Progress in Mathematical Modeling of Gastrointestinal Slow Wave Abnormalities

Abstract

Gastrointestinal (GI) motility is regulated in part by electrophysiological events called slow waves, which are generated by the interstitial cells of Cajal (ICC). Slow waves propagate by a process of "entrainment," which occurs over a decreasing gradient of intrinsic frequencies in the antegrade direction across much of the GI tract. Abnormal initiation and conduction of slow waves have been demonstrated in, and linked to, a number of GI motility disorders. A range of mathematical models have been developed to study abnormal slow waves and applied to propose novel methods for non-invasive detection and therapy. This review provides a general outline of GI slow wave abnormalities and their recent classification using multi-electrode (high-resolution) mapping methods, with a particular emphasis on the spatial patterns of these abnormal activities. The recently-developed mathematical models are introduced in order of their biophysical scale from cellular to whole-organ levels. The modeling techniques, main findings from the simulations, and potential future directions arising from notable studies are discussed.

Keywords: Electrophysiology; GI; arrhythmias; multi-scale modeling; slow wave.

Figure 1

Examples of high-resolution mapping of in-vivo gastric slow waves. (i) An array of 16 × 16 electrodes (brown patch) were placed on the serosal surface of the stomach. (ii) Activation times of slow waves were identified and reconstructed into activation maps with red representing early activation and blue presenting late activation. (iii) Example slow wave recordings from six electrodes are shown in each case. (A) Normal antegrade propagation pattern of gastric slow wave activation (Angeli et al., 2015). (B) An ectopic activation (star) in the proximal stomach. (C) Simultaneous ectopic activation, conduction block and collision of slow waves in the gastric corpus. Adapted from (Angeli et al., 2015).

Figure 2

A finite state machine cell model of gastric interstitial cells of Cajal. (A) The model consists of an active state and a passive state. ST indicates if time has passed the startTime, which is set as a parameter and which determines initial excitation when there is no threshold voltage. AT indicates if the non-refractory period has been passed and signals transition from passive state to active state. DC identifies if the change in concentration of intracellular Ca²⁺ has returned to quiescent state. VP variable is set to true if there is a voltage which is greater than the threshold of the cell. (B) Simulated gastric slow waves and the associated intracellular calcium. Adapted from (Sathar et al., 2014).

Figure 3

Mathematical models of intestinal slow wave propagation. (A) Entrained slow waves over an intrinsic frequency gradient of 17–14.6 cpm (entrained to 17 cpm) were simulated over a 2D model. (B) A functional rotor was invoked in the middle of the 2D model using a 30 s long prolonged temporary conduction block. The rotor could be sustained with entrained waves propagating in both antegrade and retrograde directions, with an elevated frequency of 21 cpm. Adapted from (Du et al., 2017).

Figure 4

Simulation of whole-organ gastric slow waves. (A) Gastric slow waves originate from a pacemaker region along the greater curvature in the proximal stomach. Simultaneous and multiple wavefronts occur in the stomach, with each propagating wavefront taking up to 60 s to reach to the terminal antrum. (B) The existence of resting membrane potential gradient in the stomach plays a key role to the recovery component in the extracellular signals, when calculated as a difference between membrane potential (V_m) and a spatially invariant term (V_r). Adapted from (Paskaranandavadivel et al., 2017).

Figure 5

Whole-organ gastric slow wave dysrhythmias and electrogastrography (EGG) simulations (Calder et al., 2016). (A) Four instance of gastric slow wave activation (normal, re-entry, conduction block in the antrum and ectopic pacemaker in the proximal stomach). (B) Corresponding EGG simulations are calculated using a forward approach over an anatomically realistic torso, with the EGG potentials normalized (U).