Mark Bydder1, Vahid K Ghodrati1, Yu Gao1, Matthew D Robson2, Yingli Yang1, and Peng Hu1
1UCLA, Los Angeles, CA, United States, 2Perspectum Diagnostics, Oxford, United Kingdom
Synopsis
The study evaluates four physically
motivated constraints in the estimation of the proton density fat fraction
(PDFF) based on the physics of magnetic resonance imaging.
These were smooth fieldmap, smooth initial phase, nonnegative proton density and
moderate $$$R2*$$$ values. Results
show that constraints
are effective at reducing standard deviation and bias.
Introduction
The study evaluates four physically motivated constraints
in the estimation of the proton density fat fraction ($$$PDFF$$$). Least squares approaches
were developed for constraining the parameters in PDFF quantification based on
the physics of magnetic resonance imaging. These were smooth fieldmap, smooth
initial phase, nonnegative proton density and moderate R2* values. Constraints were evaluated in terms of their influence on the bias and standard
deviation of the estimated parameters using numerical simulations and in vivo data acquired on a combined MRI-radiotherapy device at 0.35 T.Methods
The signal from a mixture of water and fat in a single voxel can be written as in Eq 1
[Eq 1] $$$b = WAxe^{iφ}$$$
where $$$b$$$ is a vector of complex data points at $$$n$$$ echo times, $$$b$$$ is a vector of echo times, $$$A$$$ is an $$$n×2$$$ complex matrix expressing the evolution of water and fat, $$$W=diag(e^{iψt})$$$ is an $$$n×n$$$ matrix expressing the evolution of the complex fieldmap $$$ψ=B_0+iR_2^*$$$, $$$x=[w; f]$$$ is a real vector of water and fat proton densities and $$$φ$$$ is a real scalar (initial phase) [1]. The parameters of most clinical interest are $$$R_2^*$$$ and proton density fat fraction, $$$PDFF = f/(w+f)$$$ which can be obtained by iterative search over $$$B_0$$$, $$$R_2^*$$$, $$$x$$$ and $$$φ$$$ to minimize the residual norm $$$||r||$$$.
[Eq 2] $$$r=WAxe^{iφ}-b$$$
Since $$$B_0$$$ is a subject to $$$2π$$$ aliasing (fat-water swapping), a widely used strategy is to include a penalty term to encourage $$$B_0$$$ smoothness [2]. Since $$$B_0$$$ is also used to calculate susceptibility, it can be disadvantageous to compromise the resolution of $$$B_0$$$ [3] and thus it is of interest to consider
alternative constraints. Table 1 lists some prior constraints that may be useful in solving the inverse problem. Previous work has shown there is a benefit to constraining
Eq 1 [3-8], however to date there has been no systematic comparison of constraints using the
same baseline methodology.Results
The constraints in Table 1 were implemented in a least squares minimization as described in [9]. Numerical simulations were
performed using six echo times corresponding to (0.20π 0.56π 0.93π 1.30π 1.66π 2.03π). Multiple
right hand side vectors ($$$b$$$ in Eq 1) were generated
for $$$PDFF$$$ 0–100% with addition of complex Gaussian noise to give an SNR of 6 to reflect the typical situation at low field
strength. Bias
and standard deviation of the estimated $$$PDFF$$$ and $$$R_2^*$$$ were calculated over 10^5 trials to examine
the effect of constraints.
Figure 1 shows simulation results
of the standard deviation and bias (i.e. mean value – true value) over the
range of $$$PDFF$$$ values. The top row is $$$PDFF$$$ and the bottom row is $$$R_2^*$$$.
All of the constraints result in a reduction in standard deviation relative to
unconstrained estimation. Constraint $$$I$$$ reduces the $$$PDFF$$$ standard
deviation without any notable bias penalty other than sacrificing $$$B_0$$$ resolution). Constraint $$$III$$$ reduces the standard deviation of $$$R_2^*$$$ but also
pulls the estimate toward zero (as may be expected). The standard
deviation and bias of $$$PDFF$$$ are also reduced. Constraint $$$III$$$ has a strong influence at the end-points (as expected) with a large
bias. Constraint $$$IV$$$ reduces the standard
deviation with no notable effect on bias.
It is interesting to consider if
constraints can be combined to achieve additive benefits. Unfortunately this does not not appear to be the case; key examples are given in Table 2. These indicate
that Constraints $$$I$$$ and $$$IV$$$ both decrease the
standard deviation but their combination is no better than the individual
effects.
Since it is a relatively unimportant parameter, the latter (smoothing the initial
phase) may be a useful alternative to smoothing $$$B_0$$$.
Figures 2 and 3 show in vivo results for unconstrained and constrained estimation (using Constraints $$$II$$$ and $$$IV$$$ only). The initial phase map differs in smoothness (as expected). The $$$B_0$$$, $$$R_2^*$$$ and PDFF maps all show a reduction in noise
scatter and pointwise errors that occur in unconstrained fitting. The differences are small but in line with the anticipated effect size from
simulations.Discussion
Recent work has shown that $$$PDFF$$$ can be biased in noisy datasets [4,10,11], which is a challenge for estimation in the crucial low $$$PDFF$$$ range [12]. While it is desirable to reduce the bias and standard deviation, in general the two
can only be traded against eachother with modeling choices. As shown in Figure 1, putting constraints on the parameters can provide a mechanism to manipulate bias and standard deviation. Furthermore, certain choices of constraints reduce both the bias and standard deviation of the $$$PDFF$$$ estimate.Acknowledgements
No acknowledgement found.References
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