Dynamic Spatial Treatment Effects in Neurotransmitter Diffusion: Applications to Movement Disorders
Abstract
Traditional spatial treatment effect methods impose arbitrary boundaries between treated and control regions, obscuring how treatments spread through neural tissue. We develop a continuous functional framework deriving explicit treatment boundaries from diffusion physics, eliminating discretization artifacts while providing testable predictions. Our approach applies partial differential equations to neurotransmitter diffusion, unifying spatial scales from synaptic spillover (micrometers) to volume transmission (centimeters). We validate using synthetic data calibrated to established neuroscience parameters across five conditions: healthy controls, dystonia, Parkinson's disease, Alzheimer's disease, and acute ischemia. Results demonstrate systematic disease-induced boundary alterations. Dystonia reduces treatment reach by 16.3\% (p $<$ 0.0001), requiring 24\% dose increases for equivalent coverage. Parkinson's shows 31.8\% reduction, ischemia 34.3\%. Spatial decay parameters evolve from 483 mm$^{-1}$(1 hour) to 54 mm$^{-1}$ (72 hours), matching theoretical $1/\sqrt{t}$ predictions. Compared to difference-in-differences methods, our framework achieves superior fit ($R^2$ = 0.92 vs. 0.76) with explicit boundary detection. Non-parametric approaches achieve higher in-sample fit ($R^2$ = 0.99) but lack physical interpretability and cannot identify boundaries. Clinical applications include: (1) data-driven determination of injection sites and stimulation parameters, (2)…
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