Table of Contents
Overview
Modeling plays a central role in our project, enabling us to develop and integrate all our innovative components effectively.
Our modeling efforts focus on two key areas. First, to predict the therapeutic performance of the smart bandage without conducting human trials, the Dry Lab team constructed a numerical model to simulate the treatment process. We then used these simulation results, which capture complex bacterial dynamics, as input data for neural networks. This data-driven approach allows us to optimize drug release strategies individualized to specific wound conditions or clinical needs. Second, to meet the performance requirements of our toolkit, we used physical environment simulations to guide the design and algorithm development, ensuring practical and efficient engineering solutions.
INTELLIGENT TREATMENT MODEL
The Pivot of Pandage Development
- In-depth analytical method on probiotics growth and bacterial inhibition dynamics with AMP released
- Prey-Predator based simulation of the entire treatment course with complex conditions
- An innovative approach to generate individualized treatment methods with machine learning while lacking clinical trial
UNDERSTANDING THE DYNAMICS
ODE Framework from Experimental Data
We have constructed an ODE framework to simulate the effects of different AMP releasing methods on the dynamics of Gram bacteria and L.lactis. By fitting bacterial population experimental data (from WL), we built a treatment course simulation integrated by ODEs of Gram bacteria (replaced by E.coli in WL exp), L.lactis and AMP interaction. The simulation comprehensively provides numerical data during a 96 day-course of treatment dependent on different bacterial initial conditions and drug release methods conducted by our automatic systems.
To model the dynamics of bacterial populations in the presence of antimicrobial peptides (AMPs), we employed ordinary differential equations (ODEs) to describe the growth and inhibition processes observed in WL experiments. WL performed controlled experiments where different concentrations of AMP (0, 25, and 50 µg/ml) were released at t=0 and monitored the growth of E. coli (starting from about 550000 CFU/ml) over 38 hours. The framework was developed using experimental data on E. coli (representing Gram-negative bacteria) and L.lactis probiotics independently subjected to different concentrations of AMP. This approach allowed us to quantify the interaction strength and growth dynamics of E. coli under varying AMP levels and then extend this modeling strategy to L.lactis.
The ODE chosen for bacterial growth is based on the Logistic Growth Model with an additional inhibition term to account for the AMP’s effect. This choice is supported by standard population dynamics theory, where logistic growth is used to model limited resource environments [1] and the inhibition is modeled using a decay-modified interaction term [2]. The chosen ODE framework is inspired by the Lotka-Volterra predator-prey model, adapted for antimicrobial interactions.This type of ODE is suitable for our system since:
- Logistic Growth for Prey (Bacteria): Bacterial populations typically follow logistic growth patterns, where the growth rate decreases as the population approaches its carrying capacity. And the natural growth curve of E.Coli (AMP=0) from exp data shows similar characteristics. This is expressed as:
G : Gram bacteria population at time.
k_growth : Growth rate of Gram bacteria.
G_max: Maximum carrying capacity of Gram bacteria.
2. Interaction with AMP (Predator Effect): The AMP acts as a predator, reducing bacterial concentration through a reaction-like interaction. The minusing term is a first-order interaction based on mass-action principles. This type of interaction term is well-suited for modeling inhibition effects[3].
The fitting was conducted using the `scipy.integrate.odeint` function in Python, which numerically integrates the ODE system. The optimization targeted minimizing the sum of squared errors (SSE) between experimental and simulated data for each AMP concentration.
C_AMP: Concentration of AMP at time.
lambda_G : Decay rate of AMP for Gram bacteria model.
G: Gram bacteria population at time.
k_growth: Growth rate of Gram bacteria.
G_max: Maximum carrying capacity of Gram bacteria.
gamma_G: Interaction coefficient indicating AMP inhibition effect on Gram
The fitting with ODE of our Lotka-Volterra inspired dynamic model shows high accuracy, convincing the validity of applying the model to further simulation. We then apply the same model to the WL performed controlled experiments where different concentrations of AMP (0, 25, and 50 µg/ml) were released at t=0 and monitored the growth of L.lactis (starting from about 550000 CFU/ml) over 38 hours.
The fitting with the same dynamic model from E.coli shows low error as well, but with different parameter values. Picking up the parameters in each curve fitting, the ODE framework of “Gram bacteria and L.Lactis dynamics with AMP released” is completed and available for further investigation on complex situations.
C_AMP: Concentration of AMP at time.
lambda_L: Decay rate of AMP for Lactis model.
L : Lactis population at time.
k_growth_L: Growth rate of Lactis.
L_max: Maximum carrying capacity of Lactis.
gamma_L: Interaction coefficient indicating AMP inhibition effect on Lactis.
SIMULATING OUR SYSTEMS OPERATION
Dynamical Model with AMP Released Control
With the ODEs built from experimental data. We can build models to simulate and predict how Gram Bacteria and L.lactis interact with AMP in Pandage’s automatic treatment system. Integrating the Pandage automatic AMP release mechanism (when bacteria>threshold, release C concentration of AMP) with the dynamical models fitted, and established the simulation framework.
C_AMP(t): \text{Concentration of AMP at time
Delta T: Time step size for the simulation.
lambda_G, lambda_L : Decay rates for AMP in the presence of Gram bacteria and Lactis.
G(t): Gram bacteria population at time.
threshold_G: Threshold Gram bacteria population triggering AMP dosing.
C_dose: Additional AMP concentration added at each dosing event.
The goal was to simulate various scenarios by adjusting parameters such as initial Gram levels, growth rates, and AMP dosing thresholds to assess two critical performance metrics:
Healing Efficiency – Defined as the total time during which L.lactis is active and secreting healing factors total time of which “concentration of L.lactis” > 7.5E5 (from WL exp)
Bacterial Suppression – Measured by the maximum Gram bacteria population reached during the course. We expected the bacteria concentration to be inhibited under 5e6. (Maximum bacterial concentration allowed without risk of infection)
By incorporating these performance indicators, we ensured a comprehensive evaluation of treatment effectiveness, balancing bacterial inhibition with healing promotion.
ANALYZING SIMULATION RESULTS
Parameter Tuning Optimization on the System’s Setup
PARADOX
While AMP is effective in suppressing Gram bacteria to prevent infections, it simultaneously hinders the secretion of growth factors from L.lactis, which are crucial for wound healing. Without an adequate amount of growth factors, the healing process can be prolonged despite effective bacterial control.
Our Solution
Running the treatment simulation, we varied key input parameters (e.g., initial Gram bacteria population, AMP threshold, and dosing concentration) to systematically explore the effect on our performance metrics with a 96-hour course simulation. This extensive simulation allowed us to establish a range of effective conditions for AMP application, which the automatic treatment ensures bacteria inhibition.
From the analyzing testing results, we had several setup parameters optimized:
- Detection Period: The sensor detection period (period of dosing activation sensing) gives optimized results at 10min.
- Dosing Logic: The concentration of AMP dosed should be kept within 10~50 ug/ml with dosing threshold < 3.2e6 CFU/ml of Gram bacteria to ensure the bacterial level being kept under 5e6 (a level we consider exists risk of infection)
- Bacterial Characteristic: We also investigated the range of growth rate and initial Gram level for which the system is capable of performing both bacteria inhibition and healing promotion. The deviation for the two parameters represents the wound and users’ physiological conditions.
- AMP Usage: We accounted for the total use of AMP in each simulation, and picked the maximum amount of AMP estimated to be released. The results give us a reference on the amount of AMP we required in the drug releasing CS hydrogel. (Interestingly, it is hard to estimate)
- L.lactis Initial Value: From the simulation results of L.lactis concentration adjusting initial L.lactis concentration, we obtained the minimal initial L.lactis value evaluated that promised healing performance during general AMP dosing treatment. Which is the fundamental setup for Pandage’s probiotics.
The results offer suggestions for not only the hardware & systems’ setup on bacterial sensor and AMP release system (automatic control logic & AMP usage), but also dictates the initial L.lactis population needed initially in the treatment course.
Demo of Our Treatment Simulation
Since we have some parameters for system’s setups set as fixed values after optimization testing, the resting parameters are the ones that would depend on different scenarios of Pandage application. Initial Gram concentration and growth rate of Gram varies with different users, while dosing parameters (C_dose & threshold_G) are adjustable in every Pandage product as we expect to define the values with individualized approaches for each treatment course. Testing the resting parameters of inputs provides us in-depth information on how healing and bacterial inhibition performs in varying scenarios.
TREATMENT SIMULATION
PERFORMANCE
ESTABLISHING INDIVISUALIZED TREATMENT
In search of the optimal AMP releasing logic individualized to different users’ condition
Having just experimented with the treatment simulation, we observed how variations in initial bacterial populations and dosing parameters significantly influence the overall treatment performance. Through adjusting these parameters, we discovered that even minor changes in the “initial Gram bacteria level” or the “threshold for AMP activation” can result in dramatically different healing times and infection control outcomes. For instance, a slight increase in the AMP threshold may reduce the maximum Gram bacteria load but at the cost of inhibiting L.lactis activity, delaying healing.
These observations highlight the critical need for an individualized treatment strategy that can adapt to the unique conditions of each wound. The “performance metrics” you’ve evaluated—such as “maximum Gram bacteria population” and “L.lactis efficient time”—are not merely static measurements but serve as dynamic indicators of how well the system is balancing infection suppression and healing promotion.
The key to effective individualization is defining performance metrics that align with the patient’s healing and bacteria inhibition demands. By adjusting the simulation parameters, we gain a detailed understanding of how to balance infection control and healing under different scenarios. We took 2 possible scenarios into consideration.
SCENARIO TYPE 1
In the case of diabetic patients, wounds tend to have higher bacterial loads and slower healing rates due to impaired immune function and poor circulation. For these patients, it is critical to minimize the risk of infection spread while ensuring that the healing process is not excessively inhibited. This might involve using a lower AMP activation threshold to quickly suppress initial bacterial growth, followed by a gradual tapering of AMP concentration to promote L.lactis activity and encourage tissue regeneration.
In contrast, military personnel operating in harsh, dusty environments face unique challenges. Their wounds are often exposed to a variety of external contaminants, leading to rapid bacterial growth and an increased risk of infection. For these individuals, the primary concern is rapid and robust infection control to prevent any small cuts or abrasions from developing into serious, life-threatening infections. Here, the simulation might recommend a more aggressive AMP dosing strategy with frequent dosing intervals to maintain a sterile wound environment, even if it means sacrificing some L.lactis activity temporarily. Once the wound is secured against infection, the dosing parameters can be adjusted to support healing.
SCENARIO TYPE 2
For pediatric patients, the focus shifts again. Children often have more sensitive skin and react strongly to antimicrobial agents, making it crucial to find the lowest effective dose that maintains bacterial control without causing irritation or disrupting the natural microbiome balance. For these scenarios, the model might suggest using a high AMP threshold and micro-doses at extended intervals, allowing the body’s natural defenses to engage before applying any external agents.
Elderly patients with chronic wounds present yet another unique challenge. These wounds typically have a low bacterial load but are highly resistant to healing, necessitating a strategy that prioritizes prolonged L.lactis activity to enhance growth factor secretion and support tissue regeneration. For this group, AMP should be applied sparingly, only when bacterial levels show signs of rising, to ensure that L.lactis remains active for as long as possible.
IMPLEMENTING MACHINE LEARNING
Key Innovations on Treatment Individualization to Complex Systems
After analyzing various wound scenarios, it becomes clear that the optimal treatment strategy cannot rely on a fixed set of parameters. Instead, a dynamic approach is required, where the dosing decisions are guided by individualized inputs and expected outcomes.
To effectively customize treatment for diverse patient conditions, we constructed a comprehensive model, combining simulation with a neural network, that explores how varying combinations of key parameters—initial bacterial population, growth rate, AMP activation threshold, and AMP concentration—affect healing performance and bacterial suppression.
The performance metrics need to be aligned with the specific wound conditions. By balancing these performance indicators with the wound’s bacterial parameters, our goal is to recommend the most effective dosing method that suppresses harmful bacteria while promoting rapid and sustained healing.
To achieve this optimization, a direct analytical approach would struggle due to the complex, nonlinear relationships between these parameters and the desired outcomes. Thus, we require a method that can learn and generalize from diverse input-output scenarios, predicting the optimal dosing strategy for any given set of wound conditions. This is where a neural network becomes indispensable, as it excels at capturing hidden patterns and nonlinear dependencies in large, complex datasets.
The correlation matrix and mutual information table shown above illustrate why traditional linear approaches fall short. From the correlation matrix, it’s clear that while some parameters like initial Gram bacteria load and AMP threshold strongly influence the maximum bacteria population, their effects on healing time are more subtle and not easily captured through straightforward correlations.
These relationships are not straightforward to model using conventional methods, as they involve nonlinear interactions and context-specific dependencies that vary across different scenarios. The neural network can learn these complex interactions from the dataset generated through simulation, enabling it to make accurate dosing predictions even for unseen wound conditions. This capability allows us to provide an individualized, data-driven dosing strategy that dynamically adapts to diverse wound environments and healing requirements.
We begin by systematically varying four input parameters across defined ranges to capture their influence on two critical performance indicators: total healing time (total duration of L.lactis activity) and the maximum Gram bacterial population level. The simulation generates 3,639 unique data sets, each representing a different combination of conditions and outcomes. This data is then structured into a comprehensive matrix, forming the foundation for our Multi-Layer Perceptron (MLP) Neural Network model.
The neural network is trained using the simulation data to learn the complex relationships between wound conditions and optimal dosing strategies. With this trained model, we can predict the ideal AMP concentration and activation threshold for any new patient scenario, ensuring that each treatment plan is tailored to meet the unique demands of the wound condition. The resulting neural network model acts as an intelligent decision-making tool, guiding the Pandage system to deliver precise, data-driven care for a wide range of wound environments.
Network Architecture:
Input Layer: 4 neurons (one for each input feature).
Hidden Layers: Two hidden layers with 10 and 8 neurons, respectively, using ReLU activation functions.
Output Layer: 2 neurons predicting:
Training Details:
- Loss Function: Mean Squared Error (MSE).
- Optimization: Adam Optimizer.
- Epochs: 2000.
- Training-Testing Split: 80% training, 20% testing.
Kit Performance
Diffusion Simulation
In this study, we utilized Ansys Fluent Student 2024R to simulate the diffusion of the biomarker lipopolysaccharide (LPS) to evaluate its ability to effectively reach the aptamer binding site on the gold working electrode. Given the dynamic growth of bacteria and the non-linear behaviors observed by our wet lab team, we selected Fick’s Second Law as our primary model for diffusion simulation.
FIck’s law of diffusion
Fick’s First Law describes the diffusive flux, which is the flow of substance from regions of high concentration to regions of low concentration, directly proportional to the concentration gradient. This law is particularly applicable to steady-state conditions where the concentration does not change over time. Mathematically, it is expressed as:
Where:
- J represents the diffusion flux, defined as the amount of substance flowing through a unit area per unit time.
- D is the diffusion coefficient, also known as diffusivity, which depends on the squared velocity of the diffusing particles. This velocity is influenced by factors such as temperature, the viscosity of the fluid, and particle size, as described by the Stokes-Einstein relation.
Fick's Second Law
Fick’s Second Law is utilized to model the changes in concentration over time due to diffusion. This law is derived from the first law and provides a more comprehensive description of diffusion in systems where concentration changes dynamically.
It can be derived from the first law and the conservation equation, which states the concentration changes with time.
Plug in the first law gives
If the diffusivity D is constant (for simplicity), the equation simplifies further as
Notice that at steady state
gives the first law.
So, Fick’s first law can be considered as a specific condition of the second law when applied to a steady state.
Simulation
We simulated the diffusion of LPS from the wound surface using Ansys Fluent 2024R, a robust computational fluid dynamics (CFD) tool for modeling fluid flow, heat transfer, and chemical reactions. Ansys Fluent is part of the comprehensive Ansys software suite, which supports various engineering simulations. The simulation process consists of four key stages: Geometry, Mesh, Solution, and Results.
Geometry
The central cylinder represents the drug-releasing hydrogel, while the two side cylinders represent the gold electrodes with aptamers bound to their surface.
Mesh
Solution
We employed a transient solution method to track the LPS diffusion over time. The hydrogel’s properties, including a density of 1.02 g/mL and a viscosity of 4000 CPS at room temperature, were modeled according to the study by Jun-Kyu Park et al.[1]. The diffusion coefficient of the LPS is .02 μm^2/sec. Boundary conditions were set to zero LPS concentration at the edges of the domain to replicate the surrounding environment.
Results
The concentration gradient showed a steady release from the hydrogel, with diffusion progressing toward the Sensor. This simulation offers an understanding of the diffusion process’s efficiency and potential areas for optimization in drug delivery. The time-dependent concentration plots allowed for precise evaluation of LPS behavior under different conditions, contributing to the overall design of the microneedle system.
Refined Model for Drug Release: Fixed Volume Decrease per Event
After implementing the initial models, we quickly realized that the contraction speed was too fast when we attempted to fit the model to a 96-hour treatment period. Thus, we introduced a new control variable: fixed volume decrease per event (μ\muμ). This parameter allowed us to define the amount of volume reduction for each drug release event, rather than relying on proportional decay.
For a given initial volume VVV, the hydrogel’s volume changes by a fixed amount μ\muμ each time the electric field is applied. To ensure accurate dosing, we calculated the duration TTT of each electric field application as follows:
This formula was adapted from the exponential decay model, taking into account a fixed volume decrease per event instead of continuous decay. This adaptation was inspired by research on controlled drug release systems using hydrogels by Chen and Chang, who demonstrated that a constant volume decrease is more practical in scenarios where precision dosing is required【6】. By selecting an appropriate value for μ\muμ, we ensured that the hydrogel volume gradually approached zero by the end of the 96-hour treatment period.
Discrete Volume Decay with Fixed Volume Release Events
After refining the initial exponential decay model, we tested the discrete volume decay model under various drug dosages. Our goal was to validate whether the chitosan (CS) hydrogel could achieve controlled drug release based on the model derived from fixed volume decrease per event. Below, we present the results of three distinct dosage cases: 10, 23, and 40 mm³ per event.
Key Equations Utilized
In this experiment, we implemented the following mathematical models:
- Exponential Decay Model: Derived from the Nernst-Planck equation and applied for continuous volume change under electric field stimulation:
This differential equation was solved and resulted in the formula for volume over time:
The formula provides a general trend for volume decay, but it was not suitable for discrete, event-based volume change.
2. Duration of Each Release Event: For a given fixed volume decrease μ\muμ, the duration of each electric field application TTT is given by:
This equation allows us to calculate the duration of each release event precisely, ensuring that a specified volume reduction μ\muμ is achieved at each administration point.
Simulation Parameters
- We selected dosage = 23 mm³ as our reference case, adjusting the corresponding volume decrease parameter μ\muμ until the system achieved a controlled volume decay that matched the expected behavior over a 96-hour period.
-
We found the optimal value for μ\muμ to be 7.61 mm³ per event for dosage = 23. We then linearly scaled this value for the other two cases:
- Dosage = 10 mm³: μ=3.31\mu = 3.31μ=3.31
- Dosage = 40 mm³: μ=13.23\mu = 13.23μ=13.23
Experimental Conditions
- The drug administration times (drug_admin_times_hr) used in this experiment were derived from previous simulations that focused on optimizing the concentration release profile based on diffusion dynamics and concentration gradients inside the hydrogel. These simulations helped establish an optimal sequence of drug release timings to ensure therapeutic effectiveness. This sequence of administration times was kept consistent across different dosage levels to isolate the effect of varying μ\muμ values.
Results
-
Dosage = 23 mm³ per Event
In this reference case, we observed a steady volume reduction, and the hydrogel volume approached zero after approximately 96 hours. As shown in Figure 1, each red dot corresponds to a drug administration point, and the blue line represents the overall volume change over time. The behavior closely follows our expectations, validating the derived model.
2. Dosage = 10 mm³ per Event
By reducing the dosage to 10 mm³, we applied a smaller μ\muμ value of 3.31, resulting in a slower and more gradual decay pattern. This adjustment allowed us to achieve a controlled volume reduction over a longer period, as seen in Figure 2. The resulting curve still reached near-zero volume at around 96 hours, indicating successful adaptation of the model for lower dosages.
3. Dosage = 40 mm³ per Event
For the high-dosage case, we increased the value of μ\muμ to 13.23 to reflect the larger drug amount released at each event. As expected, the hydrogel exhibited a steeper and more rapid contraction pattern, with more significant volume reductions at each administration point. Nonetheless, the overall volume still approached zero in a controlled manner by the end of the treatment period, as shown in Figure 3.
Summary
Our refined model successfully captured the distinct volume contraction behavior of CS hydrogels across various dosage levels. By using the fixed-volume release equation and adjusting μ\muμ linearly based on the desired dosage, we ensured that the hydrogel achieved controlled volume decay over a 96-hour period. This consistency across different conditions confirms the robustness of our model and its applicability to various drug administration scenarios.
Reference
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- Chou, T., Wang, J. (2006). Effects of antimicrobial actions on population dynamics of susceptible and resistant bacteria. Journal of Theoretical Biology, 238(2), 260–275.
- Jorge, A., Diniz-Filho, F. (2013). Modeling the Dynamics of Pathogen-Host Interaction Using Differential Equations. Journal of Mathematical Biology, 67(6), 1359–1375.
- Horkay, F., and Basser, P. J. “Electro-Responsive Hydrogels: Anisotropic Deformation and Dynamic Swelling Behavior.” Journal of Polymer Science Part B: Polymer Physics, vol. 50, no. 3, 2012, pp. 213-222.
- Tanaka, T., et al. “Collapse of Gels in an Electric Field.” Science, vol. 218, no. 4571, 1982, pp. 467-469.
- Chen, J., and Chang, Y. “Controlled Drug Release from Chitosan Hydrogels Under External Electric Fields.” Carbohydrate Polymers, 2011, pp. 628-634.
- Kennedy, S., et al. “Rapid and Extensive Collapse from Electrically Responsive Macroporous Hydrogels.” Advanced Healthcare Materials, 2013.
- Liu, Y., et al. “Electrically Induced Anisotropic Assembly of Chitosan-Based Hydrogels.” Journal of Physical Chemistry B, 2015, pp. 2185-2193.