Editing Iowa Mutation Modeling Program (section)

==== 2.5 Minimal “brain-sized” production–clearance model (first-pass) ====

Let:
* <math>M(t)</math> = local monomer (or soluble Aβ) concentration (mol·L<sup>−1</sup>)
* <math>F(t)</math> = fibril mass concentration (or total fibril mass in a compartment)
* <math>A(t)</math> = total exposed fibril surface area (m<sup>2</sup>) in the relevant compartment (e.g., vascular wall + perivascular region)
* <math>O(t)</math> = soluble oligomer concentration (mol·L<sup>−1</sup>)
* <math>f_{\mathrm{iso}}(t)</math> = fraction of deposited molecules containing isoAsp23

'''IsoAsp formation (phenomenological):'''
<math>
\frac{df_{\mathrm{iso}}}{dt} = k_{\mathrm{iso}}(1-f_{\mathrm{iso}}) - k_{\mathrm{repair}} f_{\mathrm{iso}}
</math>

'''Aggregation / fibril growth (minimal):'''
<math>
\frac{dF}{dt} = k_n M^n + k_2 M^m A - k_{\mathrm{clear},F} F
</math>

Here <math>k_2</math> captures surface-catalyzed addition / secondary nucleation terms; <math>k_{\mathrm{clear},F}</math> is effective clearance of fibrillar material.

'''Surface area mapping (“morphology parameter”):'''
<math>
A(t) = \alpha_{\mathrm{morph}} \, F(t)
</math>

where <math>\alpha_{\mathrm{morph}}</math> has units of (surface area)/(fibril mass) and encodes the difference between compact plaques vs diffuse CAA fibril clouds. The key claim is:
<math>
\alpha_{\mathrm{CAA}} \gg \alpha_{\mathrm{plaque}}
</math>

'''Oligomer production on fibril surfaces:'''
<math>
\frac{dO}{dt} = k_{\mathrm{cat}} M^p A - k_{\mathrm{clear},O} O
</math>

At quasi–steady state (fast oligomer dynamics relative to years-scale deposition), this gives:
<math>
O^{*} \approx \frac{k_{\mathrm{cat}}}{k_{\mathrm{clear},O}} M^p A
</math>

Thus the “surface-area amplification” prediction is:
<math>
\frac{O^{*}_{\mathrm{CAA}}}{O^{*}_{\mathrm{plaque}}} \approx
\frac{\alpha_{\mathrm{CAA}}}{\alpha_{\mathrm{plaque}}}
\cdot
\frac{F_{\mathrm{CAA}}}{F_{\mathrm{plaque}}}
\cdot
\left(\frac{M_{\mathrm{CAA}}}{M_{\mathrm{plaque}}}\right)^p
\end{math}

'''Toxicity / neuronal loss (minimal hazard model):'''
Let <math>N(t)</math> denote viable neurons (or synapses) in a region. A minimal coupling is:
<math>
\frac{dN}{dt} = -k_{\mathrm{tox}} \, O^q \, N
</math>

which implies an exponential survival curve with an oligomer-dependent hazard.

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