Project 1: Linear Algebra-Based Myoelectric Pain Sensor
Design a low-cost EMG biosensor capable of estimating muscular pain through linear algebra-based signal processing.
Acquire EMG signal.
Build a feature vector.
Compute covariance matrix.
Perform eigenvalue decomposition.
Project noisy components.
Estimate pain index.
Pain Index
0–20 Normal
20–50 Fatigue
50–80 Acute pain
80–100 Severe pain
PCA
SVD
Machine Learning
Adaptive filtering
Project 2: DIY Bioinformatic Pipeline Recipe: Low-Cost Disease Detection
Ingredients (The Analytical Framework)
Raw Data (\(I(\lambda)\)): Light intensity spectrum from a DIY smartphone spectrometer.
Error Matrices (\(\sigma _{i}\)): Experimental uncertainties (pixel noise, light fluctuations).
Biological Kinetics: Differential equations governing cellular/bacterial growth.
Step 1: Background Subtraction & Signal Isolation
Before analyzing the biomarker, we must isolate the true signal from the baseline matrix (healthy saliva background).
Action: Subtract the baseline healthy spectrum (\(I_{bg}\)) from the raw sample spectrum (\(I_{raw}\)).
Equation:
\(S(\lambda )=I_{raw}(\lambda )-I_{bg}(\lambda )\)
Statistical Filter: Apply a smoothing filter (e.g., Savitzky-Golay) driven by pure mathematical analysis to eliminate high-frequency white noise without destroying the physical peaks.
Step 2: Linear Fitting & Concentration Extraction
We map the isolated absorbance peak to a physical concentration using the linear regime of the Beer-Lambert Law.
Action: Perform a Weighted Linear Fit (Least Squares) using the experimental uncertainties as weights (\(w_i = 1/\sigma_i^2\)).
Equation:
\(A(\lambda )=\epsilon \cdot c\cdot l+b\)
Where \(A\) is absorbance, \(\epsilon \) is the molar attenuation coefficient, \(l\) is the path length, and \(c\) is the unknown concentration.
Output: Extract the precise concentration (\(c\)) of the pathogen/biomarker.
Step 3: Error Propagation & Confidence Intervals (The Particle Physics Core)
Every measurement has noise. We must calculate if our detected concentration (\(c\)) is a real signal or a statistical fluctuation.
Action: Propagate the variances of the raw data through the fitting parameters using the covariance matrix.
Equation:
\(\sigma _{c}^{2}=\sum _{i}\left(\frac{\partial c}{\partial I_{i}}\right)^{2}\sigma _{I_{i}}^{2}\)
Discovery Threshold: A disease is flagged only if the concentration signal exceeds the background fluctuation by a threshold of \(3\sigma\) or \(5\sigma\) (Confidence Level \(> 99.7\%\)), utilizing Bayesian updates to include demographic priors.
Step 4: Biological Dynamics Integration (Cellular & Bacterial Growth)
Once the initial concentration (\(c_{0}\)) is extracted, the pipeline connects it to the time-domain evolution of the infection to predict the severity of the disease.
Option A: Bacterial Infection (Exponential / Logistic Growth)
If the biomarker belongs to a bacterial pathogen, the pipeline plugs \(c_{0}\) into the Logistic Growth Equation to simulate population dynamics under finite resource constraints:
\(\frac{dc}{dt}=r\cdot c\left(1-\frac{c}{K}\right)\implies c(t)=\frac{K\cdot c_{0}\cdot e^{rt}}{K+c_{0}(e^{rt}-1)}\)
Where \(r\) is the growth rate and \(K\) is the carrying capacity of the host environment.
Option B: Cellular / Viral Damage (Kinetic Decay or Monod Model)
If the disease involves host cellular destruction or viral replication, we couple the concentration to a system of coupled differential equations tracking cell substrate depletion:
\(\frac{dx}{dt}=\mu _{max}\frac{c}{K_{c}+c}x\)
Where \(x\) is the damaged cell density, allowing us to map the precise analytical trajectory of the pathology over time.