The Impact of Novel Treatments on Aβ Burden in Alzheimer’s Disease: Insights from A Mathematical Model

  • David L. Craft
  • Lawrence M. Wein
  • Dennis J. Selkoe
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 70)


Motivated by recent therapeutic initiatives for Alzheimer’s disease, we developed a mathematical model of the accumulation of amyloid β-protein (Aβ) in the brain. The model incorporates the production and clearance of monomers, and the elongation and fragmentation of polymers by monomer aggregation and break-off, respectively. Our analysis suggests that dynamics are dictated by a single unitless measure referred to as the polymerization ratio, which is the product of the production and elongation rates divided by the product of the clearance and fragmentation rates. Cerebral burden (i.e., the total number of molecules, whether they exist as monomers or polymers) attains a finite steady-state level if this ratio is less than one, and undergoes sustained growth if this ratio is greater than one. The highly nonlinear relationship between the polymerization ratio and the steady-state burden implies that a modest reduction in the polymerization ratio achieves a significant decrease in the burden. Our model also predicts that after initiation or discontinuation of treatment, it may take months to reach a new steady-state burden. Taken together, our findings suggest that the research community should focus on developing agents that provide a modest reduction of the polymerization ratio while avoiding long-term toxicity. Finally, our model can be used to indirectly estimate several crucial parameters that are difficult to measure directly: the production rate, the fragmentation rate and the strength of treatment.

Key words

Alzheimer’s disease Smoluchowski equation 


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Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • David L. Craft
    • 1
  • Lawrence M. Wein
    • 2
  • Dennis J. Selkoe
    • 3
  1. 1.Operations Research CenterMassachusetts Institute of TechnologyCambridge
  2. 2.Graduate School of BusinessStanford UniversityStanford
  3. 3.Center for Neurologic DiseasesHarvard Medical School Brigham and Women’s HospitalBoston

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