A sensitivity analysis of a mathematical model for the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer's disease
Michiel Bertsch, Bruno Franchi, Valentina Meschini, Maria Carla Tesi, Andrea Tosin
AA sensitivity analysis of a mathematical model for thesynergistic interplay of Amyloid beta and tau on thedynamics of Alzheimer’s disease
Michiel Bertsch , Bruno Franchi , Valentina Meschini , Maria Carla Tesi , andAndrea Tosin Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Roma, Italy Department of Mathematics, University of Bologna, Bologna, Italy Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Torino, Italy
Abstract
We propose a mathematical model for the onset and progression of Alzheimer’s diseasebased on transport and diffusion equations. We treat brain neurons as a continuous mediumand structure them by their degree of malfunctioning. Three different mechanisms are as-sumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomerationof soluble Amyloid beta, ii) effects of phosphorylated tau protein and iii) neuron-to-neuronprion-like transmission of the disease. We model these processes by a system of Smoluchow-ski equations for the Amyloid beta concentration, an evolution equation for the dynamics oftau protein and a kinetic-type transport equation for the distribution function of the degreeof malfunctioning of neurons. The latter equation contains an integral term describing therandom onset of the disease as a jump process localized in particularly sensitive areas of thebrain. We are particularly interested in investigating the effects of the synergistic interplay ofAmyloid beta and tau on the dynamics of Alzheimer’s disease. The output of our numericalsimulations, although in 2D with an over-simplified geometry, is in good qualitative agreementwith clinical findings concerning both the disease distribution in the brain, which varies fromearly to advanced stages, and the effects of tau on the dynamics of the disease.
Keywords:
Alzheimer’s disease, transport and diffusion equations, Smoluchowski equations,numerical simulations
Mathematics Subject Classification:
Alzheimer’s disease (AD) is a progressive neurodegenerative disease characterized by the forma-tion of insoluble protein aggregates and loss of neurons and synapses. This causes a progressivedecline in memory and other cognitive functions, and ultimately dementia. The processes leadingto protein aggregation and neurodegeneration are only partially understood [1]. AD was first de-scribed in 1907 by Alois Alzheimer who reported two pathological hallmarks in the brain: amyloidplaques in the extracellular medium and neurofibrillary tangles (NFTs) inside neurons. It was notuntil eight decades later that the major proteinaceous components of these lesions were identified.Amyloid plaques consist primarily of aggregates of Amyloid beta peptides (A β ) [8], whereas themain constituent of neurofibrillary tangles is tau protein ( τ ) in a hyperphosphorylated form [10].Up to now, A β and τ remain the major therapeutic targets for the treatment of AD. According tothe 2019 World Alzheimer Report, it is estimated that there are presently 50 million people livingwith AD and related disorders, and this figure is expected to increase to 152 million by 2050 due to1 a r X i v : . [ phy s i c s . b i o - ph ] J un n increasingly aged population. As current treatments are purely for modest symptomatic relief,there is an urgent need for realiable and efficient computational models able to provide insightsfor effective therapies for the disease.How A β and τ interact to cause neurodegeneration remains a major knowledge gap in thefield. There is substantial evidence that oligomeric A β has a role in synapse degeneration, bothin computational models and in human postmortem tissues, and that pathological forms of τ aresufficient to induce synapse loss and circuit dysfunction in models of tauopathy. But there is alsoincreasing evidence that the progression of AD dementia is driven by the synergistic interactionbetween A β and τ , which is responsible for synaptic dysfunction, neurofibrillary tangle mediatedneuron loss, and behavioral deficits [22, 17, 7]. β Sequential cleavage of the amyloid precursor protein (APP) by β - and γ -secretase results in thegeneration of a range of A β peptides from 39 to 43 amino acid residues in length, although A β andA β are the predominant species in vivo. The hydrophobic nature of the peptides, particularlyA β and A β , allows them to self-aggregate and form a myriad of species from dimers to smallmolecular weight oligomers, to protofibrils, to fibrils, ultimately leading to their deposition asamyloid plaques. Furthermore, A β peptides can also undergo pyroglutamate modification atamino acid position three (A β β . The mechanism by which excessive A β accumulation occurs in sporadic AD remains unclear. Reduced A β clearance or small increases inA β production over a long period of time are potential mechanisms that result in the accumulationof A β in the brain. There is considerable debate regarding which of the A β species is mostneurotoxic. Increasing evidence suggests that small molecular weight oligomers correlate bestwith the disease and that insoluble amyloid plaques are not toxic [19]. It seems that the mosttoxic A β is identified in A β . τ The τ protein is a microtubule-associated protein (MAP) that is encoded by the MAPT gene. Theprotein contains an amino-terminal projection domain, a proline-rich region, a carboxy-terminaldomain with microtubule-binding repeats, and a short tail sequence. Tau has been reported tointeract with many proteins, serving important scaffolding functions [16]. In particular, it acts inconcert with heterodimers of α - and β -tubulin to assemble microtubules and regulate motor-drivenaxonal transport. In the adult human brain τ occurs in six isoforms, with either three or fourmicrotubule-binding domains, which result from alternative splicing of exons 2, 3 and 10 of theMAPT gene. Tau is enriched inside the neurons. In nature neurons it is largely found in axons;in dendrites it is present in smaller amounts. The distinguishing factor that separates normal τ from that observed in patients with AD is its hyperphosphorylation. The longest isoforms ofhuman τ contain 80 serine and threonine residues and five tyrosine residues, all of which canpotentially be phosphorylated. In the disease state, the amount of hyperphosphorylated τ is atleast three times higher than that in the normal brain [14]. It is not entirely clear, however, whetherphosphorylation of τ at specific sites results in the pathogenicity observed in AD or whether itonly requires a certain overall level of phosphorylation. Nevertheless, hyperphosphorylation of τ negatively regulates the binding of τ to microtubules, which compromises microtubule stabilizationand axonal transport. Hyperphosphorylation also increases the capacity of τ to self-assembleand form aggregates from oligomers to fibrils, eventually leading to its deposition as NFTs .In addition, hyperphosphorylated τ has been shown to interfere with neuronal functions, causingreduced mitochondrial respiration, altered mitochondrial dynamics and impaired axonal transport.Tau pathology progresses through distinct neural networks, and in AD NFTs are prominent inthe cortex in an early stage, and later appear in anatomically connected brain regions.Furthermore, it has been demonstrated that excess τ aggregates can be released into theextracellular medium, to be internalized by surrounding neurons and induce the fibrillization of2ndogenous τ ; this suggests a role for τ seeding in neurodegeneration [11]. Increasing clinical evidence suggests that A β and τ do not act in isolation and that there issignificant crosstalk between these two molecules [18]. Experiments using primary neurons orneuronal cell lines have shown that application of A β oligomers increases τ phosphorylation [21].This suggests a link between A β toxicity and τ pathology, but it is unclear how A β and τ interactin pathological cases. Often τ is placed downstream of A β in a pathocascade, providing supportfor the amyloid cascade hypothesis [9]. Having these considerations in mind the main purposeof the present paper is to provide a reliable mathematical tool for better investigating mutualinteraction mechanisms between A β and τ and assess the resulting effect on the dynamics ofAlzheimer’s disease. In this context we present a multiscale model for the onset and evolutionof AD which accounts for the diffusion and agglomeration of A β peptide, the protein τ and thespreading of the disease. We stress that different spatial and temporal scales are needed to capturethe complex dynamics of AD in a single model: microscopic spatial scales to describe the role ofthe neurons, macroscopic spatial and short temporal (minutes, hours) scales for the descriptionof relevant diffusion processes in the brain, and large temporal scales (years, decades) for thedescription of the global evolution of AD. The mathematical model employed in the present study derives from that extensively described,analyzed and motivated in [2, 4, 5]. Modifications to the original model are made to cope withthe main focus of the present study: the interplay between A β and τ .We identify a portion of cerebral tissue by an either two- or three-dimensional bounded setΩ. The space variable is denoted by x . Two time scales are needed to describe the evolution ofthe disease over a period of years: a short (i.e., rapid) s -scale, where the unit time coincides withhours, for the diffusion and agglomeration of A β [16]; and a long (i.e., slow) t -scale, where theunit time coincides with several months, for the progression of AD. Introducing a small constant0 < ε (cid:28)
1, the relationship between these two time scales can be expressed as t = εs. (1)On the whole, the model that we propose for the present study reads as follows: ∂ t f + ∂ a ( f v [ f ]) = J [ f ] in Ω × [0 , × (0 , T ] ε∂ t u − d ∆ u = − αu U + F [ f ] − σ u in Q T = Ω × (0 , T ] ε∂ t u − d ∆ u = α u − αu U − σ u in Q T ε∂ t u = α (cid:88) ≤ j + k< u j u k in Q T ∂ t w = C w ( u − U w ) + + (cid:90) Ω h w ( | y − x | ) w ( y, t ) dy in Q T , (2a)(2b)(2c)(2d)(2e)where in (2b), (2c) we have set U := (cid:88) j =1 u j for conciseness.Equations (2b), (2c), (2d) are meant to describe the aggregation of A β . In particular, (2b)and (2c) are compartmental Smoluchowski-type equations with diffusion, agglomeration and cleav-age. In particular: 3i) u ( x, t ) is the density of monomers in the point x ∈ Ω at time t > u ( x, t ) is the cumulative density of soluble oligomers, which are regarded collectively as asingle compartment;(iii) u ( x, t ) is the density of senile plaques.Such a compartmental model, which in particular does not distinguish the densities of thesoluble oligomers based on their length, is justified by the fact that, according to the literature,there is no clinical evidence on the maximum length of toxic oligomers [12]. Therefore, any precisevalue of such a length would be partly arbitrary. Although this may seem an over-simplification,in fact it is not because, as far as the results of the model are concerned, considering a moredetailed description of the soluble oligomers with their precise lengths, such as e.g. in [2], hasproved not to add significant information [3, 6]. The coefficient ε in front of the time derivativesis due to the relationship (1), in particular to the fact that, on the longer time scale t , the rateat which agglomeration and diffusion of A β take place is as high as ε . Equation (2d) for thefibrils is written in the same spirit as the previous two ones, except for the fact that fibrils areassumed not to move. Thus the equation for their concentration u does not feature a diffusionterm. We include in this compartment all the combinations of monomers and oligomers producingA β entities other than those comprised in the compartments 1 and 2, taking further into accountthat senile plaques do not aggregate with each other. Finally, consistently with the compartmentalnature of the model, we take the coagulation parameters constant and equal to α >
0, neglectingthe fact that they may feature a dependence on the specific lengths of the aggregating oligomers,see [2].Equation (2a) is instead a kinetic-type equation which describes the progression of the disease.Roughly speaking, f = f ( x, a, t ) is the probability density of the degree of malfunctioning a ∈ [0 ,
1] of neurons located in x ∈ Ω at time t > f ( x, a, t ) da represents the fractionof neurons in x which at time t have a degree of malfunctioning comprised between a and a + da .For a precise mathematical formulation in terms of probability measures, see [5].We assume that a close to 0 stands for “the neuron is healthy” whereas a close to 1 stands for “the neuron is dead”.The progression of AD occurs on the longer time scale t , over decades, and is determined bythe deterioration rate v [ f ] = v [ f ]( x, a, t ), which we assume to have the following form: v [ f ]( x, a, t ) = C G (cid:90) ( b − a ) + f ( x, b, t ) db + C S (1 − a ) (cid:0) u ( x, t ) − U (cid:1) + + C W (1 − a ) w ( x, t ) . (3)The integral term describes the propagation of AD among close neurons. The second term modelsinstead the action of toxic A β oligomers, leading ultimately to apoptosis. The threshold U > β needed to damage neurons. Finally, C G , C S > τ , whosedensity in the point x ∈ Ω at time t > w ( x, t ). Specifically, this term assumes thatsuch a toxicity is proportional to the concentration w through a proportionality constant C W > τ on them.The term J [ f ] = J [ f ]( x, a, t ) on the right-hand side of (2a) describes the possible randomonset of AD in portions of the domain Ω as a result of a microscopic stochastic jump process. Thelatter takes into account the possibility that the degree of malfunctioning of neurons randomlyjumps to higher values due to external agents or genetic factors. The explicit expression of thisterm is J [ f ]( x, a, t ) = η (cid:90) P ( t, x, a ∗ → a ) f ( x, a ∗ , t ) da ∗ − f ( x, a, t ) , (4)where P ( t, x, a ∗ → a ) denotes the probability that the degree of malfunctioning of neurons in thepoint x ∈ Ω jumps at time t > a ∗ to a > a ∗ . The coefficient η > v [ f ], J [ f ], may be obtained from a mesoscopic description of a microscopic model of neuron-to-neuroninteractions as shown in [4].Equation (2e) models the production and diffusion dynamics of the phosphorylated τ withdensity w ( x, t ). Specifically, its production is linked to the presence of toxic A β oligomers andfurthermore its diffusion in the brain is assumed to happen according to a sort of prion-likemechanism. The first term on the right-hand side models the activation of the phosphorylated τ induced by toxic A β oligomers, as soon as their concentration is above a certain threshold U w >
0. If it is not, the phosphorylated τ is known not to activate [15]. The parameter C w > τ in possibly distant points of the brain according tothe spatial kernel h w . In the next Section 2.2 we will investigate in detail some properties of thisintegral term, which will further elucidate its physical meaning. We assume that the dynamics of τ take place on the slow time scale t . In the absence of precise indications from the biomedicalliterature, this choice seems reasonable in view of the fact that τ is especially involved in theprogression of the disease rather than in the A β agglomeration and the consequent formation ofsenile plaques.To conclude the presentation of the model, we mention that the term F [ f ] = F [ f ]( x, a, t )in (2b) describes the production of A β monomers by neurons, taking into account that, up to acertain extent, damaged neurons increase such a production. In view of these considerations, wechoose F [ f ]( x, a, t ) = C F (cid:90) ( µ + a )(1 − a ) f ( x, a, t ) da. Here, the small constant µ > β production by healthy neurons while the factor1 − a expresses the fact that dead neurons do not produce amyloid. As usual, C F > As far as boundary conditions are concerned, we assume that ∂ Ω consists of two smooth disjointparts, say ∂ Ω and ∂ Ω , being ∂ Ω the outer boundary which delimits the considered portionof cerebral tissue and ∂ Ω the inner boundary of the cerebral ventricles. On ∂ Ω we prescribeclassical no-flux conditions for all the concentrations. Conversely, on ∂ Ω we prescribe a Robincondition for the concentrations of the A β oligomer mimicking their removal by the cerebrospinalfluid through the choroid plexus [13, 20]. On the whole, we have then: ∇ u i · n = 0 on ∂ Ω , i = 1 , ∇ u i · n = − βu i on ∂ Ω , i = 1 , ∇ w · n = 0 on ∂ Ω , (5)where β > n the outward normal unit vector to ∂ Ω. We alsocomplement system (2) with a proper set of initial conditions: f ( x, a,
0) = f ( x, a ) , u i ( x,
0) = u ,i ( x ) ( i = 1 , , , w ( x,
0) = 0 . (6)A numerical discretisation of the initial/boundary-valued problem (2)-(5)-(6) can be set upstraightforwardly by adapting the one described in detail in [2]. τ In this section, we want to explore the effect of the non-local term I ( x, t ) := (cid:90) Ω h w ( | y − x | ) w ( y, t ) dy
5n (2e) with Ω ⊂ R n ( n = 2 , h w : R + → R + is such that h (cid:48) w (0 + ) = 0. Furthermore, we assume that h w is compactly supported in [0 , R ] ⊂ R , where R > τ is supposed to be effective. Hence I ( x, t ) = (cid:90) B R ( x ) h w ( | y − x | ) w ( y, t ) dy. Assume that R is sufficiently small, so that y is close to x , and that the functions h w , w aresmooth enough about 0 + and x , respectively. By Taylor expansion we find: h w ( | y − x | ) = h w (0 + ) + 12 h (cid:48)(cid:48) w (0 + ) | y − x | + o ( | y − x | ) w ( y, t ) = w ( x, t ) + n (cid:88) h =1 ∂ x h w ( x, t )( y h − x h )+ 12 n (cid:88) h,k =1 ∂ x h x k w ( x, t )( y h − x h )( y k − x k ) + o ( | y − x | ) , thus h w ( | y − x | ) w ( y, t ) = h w (0 + ) w ( x, t ) + h w (0 + ) n (cid:88) h =1 ∂ x h w ( x, t )( y h − x h )+ 12 h w (0 + ) n (cid:88) h,k =1 ∂ x h x k w ( x, t )( y h − x h )( y k − x k )+ 12 h (cid:48)(cid:48) w (0 + ) w ( x, t ) | y − x | + o ( | y − x | ) . Neglecting the remainder o ( | y − x | ), we can therefore approximate I ( x, t ) locally as I ( x, t ) ≈ h w (0 + ) w ( x, t ) (cid:90) B R ( x ) dy + h w (0 + ) n (cid:88) h =1 ∂ x h w ( x, t ) (cid:90) B R ( x ) ( y h − x h ) dy + 12 h w (0 + ) n (cid:88) h,k =1 ∂ x h x k w ( x, t ) (cid:90) B R ( x ) ( y h − x h )( y k − x k ) dy + 12 h (cid:48)(cid:48) w (0 + ) w ( x, t ) (cid:90) B R ( x ) | y − x | dy. Let ω n denote the volume of the unit ball in R n . By switching to polar coordinates, we find (cid:90) B R ( x ) dy = ω n R n , (cid:90) B R ( x ) ( y h − x h ) dy = 0 , (cid:90) B R ( x ) ( y h − x h )( y k − x k ) dy = ω n n + 2 R n +2 if h = k h (cid:54) = k, (cid:90) B R ( x ) | y − x | dy = nω n n + 2 R n +2 , whence finally I ( x, t ) ≈ ω n R n (cid:18) h w (0 + ) + n n + 2) h (cid:48)(cid:48) w (0 + ) (cid:19) w ( x, t ) + ω n R n +2 n + 2) h w (0 + )∆ w ( x, t ) . I ( x, t ) to the spreading of the phosphorylated τ consists in a source term proportional to the quantity of τ already present in the site x and in alinear diffusion.The local approximation of I ( x, t ) elucidates also the role of the pointwise values of h w , h (cid:48)(cid:48) w in r = 0. If, for instance, we take h w ( r ) = 1 ω n R n χ [0 , R ] ( r ) , which makes I ( x, t ) a uniform average of w in a neighbourhood of x of radius R , then h w (0 + ) = ω n R n and h (cid:48)(cid:48) w (0 + ) = 0, whence I ( x, t ) ≈ w ( x, t ) + R n + 2) ∆ w ( x, t ) . This implies that, at the leading order in R , I consists in a pointwise source and, at higher order,in a diffusion term responsible for the spatial spreading of the phosphorylated τ . In this section we present and discuss the numerical results of the mathematical model forAlzheimer’s disease described in Section 2. We show the evolution of AD in two different modelingschemes: the first does not take into account the role of τ , whereas the second includes τ . Weshow spatial plots of the evolution of the disease, expressed by the quantity f in (2a), as well asspatial plots of the concentration of the toxic polymers, plaques and (phosphorylated) τ .As a consequence of the compartmental description of the A β oligomers, we find that the nu-merical results are quite sensitive to the constant U in (3), which we will therefore tune accuratelyso as to observe in practice an evolution of the disease. In particular, a too high value of U mayeasily hinder the effective toxicity of the cumulative density u of A β , thereby preventing theoccurrence of the dynamics of AD. As a matter of fact, since the phosphorylated τ contributes tothe overall toxicity [17], the spread of the disease may also occur for a relatively high value of U .In order to compare our numerical results corresponding to the two set-ups with and without τ ,we test different cases with U = 0 .
01 and U = 0 . U , the equation for τ (see (2e)) contains another threshold constant U w to whichour model results very sensitive, since, out of a small range around U w = 0 .
01, the numericalsimulations suggest that the interaction between A β and τ is not significant for the evolution ofthe disease.. Indeed, below U w = 0 . τ is excessively predominant, whereas, above the same orderof magnitude the phosphorylation process is not triggered. Thus in all our simulations we keep U w fixed, taking U w = 0 .
01. In both simulation set-ups (with and without τ ) we decide to carryout a sort of sensitivity analysis of the model outcomes to some parameters which we identify asrepresentative and in particular with clinical implications. First of all, we consider the constant β , which enters the model through condition (5) at the boundary of the cerebral ventricles. Smallvalues of β correspond to the assumption that a small amount of A β is removed from the CSFthrough the choroid plexus. To test the sensitivity of the outcomes, we take for β two values: β = 0 .
01 and β = 1. Successively, another significant parameter is α (the coagulation parameterin (2b), (2c)). We decide to set α at different order of magnitude to observe how it affects thedynamics of A β in the brain. We run a simulations campaign (both with and without inclusion of τ ) varying α , β and U but keeping fixed all the other parameters (see the captions in the sequel)which chosen values are reported in Table 1. We wish to point out that these values represent anorder of magnitude and have a proper unit measure that can be deduced from the equations inwhich they are involved.In Table 2 we give a schematic and complete overview of all the cases that have been testedin numerical simulations. Based on the resulting outcomes concerning how parameters and τ inclusion affect the overall dynamics of the disease, we aim to obtain insights on the disease,hopefully with applications to treatments and clinical procedures.7able 1: Values of the fixed parameters C G C S D σ µ C F C W U w − − − −
10 10 10 − Table 2: All the simulated cases and involved parameters with respective valuesInclusion of τ α β U
Case 1 No 10 1 10 − Case 1.1 No 10 1 10 − Case 2 Yes 10 1 10 − Case 2.1 Yes 10 10 − − Case 2.2 Yes 1 1 10 − Case 2.3 Yes 10 1 10 − α = 10, β = 1 and no inclusion of τ (Figure 1) The first case we simulate employs the same mathematical model for Alzhaiemer’s disease presen-ted in [2] without τ in the model but, as mentioned above, assuming N = 3. As a consequence,the only toxic elements in the model are the dimers and the overall toxicity level is lower withrespect to the same case in [2], in which N = 50. The toxicity level enters the formula for thedegree of malfunctioning of the brain thus providing information on the health state of the brain.The parameter α , which regulates the coagulation of monomers and dimers, is taken equal to 10,while β , which refers to portion of toxic oligomers which is extracted from the brain, is equal to1. As previously pointed out we choose the threshold value for the toxic u equal to 10 − whichturns out to be with these set-up values the minimum order of magnitude to allow the disease totake place. The numerical outputs are shown in Figure 1, where the spatial plots of the degreeof malfunctioning f at different times are plotted. The rectangle, which is the numerical domain,represents a two dimensional section of the brain where the upper part the frontal region, thelower part is the the occipital region and the two inner rectangles are the ventricles. The diseasestarts from some random sources (the blue spot) and as time passes, owing to the neural infection,it propagates on other portions of the brain. It continues to spread until we reach a numericalequilibrium configuration in the sense that, even if we allow the time to increase, the spatial plotof f does not change. In this “stable” state we can see that there is a blue zone (dead part)but the brain is not completely ill. This result is clearly different from that obtained in [2], inwhich the whole brain is blue thus meaning basically that the patient is dead. The first questionis: why does this happen in this case? The answer is that, having a coagulation factor α = 10,the monomers aggregate to form dimers but remain briefly in this toxic state and then evolve toplaques, which are non toxic. Therefore the toxicity of the whole system is much lower. Thisis linked also to another question that is: why with this set-up the system reaches a “numericalsteady state” that in [2] is not observed? This is because at a certain point the concentrationof toxic oligomers u goes below the threshold level, because toxic oligomers quickly agglomerateinto plaques.This prevents the dynamics of the degree of malfunctioning from evolving further.8 ) b)T=100 T=600 Figure 1: Spatial plots of the degree of malfunctioning of the brain for α = 10, β = 1, U = 10 − and without τ at two time instants: a) T = 100, b) T = 600 a) b)T=100 T=10000 Figure 2: Spatial plots of the degree of malfunctioning of the brain for α = 10, β = 1, U = 0 . τ at two time instants: a) T=100, b) T=10000. α = 10 , β = 1 , no inclusion of τ and increased toxicitythreshold (Figure 2) The same numerical set-up of Case 1 of Table 1 is considered here, while increasing the value of U to 0 . τ and with U = 10 − the degree of malfunctioning of the brain does not evolvein time meaning that the disease is not able to spread, as shown in Figure 2. α = 10, β = 1 and inclusion of τ (Figures 3, 4) Here we exploit the novelty of this paper with respect to [2], since we include in the model thetoxic effects of τ . All parameters are the same as in Case 1 of Table 1, i.e. we keep α = 10,9 ) b)T=100 T=600 Figure 3: Spatial plots of the degree of malfunctioning of the brain for α = 10 , β = 1 and U = 0 . τ at two times: a) T=100, b) T=600. a) b)T=100 T=600 Figure 4: Spatial plots of τ for α = 10, β = 1 and U = 0 .
01 at two time instants: a) T=100, b)T=600. β = 1 and U = 0 .
01. In terms of the mathematical model we add an equation describing thediffusion in time of τ and also its toxic effects on the degree of malfunctioning of the brain asdescribed in section 2.1. In figure 3 the spatial plots of the degree of malfunctioning of the brainshow that a ”numerical steady state” is reached. The dynamics is different from that of Case 1since it starts as in Case 1 from some random sources but then evolves following the spreadingtrend of τ concentration. The evolution of the disease appears faster and the spatial localizationof completely ill regions (the blue regions) is larger with respect to the disease dynamics analyzedin Case 1.Also in this case we observe a sort of ”localization” of Alzheimer’s disease at the steady state.Looking at the spatial plots of f and τ (see Figures 3 and 4), the disease is not diffused in all thebrain but only on a confined part of it. In fact, in this case the level of toxic oligomers u goesbelow the threshold level U = 0 .
01, thus avoiding the disease to continue spreading and damaging10 ) b)T=100 T=600
Figure 5: Spatial plots of the degree of malfunctioning of the brain for α = 10, β = 1, U = 0 . C G = 0 at two times: a) T=100, b) T=600.the whole brain. α = 10, β = 1, inclusion of τ and no neural infection (Figure 5) In addition to what observed in Case 2 we aim to isolate the effects of the inclusion of τ onthe numerical results. Thus we ignore the neuron-to-neuron infection mechanism, i.e we put theconstant C G of Section 2 equal to zero. We decide to select the same set-up of the parameters as inCase 2.The spatial plot of the degree of malfunctioning, shown in Figure 5, turns out to be differentfrom that of Case 2 : the disease is less diffused in the cerebral parenchyma, and brain damagesturn out to be localized in some specific portion of the brain in which τ is concentrated. Moreover,the brain is divided into two separated regions: the occipital part that is totally damaged (blueregion) and the rest that is completely healthy (red region). In conclusion avoiding the prion-likeinfection mechanism between neurons strongly reduces the brain damages, thus underlying thehigher toxic effects of synergistic interactions between A β and τ when compared to their singletoxic effects. α = 10, β = 0 .
01 and inclusion of τ (Figures 6, 7) In this simulation we use the same set of parameters values as in Case 2, except for the value of theparameter β that we set here equal to 0 .
01, meaning that less toxic oligomers are extracted fromthe brain with respect to Case 2. As for the cases described before we report in Figure 6 somesnapshots of spatial plots of the degree of malfunctioning at two different instants. The overalldynamics of the spreading of the disease is similar to that described in Case 2, but the numericalsteady state is reached faster and the completely damaged region of the brain is larger. This resultappears reasonable since if less toxic elements are removed, then the degree of malfunctioning turnsout to be higher. In agreement to what observed in Case 2, it is found that the spatial plots ofthe degree of malfunctioning reflects that of τ (see Figure 7). Indeed, τ in the present simulationis spread in a wider region. α = 1, β = 1 and inclusion of τ (Figures 8, 9) Here we use the same simulation set-up and parameters values as in Case 2, except for the coagu-lation parameter α that is set equal to 1. This means that the aggregation of monomers is slower11 ) b)T=10 T=100 Figure 6: Spatial plots of the degree of malfunctioning for α = 10 , β = 0 . U = 0 .
01 andinclusion of τ at two times : a) T=10, b) T=100. a) b)T=10 T=100 Figure 7: Spatial plots of τ for α = 10, β = 0 . U = 0 .
01 at two time instants: a) T=10, b)T=100.with respect to Case 2. The numerical outputs of this case are shown in Figure 8, where the spatialplots of the degree of malfunctioning of the brain are plotted at two different instants. Unlike allthe cases seen so far, here the disease spreads faster into the whole cerebral parenchyma, until thebrain is completely damaged. Moreover, looking at the spatial distribution of τ , see Figure 9, it isclear that, unlike the previous simulations, τ spreads very early on the whole brain, acceleratingthe spread of the disease and the complete damaging of the brain. α = 10, β = 1, inclusion of τ and increased toxicity threshold(Figures 10, 11) In all previous simulations the threshold value U for the toxic oligomers u remains constant andequals 0 .
01 which, as stressed in section 3, is the minimum value which allows the disease to start12 ) b)T=10 T=150
Figure 8: Spatial plots of the degree of malfunctioning for α = 1 , β = 1, U = 0 .
01 and inculsionof τ at two times : a) T=10, b) T=150. a) b)T=10 T=150 Figure 9: Spatial plots of τ for α = 1, β = 1 and U = 0 .
01 at two time instants: a) T=10, b)T=150.spreading when we do not include τ in the model. In this section we want to assess the effectson the disease evolution when we increase the threshold U including τ in the model. In the nextsimulation we use the same parameters set-up as in Case 2, except for the toxicity threshold thatis set to U = 0 .
1. We make this choice of the parameters since it is the one yielding a numericalsteady state in which the brain is not completely damaged, thus possibly giving clinical insightson intervention measures against the disease.The numerical outcomes of this simulation at the numerical stationary state are shown inFigures 10b) and 11b), respectively for the spatial plot of the degree of malfunctioning and thespatial plot of τ . Both pictures display a trend that is similar to that observed in Case 2 with alocalization of the portion of damaged brain. However, since the threshold value U = 0 . u is higher than the threshold value U = 0 .
01 of Case 2, it happens that u goes under U muchearlier than in Case 2. This finding translates into a spatial plot for the degree of malfunctioning13 ) b)T=100 T=10000 Figure 10: Spatial plots of the degree of malfunctioning for α = 10 , β = 1, U = 0 . τ at two times: a) T=100, b) T=10000. a) b)T=100 T=10000 Figure 11: Spatial plots of τ for α = 1, β = 1 and U = 0 . In this section we compare and discuss the numerical results of the simulated cases extensivelydescribed in Section 3 with the aim of pointing out the effects of varying the parameters focusof performed sensitivity analysis. The first insight we extrapolate from our analysis concernsthe coagulation parameter α standing for the velocity at which the monomers and oligomersaggregate. We find out that setting α at least equal to 10 leads to a numerical stationary stateof the system that does not correspond to a full damage of the brain, as in [2]. Depending onthe remaining parameters of the model, we find different spatial localization of the damages,14anging from small regions to larger ones, but in none of the cases considered the whole brain getscompletely damaged. This result can have relevant clinical implications: it suggests the possibilityof a drug therapy aiming to increase the aggregation rate of monomers and oligomers in order tocontrol and localize the deterioration of the white matter. Another finding concerns the role of β : keeping fixed all other parameters, the higher the value of β we set, the lower is the resultingdegree of malfunctioning of the brain (in both cases, with or without the inclusion of τ in themodel). This is in agreement with the clinical evidence that an efficient clearance mechanism cancontrol the brain damages produced by the combined effect of toxic A β and τ . Finally, comparingthe cases with and without τ and keeping all other parameters fixed, we clearly observe that theevolution of the disease is much faster and the damaged brain region larger when we include τ inthe mode. Looking at the the spatial plots of the degree of malfunctioning this fact is evident. Inconclusion, τ interacts with A β accelerating the pathological dynamics of AD. Acknowledgements
The authors would like to express their gratitude to MD Norina Marcello for many stimulatingand fruitful discussions over several years.B.F. and M.C.T. are supported by the University of Bologna, funds for selected research topics,by MAnET Marie Curie Initial Training Network and by GNAMPA of INdAM (Istituto Nazionaledi Alta Matematica “F. Severi”), Italy.M.B. thanks the project Beyond Borders (CUP E84I19002220005) of the University of RomeTor Vergata. M.B. and V.M. acknowledge the MIUR Excellence Department Project awarded tothe Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.A.T. acknowledges support by the Italian Ministry of Education, University and Research(MIUR) through the “Dipartimenti di Eccellenza” Programme (2018-2022) – Department of Math-ematical Sciences “G. L. Lagrange”, Politecnico di Torino (CUP:E11G18000350001) and throughthe PRIN 2017 project (No. 2017KKJP4X) “Innovative numerical methods for evolutionary par-tial differential equations and applications”. This work is also part of the activities of the StartingGrant “Attracting Excellent Professors” funded by “Compagnia di San Paolo” (Torino) and pro-moted by Politecnico di Torino. Moreover, A.T. acknowledges membership of GNFM (GruppoNazionale per la Fisica Matematica) of INdAM (Istituto Nazionale di Alta Matematica), Italy.
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