Robust Regression and Optimal Transport Methods to Predict Gastrointestinal Disease Etiology from High Resolution EGG and Symptom Severity

Abstract

Objective:

Gastric functional and motility disorders are highly prevalent, with gastroparesis (GP) and functional dyspepsia (FD), affecting 1.5-3% and 10% of the population, respectively. Multiple disease etiologies with overlapping symptoms, such as antral hypomotility, pylorospasm, autonomic dysfunction, and gastric myoelectric dysfunction underlie GP and FD. There is an unmet need to differentiate these etiologies non-invasively to tailor treatment strategies and predict treatment response.

Methods:

We performed cutaneous high-resolution electrogastrogram (HR-EGG) recordings on 32 human subjects (controls, GP, and FD) and computed gastric slow wave propagation patterns. We implemented robust regression and clustering methods to identify one group of patients with symptoms well explained by spatial slow wave features and another with symptom severity significantly exceeding predictions from spatial slow wave features. Five patients were re-assessed with validated symptom questionnaires after pyloric and prokinetic interventions.

Results:

A group of seven patients was identified whose spatial slow wave features lie within the same range as control subjects but whose symptom severity significantly exceeded what is predicted from spatial slow wave features. We hypothesize that gastric myoelectric dysfunction is not a prominent disease etiology in this group. A highly accurate regression holds in the other group of patients (r=0.8). Of the patients with repeat questionnaires, patients with symptom severity exceeding the regression line reported symptom improvement, whereas patients with symptoms in close proximity to the regression line experienced no improvement.

Conclusion:

These findings suggest that patients with symptom severity significantly exceeding the robust regression line have symptoms that cannot be explained by gastric myoelectric dysfunction alone, and vice versa.

Significance:

This methodology may provide clinicians with an opportunity to screen patients to determine when existing interventions will be effective, and on the flipside, when slow wave restoration interventions, such as gastric neuromodulation, may be most effective in improving symptoms and quality of life.

I. Introduction

A. Background and Unmet Need

Gastrointestinal (GI) problems are the second most common reason for someone to miss school or work, after the common cold [1]. Two of the most prevalent GI functional and motility disorders are functional dyspepsia (FD) and gastroparesis (GP), which affect 10% and 1.5-3% of the U.S. population, respectively [2], [3]. Multiple disease etiologies underlie the umbrella diagnoses of GP and FD, including but not limited to i) antral hypomotility, ii) pylorospasms, iii) gastric myoelectric dysfunction [4], and iv) autonomic dysfunction. Tailored treatments for such causes include i) prokinetic medications, ii) botulinum toxin injection of the pylorus [5], gastric neuromodulation [6], [7], [8], [9], [10], [11], [12], [13] and iv) vagus nerve stimulation [14], respectively. The common occurrence of treatment regimens tailored to the wrong etiology or to symptom mitigation results in repetitive visits to the clinic and adverse effects on quality of life with vast economic, social, and healthcare consequences. There is a desire and unmet need to sub-type GP and FD in terms of underlying etiologies to improve prediction of treatment response and enable targeted interventions [15], [16]. GP and FD are analyzed together as one group in this work, since subtyping is geared toward distinguishing between underlying etiologies (i.e. pylorospasms vs gastric myoelectric dysfunction), as opposed to subtyping between GP and FD (which is increasingly being questioned by key opinion leaders due to the indistinguishable nature of the two diagnoses [17]).

B. Previous Work

Current diagnostic tools are typically not etiology-specific. For instance, the PAGI-SYM questionnaire on foregut symptoms, which includes the gastroparesis cardinal symptom index, is used to manage treatments but shows a high degree of symptom overlap with underlying disease etiologies [4]. Alternatively, the gastric emptying test, a clinical gold standard that entails post-fasting ingestion of a meal containing a radioactive tracer and measuring the amount of tracer that has left the stomach at fixed intervals, shows inconsistent correlation with symptom severity [18], [19]. Further, some medications improve symptoms but not gastric emptying, and vice versa [20], [21], [22]. As such, there is an unmet need to develop widely-deployable screening tools that correlate with symptom severity and sub-type GP/FD in terms of disease etiology.

Gastric neuromuscular activity involves the interplay between electrical activity on the stomach’s surface and smooth muscle contractions that propel food from the esophagus to the intestines. Gastric pacemaker cells, or interstitial cells of Cajal (ICCs), receive modulatory inputs from enteric neurons and undergo spontaneous cyclical depolarization, similar to cardiac pacemaker cells in the heart. These oscillatory membrane potentials excite neighboring smooth muscle cells, causing contractions at regular intervals, and excite adjacent downstream ICCs in their network [23], [24]. This results in a traveling electrical wave, called the gastric slow wave [25], [26].

In healthy individuals, slow waves oscillate at 3 cycles per minute (0.05Hz), originate in the pacemaker region of the stomach, and propagate along the outer surface of the stomach in an anterograde direction toward the duodenum [26], [27]. Abnormalities in slow wave initiation and propagation patterns arise from loss or deterioration of ICCs [28], [29]. These pathological slow waves can spontaneously arise from locations other than the pacemaker region (i.e. the antrum) and/or propagate in aberrant directions (i.e. in a retrograde direction towards the esophagus or a bifurcated combination of anterograde/retrograde propagation). These abnormal propagation patterns disrupt the spatial coordination of smooth muscle contractions and have been observed in GP/FD patients [30].

High-resolution spatiotemporal mapping of the gastric slow wave has been performed with electrodes placed directly on the surface of the stomach during open-abdominal surgery. These findings indicated normal slow wave behavior in control subjects (n=12) [31] and abnormalities in slow wave initiation and propagation in > 80% of GP subjects (n=12) [30]. Whereas these studies demonstrated spatial slow wave abnormalities as possible explanations for foregut disorders, the involvement of surgical procedures limits the technique’s scalability.

A non-invasive alternative is the electrogastrogram (EGG), which involves electrodes placed on the surface of the abdomen (as opposed to the stomach) and measures electrical activity that reaches the abdominal surface via volume conduction. Conventional EGG is comprised of 3-4 electrodes and is analyzed via spectral methods (i.e. power-frequency) [26], [32]. However, it has been shown that i) spectral features do not reliably correlate with symptoms or disease [33], [34] and ii) spatial slow wave abnormalities (which do have high correlation with disease and symptoms) are undetectable by spectral analyses [30], [35], [29], [36].

The high-resolution EGG (HR-EGG) has recently been established to overcome the pitfalls of traditional EGG. This approach is comprised of a dense cutaneous electrode array and subsequent signal processing methods for artifact rejection [37] and estimation of spatial slow wave directions [38]. It has been shown, both in silico and in human subjects, that estimated wave directions orient in an anterograde direction towards the duodenum in healthy controls [38], [39].

HR-EGG was also recently used in human subjects with GP (n=18) and FD (n=7) to establish a correlation between incidence of non-anterograde slow wave propagation and symptom severity [35]. Specifically, a linear correlation (r=0.57) was established between the percentage of non-anterograde slow wave directions and the mean Gastroparesis Cardinal Symptom Index (GCSI) symptom score, a standardized foregut symptom severity score that takes into account nausea, vomiting, early satiety, bloating, and abdominal distention [40]. A more recent finding showcased that gastric slow wave spatial patterns derived from HR-EGG distinguish children with symptomatic chronic nausea from healthy subjects [41].

Although these findings suggest that gastric slow wave features determined from cutaneous multi-electrode recordings may provide an explanation for GP and FD symptoms, some key limitations arise. Since multiple disease etiologies underlie GP and FD, a ‘one size fits all’ regression paradigm is likely inadequate to explain symptom severity solely from spatial slow wave features. For instance, in patients with high symptom severity arising from etiologies other than gastric myoelectric dysfunction, spatial slow wave features would likely resemble those of healthy controls although their symptom severity score would exceed that of control subjects [35]. We hypothesize that if a patient’s symptom severity score exceeds what would be ‘expected’ from slow wave patterns, then the underlying cause of symptoms is other than gastric myoelectric dysfunction. In line with this reasoning, if we can isolate and exclude groups of patients for which symptoms exceed regression predictions, one can reasonably hypothesize that a higher-accuracy correlation can be drawn between abnormal wave propagation and symptoms in the remaining subjects.

C. Our Proposed Approach

The aforementioned drawbacks call for the implementation of a “robust regression” method which involves a mathematically rigorous approach to determine an inclusion criterion of data to be used in regression, and an exclusion criterion of data to be considered outliers. Robust regression has been widely used in the current literature to fit highly-accurate regressions in the presence of outliers by assuming the probability distribution of residuals has multiple modes [42], [43], [44], [45]. Optimization of model-fitting parameters typically involves maximizing a multi-modal likelihood function over the residual errors, with at least one mode near zero [42], [43], [44]. This method optimizes regression fit over the data assumed to represent ‘signal’ rather than ‘noise.’

We use robust regression in our work presented here in order to identify and exclude an “outlier” class of patients for which we hypothesize that their underlying GP or FD etiology is not gastric myoelectric dysfunction. This separation of patients whose symptoms are adequately explained by slow wave features, and those whose symptoms are not, may have the potential to predict treatment response to, for instance, gastric neuromodulation (a more invasive intervention) or medications (a more conservative approach). Prediction of treatment response has the potential to rapidly guide effective treatments, as well as reduce the risk of performing unnecessary procedures.

Another potential improvement to the preliminary correlation study in [35] involves estimating an entire probability distribution of wave directions for each recording. This provides more information, pertaining to modality and overall distributional structure, that may lead to improved characterizations of recordings. Furthermore, utilizing the full probability distributions enables the use of an optimal transport framework to build metrics between distributions. ‘Optimal transport’ refers to the process of minimizing the cost of transporting all of the mass in one probability distribution to another, and has powered state-of-the-art studies in broad applications of mathematics, data science, and economics [46], [47], [48], [49]. In this study, we make use of the Wasserstein metric between probability distributions to develop unique spatial slow wave features to be used downstream for robust regression.

We calculate Wasserstein barycenters [50] between probability distributions of control subjects, which serves as a ‘model’ of normalcy. We then compute the optimal transport cost between each subject’s probability distribution and the controls’ barycenter to quantify the deviation of subject’s probability distribution from ‘normalcy’. In addition, we compute distances between each subject’s probability distribution and a uniform distribution over directions as a secondary feature to quantify the deviation from chaotic or uncoordinated propagation. As previously mentioned, abnormal slow wave propagation can take on several forms. One such form is an abnormal initiation near the antrum with a large degree of coordinated retrograde propagation, in addition to some anterograde propagation [30]. Probability distributions of wave directions in these types of cases would differ in shape and modality from recordings of healthy controls, which are typically dominated by anterograde propagation. In addition, this physiology is distinct from chaotic or uncoordinated propagation, which would result in wave directions having probabilistic signatures more akin to a uniform distribution. As such, these features derived from a sophisticated optimal transport framework follow insights from slow wave physiology.

Lastly, previous HR-EGG methods [38], [35] required computed tomography (CT) scans for both accurate placement of electrodes over the stomach, as well as measurements related to anatomical landmarks in order to define ‘normal’ propagation. The need for a CT presents a current bottleneck in these existing HR-EGG methods because CT is costly for patients and requires otherwise unnecessary exposure to radiation. We have addressed this in our present work by taking into account the features and modalities of full probability distributions, which results in a method that is robust to an estimation of these anatomical measurements.

D. Key Findings

In summary, our contributions in this manuscript involve building a full probability distribution from measured HR-EGG wave directions, developing an optimal transport framework for features pertaining to distances between probability distributions, and performing robust regression to a) isolate a group of patients in which we hypothesize that gastric myoelectric dysfunction is not a prominent etiology in their disease and b) form a more accurate correlation between symptoms and features of wave directions in the remaining group of subjects. Our methods are robust to a barycenter-type estimate of the angle that defines ‘normal’ propagation and, if used in conjunction with recent methods in stomach identification with portable ultrasound [51], would eliminate the need for a CT scan. Overall, we propose a method to screen patients for which aggressive intervention, such as gastric neuromodulation, would be most effective in promoting good clinical outcomes.

II. Methods

A. Data Acquisition and Signal Processing

We enrolled healthy control subjects with no prior GI disease (n=7), GP patients (n=18), and FD patients (n=7), for a total of 32 subjects. We performed a PAGI-SYM questionnaire on all subjects, with repeat PAGI-SYM questionnaires on 5 patients (4 GP, 1 FD) after non-electrical treatments (pyloric or prokinetic interventions). Since both treatments do not directly target gastric myoelectric dysfunction, the distinction between pyloric and prokinetic interventions is not explored here. The PAGI-SYM questionnaire is a standardized, validated patient survey to rate symptoms such as nausea, bloating, early satiety, etc [52]. We performed 25-channel HR-EGG recordings on all subjects for 90 minutes (approximately 30 minutes preprandial and approx. 60 minutes postprandial). Each subject was asked to fast for 12 hours prior to ingestion of the meal at the 30 minute time point.

The ingestion of food activates excitatory neural responses that interface with the stomach to increase the magnitude of gastric smooth muscle contractions [53]. Given this fact, and since gastric contraction strength correlates with EGG power [54], [55], we solely use the postprandial portion of the recording the remainder of this section to operate in a regime of high signal-to-noise ratio (SNR) for our HR-EGG recordings.

HR-EGG recording and prepossessing methods were identical to those published here [35], which included filtering in the range of 0.05 Hz, artifact rejection, etc. For generalizability, we refer to the number of subjects as $S$ and the number of electrodes as $N$ throughout this manuscript. Ethical approval for this work was obtained from the institutional review board at the University of California, San Diego, USA, Protocol Number: 141069, Date of Approval: June 19, 2014.

B. Instantaneous Wave Direction at Every Sensor

We performed the Hilbert transform on mean-centered HR-EGG waveforms to extract the instantaneous wave phase, ${\varphi }_n^{(s)}(t)$, at the $n^{th}$ channel $(n=1,\dots N)$ and the $s^{th}$ subject $(s=1,\dots S)$.

$$V_n^{(s)}(t)+iHb[V_n^{(s)}(t)]=a_n^{(s)}(t)e^{i{\varphi }_n^{(s)}(t)}$$ (1)

In accordance with current HR-EGG analysis studies [35], [38], [39], we calculated the instantaneous wave direction, ${\Lambda }_n^{(s)}(t)$, at each channel by taking the inverse tangent of phase gradients.

$${\Lambda }_n^{(s)}(t)={\tan }^{-1}\left(\frac{\nabla {\varphi }^{(s)}(y_n,t)}{\nabla {\varphi }^{(s)}(x_n,t)}\right)$$ (2)

In order to exclude segments of recordings dominated by noise, we defined a notion of a ‘sustained wave’ and compiled subsets of the recording in which the ‘sustained’ criterion was met, explained as follows. We defined the instantaneous phase gradient directionality (PGD) for the $s^{th}$ subject as:

$${\text{PGD}}^{(s)}(t)=\frac{{\Vert \frac{1}{N}{\sum }_{n=1}^N\nabla {\varphi }^{(s)}(x_n,y_n,t)\Vert }_2}{\frac{1}{N}{\sum }_{n=1}^N{\Vert \nabla {\varphi }^{(s)}(x_n,y_n,t)\Vert }_2}$$ (3)

where $t=1,\dots T^{(s)}$. The PGD has been used in neuroscience [56] as well as previous HR-EGG analyses [39], [38], [35] as a measure of cohesivity among wave directions. From Jensen’s inequality, it is upper bounded by 1 which occurs when a plane wave, which has the highest degree of direction agreement, is present. In order to reduce the false discovery of noise to below 1% with 25 channel arrays, we categorize a wave as satisfying the ‘sustained’ criterion if the PGD exceeds 0.5 for longer than 2 seconds [38]. As such, direction values calculated in (2) that indicate sporadic or otherwise uncoordinated wave propagation must have some inherent pattern and coordination in order to meet the sustained criterion, as is consistent with physiology (i.e. an ectopic event results in local coordination distinct from patterns elsewhere on the stomach [30]).

C. Direction Values Relative to the Duodenum

The aforementioned wave direction values are measured with respect to the cutaneous recording array. However, the stomach has a characteristic ‘J’ shape, with subject-to-subject anatomical variability, and further is not always perfectly in plane with the sensor array. Previous studies have addressed this by translating direction values by the subject-specific angle to the duodenum, derived from CT measurements [35], [38]. Thus, a translated direction value of zero radians would indicate wave propagation toward the duodenum (or anterograde propagation).

Instead of translating each subject’s direction values by their unique duodenum angle, we translated direction data by an average of all $S$ subjects’ duodenum angle. We then slightly adjusted the translation, such that the location of maximum incidence of the histogram of translated directions within a range of $\pm \pi ∕6$ occurred at zero radians. There is precedent for this approach within the literature that gives a $\pm \pi ∕3$ degree window of tolerance around the subject-specific angle to the duodenum in order to define ‘normal’ wave propagation [35]. Our adjustment to the translation operated on a more conservative window while still accounting for anatomical variability. Translating directions in this way eliminates the need for a CT going forward, since we can utilize measurements from CTs that have already been performed in calculation of averages, as well as incorporate data from large imaging databases to further improve the average if desired. Because the downstream analysis methods presented in this work estimate the entire probability distribution and exploit characteristics of the shape and modalities of the distribution, we obviate the need for high precision with the angle by which we translate direction values.

We collapsed sustained wave direction values across both spatial and temporal axes, giving rise to set ${\mathcal{D}}^{(s)}$, for the $s^{th}$ subject, as seen in (4). The set ${\mathcal{D}}^{(s)}$ is shown for two human research subjects as a histogram in the circular domain in Fig. 2A.

Fig. 2.

A) Histograms of wave directions, ${\mathcal{D}}^{(s)}$. B) Estimated probability density function, from methods described in Section II-D, shown in polar axes. C) Estimated probability density function, from methods described in Section II-D, shown in Euclidean axes. Panels A-C are shown for a healthy control subject with normal slow wave physiology (top) and a symptomatic patient with indications of abnormal gastric slow wave function (bottom).

$${\mathcal{D}}^{(s)}=\{{\Lambda }_n^{(s)}(t)\}\forall n\in \{1,\dots N\},\forall t\in \{1,\dots T^{(s)}\}$$ (4)

D. Probability Density Estimation with von Mises Mixtures

Since direction data in set ${\mathcal{D}}^{(s)}$ lie on a circle, a natural family of probability density functions on which we can perform density estimation is the family of von Mises distributions, which are commonly referred to as the “Gaussians” of circular statistics [57], [58]. There is precedent established in the literature across multiple scientific fields for the von Mises mixture model approach used in this work [59], [60]. The von Mises probability density function is given by:

$$f(\theta \mid \mu ,\kappa )=\frac{e^{\kappa \cos (\theta -\mu )}}{2\pi I_0(\kappa )}$$ (5)

where $\mu$ is the mean of the distribution, $\kappa$ is the inverse variance, and $I_0(\kappa )$ is the $0^{th}$ order Bessel function. See Fig. 1A for examples.

Fig. 1.

A) Probability density function of a unimodal von Mises distribution (5) . b) von Mises mixture of two distributions (6). Both unimodal and multimodal distributions shown in polar (left) and Euclidean (right) axes.

We performed maximum likelihood density estimation of von Mises mixtures (vMM), given by (6), in order to capture multiple modalities in ${\mathcal{D}}^{(s)}$, which could indicate multiple modes of wave propagation (i.e. bifurcated anterograde and retrograde propagation). An example of a von Mises mixture is shown in Figure 1B.

$$f_{vMM}(\theta \mid \Theta )=\sum _{k=1}^K{\lambda }_k\frac{e^{{\kappa }_k\cos (\theta -{\mu }_k)}}{2\pi I_0({\kappa }_k)}$$ (6)

where $\Theta =({\lambda }_1,\dots {\lambda }_K,{\mu }_1,\dots {\mu }_K,{\kappa }_1,\dots {\kappa }_K)$, and

$$\sum _{k=1}^K{\lambda }_k=1.$$ (7)

The log likelihood, given the set of vMM parameters, $\Theta$, and set ${\mathcal{D}}^{(s)}$ is:

$$L(\Theta )=\sum _{r=1}^R\log \sum _{k=1}^K{\lambda }_k\frac{e^{{\kappa }_k\cos ({\mathcal{D}}_r^{(s)}-{\mu }_k)}}{2\pi I_0({\kappa }_k)}.$$ (8)

where $R$ is the length of ${\mathcal{D}}^{(s)}$, or $Nx{T}^{(s)}$. Optimal parameters, $\widehat{\Theta }$, are those which maximize (8).

$$\widehat{\Theta }=\text{arg}\underset{\Theta }{\max }L(\Theta ;{\mathcal{D}}^{(s)})$$ (9)

E. Model Selection Procedure

Solving (9) with no restriction on the number of mixture distributions, $K$, may cause the model to over-fit to variants arising from noise or human subject-to-subject variability, which results in a loss of generalizability. Adding a penalty to the number of distributions prevents over-fitting while still capturing multiple modalities in ${\mathcal{D}}^{(s)}$. Since all weights are non-negative and sum to 1, by virtue of (7), traditional ${\ell }_1$ penalized model fitting procedures are not applicable to this context. We implemented a pruning method to iteratively reduce the number of mixture distributions, $K$, and subsequently perform dimensionality-based penalization.

We began the iterative pruning method by executing maximum likelihood optimization in (9), given the set ${\mathcal{D}}^{(s)}$ and a large dictionary of $‘‘M’’$ pairs of von Mises density parameters, ${(\mu ,\kappa )}_m$, to over-represent the problem space [61]. The parameter dictionary consisted of samples of $\mu$ within a range of $[-\pi ,\pi ]$, sampled in increments of $2\pi ∕9$. Similarly, $\kappa$ was sampled within a range of [0, 15], in increments of 15/9. The complete dictionary consisted of a mesh grid of all possible combinations of $\mu$ and $\kappa$, with size $M=100$. Since parameters ${\mu }_k$ and ${\kappa }_k$ were specified by this dictionary, optimization in (9) solves for the mixture probabilities, ${\lambda }_k(k=1,\dots K)$, which maximize (8).

We then eliminated the set ${(\mu ,\kappa )}_{k^{\prime }}$ which was lowest-weighted, or

$$k^{\prime }=\text{arg}\underset{k=1,..K}{\text{min}}{\lambda }_k,$$ (10)

since ${(\mu ,\kappa )}_{k^{\prime }}$ contributed the least to the overall log likelihood. We repeated this pruning process until the dimensionality of the parameter dictionary, $K$, was equal to 1. We computed the negative normalized log likelihood (NNLL) at each pruning iteration, given by:

$$\text{NNLL}(K)=-\frac{1}{R}L({\widehat{\Theta }}_K)$$ (11)

where ${\widehat{\Theta }}_K$ is the set of parameters that maximizes the log likelihood in (8) when the parameter dictionary is pruned to $K$ specific $(\mu ,\kappa )$ pairs. We implemented Bayesian Inference Criterion (BIC) [62] penalization where the penalized normalized negative log likelihood (PNNLL) is given below.

$$\text{PNNLL}(K)=\text{NNLL}(K)+\frac{K\log R}{2R}$$ (12)

We obtained the optimal number of von Mises densities $K^{\ast }$. which minimizes (12), and then obtained the corresponding set of optimal $(\mu ,\kappa )$ density parameters, to construct the density estimate.

F. Maximum Likelihood Estimation with the EM Algorithm

During each step of the pruning method, we used the Expectation Maximization (EM) for mixtures [63] to carry out the maximum likelihood estimation in (9).

The EM algorithm makes use of a latent variable, ${\underset{¯}{Z}\in \{1,2,\dots K\}}^R$, for which $Z_r=k$ encodes which distribution ${\mathcal{D}}_r^{(s)}$ is drawn from in the von Mises mixture model. The complete log likelihood (conditioned on $\underset{¯}{Z}$) is given by:

$$p(\underset{¯}{Z},{\mathcal{D}}^{(s)};{\Theta }_K)=\sum _{r=1}^R\sum _{k=1}^K{1}_{\{Z_r=k\}}\log [{\lambda }_{k}f({\mathcal{D}}_r^{(s)}\mid {\mu }_k,{\kappa }_k)]$$ (13)

where $1_{\{A\}}$ equals 1 if event $A$ occurs and 0 otherwise. For the sake of brevity we will use the notation ${\mu }_k$, ${\sigma }_k$, ${\lambda }_k$ to index specific parameters in the iteration of the pruning method in which there are $K$ mixture distributions. It is implied that ${\mu }_k$ refers to the specific ${\mu }_k\in {\Theta }_K$, for instance. The posterior probabilities of latent variable, $\underset{¯}{Z}$, conditioned on set ${\mathcal{D}}^{(s)}$ are given by:

$${\Gamma }_{r,k}^{(m)}\triangleq P(Z_r=k\mid {\mathcal{D}}_r^{(s)};{\Theta }_K^{(m)})$$ (14)

$$=\frac{{\lambda }_{k}f({\mathcal{D}}_r^{(s)}\mid {\mu }_k^{(m)},{\kappa }_k^{(m)})}{f_{vMM}({\mathcal{D}}_r^{(s)}\mid {\Theta }_K^{(m)})}$$ (15)

at the $m\text{th}$ iteration of the EM algorithm. We define the expectation of the complete log likelihood given by (13) as:

$$Q({\Theta }_K^{(m+1)},{\Theta }_K^{(m)})=\sum _{r=1}^R\sum _{k=1}^K{\Gamma }_{r,k}^{(m)}\log [{\lambda }_{k}f({\mathcal{D}}_r^{(s)}\mid {\mu }_k,{\kappa }_k)]$$ (16)

where ${\Theta }_K^{(m)}$ is the vector of parameters used to compute the posterior probability, ${\Gamma }_{r,k}$, and vector ${\Theta }_K^{(m+1)}$ houses each ${\lambda }_k$, ${\mu }_k$, ${\sigma }_k$ in (16). Maximizing this expectation yields parameter updates:

$${\lambda }_k^{(m+1)}=\frac{{\sum }_{r=1}^R{\Gamma }_{r,k}}{R}$$ (17)

for $k=1,\dots K$ where $R$ is a normalization constant to guarantee the sum of ${\underset{¯}{\lambda }}^{(m+1)}$ is equal to 1.

We performed 20 iterations of parameter updates.

G. Wasserstein Distances: Optimal Transport Theory

We used an approach based in Optimal Transport Theory, namely the Wasserstein distance, to define a metric between probability distributions. Such an approach is standard across multiple disciplines [46], [48], [49]. In this setting, to account for the circular nature of directional information, we utilize a Wasserstein distance with the base distance being a geodesic defined on the unit circle [64], [65]. Assuming that probability distributions $P$ and $Q$ have densities $p$ and $q$ respectively, the Wasserstein squared distance between probability distributions $P$ and $Q$ is defined as:

$$W_2{(P,Q)}^2=\underset{\pi \in \Pi [P,Q]}{\text{min}}{\int }_{u\in U}{\int }_{v\in V}c(u,v)\pi (u,v)dudv$$ (18)

where $\Pi [P,Q]$ is the set of all possible joint probability densities with marginals $P$ and $Q$, $U$ is the support of $P$ and $V$ is the support of $Q$. The function $c(u,v)$ is a cost function pertaining to the squared distance between $u$ and $v$.

In the work presented here, data are direction values, which lie on a circular manifold. As such, we computed a quadratic cost

$$c(u,v)=d_c{(u,v)}^2$$ (19)

operating on $d_c(u,v)$

$$d_c(u,v)=\text{min}\{\mid u-v\mid ,2\pi -\mid u-v\mid \}$$ (20)

which is the geodesic distance on the unit circle. We formulated the optimization in (18) as a linear program, discretized over the compact set with boundary points $-\pi$ and $\pi$ subject to:

$$\begin{gathered} \pi (u,v)\ge 0 \\ {\int }_{u\in U}{\int }_{v\in V}\pi (u,v)dudv=1 \\ {\int }_{v\in V}\pi (u,v)dv=p(u) \\ {\int }_{u\in U}\pi (u,v)du=q(v) \end{gathered}$$

where the first two constraints hold because $\pi (u,v)$ is a joint probability density, and the latter two hold because $\pi \in \Pi [P,Q]$ and thus it must preserve marginal densities $p$ and $q$.

H. Wasserstein Barycenter of Probability Distributions from Control Subjects

We constructed a Wasserstein barycenter (denoted $P_B$) between healthy control subjects. This barycenter represents an ‘average’ distribution for the control group. Other aggregate methods, such as ${\ell }_2$ averaging of probability distributions, fail to capture modality and shapes of individual distributions (see Figure 3). This motivated our use of a Wasserstein barycenter approach, as distributional shape and modality are paramount to the downstream feature space in this work.

$$P_B=\text{arg}\underset{P\in \Omega [P,P{}_c]}{\text{min}}\frac{1}{C}\sum _{c=1}^C{W}_2{(P,P_c)}^2$$ (21)

where $C$ is the number of control subjects (n=7) and $P_c$ refers to individual probability distributions of control subjects, solved for in Section II-D. For each subject (including controls) we computed Wasserstein squared distances, via (18), between i) each probability distribution, $P_s$ and the control barycenter, $P_B$ and ii) each probability distribution and a uniform distribution, denoted as $P_U$, over the support $[-\pi ,\pi ]$. This first feature is a metric of how close each distribution is to an approximation of normalcy, whereas the second feature measures closeness to complete randomness, as a uniform distribution models the general distributional shape of aberrant wave propagation (see Fig. 4 for such an outlier). These features populate the regressor matrix, defined in the following section, and will be referred to throughout the remainder of this work as ‘slow wave features.’

Fig. 3.

Example highlighting the importance of using a Wasserstein barycenter to represent a model of healthy controls in Section II-H. A) Two unimodal von Mises probability distributions. B) The L-2 average of the two distributions is bi-modal, with one mode centered at each individual distribution’s mean. C) The Wasseerstein barycenter between the two distributions is unimodal and therefore better represents the structure of both of the original distributions. Panels A-C shown in uniform-radius polar (top) and Euclidean (bottom) axes.

Fig. 4.

Probability density function (found in Section II-D) of outlier within the control group, with uniform distribution along the support $[-\pi ,\pi ]$ for reference. The outlier distribution lacks the high-energy mode near zero that is expected of healthy controls exhibiting primarily anterograde wave propagation (i.e. as seen in Fig. 2C), and instead exhibits overall structure similar to a uniform distribution. Shown in A) polar axis and B) Euclidean axis.

I. Robust Regression and EM Clustering

We identified an outlier in the control group. This outlier probability distribution deviated in modality from both a) the expected probability distribution of healthy controls with respect to physiology and b) the other control subjects (see Fig. 4). In fact, this probability distribution showed aberrant, chaotic slow wave propagation similar to distributions estimated from from GP and FD HR-EGG recordings. In this section, we have removed the outlier from calculation in the robust regression algorithm. It is plotted in the results figures but not used for computation of regression coefficients and residuals. As such, the indexing notation of human research subjects going forward is $d\in 1,\dots D$, where $D=31$.

To carry out robust regression in this work, we defined a regressor matrix $F$, of size $D\times 3$, with the $d^{th}$ row given by:

$${\underset{¯}{F}}_d^T=[1,W_2{(P_d,P_U)}^2,{W_2(P_d,P_B)}^2].$$ (22)

We defined the response vector, $\underset{¯}{Y}\in {\mathcal{R}}^D$ as the Mean GCSI (Gastroparesis Cardinal Symptom Index) score for all subjects, which is a subset of the PAGI-SYM questionnaire [52], [40]. We did not study the effect of gastric myoelectric features on improvement of specific symptoms, as these changes may not have sufficient statistical power with the limited patient cohort size. This is a subject of future work and is fitting for a large cohort study. We modeled $\underset{¯}{Y}$ to linearly relate to $F$ such that:

$$\underset{¯}{Y}={\underset{¯}{\beta }}^{T}F+\underset{¯}{N}$$ (23)

where $\underset{¯}{N}$ is the vector of residual errors. Since spatial gastric slow wave abnormalities are only one type of etiology of gastrointestinal foregut disorders, we hypothesize that this linear model will fit a subset of the data for which spatial slow wave abnormalities are a dominant cause of symptoms. For the rest of the subjects, however, we expect their symptom and HR-EGG features to be anomalous to the linear trend. For instance, this anomalous class would encompass patients who have normal slow waves, with probability distributions indistinguishable from healthy controls, but high symptom severity scores. Following this logic, we assume residual errors from the linear regression, $\underset{¯}{N}$, will follow a bi-modal distribution: with one mode near zero and another mode near some $\alpha >0$. This bi-modal distribution can be modeled as a Gaussian mixture model (GMM)

$$f_{GMMM}(x;\underset{¯}{\eta },\underset{¯}{\sigma },\underset{¯}{\varphi })=\sum _{i=1}^2{\varphi }_i\mathcal{N}(x;{\eta }_i,{\sigma }_i)$$ (24)

where ${\varphi }_1$ and ${\varphi }_2$ are non-negative weights that add to 1 and $\mathcal{N}(x\mid \eta ,\sigma )$ is the density of a Gaussian random variable with expectation $\eta$ and variance ${\sigma }^2$ evaluated at $x$:

$$\mathcal{N}(x;\eta ,\sigma )\triangleq \frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{{(x-\eta )}^2}{2{\sigma }^2}}$$ (25)

Since we hypothesize that the residuals $\underset{¯}{N}$ follow a bi-modal GMM, $\underset{¯}{Y}$ follows a bi-modal GMM, with one mode at ${\underset{¯}{\beta }}^{T}F$ and the other mode at ${\underset{¯}{\beta }}^{T}F+\alpha$. Because we hypothesize that the outlier class will be comprised of patients for which symptoms exceed what would be ‘expected’ from features of the slow wave, we aimed to identify outliers that lie above the regression line. As such, we defined $\alpha >0$. To find the optimal offset, ${\alpha }^{\ast }$, we created a dictionary of offset values, with entries ${\alpha }_h$ (for $h=1,\dots H$), on the interval (0, 1] in increments of $\frac{1}{H}$. The EM algorithm was used to solve for the remainder of the density estimation parameters in (26). EM-model fitting was performed $H$ times, for each ${\alpha }_h$ in the dictionary. The optimal offset, ${\alpha }^{\ast }$, was chosen to be the value of $\alpha$ which maximized the log likelihood given in (26). We implemented this method with $H=10$

We used methods similar to those presented in [44]. However, deviating from the approach in [44], where regression coefficients, $\underset{¯}{\beta }$, for each mode were distinct from one another, we solved for the regression coefficients only for the non-outlier class. This decision rests upon our hypothesis that the non-outlier class corresponds to patients where slow wave patterns are the dominant etiology for their symptoms, which implies that the outlier class pertains to patients where the dominant cause of symptoms is an etiology other than slow wave abnormalities (i.e. dysfunction of the pylorus, or autonomic dysfunction). Because there are multiple such etiologies, and given that our dataset contained $n=31$ subjects, we determined that this approach minimizes opportunities for overfitting or spurious calculations.

Similar to the von Mises mixture density estimation procedure discussed in Section II-D, optimization of the GMM problem can be solved with the EM algorithm. We initialized the weights as ${\varphi }_1=0.9$, ${\varphi }_2=0.1$ since we can reasonably hypothesize that the majority of the data fits within Class 1. We initialized $\underset{¯}{\beta }$ as the coefficients of the linear least squares regression of the entire dataset with symptoms. We initialized ${\sigma }_1^2$ and ${\sigma }_2^2$ as the variance of the residuals given by (23) with $\underset{¯}{\beta }$ given by its initialized value.

We define a latent binary random vector ${\underset{¯}{Z}\in \{1,2\}}^D$ for which $Z_d=1$ if $N_d$ is drawn from a $\mathcal{N}(0,{\sigma }_1^2)$ distribution and $Z_d=2$ if $N_d$ is drawn from a $\mathcal{N}(\alpha ,{\sigma }_2^2)$ distribution. As such, the joint distribution on $(\underset{¯}{Z},\underset{¯}{Y})$ for parameters $\theta =(\underset{¯}{\eta },\underset{¯}{\sigma },\underset{¯}{\varphi })$ is given by:

$$p(\underset{¯}{Z},\underset{¯}{Y};\theta )=\sum _{d=1}^D\sum _{i=1}^2{1}_{\{Z_d=i\}}\log [{\varphi }_i\mathcal{N}(Y_d\mid {\eta }_i,{\sigma }_i)]$$ (26)

where $1_{\{A\}}$ equals 1 if event $A$ occurs and 0 otherwise.

For the E step, we define ${\gamma }_{d,i}^{(m)}$ to be the posterior probability at the $m\text{th}$ iteration that the $d\text{th}$ sample is generated from the $i\text{th}$ Gaussian component, given by [66]:

$${\gamma }_{d,i}^{(m)}\triangleq P(Z_d=i\mid Y_d;{\theta }^{(m)})$$ (27)

$$=\frac{{\varphi }_i\mathcal{N}(Y_d\mid {\eta }_i,{\sigma }_i)}{{\varphi }_1\mathcal{N}(Y_d\mid {\eta }_1,{\sigma }_1)+{\varphi }_2\mathcal{N}(Y_d\mid {\eta }_2,{\sigma }_2)}$$ (28)

where

$${\eta }_1=\underset{¯}{{\beta }^T}F$$ (29)

$${\eta }_2=\underset{¯}{{\beta }^T}F+{\alpha }_h$$ (30)

where $h\in 1\dots H$. For the $M$ step, the standard EM parameter updates for the GMM problem at iteration $m+1$, ${\theta }^{(m+1)}=({\eta }^{(m+1)},{\sigma }^{(m+1)},{\varphi }^{(m+1)})$, are given as follows [66]. We first define $n_D^{(m)}$ as $n_D^{(m)}={\sum }_{d=1}^D{\gamma }_{d,i}^{(m)}$ and then

$${\varphi }_i^{(m+1)}=\frac{n_D^{(m)}}{D}$$ (31)

$${\eta }_i^{(m+1)}=\frac{1}{n_D^{(m)}}\sum _{d=1}^D{\gamma }_{d,i}^{(m)}(i=1)Y_d$$ (32)

$${\sigma }_i^{2(m+1)}=\frac{1}{n_D^{(m)}}\sum _{d=1}^D{\gamma }_{d,i}^{(m)}{\left(Y_d-{\eta }_i^{(m+1)}\right)}^{{}^{{}^2}}$$ (33)

We used (32) to find the regression coefficients by virtue of (29). We performed 100 iterations of the EM algorithm (for each $h\in 1,\dots H$) and found the optimal offset, $\alpha \ast$ to be 0.8. We performed maximum likelihood classification [67] to assign each subject to either ‘Class 1’ or ‘Class 2,’ where Class 2 included data that lies far above the regression line. Subject $p$ was assigned to Class 2 if:

$$\frac{\mathcal{N}(Y_d\mid {\eta }_1,{\sigma }_1)}{\mathcal{N}(Y_d\mid {\eta }_2,{\sigma }_2)}\le 1$$ (34)

We performed clustering to isolate a subset of Class 2 with features that most closely resembled healthy controls. We refer to this subset of Class 2 as the ‘anomalous class.’ In total, this anomalous class involved subjects which a) lie far above the regression line (i.e. high symptom severity with respect to regression predictions from slow wave features) and b) had slow wave features that were indistinguishable from controls. We used two different clustering algorithms: i) DBSCAN [68], [69] with parameters: epsilon=0.2, minimum samples=2 and ii) K-Means [70] with K=3, to cluster the transformed features ${\underset{¯}{\beta }}^{T}F$ within Class 2. The DBSCAN clustering algorithm was not found to be sensitive to small perturbations in epsilon. As such, implementation of a more sophisticated algorithm, such as HDBSCAN [71], was not explored at this time but could be a source of future work to boost robustness. As shown in Fig 5, both algorithms isolated the same group of seven data points which met criteria i) and ii). In addition, these seven data are the only data whose linear combination of slow wave features fit within a range of the controls’ linear combination of slow wave features. Thus, we labeled these seven subjects as “anomalous” and hypothesize that their prominent disease etiology is not gastric myoelectric dysfunction. Lastly, we performed a final linear least squares regression on the complete dataset labeled as ‘non-anomalous’ (post anomaly-exclusion) to adjust the regression coefficients and compute r values.

Fig. 5.

DBSCAN (A) and K-Means (B) clustering of linear transformation of features. K-Means centers plotted in magenta (diamonds). Both clustering algorithms identify the same anomalous class (orange). The upper limit of transformed features within the control group (outlier excluded) is plotted as a dashed line to show that all anomalous class features fit within a boundary established by controls.

III. Results

Figure 6 shows the result of the robust regression and identification of the anomaly class. We show, in orange, the patients for which we hypothesize that spatial gastric slow wave abnormalities are not a prominent etiology of their disease. These patients exhibit slow wave patterns that closely resemble healthy controls, as is seen with feature overlap, but exhibit high symptom scores, which is not a characteristic of controls. We measured an r value of 0.8 from regression on only non-anomalous data.

Fig. 6.

Identification of anomalous class (orange) for which we hypothesize that the underlying GP/FD etiology was not spatial gastric slow wave abnormalities. Regression with non-anomalous data (blue). We show 95% confidence interval (shaded) and report r=0.8. Outlier in the control group is shown in red but was not used in regression.

Figure 7 shows the symptom prognosis of five patients after treatment. Two patients, shown in triangle markers, had no improvement, and three patients, shown in hexagon markers, did see improvements. Interventions included treatments of the pylorus and prokinetic medications. The three patients that experienced symptom improvement lie farther away from the regression line than the patients that did not improve. A natural hypothesis is that these treatments helped mitigate other etiologies responsible for the patients’ GP and FD. However, for patients that lie directly on the regression line, it is hypothesized that their symptoms are a direct cause of spatial gastric slow wave abnormalities and as such treating other etiologies may not result in improvement of symptoms.

Fig. 7.

Symptom progression after pyloric and prokinetic interventions (n=5). These interventions did not directly target spatial gastric slow wave abnormalities. Three patients reported improvement of symptoms (marked with a hexagon), and two patients saw no improvement (marked with a triangle). Patients who did not improve after treatment were in close proximity to the regrssion line, whereas patients who did improve were far above the regression line, with post-treatment scores lying closer to the regression line.

Figure 8 compares methods to previous work that pioneered linear regression of spatial gastric slow wave features with patient symptom scores [35]. These previous findings utilize raw direction values, where wave directions are calculated analogously to those described in (2), have an inclusion criteria given by (3) and are translated by the patient-specific angle to the duodenum, derived from the CT. ‘Normal’ wave directions are defined to be those which lie with in $\pm \pi ∕3$ of the angle to the duodenum., and direction values are labeled as ‘abnormal’ otherwise. To create a comparison plot, we integrated the probability distributions found in Section II-D on the closed interval $\pm \pi ∕3$, and defined this as the percentage of ‘normal’ slow wave activity. Since the integral of a probability density function over its entire support is equal to one, this integration technique captures a percentage of ‘normal’ and ‘abnormal,’ and yields very similar results to [35, Fig. 4]. Fig. 8A shows the linear least squares (LLSQ) regression (blue line) with the shaded 95% confidence interval. Data closely followed trends from [35]. We report a r value of 0.56 and previous findings report r=0.57 [35, Fig. 4]. This firstly validates our approach of probability density estimation because it shows that probability distributions are able to capture signatures of the histogram well. Secondly, this provides a good basis to make comparisons between methods in [35] and the design of our new methods presented here.

Fig. 8.

Linear least squares (LLSQ) regression of percentage of slow waves propagating in an abnormal direction with mean GCSI symptom score (shown as blue line in panels A-D with 95% confidence interval shaded in lighter blue). Percentage of slow waves exhibiting abnormal propagation was found by integrating each probability distribution on the interval $[-\pi ∕3,\pi ∕3]$ (same interval as in [35]). Panels A-D serve as a comparison to results published in [35]. Panels A-C are computed from a 90 minute recording, with approx. 30 minutes preprandial and approx. 60 minutes postprandial. (A) We calculated an r value of 0.56 if all 32 subjects were included (consistent with [35]) in the regression and an r value of 0.59 if control outlier was excluded. (B) Pink data show GP and FD subjects whose percentage of slow waves overlap with the range of controls (excluding the outlier). (C) Orange data show the seven subjects labeled as “anomalous” from the robust regression framework presented here. It is hypothesized that these subjects have GP/FD arising from an etiology distinct from gastric myoelectric dysfunction. (D) Percentage of slow waves propagating in an abnormal direction was calculated for only postprandial recording and subsequent regression was performed. Analogous to panel B, pink data show GP and FD subjects whose percentage of slow waves overlap with the range of controls (excluding the outlier).

Figure 8 B shows, in pink, the GP and FD subjects with percentages of abnormal slow waves that coincide within the range of healthy controls, excluding the control outlier. It is unlikely that a cluster of patients which have normally functioning slow waves but high symptom scores could be identified with these types of methods because the control group spans a large portion of the total data. For example, data marked as pink (within range of controls) are directly adjacent to data marked in purple (out of range of controls) and as such, there is no clear separation between a healthy and pathological percentage of abnormal slow waves.

Figure 8 C replicates the approach shown in Panel A, but we have marked the seven subjects that were flagged as ‘anomalous’ in the methods presented in this work. Some of these subjects lie either close to or directly on the regression line and thus do not appear anomalous at all in this previous approach. This shows that using the previous methodology, this group of patients for which it is hypothesized that spatial slow wave abnormalities are not a likely cause of disease would be missed.

In Fig. 8 D, we replicate panel B with only the postprandial period of the recordings. This was done to show that the paradigm shift within our methodology (i.e. looking at signatures of the full probability distribution and performing robust regression to account for different disease etiologies) was the key driver of our novel findings presented in this work. While we assert that performing HR-EGG studies during an immediate postprandial period is a best practice moving forward (with justification in II-A), isolating the postprandial portion of the recording alone was not the mechanism behind our results.

We show in Fig. 8, in red, the outlier within the control group that we identified after estimating probability distributions. Interestingly, this outlier does not appear to separate from the remainder of the control subjects in Fig. 8 panels A-D. We show the probability density function for this outlier subject in Fig. 4. This density lacks a cohesive single mode centered near zero and more closely resembles a probability distribution that would be expected from chaotic propagation. However, summing energy across the multiple modes within the range of $\pm \pi ∕3$, gives a false indication of normalcy since this total energy is closer to a range of the other controls. As such, investigating signatures and modalities of the entire probability distribution allowed us to recognize an outlier that would have otherwise gone undetected.

IV. Discussion

In this work, we were able to show a strong correlation (r=0.8) between features relating to spatial gastric slow wave abnormalities and standardized patient symptom scores, improving upon previous published findings’ r value by 0.23 [35]. We were also able to isolate a group of patients for which we hypothesize spatial gastric slow wave abnormalities are not likely a predominant etiology of their disease. We have shown results that allow us to hypothesize that treating a patient’s GP or FD with a treatment regimen that does not directly target gastric myoelectric dysfunction may only produce results in the case where other underlying etiologies are present, and patients who lie on the regression line (i.e. we hypothesize that their symptoms can be explained entirely based on spatial gastric slow wave abnormalities) will see limited success in such treatments. All patients in the anomalous group were diagnosed with GP. We hypothesize this was coincidental due to the larger population of GP patients in the cohort and further the 4:1 GP:FD ratio of patients that were evaluated after treatment. However, further investigation is warranted to explore this hypothesis, and its possible refutation could lead to findings of new distinction between GP and FD.

One limitation to this study is the size of the patient cohort evaluated after treatment. These preliminary results motivate the continuation of this work with a larger cohort of human subjects. In addition, this study was limited in that the evaluation of patients with treatment lacked a placebo group and double-blind control. However, it should be noted that the treatments given to the patients were not themselves experimental and not specific to diagnostic research [15], [16]. Another limitation of this work is the imbalance in the number of GP/FD patients, as well as control subjects. Though GP and FD subgroups were intentionally analyzed as one group, with justification provided earlier in this manuscript, this imbalance can nonetheless be mitigated in future work to strengthen the conclusions drawn.

Previous studies have used a patient CT to guide electrode array placement over the antrum of the stomach and determine the angle towards the duodenum [35], [38]. The advent of portable ultrasound devices and their usage to locate the antrum [72], [51] suggest placement of electrodes to cover the stomach need not require a CT. The methods presented here do not make use of the exact angle to the duodenum, from CT reconstructed geometry, but rather are robust to a barycenter of duodenum angles across several human subjects. This lays the groundwork for future studies to investigate the dependence of EGG-analysis methods on imaging. In addition, even if the electrode placement is less precise with the ultrasound localization method, we hypothesize, based on what has been shown here, that designing analysis methods to find patterns within the overall shape and modalities of probability distributions may provide robustness to slight deviations from optimal electrode placement, as these perturbations would not likely change distribution modalities. Since the HR-EGG device is non-invasive, eliminating the need for a CT will eliminate a significant barrier in bringing this technology to the clinic as a widely-deployable screening tool.

By virtue of using histograms and performing density estimation, we made the implicit assumption in this work that wave direction values can be treated as independent, identically distributed samples collapsed across temporal and spatial axes (see Section II-B). A subject of future work would be to solve for conditional probabilities in both the temporal and spatial axes using Markov modeling techniques.

It is well documented that the vulnerabilities of the EM algorithm arise from the non-convexity of the log likelihood surface and thus the presence of local minima/ maxima [73], [74]. Two of these vulnerabilities are: the need for appropriate parameter initialization as well as a proper stopping criterion to limit the number of extra iterations performed once convergence has been reached [73], [74], [75]. Failure to implement these regulations can result in parameters converging to one of their most extreme values. As such, it is typically standard to apply targeted parameter initializations [74]. In our work presented here, we initialized parameters with insights specific to the problem in the GMM implementation of the EM algorithm. However, when using the EM algorithm for the vMM problem, we initialized distribution weights randomly and we observed that convergence did not depend on proper initialization of weights, as was expected since this problem is convex. In addition, published studies have established precedent for enforcing a ‘stopping criterion’ for the EM algorithm [76]. As such, in both implementations of the EM algorithm (vMM and GMM problems), we chose a maximum number of parameter updates that allowed the algorithm to achieve convergence. More robust parameter initializations have been proposed [74], as well as competitive EM (CEM) [75] and other ‘split and merge’ algorithms [77], which offer an approach to address both the challenges of proper parameter initialization and developing a stopping criterion. A subject of future work could entail implementing one of these algorithms in lieu of the standard EM algorithm presented here.

Through the methods presented in this work, we identified an outlier within the control group. As discussed previously, identification of this outlier was only made possible once we estimated the full probability distribution of wave directions. The outlier, which deviates from from the ‘normal’ slow wave propagation pattern typically seen in controls, may be a result of recording noise or artifacts. However, this finding could instead suggest discovery of an asymptomatic class of patients that have spatial slow wave abnormalities. This suggestion is not meant to imply certainty of any kind, but rather to motivate a future inquiry in the GP and FD research community. Future work could entail enrolling a large cohort of healthy controls and performing HR-EGG with analysis methods analogous to those presented here to assess probability distributions of wave directions. Given that the cerebral cortex exerts influence over the stomach [78] and vice versa [79], it may also be sensible to perform simultaneous brain and gastric recordings to associate the two modalities, for instance from wave propagation [80], [81] or phase coupling [79], [82] perspectives.

V. Implications of This Work

Although recent findings suggest that gastric neuromodulation is able to entrain the slow wave, there is not a reliable method to determine which GP and FD patients may benefit from such treatment. Furthermore, we may be able to identify patients for which existing conservative treatments would yield symptom improvement, and for which patients may benefit from neuromodulation. Finally, we may be able to provide an explanation as to why certain treatments either only partially alleviate symptoms or fail to do so completely.