 Research
 Open Access
Novel EM based ML Kalman estimation framework for superresolution of stochastic threestates microtubule signal
 Vineetha Menon†^{1}Email author,
 Shantia Yarahmadian†^{2} and
 Vahid Rezania^{3}
https://doi.org/10.1186/s1291801806315
© The Author(s) 2018
 Published: 22 November 2018
Abstract
Background
Recent research has found that abnormal functioning of Microtubules (MTs) could be linked to fatal diseases such as Alzheimer’s. Hence, there is an imminent need to understand the implications of MTs for disease diagnosis. However, studies of cellular processes like MTs are often constrained by physical limitations of their data acquisition systems such as optical microscopes and are vulnerable to either destruction of the specimen or the probe. In addition, study of MTs is challenged with nonuniform sampling of the MT dynamic instability phenomenon relative to its timelapse observation of the cellular processes. Thus, the above caveats limit the overall period of time that the MT data can be collected, thereby causing limited data availability scenario.
Results
In this work, two novel superresolution frameworks based on Expectation Maximization (EM) based Maximum Likelihood (ML) estimation using Kalman filters (MLK) technique are proposed to address the issues of nonuniform sampling and limited data availability of MT signals. The proposed MLK methods optimizes prediction of missing observations in the MT signal through information extraction using correlationbased patch processing and principal component analysis based mutual information. Experimental results prove that the proposed MLKbased superresolution methods outperformed nonlinear interpolation and compressed sensing methods.
Conclusions
This work aims to address limited data availability and data/observation loss incurred due to nonuniform sampling of biological signals such as MTs. For this purpose, statistical modelling of stochastic MT signals using EM based ML driven Kalman estimation (MLK) is considered as a fundamental framework for prediction of missing MT observations. It was experimentally validated that the proposed superresolution methods provided superior overall performance, better MT signal estimation using fewer samples, high SNR, low errors, and better MT parameter estimation than other methods.
Keywords
 Superresolution
 Kalman filtering
 Expectation Maximization
 Wavelets
 Principal component analysis
 Mutual information
 Missing data
Background
Research on Microtubules (MTs) have recently garnered a great level of interest due to its anomalous functioning being associated with the onset of several lethal diseases including Alzheimer’s disease [1], Parkinson’s disease [2], and various forms of cancer [3]. Essentially, MTs are intracellular polymers made of tubulin protein dimers [4] which are found in all eukaryotic cells, and play major roles in many intracellular activities such as trafficking, mitosis, cell motility [4], and chromosome segregation [5]. Under fixed external conditions, it was first noted that the MTs reached a steady state while randomly switching between polymerization (growth) and depolymerization (shrinking) states [4]. This unique phenomena of MTs randomly switching between the two states was named as “dynamic instability” by Mitchison and Kirschner [4, 6–8]. It was later reported in [9, 10], that the MTs were in fact switching spontaneously between three states, namely, growth (g), pause (p) and shrinkage (s). This threestate dynamic instability model involves eight parameters: six transition rates between the states of growth, pause and shrinkage and growth and shrinkage velocities v_{g} and v_{s}. The average length and mean lifetime of a MT depends on the threshold quantity of these parameters.
Optical microscopes have been utilized for decades as a mainstay for data collection of various cellular processes, such as MTs. Even after modern technological advances in the area of microscopy like single molecule sensitivity, and frame rates in microseconds, optical microscopes are still vulnerable in either damaging the specimen of interest, or the probe during the course of experiment. Moreover, the probe can only be illuminated for a certain period of time, making data collection an arduous task, and data availability is renderedsparse with respect to the time scale of intracellular processes [11, 12]. Especially, in case of MTs, where its high temporal resolution is imperative to estimate the dynamic instability parameters, understand the MT behavior, and other cellular activities. In particular, estimation of MT dynamic instability parameters is crucial to understand the state of the MT system and study its relationships to the occurrence of fatal diseases like Alzheimer’s. Consequently, it signifies the need for stochastic methods that can analytically overcome the challenges such as scarce data availability, and equipment limitations through better signal reconstruction and estimation techniques.
Many statistical methods for modeling stochastic signals have been suggested in literature such as hidden Markov models (HMMs) for capturing the underlying signal variability [13]. As a variation of traditional HMMs, many waveletbased HMMs have evolved because of their capability to model realworld nonGaussian signals [14, 15]. Compressed sensing (CS) [16, 17] and other subspace learning methods [18] have also been proposed to reconstruct signals using fewer samples than the Nyquist rate. In literature, it has also been proposed to use maximum likelihood (ML) technique to estimate parameters of a linear dynamic system from the observed data [19, 20]. Typically, this involves formulation of a timevarying Kalman predictor with a likelihood function to minimize the prediction error. In this case, a convergence assumption is made for sufficiently large observations which transforms the minimization of likelihood function to a nonlinear programming problem [21–23]. As an alternative, it was proposed to use ExpectationMaximization (EM) based maximum likelihood (ML) estimation in Kalman filters with the assumption that the state is now observable. This assumption simplifies the convergence criteria by eliminating the need for a large observation limit and facilitates to solve cases with missing observations [24]. Kalman filters have been proved to be optimal in minimizing mean square error sense, under the assumption that all noise is Gaussian. Therefore, the use of EM based ML estimation in Kalman filters as a means to study MTs is very relevant, since we want to minimize the residual error between the original MT signal, and the predicted MT signal.
In this paper, the MT signal is modelled as a threestate stochastic random evolution signal on which nonuniform sampling is performed to emulate the data loss in realworld scenario [11, 12]. Our motivation is to improve prediction of the missing timelapse intracellular observations analytically; in an effort to minimize the effect of external dependencies such as spatiotemporal resolution, and experimental equipment precision. For this, we propose two novel methods for superresolution of MT signals based on nonuniform sampling, using EM based ML estimation Kalman filter for better prediction of the interpolated MT signal, which is followed by correlation coefficient (R) based patch processing (MLKR) [25] and principal component analysis (PCA)based mutual information (MI) criterion (MLKMI) for information extraction to further optimize our final predicted MT signal. We estimate the dynamic instability parameters for three states in MTs using waveletbased peak detection and compare it with our previous work using CS [16]. Our proposed methods MLKR and MLKMI gave superior overall performance compared to all other methods, better MT signal recovery and gave high SNR with low errors validating the efficacy of our approaches.
Methods
Nonuniform sampling and interpolation
where i=2,…,d+1, and x_{l1}(n)=x_{l}(n), mod=modulo operator. For simplicity, we choose ε_{i}=i+1, and d is the resolution factor and it also denotes the level of downsampling performed. For e.g. d=2 implies that the signal was downsampled by a factor of 2. s is the number of samples in each frame, calculated as \( s= \left \lfloor \frac {N}{d}\right \rfloor \). For consistency, nonlinear interpolation using piecewise linear interpolation method is performed on the d new low resolution MT signals x_{li}(n) to get their corresponding high resolution MT signals y_{hi}(n).
Expectationmaximization based maximum likelihood estimation of a stochastic MT system
Where e_{n}, \(\Sigma _{e_{n}}\) are the prediction error and its covariance, and can be obtained from Kalman filter time and measurement update equations as in [20, 22] below:
Using fixed interval form of Kalman filter (RTS smoother), the new system recursion updates can be computed as in [22]:
Note that the formulation of d series estimation of the state vector x_{n} is made with the assumption that available data is the observed vector y_{n} and it provides d unique estimated MT signals \(\hat { \boldsymbol {x}}_{hi}(n)\). In each iteration, EM algorithm computes the datasufficient statistics in recursions (8), (9) and estimates previous model parameters (Estep). New system parameters are obtained from these statistics in the maximization step (Mstep).
Correlation coefficientbasedpatch processing
where p, q=1,2,…,d. For d Kalmanpredicted MT signals \(\hat { \boldsymbol {x}}_{hp}(n)\), we will need ^{d}C_{2} signal comparisons to be made. All missing values in the final MT signal \( \hat {\boldsymbol {x}}_{f}(n)\) (for patches < threshold R) are computed by taking the mean of the d signals values available for that particular time instance. This step is done as a tradeoff to minimize the prediction error for missing values in the original low resolution MT signal. The final MLK−Rpatch processed MT signal \( \hat {\boldsymbol {x}}_{f}(n)\) had better SNR and lower errors than Kalmanpredicted (MLK) (\(\hat { \boldsymbol {x}}_{hi}(n)\)) and interpolated (NLI) (y_{hi}(n)) signals. This final reconstructed MT signal \(\hat {\boldsymbol {x}}_{f}(n)\) was used for estimation of the threestate MT dynamic instability parameters in the wavelet domain using peak detection as in our prior works [16, 26].
Principal component analysisbased mutual information criterion
Wavelet transforms
Note that j_{0} is an arbitrary starting scale. In this paper, the maximum scale j also represents the wavelet decomposition levels. We later perform peak detection of MTs in the wavelet domain using the energy packing density (EPE) criterion [32] to estimate the threestate MT parameters. In particular, simultaneous timefrequency resolution is the key wavelet property that will be used for detection of the threetransition states in MTs.
Trichotomous Markov Noisebased threestates random evolution model
Where V gives the equilibrium point of the system and L denotes the average length of the MT signal.
Results and discussion
In this section, we experimentally validate the effectiveness of our proposed methods for superresolution of MT signals. The MT data used in this work are results of experiments on MTs (composed of purified αβ_{II} isotopes from bovine brain tubulin) performed by O. Azarenko, L. Wilson and M.A Jordan at the University of California, Santa Barbara. Tubulin proteins were first purified from the bovine brain and then seeded to polymerize at 37°C. The growth and shrinkage dynamics of individual purified MTs were then recorded at their plus ends using the differential interference contrast video microscopy. Data points representing MT lengths were collected at 2−6s intervals. MT lengths were analyzed using the Real Time Measurement program. Growth and shrinkage rates were calculated by leastsquares regression analysis of the data points. Growth and shrinkage thresholds are set to an increase in length by 0.2μm at a rate of 0.15μm/min and a decrease in length by 0.2μm at a rate of 0.3μm/min, respectively. Any length changes equal to or less than 0.2μm over the duration of six data points were considered attenuation phases (phases in which length changes were below the resolution of the microscope). It should be noted that the experimental detection limit for length changes corresponds to about 400−800 tubulin dimers. The supplied data, however, was in the form of a hard copy graphs, that they were scanned and then digitized using the software “DigitizeIt” (http://www.digitizeit.de/) [34].
In this paper, we introduce two novel methods for enhanced MT signal prediction in the new superresolution framework for MTs. Both the proposed methods have nonuniform sampling performed on the MT signals to emulate realworld data loss scenario. This is followed by EMbased ML Kalman prediction (MLK) of the MT signals as a basis framework for superresolution. The first proposed enhanced prediction method involves using an image processinginspired correlation (R)basedpatch processing (MLKR) method to further optimize the final MT predicted signal by extraction of information through identification of similar patches from the dictionary of MLKpredicted MT signals. Second method involves PCA based MI criterion (MLKMI) method that performs extraction of similar information present in principle components of the MLKpredicted MT signals using MI criterion for elimination of data redundancy and further enhancement of the predicted MT signal. Comparison analysis of the proposed superresolution methods is performed with respect to two other methods, namely, nonuniform sampling of MT signal followed by nonlinear (piecewise linear) interpolation technique (NLI) and our previous work using compressed sensing (CS) to reconstruct MT signals with fewer samples than the Nyquist rate [16]. Since, the nonuniformly sampled low resolution MT signal was sampled at a factor of d=3, this implies that it has \(s= \left \lfloor \frac {N}{d}\right \rfloor = \left \lfloor \frac {165}{3}\right \rfloor =55\) samples. This is equivalent to CS subrate of \( \frac {s}{N}= \frac {55}{165}= 0.33\). Therefore, we included results of CS rate =0.3 (CS0.3) from our prior work [16] for comparison.
Comparison of SNR and RMSE for all methods
MT Parameters  MLKR  MLKMI  NLI  CS0.3 

SNR (indB)  15.88  12.71  12.09  4.78 
RMSE  0.42  0.44  0.49  1.10 
The main difference between our proposed superresolution framework in this study and CS method from previous work [16] is that in the former case we assume that only the low resolution MT signal is available to us and using statistical modelling we try to predict and estimate the missing values in the original MT signal with minimum error possible. Whereas in the latter case, we assume that we have already acquired the original MT signal and want to compress it at the sensor side for storage or transmission purposes such that at the receiver side both the compressed signal and the transformation basis is known. The main reason for wide acceptance of CS is that we can preserve the signal details using a random measurement matrix as Gaussian, thereby eliminating the need for complex processing and transmission of transformation basis. But CS makes no effort in learning the underlying signal characteristics which can be critical in understanding biological processes as in case of MTs. On the other hand, statistical modeling methods like EM based ML estimation based Kalman prediction learns the signal structure and hence provides better signal prediction. This is demonstrated through Fig. 3 and Table 1, wherein MLKR and MLKMI methods had higher SNR, lower RMSE and performed better than the CS methods, due to the datalearning and information extraction processes involved. Thus, both our current work and previous work [16] makes an effort to analytically address various facets of the data acquisition problem in MTs and biomedical signals in general, such as low sample availability scenario, spatiotemporal resolution and compression of biomedical signals.

Step 1: Sort the lowest level wavelet coefficients in descending order (i.e. larger significant wavelet coefficients are most likely peaks).

Step 2: Compute the total energy present in the wavelet coefficients (swc) in the lowest level. Where the total energy is calculated by:$$ E_{TOT} = \sum{\mathbf{swc}^{2}} $$(21)

Step 3: Fix a desired threshold to indicate the percentage of significant wavelet coefficients to be retained. In our case, we retain values such that 85% of total energy of the coefficients in the lowest level is preserved. That is:$$ E_{TH} \geq 0.85*E_{TOT} $$(22)

Step 4: Compute the total energy of the sorted wavelet coefficients (E_{TH}) as in (21), until the threshold condition in (22) is met.
Comparison of the original and estimated transition frequency MT parameters for all methods
MT Parameters  OrigMT  MLKR  MLKMI  NLI  CS0.3 

f _{ sg}  0.22  0.22  0  0.22  0.22 
f _{ gs}  0.22  0.44  0.22  0.22  0.22 
f _{ sp}  8.76  8.76  5.47  9.53  12.04 
f _{ ps}  0.11  0  0  0.11  0 
f _{ gp}  0.11  0  0.22  0  0 
f _{ pg}  8.54  8.54  11.59  7.88  5.48 
Comparison of the original and estimated velocity and length MT parameters for all methods
MT Parameters  OrigMT  MLKR  MLKMI  NLI  CS0.3 

v _{ s}  45.17  45.17  42.16  72.27  63.24 
v _{ g}  40.65  42.16  40.11  36.14  21.68 
avg L  5.24  5.31  5.27  7.87  5.44 
Comparison of the estimated MT error parameters for all methods
MT Parameters  MLKR  MLKMI  NLI  CS0.3 

f _{ sg}  0  0.22  0  0 
error  
f _{ gs}  0.22  0  0  0 
error  
f _{ sp}  0  3.28  0.77  3.28 
error  
f _{ ps}  0.11  0.11  0  0.11 
error  
f _{ gp}  0.11  0.11  0.11  0.11 
error  
f _{ pg}  0  3.05  0.66  3.07 
error  
v _{ s}  2.00  3.00  4.00  3.00 
error  
v _{ g}  2.66  3.00  3.00  3.00 
error 
Conclusion
In this paper, we propose two novel frameworks for superresolution of MT signals to address the limited data availability scenario and emulate the data loss due to nonuniform sampling of biological/natural signals that we often encounter in the realworld. This work exploits the stochastic nature of MT signals through statistical modelling using EM based ML driven Kalman estimation (MLK) for better prediction of the nonuniformly sampled MT signal as a basic superresolution framework. The MLKpredicted MT signals were further optimized through information extraction using correlation (R)basedpatch processing (MLKR) and PCAbased mutual information criterion (MLKMI) methods. We perform comparison analysis of our proposed methods MLKR and MLKMI with respect to nonlinear interpolation (NLI) and CS methods. It was experimentally found that both the proposed methods MLKR and MLKMI achieved the best overall performance for MT signal estimation. Specifically, MLKR outperformed all the methods, and had better reconstruction performance using fewer samples, gave high SNR, low errors, as well as better MT parameter estimation than other compared methods. This work aims to demonstrate the effectiveness and significance of statistical modelling and data learning in MTs, and biomedical paradigm. Our goal is to provide an analytical solution to overcome the equipment/hardware fallacies that might occur during the signal acquisition process.
Notes
Declarations
Acknowledgements
Nothing to declare.
Funding
Publication costs were funded through faculty startup funds of author VM offered by University of Alabama in Huntsville.
Availability of data and materials
Statistical and computational models used are fully detailed in the main text. Data will be made available upon personal requests to authors.
About this supplement
This article has been published as part of BMC Systems Biology Volume 12 Supplement 6, 2018: Selected articles from the IEEE BIBM International Conference on Bioinformatics & Biomedicine (BIBM) 2017: systems biology. The full contents of the supplement are available online at https://bmcsystbiol.biomedcentral.com/articles/supplements/volume12supplement6.
Authors’ contributions
VM and SY conceptualized this idea. VM implemented the MT statistical modelling and pertinent signal processing concepts. SY formulated MT mathematical modelling. VM, SY, and VR analyzed the results. VM and SY drafted 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
The authors declare that they have no competing interests.
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Authors’ Affiliations
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