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MBW:Mathematical Modeling of Alzheimer's Disease

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This is a summary of Mathematical Modeling for the Pathogenesis of Alzheimer's Disease by Ishwar K. Puri and Liwu Li [1]. The below review was written by Joseph Parks. All work below is from the paper by the Puri et al, unless otherwise cited.


Executive Summary

Alzheimer's disease is a neurodegenerative disorder that is identified by the formation of senile plaques in the brain. This formation of senile plaque by the deposition of amyloid-\beta peptides results in progressive dementia and death. It is a complex system of intercellular cross-talks occurring throughout senescence. As such, the pathogenesis of neurodegenerative Alzheimer's disease continues to elude researchers. Currently, there is no effective cure for Alzheimer's disease and limited options to slow its progression. Thus continues the search for the cause of Alzheimer's disease.


Despite the formation of senile plaques as the key morphological feature identifying the phenotype of Alzheimer's disease, increasing clinical and basic studies suggest that inflammatory activation of microglia may play an equally important role during the initiation and progression of the disease [2]. Microglia are the resident glial cells which are innate immune macrophages within the brain. When the inflammatory signals of amyloid-\beta accumulate, microglia become capable of expressing pro-inflammatory mediators and reactive oxygen species [3]. The microglia pro-inflammatory state results in astroglia proliferation and neuron death. In comparison, the microglia of a healthy brain will be at rest. Together with quiescent astroglia, the microglia can exhibit an anti-inflammatory state. This is the ideal state which minimizes astroglia proliferation and promotes neuron survival. Neuronal debris, amyloid-\beta , and/or proliferating astroglia may in turn further exacerbate the inflammatory phenotype of microglia. There is a delicate and sensitive balance among the cell populations where a cell type will be either beneficial or destructive to the neural neighborhood. This can generate multiple positive and negative feedbacks which promote neurodegeneration, thus altering neuronal structure and impairing function.


In light of repeated failure of experimental approaches to uncover the causal mechanism for Alzheimer's disease, Puri et al define a mathematical model which represents the known intercellular cross-talks as a system of seven differential rate equations between distinct states of cell populations. There have been several approaches to mathematical modeling of Alzheimer's disease; however, the current work is the first attempt to model the intercellular cross-talks between the distinct states of microglia, astroglia, and neurons to elucidate the underlying pathological effects. Analysis of the cell population sensitivities shows that the inflammatory activation of microglia is a key node in the progression of neurodegeneration. The approach of Puri et al implicates that microglia activation together with a threshold for amyloid-\beta to be a critical catalyst in the pathogenesis of Alzheimer's disease. Therefore, the targeting of microglia may hold great potential for progress in prevention and treatment of Alzheimer's disease.


Biological Background

All together.jpg


Beta Amyloid

This is a peptide composed of many amino acids. Peptides are short chains of amino acids that are covalently bonded when a carboxyl group reacts with an amino group between two amino acids. Amyloid-\beta , ({\mathbf  {A}}\beta ), is the primary component of senile plaques that are the morphological feature identifying Alzheimer's disease. The two forms of amyloid-\beta most commonly found in the composition of senile plaque are amyloid-\beta _{{40}} and amyloid-\beta _{{42}}, which are among the longest peptide amino chains. The deposition of amyloid-\beta generate inflammatory signals which, if there is enough build up, can activate microglia to a pro-inflammatory state resulting in astroglia proliferation and neuron death.

Amyloid beta 42.jpg
  • Amyloid-\beta _{{42}} is largely responsible for the formation of senile plaque on neurons.


Amyloid-\beta is formed when local enzymes metabolize the amyloid precursor protein (APP). This process leaves free amyloid-\beta particles which are either removed or tangled with others forming senile plaques. A decrease in the removal rate of amyloid-\beta leads to increasing neuropathogenesis.


Plaque formation.jpg
  • Formation of senile plaque.


Glial Cells

Glial cells are the defense and recycling system, as well as, the support structure for neurons. They are considered non-neuronal cells and are responsible for maintaining homeostasis, forming myelin, providing nutrients and oxygen to neurons, destroying pathogens, removing dead neurons, and is the insulation in-between and medium that holds neurons in place. We will consider two types of glial cells in the brain, both capable of expressing two distinct states.

Microglia

Microglia are mobile macrophages in the brain. These cells are phagocytes that ingest foreign bodies like bacteria, as well as, dead or dying cells. The brain is separated from the rest of the body by a series of endothelial cells known as the blood–brain barrier, which prevents most infections from reaching the vulnerable nervous tissue. In the case where harmful agents are directly introduced to the brain or cross the blood–brain barrier, microglia cells must react quickly to ingest the harmful agents and decrease inflammation before the sensitive neural tissue is damaged. These are of paramount importance in the brain and the only defense inside the blood-brain barrier. Microglia are smaller than macroglia, or astrocytes, and constantly sample their environment in search of harmful particles and dead cells. Microglia will be considered in two states: Normal microglia ({\mathbf  {M}}_{2}) in an anti-inflammatory state of rest, and reactive microglia ({\mathbf  {M}}_{1}) in an active pro-inflammatory state. The first promotes neuron survival while the latter instigates neuron death.


Alzheimer inflammation.jpg
  • Activation of microglia from the normal state to the reactive state.


Astroglia

Astroglia is a type of macroglia and is the most abundant cell in the brain. These are large immobile cells which help anchor neurons to their blood supply. Where microglia is the defense and recycling system, astroglia is the support structure. Astrocytes aid in the regulation of the chemical environment around neurons. They are responsible for absorbing ions to maintain the extracellular ion balance, and repairing damaged tissue in the brain [4]. The two states of this cell type considered are quiescent astroglia ({\mathbf  {A}}_{q}) and proliferating astroglia ({\mathbf  {A}}_{p}). The state of astroglia is related to the state of the microglia. Microglia in a state of rest can prevent astroglia proliferation, and thereby, promote neuron survival. Whereas, activation of microglia responding to amyloid-\beta leads to proliferating astroglia and an increase in neuron death.


Astrocyte.jpg
  • Astrocytes in a culture.


Neurons

There are many different types of neural cells in the nervous system but they are structurally and functionally similar or related. The three main types are receptors, effectors, and neurons. These cells are electrically excitable and communicate with each other by electrochemical stimulation. The receptors receive signals from outside the body and transmit the information along neural pathways, associated with that stimulation, to be processed by specific areas of the nervous system. The effectors receive signals from within the nervous system and produce an effect like initiating the contraction of a muscle. Thus, neuron networks are like wires through which many types of signals and information are propagated throughout the body.


Hodgkin-Huxley.jpg
  • Neuron as a circuit.


Every healthy neuron has a cell body, nucleus, dendrites, axon, and myelin sheath. The job of a neuron is to relay information by electrical and chemical signals. To overcome the noise inherent in the nervous system, neurons possess a characteristic action potential. This means the electrochemical stimulus must reach or surpass a particular magnitude in order to stimulate that neuron to propagate the signal to the next neuron. Neurons do not undergo cell division, but instead are formed by special types of stem and glial cells. Neurogenesis in humans ceases almost entirely around the age of twenty. The two states of neurons considered in this model are neuron survival ({\textbf  {N}}_{s}) and neuron death ({\textbf  {N}}_{d}).


Neurons.jpg
  • Network of neurons, and an awesome picture.


History

In 1906, German physician Dr. Alois Alzheimer identified abnormal deposits of plaque around the neurons of a patient he had seen for years. Inside the patients neurons were neurofibrillary tangles. Dr. Alzheimer's observation of these two brain cell abnormalities in association to this patients rapid physical and mental health deterioration became known as Alzheimer's disease. However, this was not accepted as a disease, but part of the natural aging process known as senescence. This remained the popular opinion until the 1960's when researchers discovered a link between cognitive decline and relative magnitude of plaques and tangles in the brain. It then became popular opinion that Alzheimer's was a disease and not a normal progression of senescence. It then became a popular area of research. As such, progress in understanding Alzheimer's disease quickly followed. By the 1990's, research uncovered amyloid-\beta and its role in plaque formation, and produced several drugs capable of targeting some of the symptoms caused by the disease. To date, progress has been made in understanding the complex factors in the progression of the disease, but the cause of the disease remains undetermined. While we continue to develop more potent therapies that slow the progression, there is still no known cure for Alzheimer's disease.

Current Description of Alzheimer's Disease

Characterized as a neurodegenerative disease, wherein, an increasing rate of neuron death leads to deterioration and impairment of physical and mental abilities. The first symptom of Alzheimer's is amnesia, or loss of short-term memory. As the disease progresses the following disorders become more pronounced: aphasia, apraxia, and agnosia. Respectively, the disorders are characterized by, loss of comprehension and ability to read, write, and speak languages, loss of ability to preform articulated movements despite gross physical capability, and loss of the ability for recognition and association of objects, people smells, etc.


Mathematical Model

As the specific mechanism behind the pathogenesis of Alzheimer's disease has yet to be determined, the mathematical model proposed by Puri et al was motivated by the fact that no previous research had examined the network of intercellular cross-talks between the distinct states of microglia, astroglia, and neuron populations. Despite the two brain abnormalities identifying Alzheimer's disease, this model does not account for neurofibrillary tangles. However, there is strong evidence that neurofibrillary tangles are not the cause of Alzheimer's disease and only account for a small proportion of the neuron loss.

The proposed mechanism accounts for the feedback from amyloid-\beta , and the distinct states of microglia, astroglia, and neurons. The intercellular cross-talks are represented as the 16 pathways in the mechanism.


RatesandPathways.jpg


Two important and justified assumptions must be addressed:

1. Constant risk of neuronal death.

2. Spatiotemporal influence of diffusion is negligible.


Variable Definitions

1. {\mathbf  {N}}_{{s}}= Amount of neuron population that has survived the current time step.

2. {\mathbf  {N}}_{{d}}= Amount of neuron population that has died during the current time step.

3. {\mathbf  {A}}_{{q}}= Amount of astroglia population in a state of quiescence.

4. {\mathbf  {A}}_{{p}}= Amount of astroglia population in a state of proliferation.

5. {\mathbf  {M}}_{{2}}= Amount of normal microglia population in the resting anti-inflammatory state.

6. {\mathbf  {M}}_{{1}}= Amount of reactive microglia population in the active pro-inflammatory state.

7. {\mathbf  {A}}\beta = Number of amyloid-\beta peptide molecules.


Parameter Definitions

The proposed mechanism requires 16 rates for its description, and the model and analysis call for one additional rate. The required rates for the mechanism are given implicitly as \alpha _{{i}} for i=1,...,16, which represent the cross-talks. The additional rate considered is \alpha _{{r}}, which represents the rate of amyloid-\beta removal out of the system.


Paratable1.jpg


Model Equations

The system of equations that model the intercellular cross-talks among the magnitude of amyloid-\beta molecules and the two distinct states of microglia, astroglia, and neuron populations are given by seven coupled differential rate equations. This system of equations represents how the number of amyloid-\beta molecules and the six cell populations relate and elucidates their pathological effect on each other within an arbitrary local volume.


1. {\frac  {d{\textbf  {N}}_{s}}{dt}}=\alpha _{1}{\textbf  {A}}_{q}-\alpha _{2}{\textbf  {A}}_{p}-\alpha _{3}{\textbf  {M}}_{1}


2. {\frac  {d{\textbf  {N}}_{d}}{dt}}=-{\frac  {d{\textbf  {N}}_{s}}{dt}}


3. {\frac  {d{\textbf  {A}}_{q}}{dt}}=\alpha _{4}{\textbf  {M}}_{2}-\alpha _{5}{\textbf  {M}}_{1}


4. {\frac  {d{\textbf  {A}}_{p}}{dt}}=-{\frac  {d{\textbf  {A}}_{q}}{dt}}


5. {\frac  {d{\textbf  {M}}_{2}}{dt}}=(\alpha _{6}+\alpha _{{11}}){\textbf  {N}}_{s}-\alpha _{{10}}{\textbf  {N}}_{d}+(\alpha _{7}+\alpha _{{12}}){\textbf  {A}}_{q}-\alpha _{9}{\textbf  {M}}_{1}+\alpha _{{14}}{\textbf  {M}}_{2}-(\alpha _{8}+\alpha _{{13}}){\textbf  {A}}\beta


6. {\frac  {d{\textbf  {M}}_{1}}{dt}}=-{\frac  {d{\textbf  {M}}_{2}}{dt}}


7. {\frac  {d{\textbf  {A}}\beta }{dt}}=\alpha _{{15}}{\textbf  {N}}_{s}-\alpha _{{16}}{\textbf  {M}}_{2}


The relationships between these equations will be studied by analyzing their sensitivities. To solve for the above seven first-order differential rate equations requires seven initial conditions. As the analysis will make use of the sensitivities, we will view the initial conditions in such light. With the given initial conditions for time zero, the system of equations will be solved forward in time for 20 years. From this accumulation of data the sensitivity coefficients are calculated in Table 2 below with S({\mathbf  {Population}})={\frac  {d({\mathbf  {Population}})}{d({\mathbf  {X}}(0))}}, where {\textbf  {X}}(0) are the initial populations or amyloid-\beta values.


Paratable2.jpg


The sensitivities will describe the weight with which a small or large perturbation in one population effects all of the other populations and number of amyloid-\beta molecules. The primary sensitivities of concern are with respect to neuron survival and neuron death. Thus, the equation sought is


8. S({\textbf  {N}}_{j})={\frac  {d{\textbf  {N}}_{j}}{d\alpha _{i}}},\;\;{\text{for}}\,\,j=s,d.


The greater the magnitude of \vert S({\textbf  {N}}_{j})\vert implies a greater sensitivity to the rate of change for the given cell population. For positive sensitivity coefficients of S({\textbf  {N}}_{j}) means that particular rate increases the population of {\textbf  {N}}_{j}. Thus, negative sensitivity coefficients lead to a decrease in the population of {\textbf  {N}}_{j}.

Results

From Table 1, we see that the rate \alpha _{1} associated to the {\mathbf  {A}}_{q}\rightarrow {\mathbf  {N}}_{s} pathway has a large positive sensitivity value indicating that it promotes neuron survival, thus inhibiting neuron death. Scanning down the S({\mathbf  {N}}_{s}) column in Table 1 shows if the pathway associated with \alpha _{i} increases the neuron survival population if the sensitivity coefficient is positive. If its sensitivity coefficient is negative, then the pathway \alpha _{i} decreases the neuron population, the S({\mathbf  {N}}_{d}) coefficient is positive. The larger the magnitude of the coefficient indicates that particular pathway to be more sensitive. The most sensitive pathways which promote neuron survival in decreasing order of magnitude are {\mathbf  {A}}_{q}\rightarrow {\mathbf  {N}}_{s}, {\mathbf  {A}}_{q}\rightarrow {\mathbf  {M}}_{2}, {\mathbf  {A}}_{q}\perp {\mathbf  {M}}_{1}, and {\mathbf  {M}}_{2}\perp {\mathbf  {M}}_{1}. The most sensitive pathways that promote pathogenesis of Alzheimer's disease in decreasing order of magnitude are {\mathbf  {A}}\beta \perp {\mathbf  {M}}_{2}, {\mathbf  {A}}\beta \rightarrow {\mathbf  {M}}_{1}, and {\mathbf  {M}}_{1}\rightarrow {\mathbf  {N}}_{d}.


Result2.jpg


The plots above show curves for three different rates of amyloid-\beta removal, (\alpha _{r}), for the populations of reactive microglia ({\mathbf  {M}}_{1}), proliferating astroglia ({\mathbf  {A}}_{p}), amyloid-\beta ({\mathbf  {A}}\beta ), and neuron death ({\mathbf  {N}}_{d}). It is immediately apparent that a decreasing rate of amyloid-\beta removal results in an increase of the four above populations, and therefore promotes pathogenesis of Alzheimer's disease.


The authors conclude that, by their model, an effective strategy for intervention is by promoting inhibition of reactive microglia ({\mathbf  {M}}_{1}) and decreasing the amount of amyloid-\beta ({\mathbf  {A}}\beta ) through aiding the normal microglia population together with the promotion of quiescent astroglia ({\mathbf  {A}}_{q}). This model had elucidated the importance of reactive microglia as a central component in the progression of Alzheimer's disease, which, prior to this work, was considered negligible. In the end, this is a simplified model to serve as a compass and springboard toward more complex and accurate research models for the pathogenesis of Alzheimer's disease.


References

  1. Puri IK, Li L. (2010) Mathematical Modeling for the Pathogenesis of Alzheimer’s Disease. PLoS ONE 5(12): e15176. doi:10.1371/journal.pone.0015176.
  2. Perry VH, Nicoll JA, Holmes C. (2010) Microglia in neurodegenerative disease. Nat Rev Neurol 6: 193–201.
  3. Lue LF, Kuo YM, Beach T, Walker DG. (2010) Microglia activation and anti-inflammatory regulation in Alzheimer’s disease. Mol Neurobiol 41: 115–128.
  4. Kril J. J., Patel S., Harding A. J., Halliday G. M. (2002). "Neuron loss from the hippocampus of Alzheimer's disease exceeds extracellular neurofibrillary tangle formation. [Article]". Acta Neuropathologica 103 (4): 370–376. doi:10.1007/s00401-001-0477-5. PMID 11904757.