tissue
Tissue typeYoung's modulus (kPa)Frequency (Hz)
Saravazyan et al. (1)
 Normal fat to normal glandular5 to 50Static
 Palpable nodule100 to 5000Static
Krouskop et al. (2) (5% precompression)
 Normal fat18 ± 7 to 22 ± 120.1 to 4
 Normal glandular28 ± 14 to 35 ± 140.1 to 4
 Fibrous96 ± 34 to 116 ± 280.1 to 4
 Invasive D.C.106 ± 32 to 112 ± 430.1 to 4
Krouskop et al. (2) (20% precompression)
 Normal fat20 ± 8 to 24 ± 60.1 to 4
 Normal glandular48 ± 15 to 66 ± 170.1 to 4
 Fibrous218 ± 87 to 244 ± 850.1 to 4
 Invasive D.C.558 ± 180 to 460 ± 1120.1 to 4

Kruse et al. (5) presented preliminary results from an in vivo MRE exam of a patient with a biopsy-proven carcinoma. Similarly, Sinkus et al. (6) reported isotropic and tensor elasticity measurements for a patient with confirmed carcinoma. Table 4 presents a summary of these in vivo MRE results.

Table 4. A Summary of Results From In Vivo MR Elastography Exams of Breast Tissue
Tissue typeYoung's modulus (kPa)Frequency (Hz)
Kruse et al. (5)
 Normal fat15 to 25100
 Normal glandular30 to 45100
 Carcinoma50 to 75100
Sinkus et al. (6) (isotropic elasticity results)
 Normal fat0.5 to 160
 Normal glandular2 to 2.560
 Carcinoma3.5 to 460

Quantitative analysis of the Young's modulus values from the reconstructed images shown above and from the data presented in Figure 9 shows a general agreement with the ex vivo mechanical test data compiled in Table 3, as well as with the in vivo imaging results reported by Kruse et al. (5) in Table 4. Qualitative analysis of the property images and their 95% confidence intervals shows that there is a higher degree of variance (as a percentage of average property value) within the λ modulus results vs. the shear modulus results. This may be due to the increased sensitivity the λ parameter exhibits in the nearly incompressible environment of biological tissue. Because the E and ν distributions are generated from the λ distribution and their variances are estimated according to Eq. [3], their confidence intervals reflect the high sensitivity levels found in the λ solution.

In general, fat tissue is less stiff than fibroglandular tissue based on comparisons of shear and Young's moduli. This is in agreement with common perceptions as well as the clinical results presented in Tables 3 and 4. A less intuitive result is the noticeably lower Poisson's ratio within fat relative to FBG tissue in all subjects except subject 1004, in whom Poisson's ratio was found to be slightly higher in adipose tissue. Hypotheses offering explanations for this trend range from differences in compressibility between the two tissue types to high variances in the reconstructed parameter solution, indicating poor values for comparison. A full investigation into the nature of this observation is beyond the scope of the present work, but warrants future study.

The property values reported in Figure 9 agree well with values for similar tissues shown in Table 3, as well as with those reported by Kruse et al. (5) and presented here in Table 4. This general agreement among measured mechanical properties is illustrated in Figure 10, where Young's modulus values reported for normal fat and normal glandular tissues are plotted along with the spectrum of results presented in Figure 9. It does appear that the stiffness values of certain tissue types may fall within a fairly narrow range of possible values, so any abnormally high modulus values would be of clinical interest.

Figure 10.

A bar graph showing the Young's modulus values for normal fat and normal glandular tissues compiled from various works as well as the results presented in Table 2. General agreement is evident across most of the presented measurements.

In conclusion, an initial selection of clinical images from a subzone-based MRE method has been presented, showing various images from the four basic linear elasticity parameters; μ, λ, E, and ν, as well as the magnitudes of the 95% confidence intervals of the parameter estimates. In general, the images show relatively good correlation to existing measurements of tissue mechanical properties, including both ex vivo mechanical tests and in vivo imaging exams. Calculated CIs indicate a high level of variance within the λ modulus results, possibly related to increased sensitivity within this parameter due to the near incompressibility of tissue. Separating the properties of the two major tissue types (fatty and fibroglandular) within the breast by manual segmentation indicates that fatty tissues have lower moduli and lower Poisson's ratio levels compared to fibroglandular tissues. In general, the volumetric property images appear to mirror the stiffness distributions expected within healthy breast tissue. Coupled with the numerically generated confidence levels of these parameter estimates, the mechanical property images have the potential to be of high clinical value.

APPENDIX A

Statistical Approach to Parameter Reconstruction

While the method of reducing the squared error between calculated and measured data points presents itself as a clear choice for developing a model-based parameter image from displacement data, it also falls under the guise of a statistical approach to parameter reconstruction, and therefore offers the benefits of statistical testing and measures of accuracy. If one views the imaging process from a Bayesian (17, 18) perspective of inferring information about the underlying normally distributed parameter set, Θ ∼ ��(θ0, τ2I), whose random error component is also normally distributed, based on information contained in the measured data, Y ∼ ��(f(θ), σ2I), then the probability of a particular parameter distribution θ based on a specific observation of the data y is expressed as

equation image(5)

This formulation assumes that the parameter set is normally distributed about a mean value of θ0, the a priori assumption, with variance τ2 while the data is normally distributed about the solution f(θ) with variance σ2. The optimal parameter distribution is then obtained by finding the most likely solution, so that

equation image(6)

For a multivariate, normally distributed system, the probability density functions, p(y|θ, σ2) and p(θ, τ2), are given as

equation image(7)

and

equation image(8)

respectively, where NO is the number of observations and NP is the number of parameters being reconstructed. If σ2 and τ2 are known, the constants equation image and equation image from the substitution of Eqs. [7] and [8] into Eq. [6] can be ignored in the maximization problem. Additionally, the term maxθ(ln p(y)) can be ignored as p(y) has no dependence on θ. This leaves the maximization problem

equation image(9)

which, with the appropriate sign change, leads to the equivalent minimization statement

equation image(10)

which is often rewritten as

equation image(11)

where

equation image(12)

This formulation is also known as the Tikhonov regularization problem, which is recognized for the additional a priori image regularization term it contains. The Bayesian approach to the development of Eq. [11] presented here has the advantage of defining the regularization weighting factor, γ, in terms of the variances of the data measurements, σ2, and the a priori image, τ2. Approximations to these variances, σˆ2 and τˆ2, can be calculated as

equation image(13)

and

equation image(14)

for an average a priori image value, θ. Here API represents “all previous images” from which θ was derived, while NPI represents the “number of previous images.” In this way the history of reconstructed images can be utilized to make ensuing image reconstructions more accurate and more readily convergent. By selecting a Gauss-Newton method to solve the optimization problem set out in Eq. [11], an iterative relationship is developed:

equation image(15)

where the prime superscript indicates matrix transpose, and δl adjusts the stepsize of the iterative procedure.

Expanding statistical consideration to the evaluation of the accuracy of the reconstructed parameter set, the covariance matrix, cov(θ), is seen to be

equation image(16)

with the approximation to the variance

equation image(17)

Here the sensitivity matrix, equation image is generated by operating on Eq. [15] with the assumption that θl is a constant, leading to

equation image(18)

Once the covariance matrix is developed, the variance of a particular parameter value θi is given by the diagonal term cov(θ)i,i.

Ancillary