The origins of concentric demyelination: self-organization in the human brain

Abstract

Baló's concentric sclerosis is a rare atypical form of multiple sclerosis characterized by striking concentric demyelination patterns. We propose a robust mathematical model for Baló's sclerosis, sharing common molecular and cellular mechanisms with multiple sclerosis. A reconsideration of the analogies between Baló's sclerosis and the Liesegang periodic precipitation phenomenon led us to propose a chemotactic cellular model for this disease. Rings of demyelination appear as a result of self-organization processes, and closely mimic Baló lesions. According to our results, homogeneous and concentric demyelinations may be two different macroscopic outcomes of a single fundamental immune disorder. Furthermore, in chemotactic models, cellular aggressivity appears to play a central role in pattern formation.

Figure 1

Typical aspects of Baló's concentric sclerosis. (a) Original case of Baló; several anastomoses are located in the lower half of the lesion (from Baló (1928) Arch Neurol Psychiatr 19:242–264). (b) Lesion centered by a veinule showing ring fragmentation in a constrained area (from Hallervorden et al. 1933). (c) Lesion reproduced from Castaigne et al. (1984) Rev Neurol 140:479–487. (d) Progress of the pathologic process from a center located in a constrained area, showing formation of bands (from Behr (1950) Dtsch Z Nervenheilk 164:480–489). Loyez staining (myelin in black, destroyed areas in white); scale bars: 1 cm.

Figure 2

Space laws from 12 measures made on the cases of Baló (1928) Arch Neurol Psychiatr 19:242–264 and Hallervorden et al. . (a) Damaged (dashed line) and preserved (solid line) myelin. (b) Averaged successive positions of destroyed myelin bands (dashed), compared to the classical exponential space law for the Liesegang rings (solid line) obtained from a logarithmic regression; y-axis label in cms. The space law for concentric sclerosis is linear (correlation coefficient r ² = 0,996). In all our mathematical models, the space law strongly depends on the shape of the activating front. As the biological nature of this front is unknown, selection between different scenarios (e.g. preconditioning model vs. local macrophage recruitment model) cannot be based on the space law. However the linear law for concentric sclerosis tends to indicate that an outer diffusing signal is unlikely.

Figure 3

Apparition of concentric patterns under the preconditioning model (1)-(2). The protection factor is given by Heaviside function P(φ) = (q−φ)₊ with a given threshold q. (a) We have replaced the preconditioning equation (2) with φ = K*d, the kernel K being a stiff Hill function with a range of action, . Other parameters are ε = 0,4 and q = 0,1. Interestingly the range of action of the potential φ is larger than in figure 5 whereas the size of the domain is considerably smaller. In fact, this range fully determines the width of the bands. (b) Same illustration with a degenerated potential . Destroyed oligodendrocytes are figured in black. The size of the domain is four times smaller than in figure 4 (axis labels in cms), whereas the action range is similar.

Figure 4

Transition between concentric patterns and plaques is driven by the structural parameter χ. Reduced parameters are ε̃ = 0,2, κ/λ = 4, χ̃ = 25 (a) and χ̃ = 8 (b). The pattern diameter is approximatively 4 cm (axis labels in cms), and the final time is 24 h. These results fit the biological data (see table 2). Destroyed oligodendrocytes are figured in black. The white corners in (b) are boundary artifacts.

Figure 5

(a) Bifurcation diagram for fixed reduced parameter ε̃ = 0,1. Only two alternative patterns arise: concentric rings or plaques apparition. (b) Imposed white noise perturbation with relative standard deviation σ = 0,2 to the chemical diffusion coefficient ε̃ = 0,2. Other parameters are χ̃ = 25 and κ/λ = 5. (c) White noise perturbation with relative standard deviation σ = 0,1 to the damaging factor κ/λ = 5. We set ε̃ = 0,2 and χ̃ = 25. Destroyed oligodendrocytes are figured in black. The damaging factor is more sensible to perturbation that the chemical diffusion coefficient; increasing the standard deviation breaks the symmetry of the pattern, except around the origin. Anastomoses are noted in (c) far from the regions affected by the boundary effects.