Development of Aβ and anti-Aβ dynamics models for Alzheimer’s disease

2.1 Mathematical model of the growth of Aβ

This research began with developing a model based on equations proposed by Craft et al. [8]. The Aβ aggregation process is modeled by selecting the nucleation phase, where the process occurs more slowly. This makes it possible to consider the concentrations of monomer (M ₁), dimer (M ₂), trimer (M ₃), and so on. The next process builds oligomer (O) and fibril (P) models by considering them as the average values of O and P.

The first assumption is that an oligomer forms at least n + 1 monomers, resulting in an equation containing n + 2 components, namely, monomers (M ₁), dimers (M ₂), trimers (M ₃), …, M n , oligomers (O), and fibrils (P). The second assumption is that aggregation occurs through the addition of monomers. To simplify the model, the separation of monomers from aggregates is not considered. This is because the separation process is slower than the aggregation process. The above assumptions can be expressed in the following equations:

where O a and P a are the average sizes of the oligomers and fibrils, respectively. The above equation describes the primary nucleation process with the assumption that the number of monomers composing the aggregate is stable.

Because monomers are produced in the brain, the third assumption is that the production process can be represented by the saturation function f ( M 1 ) M 1 , where f ( M 1 ) M 1 = δ M 1 1 − M 1 γ with δ indicating the average growth of monomers and γ indicating the carrying capacity. The choice of this function is because the concentration of monomers cannot increase into larger structures if the monomers are not bound to each other. This function selection applies to the general case of a nonlinear f ( M 1 ) that satisfies the conditions f ( 0 ) = δ > 0 and f ′ ( M 1 ) < 0 . The fourth assumption is the existence of Aβ stack clearance denoted as μ i .

Explanation of the assumptions used in the simulation model:

1. Oligomer formation: The formation of oligomers begins stepwise from monomers (M ₁), dimers (M ₂), trimers (M ₃), and so on, consisting of at least n + 1 monomers. The entire process involves n + 2 components, including monomers, dimers, trimers, and oligomers. This assumption is used to explain the components involved in the molecular aggregation process from single monomers to the formation of larger structures such as oligomers and fibrils.

2. Aggregation process: The aggregation into larger structures occurs through the gradual addition of monomers, such as the first monomer joining with the second to form a dimer. This dimer can then add a third monomer to form a trimer, and so on.

3. Monomer production rate: The third assumption indicates that the rate of monomer production depends on the concentration of the monomers themselves, meaning there is an initial rapid growth when the monomer concentration is low, but the growth rate slows down and eventually stops when the maximum capacity is reached.

4. Degradation factor ( μ i ): The degradation factor ( μ i ) indicates that there is a mechanism for clearing Aβ stacks during the growth of Aβ. This assumption is important to describe the dynamic balance between the formation and clearance of Aβ stacks in the system.

Meanwhile, the assumptions used can be written in the following differential equations:

The factors ( O a − n ) and ( P a − O a ) in each monomer equation represent the average number of monomers that need to be added to the aggregate M n to form an oligomer and the average number of monomers that need to be added to the oligomer to form a fibril, respectively. Both factors are added to ensure the conservation of mass.

The selection of initial conditions and parameter values is crucial in the development of the Alzheimer’s dynamic model because it can influence the simulation results. Equation (3) indicates the parameter values and initial conditions used in the simulation.

Naturally, the parameters involved in the Aβ aggregation process have a very wide range of values. Based on this, the values of K 1 until K 6 in Table 1 are determined to illustrate the aggregation growth curve of Aβ. However, the values of K 1 − K 6 in Table 1 may not represent the actual process values. The illustrative curve of the Aβ aggregation growth process is formed to understand the growth pattern of Aβ.

The estimation of the parameter K o given in Table 1 is based on the addition of one monomer ( M 1 ) because the K o elongation rate in this model involves the addition of P a − O a monomers. We modified this estimate based on the relative sizes of fibrils and oligomers (O) [7,8,26]. The estimation of O a and P a values is based on studies of monomer binding characteristics up to fibril formation.

The degradation rate ( μ n ) depends on the size of the molecule. The larger the size, the slower the degradation rate. Based on this, the degradation value is chosen from the relative size ratio between monomers ( M 1 ) and fibrils (P) [8,14].

In the initial condition, the value of δ is chosen to be 50 because it would represent the level of aggregation or polymerization of Aβ monomers into oligomers or fibrils, which is influenced by monomer concentration. The value of γ is set to 75 because the intracellular transport system is assumed to be efficient enough to move Aβ monomers from one location to another. The value n = 6 is selected because the minimum formation to become an oligomer is n = 5 or more. M ₁ is valued at 10 because it would represent the initial concentration for the Aβ growth process. For M ₂ until P, the initial value of concentration is set to zero as these molecules are yet to be produced.

2.2 Mathematical model of the immune system

Hao and Friedman [14] developed a mathematical model for the role of microglia and astrocytes in Alzheimer’s disease. There are two phenotypes of active microglia: proinflammatory ( M i pi ) and anti-inflammatory ( M i ai ) phenotypes.

The equation for the immune system includes proinflammatory ( M i pi ) and anti-inflammatory ( M i ai ) microglia can be written as follows:

So, the model of the immune system can be written as follows:

where ε 1 = T α T α + K T α , ε 2 = I 10 I 10 + K I 10 , and β ε 1 β ε 1 + ε 2 indicate the ratio of microglia becoming M a pi macrophages, and ε 2 β ε 1 + ε 2 indicates the ratio of microglia becoming M a ai macrophages. The phenotype M i pi is under the control of TNF-α and M i ai is under the control of IL-10. Table 2 shows the parameter values used by microglia as the immune system in the brain.

2.3 Mathematical model of drug therapy

Hao and Friedman [14] developed a mathematical model for the use of Alzheimer’s drug therapy. One of the drugs used is Aducanumab. Aducanumab is chosen as an anti-Aβ agent. The use of anti-Aβ drugs can be described by the below equation:

The parameter h represents the dosage of the drug, and the value is 10. We used 10 because based on the clinical test of the drug. N represents the number of live neurons, N 0 is the mass density of neurons, A represents astrocytes, and A 0 represents the mass density of astrocytes, with the values used for all parameters being 0.14 g mL − 1 (Table 3).

2.4 Connection of the model

In this subsection, the integration of the immune system model (2.2) and the drug therapy model (2.3) into the core Aβ growth model (2.1) will be shown. The equations in Sections 2.2 and 2.3 will serve as “modifier” equations for the core model; hence, the Aβ growth would also modified. The modification is performed by incorporating the equations from Sections 2.2 and 2.3 into the equation d M 2 ( t ) d t in the Aβ growth model. This modification will affect d M 3 ( t ) d t up to d P ( t ) d t . This is because the Aβ growth model equations are coupled differential equations. The reason for modifying only d M 2 ( t ) d t is that Aβ monomers (M ₁) are small in size and non-toxic and thus are not yet detected by the immune system and drugs [8,14].

The results of the modifications to the core Aβ growth model are as follows:

In this study, we aimed to integrate mathematical models derived from equations (5) and (6) into equation (2) to obtain three curves: The role of the immune system, the application of drug therapy combined with the immune system, and a model without both as a control. The simulation results include the concentrations of monomer Aβ transforming into fibrils and a comparison of the concentrations of monomeric Aβ transforming into fibrils in the immune system and drug therapy models over time.

3.1 Simulation results of the growth of Aβ

The simulation results show the concentration growth of Aβ over time, from the monomeric form (M ₁) to the fibrils form (P), as depicted in Figure 2. Figure 2 shows that the concentration of Aβ increased from M ₁ to P. The M ₁ curve indicates that on day 0, the monomer already had a high concentration of 73.8 g mL − 1 . This is attributed to the kinetics of Aβ aggregation. The transition of Aβ monomers to dimers (M ₂), trimers (M ₃), tetramers (M ₄), M ₅, M ₆, oligomers (O), and fibrils (P) can occur within a specific concentration range. From day 0 until day 180, the concentration of Aβ monomers remained constant. This finding indicates the relationship between Aβ production and degradation processes. These processes may lead to kinetic dynamics resulting in a constant concentration of Aβ monomers over a specific period. If the concentration of Aβ produced exceeds the degradation capacity, aggregation by Aβ monomers will occur. Aβ monomers join to form dimers, trimers, tetramers, oligomers, and fibrils. As a result, there will be an equilibrium between soluble monomers and aggregated forms.

Figure 2

The Aβ growth concentrations ( g mL ‒ 1 ) for M ₁, M ₂, M ₃, M ₄, M ₅, M ₆, O, and P over 180 days (red line = M ₁ and M ₅, green line = M ₂ and M ₆, blue line = M ₃ and O, and magenta line = M ₄ and P).

The laws of thermodynamics state that the Gibbs free energy will reach a minimum value. In this case, the equilibrium between Aβ monomers and aggregates results from the lowest change in free energy [3,34,44]. The concentrations of M ₂, M ₃, M ₄, M ₅, M ₆ slightly increased and reached saturation after day 150

The saturation of the Aβ concentration indicates that the amount of Aβ that can dissolve in the brain reaches the maximum. When the Aβ concentration saturation, it aggregates to form plaques in the brain [38]. Aβ plaques have a denser structure, making them difficult to clear.

Moreover, for M ₂, M ₃, M ₄, M ₅, M ₆, and O, the Aβ concentration significantly increased from day 0 to day 90, then increased slowly until day 150 and stabilized until day 180. The concentration of the fibrils (P) increased until day 180 because the fibrils tended to be more stable and more difficult to break down than the other Aβ structures. Therefore, the concentration of fibrils tends to increase in the brain over time.

Table 4 shows the Aβ growth concentration values over 180 days at intervals of 30 days. According to Table 4, there was a difference in the concentrations of Aβ in M ₂, M ₃, M ₄, M ₅, M ₆, and O on the same day. This demonstrates that the growth concentrations of M ₁, M ₂, M ₃, M ₄, M ₅, M ₆, and O increase daily over the course of 180 days, but the difference in concentration between M ₁, M ₂, M ₃, M ₄, M ₅, M ₆, and O decreases at the same time.

The kinetic process of Aβ begins with Aβ monomers, which are normal products of APP cleavage. Monomers have the potential to undergo association through noncovalent interactions. Subsequently, the monomers undergo the nucleation phase, which involves a critical concentration of monomers for nucleus formation. The initiation point of nucleation starts with the formation of nuclei, which serve the purpose of forming small aggregates. Once the nuclei are formed, Aβ monomers will continue to undergo aggregation until oligomers are formed [39].

Oligomers can evolve into protofibrils with larger structures. Protofibrils represent the transitional phase between oligomers and fibrils. Subsequently, protofibrils undergo an elongation phase in which they elongate and aggregate to form fibrils. The fibrils then continue to accumulate, forming Aβ plaques [6,17,37].

In the nucleation phase, there are sometimes two types of cleavage: primary nucleation and secondary nucleation. Primary nucleation refers to the formation of Aβ from monomers without the involvement of preexisting aggregates. Moreover, secondary nucleation involves preexisting aggregates. Generally, preexisting fibrils can act as seeds for the aggregation of new monomers. Secondary nucleation is a process that can accelerate the growth of Aβ fibrils and is more efficient [29,35,41].

Thermodynamically, secondary nucleation is more favorable and occurs spontaneously than primary nucleation. During secondary nucleation, preexisting aggregates act as templates, reducing the activation energy and facilitating the formation of new nuclei. Conversely, the formation of nuclei during primary nucleation involves the assembly of monomers into aggregates. This process requires a significant amount of energy, making it less favorable and not occurring spontaneously [5,21].

3.2 Simulation results for the immune system

The immune system within the brain includes microglia and astrocytes. Figure 3 shows a comparison of the concentration of Aβ in the Aβ growth models without the immune system and with the immune system.

Figure 3

Comparison of Aβ growth concentration values ( g mL ‒ 1 ) without the immune system (red line) and with the immune system (black line) for each of M ₁, M ₂, M ₃, M ₄, M ₅, M ₆, O, and P over 180 days.

As shown in Figure 3, the effect of the immune system on the Aβ growth model showed a pattern of concentration increase similar to that of the model without the immune system. However, the concentrations calculated for the immune system were lower. This indicates that the abilities of microglia and astrocytes, which are involved in the immune system, play a role in reducing Aβ accumulation. The role of the immune system becomes apparent from the growth of dimers (M ₂) to fibrils (P). This is expected to bring the concentration of Aβ growth from the dimers to the fibrils back to the initial monomeric form. Compared with those in the model without the immune system, the concentrations of Aβ generated by M ₂, M ₃, M ₄, M ₅, M ₆, O (oligomer), and P (fibril) were significantly lower.

Table 5 shows the changes in the Aβ concentration with the involvement of the immune system. There was a difference in the change in Aβ concentration with the immune system compared to without the immune system over 180 days. A difference in the change in the Aβ concentration was observed at the saturation state on day 180. This is because Aβ growth has reached a point where there is no further accumulation. The difference in the change in concentration for each form is 1.389 g mL − 1 for M ₂, 0.295 g mL − 1 for M ₃, 0.046 g mL − 1 for M ₄, 0.020 g mL − 1 for M ₅, 0.004 g mL − 1 for M ₆, 0.002 g mL − 1 for O, and 2.110 g mL − 1 for P.

Most Aβ peptides can be soluble in brain fluid, especially in the form of monomers or small oligomers. Under normal conditions, Aβ peptides are cleared through two mechanisms: the enzymatic pathway and the nonenzymatic pathway. The nonenzymatic pathway includes phagocytosis by microglia in small quantities and transport across the blood–brain barrier. Conversely, enzymatic clearance involves several proteases such as neprilysin and insulin-degrading enzyme, which play a role in breaking down Aβ into smaller fragments, allowing it to dissolve in the extracellular fluid [40,43]. However, Aβ peptides can become insoluble, leading to aggregation and the formation of plaques composed of larger fibrils.

The accumulation of Aβ in Alzheimer’s patients leads to inflammation in the brain, activating astrocytes and microglia, which then produce proinflammatory cytokines that damage neurons. If not addressed, more damaged and dead neurons will accumulate, disrupting communication pathways and cognitive functions. This is due to apoptosis, a programmed cell death process. Proinflammatory cytokines such as TNF-α trigger apoptosis, causing the gradual death of neurons [2,27].

Anti-inflammatory cytokines can regulate the immune system to stimulate microglia and macrophages to clear Aβ plaques. Macrophages, which exhibit phagocytic properties, engulf Aβ particles. The concentration of Aβ in growing plaques decreases over time. If the number of Aβ plaques decreases, inflammation will also decrease, allowing for the resolution of neuronal death.

The ability of microglia to clear Aβ plaques is influenced by several factors such as the concentration of Aβ oligomers, microglial performance, and individual variability. The greater the concentration of Aβ oligomers in the brain is, the longer it may take for microglia to clear Aβ plaques. Age can affect the performance of microglia, potentially slowing the clearance process. Other factors such as genetics and overall health can impact the efficiency of microglia as antibodies [22].

3.3 Simulation results of drug therapy

One approach to addressing Alzheimer’s disease is through drug therapy. Drug therapy involving the immune system is used to simulate the formation of dimers (M ₂) to fibrils (P). The rationale is similar to the application in the immune system, aiming to reduce the concentration of dimer to fibril growth in Aβ back to the monomeric form.

The drug therapy used involves anti-Aβ medications. As shown in Figure 4, the effect of drug therapy on the Aβ growth model showed a pattern of concentration increase similar to that of the immune system model and the model without the immune system. However, the concentrations calculated for drug therapy were lower than those for the immune system and the model without the immune system. This finding indicates that the effectiveness of anti-Aβ drugs is an effective therapeutic approach for reducing Aβ plaque growth.

Figure 4

Comparison of Aβ growth concentration values ( g mL ‒ 1 ) without the immune system and drug (red line), with the immune system (black line), and with a combination of drugs and immune system (blue line) for each of M ₁, M ₂, M ₃, M ₄, M ₅, M ₆, O, and P over 180 days.

Table 6 shows the concentrations of Aβ increased with the use of a combination of drugs and the immune system. According to Table 6, the Aβ concentration significantly decreased compared to the concentrations in the previous model, especially the fibril concentration. There was a difference in the change in the Aβ concentration between patients who did and did not receive drug therapy over 180 days. A difference in the change in the Aβ concentration was observed at the saturation state on day 180. This is because Aβ growth has reached a point where there is no further accumulation. The difference in the change in concentration for each form was 4.343 g mL − 1 for M ₂, 0.922 g mL − 1 for M ₃, 0.145 g mL − 1 for M ₄, 0.064 g mL − 1 for M ₅, 0.012 g mL − 1 for M ₆, 0.006 g mL − 1 for O, and 6.594 g mL − 1 for P.

Drug therapy for Alzheimer’s disease mainly involves inhibiting Aβ aggregation. The drug used in this study was Aducanumab. Aducanumab is an anti-Aβ drug classified as immunoglobulin type 1 (IgG1). Anti-Aβ drugs aim to reduce the formation of toxic Aβ oligomers and enhance the clearance of Aβ from the brain [25]. These drugs work by targeting specific molecular mechanisms, such as primary nucleation, secondary nucleation, and concentration-dependent aggregation of monomers [1].

Anti-Aβ drugs selectively target Aβ aggregates, including Aβ plaques and oligomers, but do not target Aβ monomers. This is because, in the aggregation process, Aβ has a high energy barrier for the transition state [36]. Oligomers are believed to be more toxic form than fibril assemblies [13]. As a result, Aducanumab binds to Aβ plaques and oligomers and then stimulates microglia to reduce Aβ. The effectiveness of anti-Aβ drug therapy can reduce the rate of Alzheimer’s disease decline by approximately 30% and has a significant impact on disease progression [33]. The dosage and duration of treatment significantly influence the effectiveness of anti-Aβ drugs in slowing cognitive impairment.

3.4 Determining the duration of treatment

The duration of treatment can be determined from Figure 4. Figure 4, shows that the Aβ concentration begins to increase from day 120 to day 180. The saturation values in the context of drug usage imply that the frequency of drug administration is no longer capable of significantly increasing therapeutic effectiveness. The effectiveness of drug therapy in this case refers to the ability to reduce the concentration of Aβ.

According to Honig and Boyd [16], in controlled trials, the typical duration of treatment is approximately 6 months, at which point the drug has provided significant benefits. The simulation results for determining the duration of treatment confirmed the reference, which is approximately 4 months starting from day 0 to 120th days. However, the duration of treatment for Alzheimer’s patients is influenced by several factors such as age, the patient’s condition, and the Aβ concentration. The findings regarding the treatment duration can provide insights for further research on developing more effective drugs. Until now, the use of anti-Aβ drug therapy has only been able to reduce the concentration of Aβ but has not been able to completely clear Aβ from the brain [45].

The ideal duration of treatment depends on the specific condition of the patient. According to Osborne and Schenk, the average duration of drug availability for Alzheimer’s disease patients ranges from 31 weeks until 243 weeks [9,32,36].

3.5 Simulation results of the concentration rate

The growth concentration rate indicates the kinetics of changes that can indicate how fast or slow Aβ growth is. Figure 5 shows the comparison of the rate of increase in the Aβ concentration with the immune system, drug therapy, and the absence of both agents. All three simulation models exhibit patterns of increasing and decreasing rates on the same days. However, based on Figure 5, it is evident that the concentration of Aβ in the context of drug therapy decreased compared to that in the immune system and in the absence of both agents.

Figure 5

Comparison of the growth concentration rates ( g mL ‒ 1 s ‒ 1 ) of Aβ without the immune system and drug (red line), with the immune system (black line), and with a combination of drugs and the immune system (blue line) for each of M ₂, M ₃, M ₄, M ₅, M ₆, O, and P over 180 days.

A significant difference in the Aβ concentration was detected in M ₂. This is due to the difference in the structure between Aβ monomers and dimers (M ₂). Structural differences can influence the Aβ growth concentration rate significantly by impacting how quickly dimers are formed [23]. Aβ dimers result from the aggregation of monomeric forms, increasing the density of dimers. Denser dimers tend to join other Aβ molecules more quickly, triggering an increase in the concentration. Additionally, the availability of Aβ monomers also affects the concentration of M ₂. When the concentration of Aβ monomers is very high, dimer formation increases, thereby increasing the rate of dimer formation.

At high concentrations, Aβ will interact with each other through chemical interactions to form aggregates with large molecular weights that are capable of producing larger aggregates. These aggregates are the precursors to fibril growth. The ability of monomers to aggregate also influences the concentration rate [28]. Therefore, the curve of trimer (M ₃) to oligomer (O) has different points of maximum rate increase due to the presence of a critical concentration that triggers the aggregation process [31]. If the formed nucleus is the same size, then in the nucleation stage, the aggregation rate will tend to be uniform [30].