Mathematics Behind Poseida Therapeutics’ Therapy Science
Poseida Therapeutics uses advanced mathematics in gene editing and CAR-T cell therapy to develop therapies that effectively target cancer and other conditions. This post breaks down the core mathematical principles and models that underlie Poseida’s scientific approach.
1. CAR-T Cell Therapy and Differential Equations
CAR-T cell therapy models the interaction between cancer cells, CAR-T cells, and immune response through ordinary differential equations (ODEs), predicting how cancer and CAR-T cell populations evolve over time.
Cancer Cell Growth Equation:
The cancer cell growth rate depends on cell proliferation and changes with therapy:
dC/dt = rC - kTC
- C: Number of cancer cells
- T: CAR-T cell concentration
- r: Cancer cell growth rate
- k: CAR-T cell killing efficiency
CAR-T Cell Dynamics Equation:
CAR-T cells grow, die, or expand upon encountering cancer cells:
dT/dt = αT - βT + γTC
- α: CAR-T cell proliferation rate
- β: CAR-T cell death rate
- γ: Activation rate when encountering cancer cells
2. Gene Therapy and Dosage Calculations
In Poseida’s gene therapies, viral vectors deliver therapeutic genes, requiring precise dose calculations to achieve desired gene expression levels.
Viral Vector Concentration:
The viral dose is calculated based on body weight, in viral particles per kilogram (vp/kg):
Total Viral Particles = Dose (vp/kg) × Body Weight (kg)
Gene Expression Levels:
Gene expression levels are predicted using rates of transcription, degradation, and feedback mechanisms, ensuring the appropriate dosage for therapeutic effect.
3. Gene Editing Efficiency and Probability
Gene editing efficiency and accuracy in tools like CRISPR involve probability-based calculations for targeting success and reducing off-target effects.
Editing Efficiency:
The probability of successful edits depends on CRISPR binding efficiency:
P(Edit Success) = 1 - (1 - p)^n
- p: Probability of a single CRISPR complex binding successfully
- n: Number of CRISPR complexes introduced
Off-target Effects:
Off-target probabilities are calculated by assessing binding affinity to similar DNA sequences across the genome, often using statistical simulations.
4. Tumor-Immune Dynamics and Stochastic Modeling
For immune-oncology therapies targeting solid tumors, tumor-immune dynamics can be modeled with stochastic processes, predicting random immune cell interactions.
Stochastic Tumor-Immune Interactions:
Using a Poisson process, the probability of interaction within a time interval Δt depends on CAR-T cell density:
P(Interaction) = 1 - e^(-λTΔt)
- λ: Interaction rate between CAR-T and cancer cells
5. Pharmacokinetics (PK) and Pharmacodynamics (PD)
PK/PD modeling predicts how Poseida’s therapies distribute in the body and impact tumor size.
PK Model:
Therapeutic concentration decays over time due to clearance, modeled by first-order kinetics:
dC/dt = -kC
- k: Clearance rate constant
PD Model:
Therapeutic effect on cancer cells is modeled with the Hill equation:
E = (Emax × C) / (C + EC50)
- E: Therapeutic effect
- Emax: Maximum effect achievable
- EC50: Concentration achieving 50% of maximum effect
6. Risk and Uncertainty Quantification
Monte Carlo simulations help quantify risks for Poseida’s therapies. Each clinical phase has specific success probabilities, allowing the modeling of potential outcomes.
Monte Carlo Simulation for Success Rates:
Simulation involves multiple paths with success probabilities, for instance:
- Phase I: 10% success rate
- Phase II: 25% success rate
- Phase III: 50% success rate
These simulations offer insights into the likelihood of success across the development pipeline.
Summary
Mathematical modeling enables Poseida Therapeutics to optimize dosages, predict therapy responses, manage risks, and maximize therapeutic effectiveness. By applying these principles, Poseida can evaluate feasibility and guide decisions across development stages, from clinical trials to potential market approval.