Modeling the control on the formation and dissolution of amyloid plaques in Alzheimer’s disease

Alzheimer’s disease (AD) is a neurological disorder leading to progressive neuronal loss. The first AD symptoms are memory, language, and thinking deficits. As the disease progresses, it interferes with human abilities to perform even the most basic everyday tasks. The extracellular amyloid plaques, whose main component is the amyloid-beta peptide (A β ), are one of the main AD hallmarks. Overproduction in the brain and reduced clearance from the central nervous system of these peptides, which self-aggregate and accumulate, are believed to be responsible for amyloid plaque formation. We first refine a previous four-compartment model of amyloid plaque formation and A β transport between blood and the brain environment. Treating amyloid plaques as three-dimensional aggregates, we assume that their dissolution, like their formation, depends on surface area. Then, we introduce terms representing two idealized treatment mechanisms: enhancement of A β disaggregation and inhibition of A β aggregation. Their effects depend on the A β level in the cerebrospinal fluid, a clinically relevant AD biomarker, and decrease as amyloid plaques increase. Finally, we formulate an optimal control problem on a finite horizon that balances amyloid plaque reduction against treatment intensity and cost. Under the stated parameterization, increasing the treatment horizon increases the cumulative treatment intensities and the value of the horizon-dependent objective functional.