Linear MALDIToF simultaneous spectrum deconvolution and baseline removal
Abstract
Background
Thanks to a reasonable cost and simple sample preparation procedure, linear MALDIToF spectrometry is a growing technology for clinical microbiology. With appropriate spectrum databases, this technology can be used for early identification of pathogens in body fluids. However, due to the low resolution of linear MALDIToF instruments, robust and accurate peak picking remains a challenging task. In this context we propose a new peak extraction algorithm from raw spectrum. With this method the spectrum baseline and spectrum peaks are processed jointly. The approach relies on an additive model constituted by a smooth baseline part plus a sparse peak list convolved with a known peak shape. The model is then fitted under a Gaussian noise model. The proposed method is well suited to process low resolution spectra with important baseline and unresolved peaks.
Results
We developed a new peak deconvolution procedure. The paper describes the method derivation and discusses some of its interpretations. The algorithm is then described in a pseudocode form where the required optimization procedure is detailed. For synthetic data the method is compared to a more conventional approach. The new method reduces artifacts caused by the usual twosteps procedure, baseline removal then peak extraction. Finally some results on real linear MALDIToF spectra are provided.
Conclusions
We introduced a new method for peak picking, where peak deconvolution and baseline computation are performed jointly. On simulated data we showed that this global approach performs better than a classical one where baseline and peaks are processed sequentially. A dedicated experiment has been conducted on real spectra. In this study a collection of spectra of spiked proteins were acquired and then analyzed. Better performances of the proposed method, in term of accuracy and reproductibility, have been observed and validated by an extended statistical analysis.
Keywords
Mass spectrometry Peak picking Deconvolution BaselineAbbreviations
 BB
BarzilaiBorwein
 BHIPRO
Bayesian hierarchical inversion for mass spectrometry. Application to discovery and validation of new PROtein biomarkers
 CWT
Continuous wavelet transform
 DWT
Discrete wavelet transform
 KKT
KarushKuhnTucker
 LC
Liquid chromatography
 MALDIToF
Matrixassisted laser desorption/ionizationtime of flight
 MS
Mass spectrometry
 SG
Savitzkygolay
 SNIP
Statisticssensitive nonlinear iterative peakclipping
 UDWT
Undecimated wavelet transform
Background
Linear matrixassisted laser desorption/ionization timeofflight mass spectrometry (MALDIToF MS) has now revolutionized identification of bacteria, yeasts and molds in clinical microbiology [1]. The technology is simple, accurate, fast, and for large laboratories less expensive than conventional methods. Despite a lower resolution than other analyzers used in modern proteomics, linear ToF are preferred in microbiology because of a better sensitivity in the 220 kDa mass range, where proteins contain phylogenic information. Moreover, the lower cost of linear instruments favored a wider adoption by health institutions. In essence, identifications are performed in minutes by simply acquiring an experimental spectrum of the whole microorganism cells and comparing the resulting peak list with a database [2, 3]. In this context we propose a new method for peak extraction especially adapted to linear MALDIToF spectra. The usual approach for MALDI mass spectra processing generally consists of chaining several procedures. Most of the times we have a smoothing step, a baseline correction step and only then the final the peak extraction [4]. The main idea of our new method is to jointly perform these steps with the aim of reducing the potential unrecoverable artifacts introduced by a sequential processing. In the next sections we briefly present the three main steps of the usual approaches. We then describe our method and its detailed derivation.
Smoothing
A popular [5] smoothing technique in the spectrometry community is the use of SavitzkyGolay linear filters [6, 7]. These moving average filters perform a least squares fit of a small set of consecutive data points to a polynomial. The value of this fitted polynomial at the window central point is the filter output. One can also compute a smoothed derivative by using the derivative of the fitted polynomial to compute the central point value. This smoothed derivative can also be used by peak picking algorithms [4]. This method is versatile and efficient. However, its main drawbacks are that we have to manually choose the polynomial degree and the window length. Some studies to automatically choose the former [8] or later [9] exist but to the best of our knowledge they are not used as often as the original approach.
A more recent approach to smooth spectra is the use the wavelet transform. The Undecimated Wavelet Transform (UDWT) [10, 11] is generally preferred to the Discrete Wavelet Transform (DWT) as it produces less artifacts after coefficient thresholding. The UDWT is equivalent to an averaged DWT computed for all integer shifts of the signal and is thus a redundant and shift invariant transform. In applications it has been reported to yield better qualitative denoising [12].
Baseline correction
Baseline correction is a difficult problem that potentially also introduces artifacts [13]. There are at least two kinds of approaches for baseline correction. One category of methods is close to mathematical morphology. In these methods a lower envelope of the spectrum [14, 15] is computed. Methods of this category generally need a smoothed signal (see “Smoothing” section). The other category contains methods using an asymmetric loss function to fit spectrum baseline without being biased by peaks [16, 17]. Finally some other methods mix the two previous approaches [18].
Peak picking
After baseline removal the next step is generally a peak picking procedure. Several approaches are possible. Perhaps the most intuitive approach is to compute a regularized second order derivative (using SavitzkyGolay for instance) of the spectrum and to extract local minima [4]. The use of second order derivative minima instead of the zerocrossing of the first order derivative allows, to some extend, to detect overlapping peaks [19].
A second kind of approach, especially useful in case of overlapping peaks, is peak deconvolution. Overlapping of complex peak patterns can be deconvolved if one uses specially tuned point spread function and judicious regularizations (positivity constraint and sparsityinducing norms, like the l_{1} norm) [20, 21, 22, 23]. Further generalizations can be obtained in case of blinddeconvolution [24]. However these kinds of approaches are much more computationally intensive and are not widely used in mass spectrometry.
Finally we can mention the Continuous Wavelet Transform (CWT) [25, 26] which can be efficiently computed using the Fast Fourier Transform. The idea is to follow wavelet modulus maximum. Theses ridges characterize the regularity of the signal [27] and can be used to detect peaks.
Contributions
Computing the baseline correction and finding peaks are two strongly linked problems, it is thus natural to perform these two operations jointly. In this work we propose such an approach.
In the first part of the paper we describe a direct model with an additive noise where the spectrum is modelized by a smooth baseline plus a sparse peak list convolved by a given peak shape function. We describe how we chose our priors to enforce baseline smoothness and sparsity of the peak list. We then assume a Gaussian distribution for the noise. This allows us to use Euclidean distance to quantify the error between our model and the measured spectrum.
Next we show how the unknown baseline can be eliminated from the model. This manipulation leads to a modified problem very close to the classical deconvolution one. We underline this similarity and rigorously describe the two limiting cases, zero or infinite penalization for the baseline smoothness. As a by product we can interpret that our new deconvolution method is equivalent in some way to deconvolve a regularized second order derivative of the initial spectrum.
We then carefully examine the behavior of the computed baseline at the spectrum boundaries. We observed that when the smoothness penalty is too strong, the computed baseline can become overly flat. To avoid this effect a correction allowing to define baseline values at boundaries is proposed. With this modification the behavior of the baseline at the boundaries is no more affected by strong baseline smoothness penalty.
An effective optimization method to compute the solution of the deconvolution problem is exposed. This optimization algorithm is used twice in our twopasses deconvolution procedure. In the first pass a sparsity prior is used and a first optimization problem is solved to find peak centers. In the second pass the sparsity prior is replaced by the previously found peak positions. This second optimization problem is solved to compute peak height values.
Finally the new method is compared to one instance of the smoothing/baseline correction/deconvolution classical approach. The advantage of the joint baseline computation and peak deconvolution is demonstrated on synthetic data. An example on “real” data is shown and a reference to a more detailed comparison between our method and classical ones is given.
Method
Problem definition
where e includes measurement and model errors. It is a common linear model with additive uncertainties. These quantities are represented by vectors of size n, where n is the number of m/z channels of the original spectrum y.

It is underdetermined since the number of unknowns is twice the number of data.

The convolution reduces the resolution due to peak enlargement and possible overlap.

Measurement noise and possible model inadequacy induce additional uncertainties.
As a consequence, information must be accounted for regarding the expected signals x_{ p } and x_{ b }. In the following developments, x_{ b } is expected to be smooth while x_{ p } is expected to be spiky and positive. This knowledge will be included in the next sections.
x _{ b } smoothness
where D is a finite differences matrix of size (n−1)×n (given in Appendix “Smoothness and convolution matrix”) and μ>0.
x _{ p } sparsity and positivity
The degree of sparsity is controlled by λ_{1}. The coefficient λ_{2} is generally set to zero or to a very small value. The reason why we have introduced this extra regularization is that a small positive value can sometimes improve convergence speed of the algorithm. In practice this only happens for spectra of several thousand of m/z channels and always has a limited impact on the obtained solution.
Datafidelity term
where the n×n band matrix L represents the convolution with the peak shape p.
Complete objective
gives the desired solution \(\left (\widehat {\mathbf {x}}_{p},\widehat {\mathbf {x}}_{b}\right)\).
Elimination of x _{ b }
where \(\mathbf {B}_{\mu }=\mathbf {I}_{b}+\mu \mathbf {D}^{t}\mathbf {D}\). This operation is always possible since B_{ μ } is invertible (sum of the identity matrix and a semipositive matrix).
and its interpretation is discussed in detail in “Analysis of the A_{ μ } matrix” section).
Interpretation of the reduced criterion
The two equations are very similar apart from the fact that the operator A_{ μ } is applied to the reconstructed peaks Lx_{ p } and to the raw spectrum y. It could be interpreted as the precision (inverse of the covariance) matrix in a correlated noise framework. Instead of this classical approach and to better understand its role in our deconvolution context we study the evolution of A_{ μ } in the two limiting cases \(\mu \rightarrow 0\) and \(\mu \rightarrow \infty \).
Analysis of the A _{ μ } matrix
Hence compared to Eq. 14, one can interpret Eq. 15 as an usual deconvolution applied on the second order derivative A_{ μ }y of the initial spectrum y. The used point spread function is also the second order derivative A_{ μ }L of the initial peak shape p introduced in Eq. 1. The overall effect of the A_{ μ } operator is to cancel the slow varying component of the signal. In another terms, the baseline is removed thanks to a derivation.
Debiasing
The l_{1} penalty acts as a soft threshold to select peaks ([29], Section 10). This leads to a bias in peak intensity estimation. These intensities are artificially reduced when the l_{1} penalty increases. We use the ideas introduced in [28] to get corrected peak intensities. This yields a resolution procedure involving two stages. The first stage selects the peaks, the second one corrects their intensities.
First stage: peak support selection
The peak intensities are going to be corrected in the second stage, but their final positions are defined by the condition Eq. 16. More complex procedures can be used to find the peak support. One such example is the procedure presented in [22], Postprocessing and thresholding. These more refined methods can be introduced in a straightforward way in our approach by using them instead of the basic condition Eq. 16.
Second stage: peak intensity correction
Despite its more complex appearance, this problem is no more complicated than Eq. 12. Solving this problem will correct peak intensities by removing the bias induced by the previously used l_{1} penalty.
Boundary conditions
We solved this problem by imposing baseline values at boundaries. This corrected solution is also shown in Fig. 2 and we can see that the corrected solution does not suffer from boundary effect anymore. Appendix “Boundary correction”, page 11, provides all the details on how to modify Eqs. 9 and 13 to introduce some constraints on the baseline values x_{ b }. These modified equations will constitute our final model formulation.
Final model formulation
As before, \(\widehat {\mathbf {x}}_{b}\) is computed from \(\widehat {\mathbf {x}}_{p}\) using Eq. 18. The explicit forms of \(\widetilde {\mathbf {B}}_{\mu }\), \(\widetilde {\mathbf {y}}\) and \(\widetilde {\mathbf {A}}_{\mu }\) are given in Appendix “Boundary correction”, respectively by Eqs. 31, 32 and 33.
Algorithm summary
For ease of reading Algorithm 1 recapitulates the main steps of the proposed method in its final formulation. The optimization procedure used to solve efficiently the two minimization problems will be described in detail in the next section.
Effective minimization
Quadratic programming with bound constraints
To solve the optimization problems Eq. 12 or Eq. 20 and their associated debiasing step Eq. 17 we use the projected BarzilaiBorwein method described in [30]. The BarzilaiBorwein method [31] dramatically improves the classical steepest descent with Cauchy step.
For its simplicity and good behavior in practice we have chosen to use this method. Our implementation is given in pseudocode in Algorithm 2.
Variants of this method with nonmonotone line search [36, 37] have been tested. Theoretically this allows to prove global convergence, but in practice we have observed a performance degradation compared to the simpler method presented in [30].
Stopping criterion
 1
Stationarity \(\nabla _{\mathbf {x}_{p}} \mathcal {F}+\mathbf {\lambda }^{\mathbf {u}}\mathbf {\lambda }^{\mathbf {l}}=\mathbf {0}\)
 2
Primal feasibility \(\mathbf {x}_{p} \in \left [\mathbf {l},\mathbf {u}\right ]\)
 3Dual feasibility

Lower bounds λ^{ l }≥0

Upper bounds λ^{ u }≥0

 4Complementary slackness

Lower bounds \(\forall i,\ \mathbf {\lambda }^{\mathbf {l}}\left [i\right ](\mathbf {l}\left [i\right ]\mathbf {x}_{p}\left [i\right ])=0\)

Upper bounds \(\forall i,\ \mathbf {\lambda }^{\mathbf {u}}\left [i\right ](\mathbf {x}_{p}\left [i\right ]\mathbf {u}\left [i\right ])=0\)

Illustration of the method
We observe the nonmonotonic convergence behavior of the BarzilaiBorwein method. As reported by [30], we observe that it is more effective to alternate between BB1 Eq. 22 and BB2 Eq. 23 steps than using only one type of step update.
Results and discussion
Synthetic data

Synthetic baseline
The analytic expression of the baseline iswhere s can take one of the following values s=−1,0,or 1.$$\mathbf{x}_{b}\left[i\right] = C(s)+s\exp{\left(3\frac{i}{n}\right)}2\frac{i}{n} $$For s=0 the baseline is a straight line, for s=−1 the baseline is concave, for s=+1 it is convex.
The constant C(s) insure a positive spectrum. Its value is C(s)=5 for s=1 and C(s)=2 otherwise.

Synthetic peak list
The analytic expression of the peak list contribution is:where we have taken n_{ p }=10 and$$\left(\mathbf{x}_{p}\ast p\right)[\!i]=\sum\limits_{k=1}^{n_{p}}\alpha_{k} p\left(\mu_{k},\sigma_{p},i\right) $$$$p\left(\mu_{k},\sigma_{p},i\right)= \exp{\left(\frac{1}{2}\left(\frac{i\mu_{k}}{\sigma_{p}}\right)^{2}\right)} $$These n_{ p } Gaussian peaks are defined by a constant shape factor σ_{ p }=10. The individual heights α_{ k } and centers μ_{ k } are tabulated in Table 1.Table 1True peak centers μ_{ k } and intensities α_{ k }. All peaks have a common shape factor σ_{ p }=10
k
μ _{ k }
α _{ k }
1
50
1
2
90
0.5
3
170
0.5
4
200
3
5
230
2
6
260
1
7
350
0.5
8
370
3
9
390
2
10
410
1

Simulated zero mean Gaussian noise
The noise ε follows a Normal law of zero mean and σ_{ noise } standard deviation:$$\mathbf{\epsilon}[\!i] \sim \mathcal{N}\left(0,\sigma_{noise}\right) $$
Algorithm implementation
We provide a reference implementation^{1} that can be used to reproduce the results of the following sections. This implementation is coded in C++ and runs under Linux. The main page of the project details all the steps to reproduce the results of the “Joint baseline computationpeak deconvolution”, “Comparison with the usual sequential approach” and “Real data” sections. Typical runtimes are one second for the synthetic data and three seconds for the examples using real spectra data. To keep this implementation as simple as possible a basic projected gradient descent is used instead of the more effective Algorithm 2. Also note that this implementation requires CSV input files. A more versatile version of our algorithm, using Algorithm 2, is used in [39]. However, this implementation is not yet publicly available.
Joint baseline computationpeak deconvolution
 1
 2a second resolution, is then performed with a null λ_{1} penalty. This penalty is replaced by the restricted support Ω for peak centers we found in the first step. The role of this step is to correct peak heights and baseline values which were biased by the presence of the strong λ_{1} penalty of the first step. Illustrations are given in Figs. 6 and 7.
The baseline approximation obtained at the end of the first step is given in Fig. 4. This figure shows a common flaw of most of the baseline removal methods which is an ascent of the baseline under the peaks.
The deconvolved peaks obtained after the first step are given in Fig. 5. We notice the negative impact of a strong λ_{1} penalty which leads to an underestimation of the peak heights. The figure also shows that the deconvolved solution is less spiky in regions of strong peak overlaps (right part of the spectrum).
The baseline computed after the second step is shown in Fig. 6. During this second stage, the λ_{1} penalty is removed and replaced by a restricted support Ω computed using Eq. 16. We see that the ascent of the baseline under the peaks has been corrected (c.g. xaxis ranges from 150 to 250 and from 350 to 400).
The new peak heights are shown in Fig. 7. The main role of this second stage is to debias peak heights. Compared to Fig. 5 we can see that this objective is quite well fulfilled.
Comparison with the usual sequential approach
 1
the λ_{1} parameter enforcing the sparsity of the solution x_{ p },
 2
the μ parameter enforcing the smoothness of the baseline x_{ b }.
 1
Baseline subtraction: the Statisticssensitive Nonlinear Iterative Peakclipping (SNIP) algorithm [40, 41, 42] is an efficient algorithm to compute spectrum baseline. It generally gives good results and is easy to implement. It uses only one parameter, the window width m_{SNIP}, but requires a smoothed spectrum. To smooth the spectrum we use a SavitzkyGolay filter [6, 7]. The idea is to locally fit the spectrum by a polynomial. This leastsquares fitted polynomial of degree d is computed using m_{SG} points of a running window. For each window position, the spectrum value at the window center is then replaced by the polynomial value. In order to have only one parameter, a fixed window width m_{SG}=39 is used for the smoothing step.
 2Peak picking: once the baseline has been subtracted from the original spectrum, there are a large panel of methods to extract peaks [4]. We can mention simple thresholding [42], second derivative computation using SavitzkyGolay filters, extraction from wavelet coefficients or deconvolution. For our comparison we are more interested by deconvolutionlike methods [21, 22] which are closer to our approach. By consequence we have decided to extract peaks thanks to the usual sparse deconvolution problem:$$ \arg\min\limits_{\mathbf{x}_{p}\ge0} \frac{1}{2}\big\\mathbf{y}  \mathbf{L}\mathbf{x}_{p}\big\^{2}_{2} + \lambda_{1} \big\\mathbf{x}_{p}\_{1} + \frac{\lambda_{2}}{2} \big\\mathbf{x}_{p}\big\_{2}^{2} $$(26)
As for our method, the coefficient λ_{2} is not critical and we set it to a constant value λ_{2}=0.1. The remaining λ_{1} is our second free parameter. To solve Eq. 26 we use the same BarzilaiBorwein solver, detailed in Algorithm 2 and the same twosteps procedure which consists in peak selection (high λ_{1}) and peak heights debiasing (λ_{1}=0).
Our method: used grid for parameter search
Parameters  Min  Max  Step 

μ  100  4000  100 
λ _{1}  0  4  0.2 
Usual method, baseline removal then peak deconvolution: used grid for parameter search
Parameters  min  max  step 

m _{SNIP}  20  40  2 
λ _{1}  1  15  0.2 
Comparison between the two methods, for convex baseline s=−1
σ _{ noise }  Sequential processing  Proposed method 

\(\mathcal {E}_{e}\times 10^{3}\)  \(\mathcal {E}_{e}\times 10^{3}\)  
0.  15.  27.6 
0.2  47.8±8.76  45.2 ±9.12 
0.4  88.6±15.6  81.6 ±11.9 
0.6  131.±21.4  118. ±18. 
0.8  176.±26.  154. ±20.1 
1.  217.±33.7  184. ±21.3 
1.2  262.±45.7  215. ±25.5 
1.4  295.±48.2  245. ±29.4 
1.6  330.±52.5  275. ±32.7 
1.8  360.±59.8  300. ±38.2 
2.  390.±70.4  322. ±44.3 
On this example the joint evaluation of the baseline and of the deconvolved peaks outperforms the sequential approach in nearly all configurations except for the noisefree case. In this condition, as we are using synthetic data, a perfect reconstruction of the “ground truth” spectrum is possible. By consequence, the reconstruction error is very small. To explain this result, we think that our iterative solver had stopped its iterative resolution too early.
As explained the SNIP algorithm needs to work on a previously smoothed spectrum. This smoothing is performed using a SavitzkyGolay filter. This sequential approach, smoothing then SNIP baseline, can introduce an important bias on the computed baseline. This error is then transferred to the peak picking algorithm. The result can be a poor peak extraction.
We see that the simultaneous baseline and deconvolution approach allows a nearly perfect reconstruction of the baseline. Unlike our approach, the SNIP algorithm suffers from initial filtering of the spectrum and presents baseline ascent below peaks.
Real data
It is generally quite difficult to evaluate peak picking methods on real spectra because we do not have the ground truth at our disposal. However, in our case we have expensively used the proposed approach in the BHIPRO project. The study [39] compares our method against a more usual one on mass spectra of spiked proteins. This study quantifies the algorithmic part of the variance of the measured protein abundances and shows a clear gain in favor of our algorithm.
Conclusions
We have introduced a new method for peak deconvolution. This new approach jointly performs baseline computation and peak deconvolution. The baseline equation can be solved in a closed form and is substituted into the deconvolution equation. The new deconvolution equation contains a linear operator A_{ μ } acting as a smoothed second order derivator. The problem of boundaries is exposed and tackled rigorously leading to a modified equation. The problem is efficiently solved by the projected BarzilaiBorwein algorithm. A comparison with a traditional approach relying on a sequential baseline removal and peak picking is detailed. The benefits of our new approach are put in evidence, including a better baseline approximation avoiding ascent below the large peaks, and a better reconstruction of the deconvolved peaks. Finally the method has been tested on real data.
Perspectives
There are two main directions we want to explore. The first direction is to go one step further in a joint processing approach by allowing automatic adjustment of the peak shape function. Ideally, with such an approach, the baseline, the peak shape and the deconvolved peaks would be computed jointly. The second direction would be to devise a procedure for automatic tuning of the two main hyperparameters. The resulting algorithm would be of great value to process a large number of spectra in batch mode.
Appendix
Smoothness and convolution matrix
In the presentation the convolution product between the peak list and peak shape function, x_{ p }∗p, is often represented by the Lx_{ p } matrixvector product. However, in practical computations the L matrix is never explicitly formed and the convolution product is computed directly using a specialized subroutine.
Derivation of reduced objective
This appendix focuses on the reduced criterion Eq. 10.
Expression of \(\underset {\mu \rightarrow \infty }{\lim } \mathbf {A}_{\mu }\)
Let’s write down two basic statements about eigenpair \(\left (\nu,\mathbf {v}\right)\) of any square matrix M:
Boundary correction
Footnotes
Notes
Acknowledgment
We thank Bruno Lacroix (bioMérieux) and Pierre Mahe (bioMérieux) for their contribution during the ANR Grant submission and execution. We thank Laurent Gerfault (CEA) for his advises for the development of the BHIPRO MALDI model, data processing and data analysis. We thank Amna Klich, Delphine MaucortBoulch, Pascal Roy (Biostatistique, Hospices Civils de Lyon) and Patrick Ducoroy (CLIPP) for their advises and comments during the development and tests of the method.
Funding
The BHIPRO project has been partially funded by the Agence Nationale de la Recherche under grant ANR 2010 BLAN 0313 and by Commisariat à l’Energie Atomique et aux Energies Alternatives (CEA) for CEA authors contribution (VP, PG).
Availability of data and materials
The datasets used to generate the figures and a basic implementation of the proposed algorithm are avalaible under the https://github.com/vincentpicaud/Joint_Baseline_PeakDeconvGitHub repository.
Authors’ contributions
VP initial idea of the method, theoretical developments and C++ implementation. The contributions of JFG and AG regard (a) the design of the criterion and its optimization and (b) the writing of the paper. CT read and approved the final manuscript and proposed valuable comments about preprocessing of spectra in the context of mass spectrometry. CM participated in the problem definition and evaluated the methods on real spectra. JPC provided input in the MALDIToF field and the result interpretation. He acquired and provided real MALDIToF spectra to test the algorithm. PG was the BHIPRO project manager. He has coordinated the conception and design of the processing algorithms, in particular the design of the acquisition chain MALDI and MRM models and inversion algorithms, and the interpretation of the data. He has revised the manuscript. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable
Consent for publication
Not applicable.
Competing interests
JPC is employed by bioMérieux. The other authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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