Michael J van Rijssel^{1}, Martijn Froeling^{1}, and Josien P W Pluim^{1,2}

The recently proposed phasor representation and associated unmixing method allow separation of multi-exponentially decaying signals. This method has achieved promising results on diffusion MRI data and boasts sub-second analysis of full datasets on regular desktop PCs. This work investigates the noise propagation properties of this method and the influence of misplacing the vertex of a component in phasor space when performing unmixing. Results indicate that the phasor method is feasible and that the influence of component misplacement is systematic, but smaller than the errors due to noise at regular diffusion MR signal-to-noise-ratio levels.

Recently,
Vergeldt et al. proposed the use of the phasor representation for multi-component
separation of decaying signals in quantitative MRI.^{1-4} The formulation of the phasor
representation is general and can therefore be applied to any one-dimensional
signal with components decaying with different decay constants.^{5,6}

^{ }As Vergeldt et al. suggested, the phasor
representation can be applied to multi b-value (multi-shell) diffusion weighted
images (DWI).^{1} Due to a
fully analytical description and efficient implementation using fast Fourier
transforms (FFT) the phasor transformation is very fast. However, phasor
analysis only allows to separate three components and noise propagation from
source DWIs into component estimates is presently unknown. In this work, we
therefore investigate the influence of component (mis)placement and noise on the
accuracy of phasor analysis.

Briefly, the phasor representation is created by taking the Fourier transform of a signal and dividing that by the sum of the same signal:$$$\frac{\mathcal{F}(s(t))}{{\int}s(t)dt}$$$. If the input is an exponential function of time t with lifetime $$$\tau$$$ ($$$s(t)=e^{-t/\tau}$$$), the Fourier transform at frequency ω is the Lorentz function: $$$D(\omega)=\frac{1}{1+j{\omega}t}$$$. Plotting the real part of this function against the imaginary part for a fixed frequency ω for all possible lifetimes $$$\tau$$$ produces a semicircle with radius 0.5 and center (0.5, 0).

For
discrete sampling, the Fourier transform is approximated using the discrete FFT
and the semicircle deforms into an ellipse with known dimensions.^{5} Figure 1 schematically shows a phasor plot of a
diffusion signal with various values for diffusivity (D) and kurtosis (K).

A
mixed signal containing 2 components (i.e. bi-exponential decay) is located on a
line connecting the locations of the two pure components in phasor space. The location
of the signal on the line is determined by the relative weight of each
component. Analogously, mixes of more components are located inside a polygon
defined by the pure component vertices.^{5} Knowing the locations of the pure components, the contribution of each component per voxel can be estimated, i.e. unmixing.^{6}

First,
a digital phantom (Figure 3, top row) was constructed consisting of three components
with diffusivity values: D_{1}= 2.9, D_{2}=0.85, D_{3}=0.18 μm^{2}/s,
respectively. The phantom contains component fractions from 0 to 1 in steps of
0.1, resulting in 66 fraction combinations summing to 1. DWI images were simulated with
21 equally spaced b-values between 0 and 2500 s/mm^{2}. Gaussian noise was
added with a signal-to-noise-ratio (SNR) level of 50. The influence of
misplacing component vertices was investigated by unmixing the simulated images
with different base-diffusivities, varied from 0.75*D_{true}-1.25*D_{true}
in steps of 0.05*D_{true}.

Secondly,
a diffusion dataset of one healthy male volunteer (aged 25) was obtained with the
same b-values. For each b-value, three orthogonal gradient directions were
measured and subsequently averaged. To determine
the location of the unmixing triangle’s vertices in phasor space, the average signal of the whole brain was
fitted to a triple exponential: $$S=S_{0}(F_1e^{-b{\cdot}D_1}+F_2e^{-b{\cdot}D_2}+(1-F_1-F_2)e^{-b{\cdot}D_3+\frac{1}{6}{\cdot}b^2{\cdot}D_3^2{\cdot}K}),$$ where S is the measured MR signal, S_{0}
a scaling constant, F_{x} component x’s signal fraction, b the b-value,
D_{x} component x’s diffusivity and K the kurtosis which was allowed
only for the most strict component.

Figure
2 shows the phasor representation of the digital phantom data at SNR=50. The
spacing between increments in fraction is nonlinear and the effect of noise
seems to differ per component. Figure 3 demonstrates that the amount of noise
propagating into the fraction estimations is largest for D_{2}: the standard
deviation of the error was 0.047, 0.060 and 0.022 for D_{1},D_{2}
and D_{3} respectively. Figure 4 shows the errors due to vertex
misplacement, both with and without noise. The maximum interquartile range due
to a 10% error in diffusivity was 0.033, the maximum range due to noise
(without misplacement error) was 0.080.

Figure 5 shows the results of the fit and 3-component unmixing on in-vivo data. Applying the phasor transform and unmixing the entire brain dataset took less than one second. The values for D and K, and the fraction maps seem physiologically plausible.

1. Vergeldt F J, Prusova A, Fereidouni F, et al. Multi-component quantitative magnetic resonance imaging by phasor representation. Sci Rep. 2017;7.

2. Clayton A H A, Hanley Q S, and Verveer P J. Graphical representation and multicomponent analysis of single-frequency fluorescence lifetime imaging microscopy data. Journal of Microscopy-Oxford. 2004;213:1-5.

3. Redford G I, and Clegg R M. Polar plot representation for frequency-domain analysis of fluorescence lifetimes. Journal of Fluorescence. 2005;15(5):805-815.

4. Digman M A, Caiolfa V R, Zamai M, et al. The phasor approach to fluorescence lifetime imaging analysis. Biophys J. 2008;94(2):L14-L16.

5. Fereidouni F, Esposito A, Blab G A, et al. A modified phasor approach for analyzing time-gated fluorescence lifetime images. J Microsc. 2011;244(3):248-258.

6. Fereidouni F, Bader A N, and Gerritsen H C. Spectral phasor analysis allows rapid and reliable unmixing of fluorescence microscopy spectral images. Opt Express. 2012;20(12):12729-12741.