Model

Helicobacter pylori infection affects over half of the population. This seemingly tiny bacterium can cause chronic gastritis, gastric ulcers, and even increase the risk of gastric cancer. Once infected, it often remains dormant in the body for a long time, severely impacting quality of life.

Traditionally, we have relied primarily on combination therapy, primarily antibiotics, which initially achieved eradication rates of approximately 90%. However, with the rise of drug resistance, efficacy has steadily declined. Furthermore, antibiotics indiscriminately damage the intestinal flora, leading many patients to experience side effects such as diarrhea and bloating. Later, probiotics were tried as adjunctive therapy or even as monotherapy, but the results have been less than ideal, with eradication rates often below 30%. The core problem is that probiotics, as large particles, have difficulty penetrating gastric mucus and reaching the deeper layers where H. pylori accumulates. To this end, we designed a highly efficient delivery system. The stomach’s highly acidic environment and thick mucus layer have always been two major obstacles for drugs to reach the site of H. pylori infection. We encapsulated the modified yeast into microspheres using a pH-responsive gel—a gel that remains stable in gastric acid (pH <4) and only dissolves and releases upon reaching the gastric mucosa (pH 6-7). Crucially, we loaded the microspheres with calcium carbonate-based micromotors, which provide propulsion and help the microspheres penetrate the viscous mucus layer and reach the “lesions” where H. pylori accumulates.

Thus, in the dry lab, we aimed to model the drug’s journey from entry into the body and gradual diffusion into the stomach wall, its penetration of the gastric mucus layer under the propulsion of the micromotors, its adhesion to the Helicobacter pylori, activation of the GPCR response pathway, and secretion of AiiA (killer protein), as well as the impact of drug intervention on the spread of H. pylori in the human population. In the following content, we will follow this logic to introduce our drylab content.

Dissolution Kinetics Modeling of pH-Sensitive Microspheres

First, we needed to ensure that our living drug would not lose its activity in the highly acidic environment of gastric fluid. Therefore, we devised a gel coating method. This prevented the Saccharomyces boulardii from dying in gastric fluid. Furthermore, to ensure that it could sense and respond properly deep within the gastric mucus layer, we also needed to ensure that the outer shell would readily degrade in a neutral environment. Therefore, we first needed to simulate the degradation of the entire gel under different pH conditions.

Mathematical Modeling

We model the dissolution of alginate-based drug-loaded microspheres using a first-order kinetic model. The main assumption is that the dissolution rate is proportional to the amount of undissolved material remaining at any given time. Mathematically, this can be expressed as:

dM(t)dt=−k⋅M(t)\begin{equation} \frac{dM(t)}{dt} = -k \cdot M(t) \end{equation}

where:

  • M(t)M(t): mass (or fraction) of undissolved microspheres at time tt
  • kk: dissolution rate constant, which depends on pH

Solving the differential equation by separation of variables gives:

dMM=−k dt∫1M dM=−k∫dtln⁡M=−kt+CM(t)=M0⋅e−kt\begin{align} \frac{dM}{M} &= -k \, dt \\ \int \frac{1}{M} \, dM &= -k \int dt \\ \ln M &= -kt + C \\ M(t) &= M_0 \cdot e^{-kt} \end{align}

Assuming M0=1M_0 = 1 ( 100% undissolved at t=0t=0 ), it further simplifies to:

M(t)=e−kt\begin{equation} M(t) = e^{-kt} \end{equation}

In real scenarios, the rate constant k varies with pH, so we further model:

k=f(pH)\begin{equation} k = f(\mathrm{pH}) \end{equation}

This allows the prediction of drug release profiles at arbitrary pH levels.

Computational Method with Simulation

Workflow:

  1. Collect experimental data of undissolved mass vs. time at different pH values.
  2. Fit each data to the exponential decay model to extract the rate constant kk.
  3. Fit the kk pH relationship using a suitable nonlinear function (see Figure 1).
  4. Use the fitted k(pH)k(pH) to simulate release curves under any target pH. Here we consider k as a function of ph.

Test data:

t (min)pH = 1.5pH = 3.0pH = 5.0pH = 7.0
01.001.001.001.00
50.950.880.400.10
100.880.750.230.05
150.820.650.100.00
200.770.500.000.00

Tabulated experimental data (The values in the table represent the percentage of undepolymerization.) Here, due to reasons such as the amount of experiments, we were unable to conduct large-scale experiments. Therefore, in order to illustrate the reliability of our method, we used simulated data as a demonstration.

Result

We used the above data to calculate and obtained the following results.

alt text

Figure 1: Dissolution rate constant kk as a function of pH.

After obtaining the functional relationship of kk with respect to pH, we can then predict the change in the percentage of undegraded gel microspheres over time at any given pH.

alt text

Figure 2: Model-predicted undissolved fraction curve at pH = 2.0.

alt text

Figure 3: Model-predicted undissolved fraction curve at pH = 6.0.

In real-world experiments, we found that in a strongly acidic environment, the microspheres remained undegraded for up to an hour. In a neutral environment, however, the gel microspheres completely degraded in about 90 seconds, which was close to our prediction.

Propulsion Model of a CaCO₃ Particle in Acidic Solution

Background

Next, we simulated the velocity of the microspheres after entering the human body, driven by the calcium carbonate reaction. We simulated the propulsive force of calcium carbonate-coated microspheres in gastric fluid and reviewed the classical Stokes solution for low Reynolds number flow around a sphere. We derived the equations of motion and simulated the velocity-time behavior to obtain the velocity of the microspheres in the liquid environment.

Parameters and Physical Assumptions

We consider a spherical CaCO₃ microsphere with:

  • Mass: m=1×10−7 kgm = 1\times 10^{-7}\,\mathrm{kg} (100 µgµg)
  • Density: ρ=3.000 g/cm3\rho = 3.000\,\mathrm{g/cm^3}

At the same time, by consulting the literature, we can also obtain the parameters of the gastric mucus layer：

Its radius is determined by:

r=(3m4πρ)1/3r = \left(\frac{3m}{4\pi \rho}\right)^{1/3}

Assume the CaCO₃ reacts with excess acid at pH 2, producing CO₂ by

CaCO3(s)+2H+(aq)→Ca2+(aq)+CO2(g)+H2O(l)CaCO3(s) + 2H^+ (aq) \rightarrow Ca^{2+}(aq) + CO2(g) + H2O(l)

Assuming the entire CaCO₃ sphere reacts over time TreactT_{\mathrm{react}}, the maximum CO₂ generation rate is:

m˙CO2=mTreact⋅MCO2MCaCO3=mTreact⋅44.01100.09\dot m_{\mathrm{CO}_2} = \frac{m}{T_{\mathrm{react}}} \cdot \frac{M_{\mathrm{CO}_2}}{M_{\mathrm{CaCO}_3}} = \frac{m}{T_{\mathrm{react}}}\cdot \frac{44.01}{100.09}

where MCO2M_{\mathrm{CO}_2} and MCaCO3M_{\mathrm{CaCO}_3} are the molar masses of CO₂ (44.01 g/mol) and CaCO₃ (100.09 g/mol), respectively.

The thrust is generated by the liberation of CO₂, imparting momentum to the particle (neglecting losses):

Fthrust=m˙CO2vexF_{\mathrm{thrust}} = \dot m_{\mathrm{CO}_2} v_{\mathrm{ex}}

where vexv_{\mathrm{ex}} denotes the effective exhaust velocity of CO₂ as it escapes (estimation: vexv_{\mathrm{ex}} can be approximated as gas flow velocity at interface, typically 10–30 m/s per microfluidics literature).

Modeling the Drag Force in Fluid

The drag force is described using Stokes law.

Stream Function in Cylindrical Coordinates

For an incompressible, axisymmetric, low-Reynolds-number flow past a fixed sphere of radius RR in a uniform stream uu, introduce the Stokes stream function ψ(r,z)\psi(r,z) satisfying:

uz=1r∂ψ∂r,ur=−1r∂ψ∂zu_z = \frac{1}{r} \frac{\partial \psi}{\partial r}, \quad u_r = -\frac{1}{r} \frac{\partial \psi}{\partial z}

which ensures ∇⋅u=0\nabla\cdot\mathbf{u}=0 by construction.

Vorticity and Governing Equations

The sole nonzero vorticity is:

ωφ=∂ur∂z−∂uz∂r=−∂∂r(1r∂ψ∂r)−1r∂2ψ∂z2\omega_\varphi = \frac{\partial u_r}{\partial z} - \frac{\partial u_z}{\partial r} = -\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial \psi}{\partial r}\right) - \frac{1}{r}\frac{\partial^2 \psi}{\partial z^2}

and it satisfies:

∇2ωφ=0\nabla^2 \omega_\varphi = 0
Exact Solutions for Stream Function and Velocities

The well-known solution is:

ψ(r,z)=−12ur2[1−3R2r2+z2+12(Rr2+z2)3]\psi(r,z) = -\frac{1}{2} u r^2 \left[1 - \frac{3R}{2\sqrt{r^2+z^2}} + \frac{1}{2}\left(\frac{R}{\sqrt{r^2+z^2}}\right)^3\right]

leading to velocity components ur(r,z)u_r(r,z) and uz(r,z)u_z(r,z).

Vorticity, Pressure, and Drag Force

The azimuthal vorticity and pressure fields are:

ωφ(r,z)=−3Ru2r(r2+z2)3/2,p(r,z)=−3μRu2z(r2+z2)3/2\omega_\varphi(r, z) = -\frac{3Ru}{2}\frac{r}{(r^2+z^2)^{3/2}}, \quad p(r, z) = -\frac{3\mu R u}{2} \frac{z}{(r^2 + z^2)^{3/2}}

In spherical coordinates:

p(r,θ)=−3μRu2cos⁡θr2p(r, \theta) = -\frac{3\mu R u}{2} \frac{\cos\theta}{r^2}

The total drag force—evaluated via surface integration of the stress tensor—is:

Fd=6πμR u\mathbf{F}_d = 6\pi \mu R\,\mathbf{u}

where μ\mu is the dynamic viscosity of the surrounding solution (for water at 25 °C, μ=5×10−3\mu=5\times10^{-3} Pa·s[4]; acidic media similar). This classical Stokes drag formula is valid for Re≤1Re \leq 1 [1].

Equation of Motion

Applying Newton’s second law:

mdvdt=Fthrust(t)−Fd=m˙CO2vex−6πμrvm \frac{dv}{dt} = F_{\mathrm{thrust}}(t) - F_{d} = \dot m_{\mathrm{CO}_2} v_{\mathrm{ex}} - 6\pi \mu r v

with FthrustF_{\mathrm{thrust}} as above. If m˙CO2\dot{m}_{\mathrm{CO}_2} is constant, the analytical solution for velocity is straightforward. If time-varying, a numerical solution is required. So we will get the numerical solution through numerical simulation.

Result

In summary, we wrote a program in Python to solve the ODE equations above. We then plotted the velocity-time curve.

alt text

Figure 4: Predicting the speed of a microsphere.

The results showed that the microspheres’ speed in the stomach was relatively stable, maintaining a constant speed of around 20 micrometers per second when only the micromotor (calcium carbonate) and acid reaction were considered. To further ensure the accuracy and reliability of our model, we consulted relevant literature. Existing articles have shown that similar methods have also achieved microsphere movement speeds of 20-150 μm/s\mu m/s micrometers per second[2][3], demonstrating the validity of our model.

At the same time, take into account the possible adverse effects of gas release，we can simply calculate the amount of gas released during the reaction of each microsphere.

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Figure 5: CO−2-2 Release Dynamics Predition.

One microsphere will only produce about 0.01 ml of gas, which means that taking 2g of the drug will only produce 80 ml of gas, which is safe enough for the human body.

Final References

  1. Stokes’ law. https://en.wikipedia.org/wiki/Stokes%27_law (Accessed on July 12, 2025).

  2. de Ávila BE, Angsantikul P, Li J, Angel Lopez-Ramirez M, Ramírez-Herrera DE, Thamphiwatana S, Chen C, Delezuk J, Samakapiruk R, Ramez V, Obonyo M, Zhang L, Wang J. Micromotor-enabled active drug delivery for in vivo treatment of stomach infection. Nat Commun. 2017 Aug 16;8(1):272. doi: 10.1038/s41467-017-00309-w.

  3. Gao W, Dong R, Thamphiwatana S, Li J, Gao W, Zhang L, Wang J. Artificial micromotors in the mouse’s stomach: a step toward in vivo use of synthetic motors. ACS Nano. 2015 Jan 27;9(1):117-23. doi: 10.1021/nn507097k.

  4. Ruiz-Pulido G, Quintanar-Guerrero D, Serrano-Mora LE, Medina DI. Triborheological Analysis of Reconstituted Gastrointestinal Mucus/Chitosan:TPP Nanoparticles System to Study Mucoadhesion Phenomenon under Different pH Conditions. Polymers (Basel). 2022 Nov 17;14(22):4978. doi: 10.3390/polym14224978.