Alzheimer's disease: a mathematical model for onset and progression.

In this article we propose a mathematical model for the onset and progression of Alzheimer's disease based on transport and diffusion equations. We regard brain neurons as a continuous medium and structure them by their degree of malfunctioning. Two different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble polymers of amyloid, produced by damaged neurons and ii) neuron-to-neuron prion-like transmission. We model these two processes by a system of Smoluchowski equations for the amyloid concentration, coupled to a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. The second equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain. Our numerical simulations are in good qualitative agreement with clinical images of the disease distribution in the brain which vary from early to advanced stages.

[1]  S. de Santi,et al.  Pre-clinical detection of Alzheimer's disease using FDG-PET, with or without amyloid imaging. , 2010, Journal of Alzheimer's disease : JAD.

[2]  D. Selkoe,et al.  Soluble protein oligomers in neurodegeneration: lessons from the Alzheimer's amyloid β-peptide , 2007, Nature Reviews Molecular Cell Biology.

[3]  C. Jack,et al.  Hypothetical model of dynamic biomarkers of the Alzheimer's pathological cascade , 2010, The Lancet Neurology.

[4]  M. Mattson.  Pathways towards and away from Alzheimer's disease , 2004, Nature.

[5]  M. Weiner,et al.  A Network Diffusion Model of Disease Progression in Dementia , 2012, Neuron.

[6]  Kenjiro Ono,et al.  Structure–neurotoxicity relationships of amyloid β-protein oligomers , 2009, Proceedings of the National Academy of Sciences.

[7]  Dariusz Wrzosek,et al.  Existence of solutions for the discrete coagulation-fragmentation model with diffusion , 1997 .

[8]  B. Piccoli,et al.  Time-Evolving Measures and Macroscopic Modeling of Pedestrian Flow , 2008, 0811.3383.

[9]  L. Edelstein-Keshet,et al.  Exploring the formation of Alzheimer's disease senile plaques in silico. , 2002, Journal of theoretical biology.

[10]  R. Mayeux,et al.  Epidemiology of Alzheimer disease , 2011, Nature Reviews Neurology.

[11]  Philippe Laurençot,et al.  Global existence for the discrete diffusive coagulation-fragmentation equations in $L^1$ , 2002 .

[12]  Agneta Nordberg,et al.  Amyloid imaging in Alzheimer's disease , 2007, Neuropsychologia.

[13]  Christoph Walker,et al.  Local and global strong solutions to continuous coagulation–fragmentation equations with diffusion , 2005 .

[14]  P. Scheltens,et al.  Brain imaging and Alzheimer's disease , 1993, European Neuropsychopharmacology.

[15]  B. Straughan,et al.  Decay bounds in a model for aggregation of microglia: application to Alzheimer's disease senile plaques , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[16]  Peter J Gianaros,et al.  Longitudinal assessment of neuroimaging and clinical markers in autosomal dominant Alzheimer's disease: a prospective cohort study , 2015, The Lancet Neurology.

[17]  C. Nicholson,et al.  Extracellular space structure revealed by diffusion analysis , 1998, Trends in Neurosciences.

[18]  Paolo Frasca,et al.  Existence and approximation of probability measure solutions to models of collective behaviors , 2010, Networks Heterog. Media.

[19]  H. Braak,et al.  Alzheimer’s pathogenesis: is there neuron-to-neuron propagation? , 2011, Acta Neuropathologica.

[20]  Laurent Pujo-Menjouet,et al.  Alzheimer’s disease: analysis of a mathematical model incorporating the role of prions , 2013, Journal of mathematical biology.

[21]  C. Jack,et al.  Mild cognitive impairment: ten years later. , 2009, Archives of neurology.

[22]  Eric M Reiman,et al.  Florbetapir PET analysis of amyloid-β deposition in the presenilin 1 E280A autosomal dominant Alzheimer's disease kindred: a cross-sectional study , 2012, The Lancet Neurology.

[23]  Kewei Chen,et al.  Brain imaging and fluid biomarker analysis in young adults at genetic risk for autosomal dominant Alzheimer's disease in the presenilin 1 E280A kindred: a case-control study , 2012, The Lancet Neurology.

[24]  M McNamara,et al.  Aggregation and disaggregation of senile plaques in Alzheimer disease. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[25]  B Urbanc,et al.  Dynamics of plaque formation in Alzheimer's disease. , 1999, Biophysical journal.

[26]  Regina M. Murphy,et al.  Probing the Kinetics of β-Amyloid Self-Association , 2000 .

[27]  D. Selkoe,et al.  Aβ Oligomers – a decade of discovery , 2007, Journal of neurochemistry.

[28]  T. Good,et al.  Effect of beta-amyloid block of the fast-inactivating K+ channel on intracellular Ca2+ and excitability in a modeled neuron. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[29]  Herbert Amann,et al.  On a quasilinear coagulation-fragmentation model with diffusion , 2001 .

[30]  C. Jack,et al.  Tracking pathophysiological processes in Alzheimer's disease: an updated hypothetical model of dynamic biomarkers , 2013, The Lancet Neurology.

[31]  D. Selkoe.  Alzheimer's disease. , 2011, Cold Spring Harbor perspectives in biology.

[32]  Herbert Amann,et al.  Coagulation-Fragmentation Processes , 2000 .

[33]  W. Griffin,et al.  Glial‐Neuronal Interactions in Alzheimer's Disease: The Potential Role of a ‘Cytokine Cycle’ in Disease Progression , 1998, Brain pathology.

[34]  J. Sérot,et al.  A possible role for CSF turnover and choroid plexus in the pathogenesis of late onset Alzheimer's disease. , 2012, Journal of Alzheimer's disease : JAD.

[35]  Yves Achdou,et al.  A qualitative model for aggregation and diffusion of $$\beta $$-amyloid in Alzheimer’s disease , 2013, Journal of mathematical biology.

[36]  G. E. Vates,et al.  A Paravascular Pathway Facilitates CSF Flow Through the Brain Parenchyma and the Clearance of Interstitial Solutes, Including Amyloid β , 2012, Science Translational Medicine.

[37]  D. Holtzman,et al.  Rapid appearance and local toxicity of amyloid-β plaques in a mouse model of Alzheimer’s disease , 2008, Nature.

[38]  B. Strooper,et al.  The amyloid cascade hypothesis for Alzheimer's disease: an appraisal for the development of therapeutics , 2011, Nature Reviews Drug Discovery.

[39]  Nick C Fox,et al.  Clinical and biomarker changes in dominantly inherited Alzheimer's disease. , 2012, The New England journal of medicine.

[40]  Regina M. Murphy,et al.  Kinetics of aggregation of synthetic β-amyloid peptide , 1992 .

[41]  Richard Rebarber,et al.  Global asymptotic stability of plant-seed bank models , 2014, Journal of mathematical biology.

[42]  G. Goodhill.  Diffusion in Axon Guidance , 1997, The European journal of neuroscience.

[43]  Paco Martorell,et al.  Monetary costs of dementia in the United States. , 2013, The New England journal of medicine.

[44]  R. Murphy,et al.  A mathematical model of the kinetics of beta-amyloid fibril growth from the denatured state. , 2001, Biophysical journal.

[45]  M. C. Tesi,et al.  A QUALITATIVE MODEL FOR AGGREGATION-FRAGMENTATION AND DIFFUSION OF β-AMYLOID IN ALZHEIMER’S DISEASE , 2014 .