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The amyloid cascade hypothesis and Alzheimer’s disease: A mathematical model

Part of: Mathematical biology in general Physiological, cellular and medical topics

Published online by Cambridge University Press:  25 September 2020

M. BERTSCH Open the ORCID record for M. BERTSCH [Opens in a new window] ,
B. FRANCHI ,
L. MEACCI Open the ORCID record for L. MEACCI [Opens in a new window] ,
M. PRIMICERIO  and
M.C. TESI Open the ORCID record for M.C. TESI [Opens in a new window]
M. BERTSCH
Affiliation:
Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via della Ricerca Scientifica 1, 00133Roma, Italy, e-mail: bertsch@mat.uniroma2.it Istituto per le Applicazoni del Calcolo ‘M. Picone’, Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185Roma, Italy
B. FRANCHI
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126Bologna, Italy, e-mails: bruno.franchi@unibo.it; mariacarla.tesi@unibo.it
L. MEACCI
Affiliation:
Instituto de Ciências Matemáticas e de Computação, ICMC, Universidade de São Paulo, Avenida Trabalhador Sancarlense, 400, São Carlos (SP), CEP 13566-590, Brazil, e-mail: luca.meacci@usp.br
M. PRIMICERIO
Affiliation:
Istituto per le Applicazoni del Calcolo ‘M. Picone’, Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185Roma, Italy Dipartimento di Matematica ‘U. Dini’, Università degli Studi di Firenze, Viale Giovanni Battista Morgagni, 67/A, 50134Firenze, Italy, e-mail: mario.primicerio@math.unifi.it
M.C. TESI
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126Bologna, Italy, e-mails: bruno.franchi@unibo.it; mariacarla.tesi@unibo.it
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Abstract

The paper presents a conceptual mathematical model for Alzheimer’s disease (AD). According to the so-called amyloid cascade hypothesis, we assume that the progression of AD is associated with the presence of soluble toxic oligomers of beta-amyloid. Monomers of this protein are produced normally throughout life, but a change in the metabolism may increase their total production and, through aggregation, ultimately results in a large quantity of highly toxic polymers. The evolution from monomeric amyloid produced by the neurons to senile plaques (long and insoluble polymeric amyloid chains) is modelled by a system of ordinary differential equations (ODEs), in the spirit of the Smoluchowski equation. The basic assumptions of the model are that, at the scale of suitably small representative elementary volumes (REVs) of the brain, the production of monomers depends on the average degradation of the neurons and in turn, at a much slower timescale, the degradation is caused by the number of toxic oligomers. To mimic prion-like diffusion of the disease in the brain, we introduce an interaction among adjacent REVs that can be assumed to be isotropic or to follow given preferential patterns. We display the results of numerical simulations which are obtained under some simplifying assumptions. For instance, the amyloid cascade is modelled by just three ordinary differential equations (ODEs), and the simulations refer to abstract 2D domains, simplifications which can be easily avoided at the price of some additional computational costs. Since the model is suitably flexible to incorporate other mechanisms and geometries, we believe that it can be generalised to describe more realistic situations.

Keywords

Smoluchowski equationsmathematical models of Alzheimer’s diseaseamyloid cascade hypothesis

MSC classification

Primary: 92C50: Medical applications (general) 92B05: General biology and biomathematics
Secondary: 92C20: Neural biology
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 5: 30th Anniversary Special Issue , October 2021 , pp. 749 - 768
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Achdou, Y., Franchi, B., Marcello, N. & Tesi, M. C. (2013) A qualitative model for aggregation and diffusion of β-amyloid in Alzheimer’s disease. J. Math. Biol. 67(6–7), 1369–1392.CrossRefGoogle ScholarPubMed
Bacyinski, A., Xu, M., Wang, W. & Hu, J. (2017) The paravascular pathway for brain waste clearance: current understanding, significance and controversy. Front Neuroanat. 101, 101.CrossRefGoogle Scholar
Ballard, C., Gauthier, S., Corbett, A., Brayne, C., Aarsland, D. & Jones, E. (2011) Alzheimer’s disease. Lancet 377, 1019–1031.CrossRefGoogle ScholarPubMed
Bateman, R. J., Munsell, L. Y., Morris, J. C., Swarm, R., Yarasheski, K. E. & Holtzman, D. M. (2006) Quantifying CNS protein production and clearance rates in humans using in vivo stable isotope labeling, immunoprecipitation, and tandem mass spectrometry. Nat. Med. 12(7), 856.CrossRefGoogle Scholar
Bertsch, M., Franchi, B., Marcello, N., Tesi, M. C. & Tosin, A. (2017) Alzheimer’s disease: a mathematical model for onset and progression. Math. Med. Biol. 34(2), 193–214.Google ScholarPubMed
Bertsch, M., Franchi, B., Meschini, V., Tesi, M. & Tosin, A. (2020) A sensitivity analysis of a mathematical model for the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer’s disease. preprint, ArXiv: 2006.01749.Google Scholar
Bertsch, M., Franchi, B., Tesi, M. & Tosin, A. (2017) Microscopic and macroscopic models for the onset and progression of Alzheimer’s disease. J. Phys. A 50(41), 414003, 22.CrossRefGoogle Scholar
Bertsch, M., Franchi, B., Tesi, M. & Tosin, A. (2018) Well-posedness of a mathematical model for Alzheimer’s disease. SIAM J. Math. Anal. 50(3), 2362–2388.CrossRefGoogle Scholar
Braak, H. & Del Tredici, K. (2011) Alzheimer’s pathogenesis: is there neuron-to-neuron propagation? Acta Neuropathol. 121(5), 589–595.CrossRefGoogle ScholarPubMed
Carbonell, F., Iturria-Medina, Y. & Evans, A. (2018) Mathematical modeling of protein misfolding mechanisms in neurological diseases: a historical overview. Front. Neurol. 9(37), 1–16.CrossRefGoogle ScholarPubMed
Chen, C. Y., Tseng, Y. H. & Ward, J. P. (2019) A mathematical model demonstrating the role of interstitial fluid flow on the clearance and accumulation of amyloid β in the brain. Math. Biosci. 317, 108258.CrossRefGoogle Scholar
Chimon, S., Shaibat, M., Jones, C., Calero, D., Aizezi, B. & Ishii, Y. (2007) Evidence of fibril-like β-sheet structures in a neurotoxic amyloid intermediate of Alzheimer’s β-amyloid. Nat. Struct. Mol. Biol. 14(12), 1157–1164.CrossRefGoogle Scholar
Cohen, A. D., Rabinovici, G. D., Mathis, C. A., Jagust, W. J., Klunk, W. E. & Ikonomovic, M. D. (2012) Using Pittsburgh compound b for in vivo pet imaging of fibrillar amyloid-beta. In: Michaelis, E. K. and Michaelis, M. L. (editors), Current State of Alzheimer’s Disease Research and Therapeutics, Advances in Pharmacology, Vol. 64, Academic Press, pp. 27–81.CrossRefGoogle ScholarPubMed
Craft, D., Wein, L. & Selkoe, D. (2002) A mathematical model of the impact of novel treatments on the a beta burden in the Alzheimer’s brain, CSF and plasma. Bull. Math. Biol. 64(5), 1011–1031.CrossRefGoogle ScholarPubMed
Cruz, L., Urbanc, B., Buldyrev, S. V., Christie, R., Gómez-Isla, T., Havlin, S., McNamara, M., Stanley, H. E. & Hyman, B. T. (1997) Aggregation and disaggregation of senile plaques in Alzheimer disease. P. Natl. Acad. Sci. USA 94(14), 7612–7616.CrossRefGoogle ScholarPubMed
Deaconu, M. & Tanré, E. (2000) Smoluchowski’s coagulation equation: probabilistic interpretation of solutions for constant, additive and multiplicative kernels. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29(3), 549–579.Google Scholar
Drake, R. (1972) A general mathematical survey of the coagulation equation. In: Topics in Current Aerosol Research (Part 2), International Reviews in Aerosol Physics and Chemistry, Pergamon Press, Oxford, UK, pp. 203–376.Google Scholar
Dubovskii, P. B. (1994) Mathematical Theory of Coagulation , Lecture Notes Series, vol. 23, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul.Google Scholar
Edelstein-Keshet, L. & Spiross, A. (2002) Exploring the formation of Alzheimer’s disease senile plaques in silico. J. Theor. Biol. 216(3), 301–326.CrossRefGoogle ScholarPubMed
Fornari, S., Schaefer, A., Jucker, M., Goriely, A. & Kuhl, E. (2019) Prion-like spreading of Alzheimer’s disease within the brain’s connectome. J. R. Soc. Interface 159, 20190356.CrossRefGoogle Scholar
Fornari, S., Scher, A., Kuhl, E. & Goriely, A. (2020) Spatially-extended nucleation-aggregation-fragmentation models for the dynamics of prion-like neurodegenerative protein-spreading in the brain and its connectome. J. Theor. Biol. 486, 110102.CrossRefGoogle ScholarPubMed
Franchi, B., Heida, M. & Lorenzani, S. (2020) A mathematical model for Alzheimer’s disease: an approach via stochastic homogenization of the Smoluchowski equation. Commun. math. sci., 18(4), 1105–1134.CrossRefGoogle Scholar
Franchi, B. & Lorenzani, S. (2016) From a microscopic to a macroscopic model for Alzheimer disease: two-scale homogenization of the Smoluchowski equation in perforated domains. J. Nonlinear Sci. 26(3), 717–753.CrossRefGoogle Scholar
Franchi, B. & Lorenzani, S. (2017) Smoluchowski equation with variable coefficients in perforated domains: homogenization and applications to mathematical models in medicine. In: Harmonic Analysis, Partial Differential Equations and Applications, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, pp. 49–67.CrossRefGoogle Scholar
Franchi, B. & Tesi, M. (2012) A qualitative model for aggregation-fragmentation and diffusion of β-amyloid in Alzheimer’s disease. Rend. Semin. Mat. Univ. Politec. Torino 70(1), 75–84.Google Scholar
Friedman, A. & Hao, W. (2016) Mathematical model on Alzheimer’s disease. BMC Syst. Biol. 108(10), 1–18.Google Scholar
Gillam, J. & MacPhee, C. (2013) Modelling amyloid fibril formation kinetics: mechanisms of nucleation and growth. J. Phys. Condens. Matter IOPScience 25(37), 373101.10.1088/0953-8984/25/37/373101CrossRefGoogle ScholarPubMed
Giuffrida, M. L., Caraci, F., Pignataro, B., Cataldo, S., De Bona, P., Bruno, V., Molinaro, G., Pappalardo, G., Messina, A., Palmigiano, A., Garozzo, D., Nicoletti, F., Rizzarelli, E. & Copani, A. (2009) β-amyloid monomers are neuroprotective. J. Neurosci. 29, 10582–10587.CrossRefGoogle ScholarPubMed
Good, T. A. & Murphy, R. M. (1996) Effect of β-amyloid block of the fast-inactivating K+ channel on intracellular Ca2+ and excitability in a modeled neuron. P. Natl. Acad. Sci. USA 93, 15130–15135.CrossRefGoogle Scholar
Greer, M., Pujo-Menjouet, L. & Webb, G. (2006) A mathematical analysis of the dynamics of prion proliferation. J. Theoret. Biol. 242(3), 598–606.CrossRefGoogle ScholarPubMed
Griffin, W., Sheng, J., Royston, M., Gentleman, S., McKenzie, J., Graham, D., Roberts, G. & Mrak, R. (1998) Glial-neuronal interactions in Alzheimer’s disease: the potential role of a cytokine cycle in disease progression. Brain Pathol. 8(1), 65–72.CrossRefGoogle ScholarPubMed
Haass, C. & Selkoe, D. J. (2007) Soluble protein oligomers in neurodegeneration: lessons from the Alzheimer’s amyloid beta-peptide. Nat. Rev. Mol. Cell. Biol. 8(2), 101–112.CrossRefGoogle ScholarPubMed
Hahn, W. (1967) Stability of Motion. Translated from the German manuscript by Arne P. Baartz. Die Grundlehren der mathematischen Wissenschaften, Band 138, Springer-Verlag New York, Inc., New York.CrossRefGoogle Scholar
Helal, M., Hingant, E., Pujo-Menjouet, L. & Webb, G. F. (2013) Alzheimer’s disease: analysis of a mathematical model incorporating the role of prions. J. Math. Biol. 69(5), 1–29.Google ScholarPubMed
Herrero, M. A. & Rodrigo, M. (2005) A note on Smoluchowski’s equations with diffusion. Appl. Math. Lett. 18(9), 969–975.CrossRefGoogle Scholar
Hurd, M. D., Martorell, P., Delavande, A., Mullen, K. J. & Langa, K. M. (2013) Monetary costs of dementia in the United States. New Engl. J. Med. 368(14), 1326–1334.CrossRefGoogle ScholarPubMed
Jack, C. R. Jr., Knopman, D. S., Jagust, W. J., Shaw, L. M., Aisen, P. S., Weiner, M. W., Petersen, R. C. & Trojanowski, J. Q. (2013) Tracking pathophysiological processes in Alzheimer’s disease: an updated hypothetical model of dynamic biomarkers. Lancet Neurol. 12(2), 207–216.CrossRefGoogle ScholarPubMed
Kametani, F. & Hasegawa, M. (2013) Reconsideration of amyloid hypothesis and tau hypothesis in Alzheimer’s disease. New Engl. J. Med. 368(14), 1326–1334.Google Scholar
Karran, E., Mercken, M. & De Strooper, B. (2011) The amyloid cascade hypothesis for Alzheimer’s disease: an appraisal for the development of therapeutics. Nat Rev. Drug Discov. 10(9), 698–712.CrossRefGoogle ScholarPubMed
Laurençot, P. & Mischler, S. (2002) Global existence for the discrete diffusive coagulation-fragmentation equations in L 1. Rev. Mat. Iberoamericana 18(3), 731–745.CrossRefGoogle Scholar
Matthus, F. (2006) Diffusion versus network models as descriptions for the spread of prion diseases in the brain. J. Theor. Biol. 240(1), 104–113.CrossRefGoogle Scholar
Meyer-Luehmann, M., Spires-Jones, T., Prada, C., Garcia-Alloza, M., De Calignon, A., Rozkalne, A., Koenigsknecht-Talboo, J., Holtzman, D. M., Bacskai, B. J. & Hyman, B. T. (2008) Rapid appearance and local toxicity of amyloid-β plaques in a mouse model of Alzheimer’s disease. Nature 451(7179), 720–724.CrossRefGoogle Scholar
Mosconi, L., Berti, V., Glodzik, L., Pupi, A., De Santi, S. & de Leon, M. (2010) Pre-clinical detection of Alzheimer’s disease using FDG-PET, with or without amyloid imaging. J. Alzheimer’s Dis. 20(3), 843–854.CrossRefGoogle ScholarPubMed
Moses, W. (2011) Fundamental limits of spatial resolution in pet. Nucl. Instrum. Methods Phys. Res. A. 648(Supplement 1), S236–S240.CrossRefGoogle ScholarPubMed
Murphy, R. M. & Pallitto, M. M. (2000) Probing the kinetics of β-amyloid self-association. J. Struct. Biol. 130(2–3), 109–122.CrossRefGoogle ScholarPubMed
Nag, S., Sarkar, B., Bandyopadhyay, A., Sahoo, B., Sreenivasan, V., Kombrabail, M., Muralidharan, C. & Maiti, S. (2011) Nature of the amyloid-β monomer and the monomer-oligomer equilibrium. J. Biol. Chem. 286(16), 13827–13833.CrossRefGoogle ScholarPubMed
O’Brien, R. & Wong, P. (2011) Amyloid precursor protein processing and Alzheimer’s disease. Ann. Rev. Neurosci. 34(7), 185–204.CrossRefGoogle ScholarPubMed
Ono, K., Condron, M. M. & Teplow, D. B. (2009) Structure-neurotoxicity relationships of amyloid β-protein oligomers. P. Natl. Acad. Sci. USA 106(35), 14745–14750.CrossRefGoogle ScholarPubMed
Pallitto, M. M. & Murphy, R. M. (2001) Mathematical model of the kinetics of beta-amyloid fibril growth from the denatured state. Biophys. J. 81(3), 109–122.CrossRefGoogle ScholarPubMed
Pareschi, L. & Toscani, G. (2013) Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, UK.Google Scholar
Plog, B. & Nedergaard, M. (2018) The glymphatic system in central nervous system health and disease: past, present, and future. Ann. Rev. Pathol. Mech. Dis. 13, 379–394.CrossRefGoogle ScholarPubMed
Prüss, J., Pujo-Menjouet, L., Webb, G. F. & Zacher, R. (2006) Analysis of a model for the dynamics of prions. Discrete Contin. Dyn. Syst. Ser. B 6(1), 225–235.Google Scholar
Raj, A., Kuceyeski, A. & Weiner, M. (2012) A network diffusion model of disease progression in dementia. Neuron 73(6), 1204–1215.CrossRefGoogle ScholarPubMed
Raj, A., LoCastro, E., Kuceyeski, A., Tosun, D., Relkin, N. & Weiner, M. (2015) Network diffusion model of progression predicts longitudinal patterns of atrophy and metabolism in Alzheimer’s disease. Cell Rep. 10(3), 359–369.CrossRefGoogle ScholarPubMed
Schfer, A., Weickenmeier, J. & Kuhl, E. (2019) The interplay of biochemical and biomechanical degeneration in Alzheimer’s disease. Comput. Methods Appl. Mech. Eng. 352, 369–388.CrossRefGoogle Scholar
Shankar, G. M., Li, S., Mehta, T. H., Garcia-Munoz, A., Shepardson, N. E., Smith, I., Brett, F. M., Farrell, M. A., Rowan, M. J., Lemere, C. A., Regan, C. M., Walsh, D. M., Sabatini, B. L. & Selkoe, D. J. (2008) Amyloid-beta protein dimers isolated directly from Alzheimer’s brains impair synaptic plasticity and memory. Nat. Med. 14, 837–842.CrossRefGoogle ScholarPubMed
Smoluchowski, M. (1917) Versuch einer mathematischen theorie der koagulationskinetik kolloider lsungen. IZ. Phys. Chem. 92, 129–168.Google Scholar
Tatarnikova, O. G., Orlov, M. A. & Bobkova, N. V. (2015) Beta-amyloid and tau protein: structure, interaction and prion-like properties. Biochemistry (Moscow) 80(13), 1800–1819.CrossRefGoogle ScholarPubMed
Urbanc, B., Cruz, L., Buldyrev, S. V., Havlin, S., Irizarry, M. C., Stanley, H. E. & Hyman, B. T. (1999) Dynamics of plaque formation in Alzheimer’s disease. Biophys. J. 76(3), 1330–1334.CrossRefGoogle ScholarPubMed
Walsh, D. M. & Selkoe, D. J. (2007) Aβ oligomers: a decade of discovery. J. Neurochem. 101(5), 1172–1184.CrossRefGoogle Scholar
Weickenmeier, J., Jucker, M., Goriely, A. & Kuhl, E. (2019) A physics-based model explains the prion-like features of neurodegeneration in Alzheimer’s disease, Parkinson’s disease, and amyotrophic lateral sclerosis. J. Mech. Phys. Solids 124, 264–281.CrossRefGoogle Scholar
Wrzosek, D. (1997) Existence of solutions for the discrete coagulation-fragmentation model with diffusion. Topol. Methods Nonlinear Anal. 9(2), 279–296.CrossRefGoogle Scholar
Zou, K., Gong, J. S., Yanagisawa, K. & Michikawa, M. (2002) A novel function of monomeric amyloid-protein serving as an antioxidant molecule against metal-induced oxidative damage. J. Neurosci. 22, 4833–4841.CrossRefGoogle ScholarPubMed