Overview: My research lies at the intersection of applied mathematics, computational modeling, and data science, with emphasis on mechanistic modeling, statistical inference, and machine learning. I apply these tools to study biological systems and complex multi-agent interactions.
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Multi-Team Interaction Game Dynamics
Objectives:
Objectives:
Develop statistical models to estimate team strength and rank teams in multi-team, multiplayer games such as PlayerUnknown's Battlegrounds (PUBG).
Validate model performance using professional PUBG tournament data.
Apply machine learning techniques to predict team strength and rankings.
Compare machine learning predictions with statistical model predictions to better understand the strengths and limitations of each approach.
Methods:
Methods:
Developed an extension of the Bradley–Terry paired comparison model that incorporates:
the probability of a team suffering a defeat relative to all competing teams,
the probability of winning pairwise encounters, and
a weighted composite of these probabilities.
Applied ridge-penalized maximum likelihood estimation to improve model stability and prevent overfitting.
Calibrated and validated the model using both simulated data and professional PUBG championship data.
Implementing and evaluating several machine learning approaches, including Random Forest (RF), Extreme Gradient Boosting (XGBoost), Feedforward Neural Networks (FNN), Light Gradient Boosting Machine (LightGBM), and Multilayer Perceptrons (MLP).
Key Findings:
Key Findings:
Sensitivity analyses indicate that team size has a significant positive influence on game outcomes.
Simulation studies demonstrate over 90% accuracy in recovering true team strengths and rankings using metrics such as Spearman rank correlation, mean absolute error (MAE), root mean square error (RMSE), and Monte Carlo leave-match-out validation.
Prediction accuracy improves with increasing data availability and greater heterogeneity among teams.
Recovery of team ranking by computing the mean absolute error and the Spearman rank correlation coefficient between the predicted and true ranks
Recovery of team strengths by computing the root mean square error between the predicted and team strengths
Ranking profile of selected teams as the data size is increased
Official matched involved in the 2021 PUBG global championship, the data used for the simulation
Mechanistic Modeling of Retinal Detachment and Photoreceptor Renewal
Objectives:
Objectives:
Investigate the progression of retinal detachment and identify factors influencing disease advancement.
Study the renewal dynamics of photoreceptor cells in detached retinal tissue.
Predict the optimal treatment window to maximize photoreceptor survival and visual outcomes.
Methods:
Methods:
Developed a fluid–structure interaction (FSI) model using the immersed boundary method to simulate retinal detachment progression.
Developed a system of ordinary differential equations (ODEs) to model photoreceptor renewal dynamics in detached retinal tissue.
Designed and implemented computational algorithms in MATLAB to simulate and analyze both models.
Key Findings:
Key Findings:
Disease progression is accelerated by increased eye rotation, larger initial detachment length, greater elevation angles, higher retinal density, and higher unbinding affinity of retinal adhesion proteins.
Disease progression is slowed by increased retinal thickness, greater retinal elasticity, stronger adhesion bonds, and higher concentrations of adhesion proteins.
In healthy tissue, photoreceptors undergo continuous renewal through the addition of new discs at the base and shedding of older discs at the tip. Our results indicate that retinal detachment either completely inhibits disc addition or substantially reduces the rate of addition, while disc shedding continues.
Photoreceptor survival is strongly influenced by the rate of disc shedding, suggesting that preserving renewal dynamics may be critical for maintaining retinal function following detachment.
Limitations and Future Directions:
Limitations and Future Directions:
The current model does not account for the curvature of the eyeball; incorporating realistic eye geometry is a key direction for future work.
Experimental validation of the disease progression model remains ongoing, and collaborative opportunities are being actively pursued.
The dynamics of photoreceptor recovery following retinal reattachment have not yet been investigated and represent an important area for future study.
The current two-dimensional framework limits the ability to model eye rotations in all directions; future work will extend the model to three dimensions.
The computational complexity of the simulations limits investigations over longer time scales. Current efforts are focused on integrating machine learning approaches, particularly Physics-Informed Neural Networks (PINNs), to accelerate simulations and enable more extensive analyses.
Flow profile showing the interation between the detached retina and the vitreous fluid
Dynamics of adhesion bond breaking during the rotation of the eye
An eye showing a tear in the retina causing rhegmatogenous retinal detachment
A closer view of the adhesion bonds between the neural ayer and the retinal pigmented epithelium (RPE)
Investigating the Role of Filopodia Dynamics in Cell Signaling
Filopodia are dynamic cellular protrusions that facilitate long-range communication between cells. This project investigates how filopodia dynamics influence Notch–Delta signaling and the spatial organization of sensory organ precursor cells during tissue pattern formation.
Filopodia are dynamic cellular protrusions that facilitate long-range communication between cells. This project investigates how filopodia dynamics influence Notch–Delta signaling and the spatial organization of sensory organ precursor cells during tissue pattern formation.
Objectives:
Objectives:
Investigate the role of filopodia dynamics in Notch–Delta signaling and sensory organ precursor (SOP) pattern formation.
Quantify the spatial distribution of SOP cells and identify factors influencing pattern development.
Methods:
Methods:
Developed a mechanistic model that captures filopodia protrusion, retraction, and pausing dynamics.
Formulated a deterministic criterion for filopodia-mediated cell interactions.
Used experimental data to determine the Notch signaling threshold that distinguishes SOP cells from epithelial cells.
Developed a quantitative measure, termed the Local Sensory Area (LSA), to characterize local SOP distribution patterns.
Developed a computational algorithm in MATLAB to simulate the coupled Notch-Delta signaling and the filopodia model.
Key Findings:
Key Findings:
Increased filopodia length enhances long-range cell–cell communication, leading to a sparser distribution of SOP cells.
Increasing filopodia lifetime, particularly the duration of pausing, has a modest effect on SOP spacing and is most influential when filopodia are sufficiently long to interact with distant cells.
SOP patterns in tissues with shorter filopodia stabilize more rapidly than those in tissues with longer filopodia, indicating that filopodia length plays a significant role in the timing and organization of pattern formation.
Some pattern observed in nature
Life cycle of a fruit fly showing time the window of formation of SOP cells
A closer view of SOP connected to the central nervous system
SOP spots observed under microscope during transition from pupal to adult stage
Simulated result replicating SOP formation formation in fruit flies
The Notch level in SOP cells decreases, allowing them to express higher levels of Delta, while the Notch level in epithelial cells increases.
Effect of filopodia lifetime on SOP spacing measured in terms of relative sensory area (RSA) and local sensory area (LSA)
Effect of G-actin concentration (which regulates filopodia length) on SOP spacing, measured in terms of Relative Sensory Area (RSA) and Local Sensory Area (LSA).
Numerical Methods for Partial Differential Equations
This project investigated the convergence, accuracy, and computational performance of finite element solutions to time-dependent partial differential equations (PDEs) under different time-integration schemes. The objective was to evaluate how various temporal discretization methods perform when coupled with the Finite Element Method (FEM) for spatial discretization.
The study focused on three widely used time-stepping approaches: Exponential Time Differencing (ETD), Implicit Backward Euler, and Implicit–Explicit (IMEX) schemes. Through theoretical analysis and computational experiments, the numerical performance of each method was assessed in terms of convergence, stability, and accuracy.
Results from benchmark test problems demonstrated that the ETD scheme achieved superior convergence and accuracy while maintaining computational efficiency compared to both the IMEX and Backward Euler methods. These findings provide insight into the selection of effective numerical solvers for time-dependent PDEs arising in scientific and engineering applications.
This work contributes to the development of robust finite element frameworks for the accurate simulation of complex dynamical systems.
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