A mathematical model of bowel sound generation

The stethoscope was invented in 1816 by Rene Laennec. However, for millennia those trying to understand the workings of the body have listened to its sounds (Markel, 2006). “On Breaths” is a Hippocratic text which describes a range of body sounds including breathing, but also mentions “eructations, flatuses, borborygmus and crepitus ventris” more commonly known as burps, farts, stomach rumblings, and groanings (Sarton, 1958).

The most commonly studied sound-generating organs are still the lungs and the heart. Mathematical models have been developed to describe the generation of respiratory sounds (Bureev et al., 2016; Earis, 1992). Models have also been generated for both the first and second heart sounds (Hearn et al., 1979), which result from vibration of the atrioventricular valves and aortic valve, respectively. There has also been significant interest in modelling speech, which is generated when fluid passes through the vocal folds causing them to vibrate. The vibration can be modelled as single and multiple spring-mass systems (Honda and Zhang, 2018; Xue et al., 2010; Zanartu et al., 2007). Investigation of gastrointestinal tract sounds has been more limited. This is probably because the irregularity in bowel sounds (BS), compared to cardiovascular sounds, has made systematic analyses more difficult.

The first scientific study of bowel auscultation was by Cannon (Cannon, 1905). He found that the gut produces rhythmic noises plus continuous random noises varying in location and intensity. More recent studies have demonstrated that BS have distinctive characteristics (Dimoulas et al., 1999) and hence probably contain much meaningful information. The first analytical studies were focused on simply extracting BS. Researchers applied signal enhancement methods to eliminate noise interference, including an adaptive filter method and wavelet transformation (Allwood et al., 2018; Hadjileontiadis et al., 2000; Mansy and Sandler, 1997). Subsequently, researchers documented different types of BS. These were classified through algorithms including a Naive Bayes model (Ulusar, 2014) and a neural network (Dimoulas et al., 2008). Differentiation between types was based on typical acoustic features including values for the spectral centroid, sub-band normalised energy, and envelope chest factor, with good recognition accuracy possible.

To improve understanding of BS and the gastrointestinal system, we developed a mathematical model of BS generation. The model was based on knowledge of the anatomy and physiology of the gastrointestinal tract and analysis of recorded BS. The BS mathematical model was constructed with four bowel activity parameters: the individual wave component (IWC), the pressure index (PI), the component quantity (CQ), and the component interval time (CIT), which has already shown some clinical potential in irritable bowel syndrome diagnosis (Du et al., 2018). Adjusting these four parameters in the model allowed us to simulate all the different types of BS documented in previous studies. A long simulated BS was also provided to demonstrate the model's accuracy.

Bowel sounds were recorded from ten participants using a modified stethoscope placed on the right lower quadrant of the abdomen, which is the most active quadrant and receives minimum sound interference from the heart and lungs (Cannon, 1905). To measure standardised BS, the participants were required to fast overnight and skip breakfast. They sat still in a quiet room for 2 h whilst BS were recorded with a sampling frequency of 44.1 kHz. A 150 Hz high pass filter was used for BS prepossessing to minimize the ambient noise and heartbeat. Two medical doctors systematically evaluated a library of extracted BS as sounds to assess if they were sounds typically regarded as GI in origin. The library was established from the participants' recordings and included 18 recordings covering all the different BS types and eight irrelevant and environmental noises. The LOGIQ E9 XDclear2.0 ultrasound image system was used to record bowel contractions to demonstrate the physiology fundamental to the model. This study was approved by the University of Western Australia's Ethics Committee (study No. RA/4/1/8893).

Physiologists believe BS are generated by peristalsis. This process involves a radially symmetrical contraction and relaxation of muscles that propagates in a wave down the gastrointestinal tract (Hall, 2015). To simulate peristalsis, four different components were proposed for the BS model, comprising the IWC, PI, CQ, and CIT. The IWC is related to the bowel dimensions, PI is associated with the pressure inside the bowel, while the CQ and CIT are linked to the muscle contraction and relaxation pattern.

Under this model, the fundamental component of the BS is the IWC, which is generated when the gut contracts and compresses the fluid (gas or liquid) through its lumen (Dimoulas et al., 2006; Ladabaum and Hasler, 1999). The muscle contraction compresses fluid into a small section of the gut. This process is illustrated in Figs. 1(a), 1(c), and 1(d). Images from an ultrasound recording [Fig. 1(b)] cover the change from the relaxed state through to the contraction and then back to the relaxed state. The wall of the gut vibrates due to fluid pressure changes. This can be represented by a spring-mass-damping system attached to the wall, as shown in Figs. 1(a) and 1(e). Equation (1) represents the equilibrium of the forces in the system. The same effect is found at gastrointestinal valves and is similar to the vocal fold vibration that generates speech (Zanartu et al., 2007). Based on this, the resonant frequency can be calculated by Eq. (3) (Morse, 1936),

where x is the motion of the wall, p is the fluid pressure, M and K are the mass and stiffness of the wall, respectively. C is the damping coefficient, which is related to the energy consumption. Therefore, IWC and its resonant frequency are given by

where f_iwc is the resonant frequency of the vibration of the bowel wall shown in Eq. (3). A_iwc is the envelope of the IWC and consists of an increasing pressure at the beginning and gradual decay due to damping. The increase and decrease of the BS can be expressed in Eq. (4) as a combination of an exponential term and a proportionality term,

where t is the time stamp, E is the envelope index that links to the pressure, and b is the shape index controlling how narrow the envelope is. p_m is the PI.

CQ and CIT are related to the contraction-relaxation pattern of the muscle and the quantity of bowel segments. The CQ, which represents the amount of IWCs within one BS, is associated with the repeat times of contraction and relaxation, or the segments quantity. The CIT is the time difference between adjacent IWC, which is linked to the contraction cycle and segments distance. In this model, N represents the CQ, and T_i is the ith CIT such that the time stamp of ith IWC can be represented by Eq. (5),

Finally we accommodate the fact that whilst a BS is commonly a single IWC it can be a superimposition of multiple IWC depending on the causal gut activity. Hence, by combining all the IWC with substituting Eq. (5) into Eq. (2), a complete BS model is given in Eq. (6), which includes the bowel configuration, pressure and contraction-relaxation pattern. n(t) represents the noise,

Four typical types of BS documented by previous researchers were frequently observed in our recordings. They were also recognised as typical BS by two clinical doctors with 100% accuracy through the sound library test. Four types of BS are shown in Fig. 2 with their waveform and spectrogram, including the single burst (SB), multiple bursts (MB), continuous random sound (CRS), and harmonic sound (HS). SB is the most frequent type of BS in the recordings, which is an isolated IWC probably caused by one single contraction of the bowel (Dimoulas et al., 2006; Ladabaum and Hasler, 1999). MB sounds can be described as repetitive IWC with an interval time smaller than 100 ms. Each IWC in the MB looks similar in the spectrogram with slight differences in bandwidth and amplitude (Dimoulas et al., 2011; Ulusar, 2014). CRS are usually continuous over long periods without any significant silent gaps, have everything clustered together, and little rhythm or pattern (Cannon, 1905; Dimoulas et al., 2011). HS are whistling-like sounds and have one to a dozen frequency components in the spectrogram.

Statistical characterizations of the key properties of these four main types were derived from the 1200 min of recordings (see Table 1). HS was the least common type, whilst SB sounds occurred most frequently. SB sounds had the shortest duration. As for the spectral flatness, which describes how tonal a sound is (Herre et al., 2002), all four types overlapped, but it was slightly larger in MB. The spectral centroid did not show any significant differences between the four types. Sounds in a transient state from one type to another were observed [see Fig. 4(a) as an example]. The existence of the transient type implies that there are unclear boundaries for the typical types of BS.

The spectral centroid (which is closely related to the fundamental frequency) showed little variation between the four BS types (see Table 1). This is probably because this feature is determined by the configuration of the gut, which is constant within one person. The spectral centroid can be estimated by comparing to the vocal fold based on Eq. (3). The gut configuration (Ferrua and Singh, 2011) is similar to the vocal fold configuration (Zanartu et al., 2007). The tissue density is also similar in these two cases. As a result, the fundamental frequency range of the BS is comparable to the typical voice frequency range, which is from 100 to 1000 Hz (Bishop and Keating, 2012). The statistical analysis from our BS recording also supported this (see Table 1).

Based on Eqs. (2) and (4), an example of IWC was simulated and presented in Fig. 1(f), where b = 5, E = 1, and p_m = 1.0. The red dashed line is the simulated A_iwc, which increases to a peak at 3.5 ms, and decreases to zero at 24.0 ms. The solid red line represents the simulated IWC, and the black dashed line represents the detected SB. They showed good agreement (correlation coefficient = 95%).

Different types of BS can be described by this mathematical model, which has CIT as a key component. This latter point is demonstrated by the link between CIT and BS types when IWC is unchanged. The CIT ratio, defined as the CIT over the duration of the IWC, was used to indicate BS type transformation as it covers the IWC duration. The spectral flatness was used to distinguish the CRS, HS, and MB due to its ability to show how tonal the signal was when IWC is unchanged. We created BS covering all types by merely changing the CIT to further demonstrate its importance. These BS were created following Eq. (6), where N = 20, n(t) generated from Gaussian distribution. IWC in Fig. 1(f) was selected and its parameters included T_d = 20 ms, b = 5, E = 1, and p_m = 0.96ⁱ(σ + 0.5) where σ is uniformly sampled from 0 to 1. The CITs T_i = r_cit,iT_d(1 + 2(σ – 0.5)) where the CIT ratio r_cit,i increases from 0 to 5.

The flatness of created BS variation against CIT ratio is shown in Fig. 3. Five 200 ms demo BS presented in Figs. 3(a) to 3(e) were created under corresponding CIT ratios (indicated by a red arrow). These BS covered HS, CRS, and MB and their transient types. Examination of the five demo BS recordings in Figs. 3(a)–3(e) and comparison to the typical types in Fig. 2, demonstrated the lack of strict boundaries between BS. The approximate boundaries between types are given by the vertical dashed lines. In general, the spectral flatness increased with increasing CIT ratio with small fluctuations caused by randomness in T_i. With increasing CIT ratio, BS started as an HS similar to a sine wave with an envelope and gradually turned into CRS with more frequency components when the CIT ratio was larger than 0.2. When the CIT ratio was larger than 0.8, IWCs started to separate and turn into an MB sound type and then gradually evolved into an SB type. Using identification criteria that set the minimum time gap between two SBs as 100 ms, the maximum CIT ratio of MB sounds was approximately $0.1/Tiwc$⁠, which is 5 for selected IWC and from 1.7 to 5 for general IWC. The details of the CIT ratio range of each BS are listed in Table 1. The CIT ratio ranges could relate to the quantities of different types. For example, the SB had the most extensive range and quantity, and HS had the smallest range and quantity. The relationship between CIT and different types of BS also explained the existence of combination BS under the transient state.

The importance of the CIT parameter is also demonstrated by its possible clinical application. Preliminary study results demonstrate that this BS feature could prove useful in the diagnosis of a gastrointestinal disorder, irritable bowel syndrome (IBS) (Du et al., 2018). This is unsurprising, given IBS is associated with atypical patterns of gut contractions and motility (Chey et al., 2001; Vassallo et al., 1992).

Finally, to show the accuracy of the mathematical model, we simulated a 2500 ms long BS consisting of MS, CRS, and HS types (see Fig. 4). The overall comparisons between the modelled BS (in red) and detected or real BS (in black) are presented in Figs. 4(a), 4(b). Details are provided in (e), (f), and (g). The parameters used were f_iwc = 400 Hz, envelope parameters E = 0.0495 and b = 11.0, CQ was 305, CIT(s) were T ∈ [0.002935 − 0.102] and PI P_m ∈ [0.04878 − 1.0].

Overall, the simulated and detected BS were in good agreement (see Fig. 4) especially for the MB, CRS, and HS sections. Minor asynchronicity is observed in Figs. 4(f) and 4(g) due to frequencies variation resulting from a small change in gut dimensions, which was ignored for simulation simplification. Figure 4(d) presents the change of CIT ratio variation over time. For the first 1.5 s, the CIT ratio was relatively high with two CIT ratio dips around 0.75 at the beginning and around 0.56 s, where the CRS slightly came in. From 1.5 to 2.5 s, the CIT ratio started at 0.16 and gradually increased to 0.2 at points where the CRS slightly occurred [see the zoom-in part of 4(d)]. The CRS also started to show up at the end of this BS, where the CIT ratio was regularly higher than 0.2. The change of CIT ratio with transitions between different BS types matched that observed in Fig. 3. It is apparent that the BS mathematical model can accurately simulate BS, and that there is a clear relationship between CIT and different types of BS.

We developed an accurate mathematical model of bowel sounds with four parameters linked to bowel activity. The parameters comprised the individual wave component, which relates to the gut segment configuration; the pressure index, which demonstrates the pressure inside the gut; the component quantity, which represents the contraction quantity or gut segment number; and the component interval time, which links to the gut muscle contraction and relaxation pattern. Essentially, the vibration of the gut wall during peristalsis and the resultant BS generation can be represented by a spring-mass-damping system. This is similar to the vibration of vocal folds that generates speech. The four typical types of bowel sounds can be constructed using our model by varying the parameter values, with the component interval time key to determining the sound type. We were able to use our model to simulate an extended and variable BS with good agreement to real sounds.

The study was funded by the McCusker Charitable Foundation, who played no role in the study design or analysis. We are also grateful to Dr. Andrisha-Jade Inderjeeth for bowel sound verification.