### 2.1 Mechanism of the electrically propelled capsule

The shape of the capsule and the mechanism of the electrical stimulus capsule are illustrated in Figure 1, which shows how the smooth muscle is contracted by the electrical stimuli that is applied by a pair of electrodes and the capsule is propelled along the lumen.

In order to move the capsule, the contraction force of the small intestine causes it to move the capsule, as shown in Figure 2. It is assumed that the capsule is moving to the right and small intestine is contracted by electrical stimuli. Since the capsule is moving toward the right, the left part of the small intestine is stimulated for a long time and causes large contraction force. The small intestine is composed of smooth muscle and it requires long time to fully contract compared with the skeletal muscle. The middle part of the small intestine is just entered stimulus region and contraction force is lower than that of the left part. Since the contraction force is perpendicular to the direction of movement, the exterior shape of the capsule determines the moving speed. In addition, the friction is proportional to the contraction force and the drag is dependent on the viscosity of the small intestine and velocity of the capsule. The relationship between the three forces and total moving force can be described as

\overrightarrow{{F}_{t}}=\overrightarrow{{F}_{m}}-\overrightarrow{{F}_{f}}-\overrightarrow{{F}_{d}}

(1)

where {\overrightarrow{F}}_{t} is the total moving force, {\overrightarrow{F}}_{m} is the moving force {\overrightarrow{F}}_{f} is friction, and {\overrightarrow{F}}_{d} is the drag force.

Figure 3 illustrates how the intestine is extended like a thin cylindrical vessel when the capsule is in the small intestine and how it applies strain to the small intestine. Since the small intestine has viscoelastic properties, the internal pressure slowly decreases as time passes. The stress value of the small intestine was measured by Baek *et al*. [27] and the stress was found to be

{\sigma}_{c}\left(t\right)={\epsilon}_{c}\left(0.7{e}^{-\frac{t}{18}}+0.63{e}^{-\frac{t}{1.6}}+0.92\right)

(2)

{\epsilon}_{c}=\frac{f\left(x\right)}{{d}_{0}-1},\phantom{\rule{1em}{0ex}}{\epsilon}_{c}>0

(3)

where *ε*
_{c} is strain, *t* is time, *f* (*x*) is the exterior shape of the capsule, and *d*
_{0} is the initial diameter of the small intestine. Since the capsule extends the small intestine, the strain depends on the exterior shape of the capsule. The exterior shape of the capsule will be discussed later.

The small intestine has similar shape of the thin cylindrical vessel and internal pressure of the thin cylin-drical vessel can be express as

{P}_{i}=\frac{2h}{d}{\sigma}_{c}

(4)

where *d* is diameter of the small intestine and *h* is the thickness of the small intestine. Therefore, internal pressure of the small intestine can be express as formula 5 and the pressure is changed by time, diameter of the capsule, and the exterior shape of the capsule.

{P}_{i}=\frac{2h}{d}\phantom{\rule{0.3em}{0ex}}\left(\frac{f\left(x\right)}{{d}_{0}}-1\right)\left(7{e}^{-\frac{t}{18}}+6.3{e}^{-\frac{t}{1.6}}+5\right)

(5)

When an electrical stimulus is applied to the small intestine, a high contraction force is generated, de-pending on the time and electrical stimulus parameters. In previous experiments, the maximum contraction force and rising time constant depended on the electrical stimuli were measured using square electrodes (5 × 6 mm) with a catheter and Figure 4 summarizes these results [28, 29]. Detail experimental processes are included in the additional file 1.

Figure 4 (a) depicts the electrical stimulus parameters, Figure 4 (b) shows the maximum contraction pressure depending on various electrical stimulus parameters, and Figure 4 (c) shows transient response of contraction. In order to reduce the number of experiments, the duration was fixed at 5 ms that is twice of chronaxie of the smooth muscle. The experimental results shows that the maximum contraction force increased nonlinearly when the voltage was increased.

From the experimental results, the maximum contraction pressure is modeled by using logistic regression and it is shown as

{P}_{m}=\frac{0.24{f}_{s}+1.18}{1+{\left(\frac{{A}_{s}}{4.97}\right)}^{-3.72}}

(6)

where *P*
_{m} is the maximum contraction pressure, *f*
_{s} is the stimulus frequency, and *A*
_{s} is the stimulus amplitude. Standard coefficient error is lower than 0.2 and the P-value showed less than the significance level (< 0.01).

After decision of the maximum contraction pressure, transient response of the contraction is modeled as simple first-order ordinary differential equation and it is shown as

\frac{d{P}_{s}\left(t\right)}{dt}+\frac{1}{{\tau}_{s}}{P}_{s}\left(t\right)=\frac{{P}_{m}}{{\tau}_{s}},\phantom{\rule{0.3em}{0ex}}{P}_{s}\left(0\right)=0

(7)

where *P*
_{
s
} (*t*) is the contract pressure, *τ*
_{s} is the rising time constant, *P*
_{m} is the maximum contraction force.

The value of the rising time (*τ*
_{s}), which is defined as a time to reach 63.2% of its maximum contraction pressure, was measured from for three different stimulus parameters (10, 20, and 40 Hz @ 6 V and 5 ms) with four different samples from two different swains (N = 20). The average (SD) rising time constant value was measured as 17.3 ± 8.3 seconds.

It is assumed as the pressure is locally distributed in Gaussian form because the pressure was highly generated from the small electrode, and the size of the balloon catheter was large enough to measure the averaged pressure data. The Gaussian equation is shown as

G\left(x\right)=1.4{e}^{-\frac{{\left(x\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{x}_{0}\right)}^{2}}{2\omega}}

(8)

where *x* is the *x-axis, x*
_{
0
} is the center placement of the electrode, and *ω* is the variance. The variance value was empirically set as 0.22.

Figure 5 (a) illustrates the importance of the exterior shape and it shows that the streamlined shape is better than the cylindrical shape. Figure 5 (b) summarizes the relationship between the contraction force and the shape of the exterior of the capsule.

The exterior shape of the capsule is modeled using the exponential function as

f\left(x\right)=\left\{\begin{array}{cc}\hfill \left(R-r\right)\left(1-{e}^{-\alpha x}\right)+r,\hfill & \hfill 0\le x<\frac{L}{2}\hfill \\ \hfill \left(R-r\right)\left(1-{e}^{\alpha \left(x-L\right)}\right)+r\hfill & \hfill \frac{L}{2}\le x\le L\hfill \end{array}\right.

(9)

where *r* and *R* are the initial and maximum radii, *α* is the dimensionless constant that determines the slope of the capsule, and *L* is the length of the capsule. Figure 5 (b) depicts the relationship between the contraction force and moving force. The moving pressure can be described as

{P}_{m}=\left\{\begin{array}{c}{P}_{1}cos\left(\theta \left(x\right)\right),0\le x<\frac{L}{2}\\ -{P}_{1}cos\left(\theta \left(x\right)\right),\frac{L}{2}\le x\le L\end{array}\right.

(10)

{P}_{1}=\left({P}_{s}+{P}_{i}\right)cos\left(\frac{\pi}{2}-\theta \left(x\right)\right)

(11)

\theta \left(x\right)=\frac{\pi}{2}-{tan}^{-1}\left({f}^{\prime}\left(x\right)\right)

(12)

Where *P*
_{
s
} and *P*
_{
i
} are the contraction pressure by the electrical stimulus and internal pressure from extension by the capsule, and *θ*(*x*) is the slope of exterior capsule. The force can be calculated from the distribution of pressure to force when integrated over the exterior area of the capsule. Therefore, the moving force can summarized as formula 13.

{F}_{m}=2\pi \phantom{\rule{0.3em}{0ex}}{\int}_{0}^{L}{P}_{m}f\left(x\right)\sqrt{1+{f}^{\prime}{\left(x\right)}^{2}}dx

(13)

In addition, the friction can be described similarly

{F}_{f}=2\pi \mu {\int}_{0}^{L}{P}_{f}f\left(x\right)\sqrt{1+{f}^{\prime}{\left(x\right)}^{2}}dx

(14)

{P}_{f}={P}_{1}sin\left(\theta \right)

(15)

Where μ is the frictional coefficient and it is set at 0.1 based on Baek's experiments [27]. The friction increases with increasing contraction force and grooves at the capsule surface.

Another force is drag, which is highly influenced by the viscosity properties of the small intestine. Since the moving speed of the capsule is very slow, it satisfies Stokes' fluid law. The Stokes' drag is

{F}_{d}=b\phantom{\rule{0.3em}{0ex}}v

(16)

where *b* is the drag coefficient and *v* is the velocity of the capsule. In order to calculate the drag coefficient of various shapes of the capsule, a computational fluid dynamics program (CFD, ANSYS) is used and the Navier-Stokes equation is calculated. For the simulation, the small intestine is assumed to be a non-compressible viscous liquid [30], laminar flow, Newtonian fluid, steady state, and no slip condition. Figure 6 (a) shows 9512 generated meshes and Figure 6 (b) shows that the drag depends on the varying slope and initial radius when the velocity of moving capsule is 0.5 mm/s. When the moving speed is slow, the exterior shape of the capsule does not highly affect.

Figure 7 depicts a block diagram for the calculation of the above equations. When the electrical stimulus is applied, the drag, friction, and moving force are calculated using the velocity and position information of the capsule. The friction, drag, and moving force were recalculated every 0.001 second to prevent a diversion problem.