Variant assumptions made in deriving equilibrium solutions to Little et al (PLoS Comput Biol 2009 5(10) e1000539)
Mark P Little, Anna Gola, Ioanna Tzoulaki, Wendy Vandoolaeghe
VVariant assumptions made in deriving equilibrium solutions to “A model of cardiovascular disease giving a plausible mechanism for the effect of fractionated low-dose ionizing radiation exposure” (
PloS Comput Biol
Mark P. Little a, b , Anna Gola a , Ioanna Tzoulaki a , Wendy Vandoolaeghe a a Department of Epidemiology and Public Health, Imperial College Faculty of Medicine, Norfolk Place, London W2 1PG, UK b To whom all correspondence should be addressed at: tel +44 (0)20 7594 3312; Fax +44 (0)20 7402 2150; Email [email protected]
Abstract
The paper of Little et al. ( PloS Comput Biol et al. (2009) indicates that: [ ( ) ] ( / ) ( / ) [( / ) M in M M
C C M ML d M M D Mt ] η ηχ ρ η η η η∂ + ∇ ⋅ ∇ = − + ∇ ⋅ ∇∂ (1) where C is the chemo-attractant (monocyte chemo-attractant protein 1 (MCP-1)) concentration, M is the macrophage concentration and η is the bound lipid concentration. ( ) M C χ is the chemotactic factor (assumed constant) associated with macrophages; the mechanism for chemotaxis is similar to that of Keller and Segel (1971a, b). M D is the rate of diffusion of the macrophages. In equilibrium let ,, eq eq eq M C η , etc. be the values of the various quantities, and let , , M C η Δ Δ Δ eq C C be the differences from these equilibrium values after perturbation – so that, for example, C Δ = + , and similarly for the other species. As in Little et al. (2009) we must have that in equilibrium: CE eq CM eq CT eq CM eq eq CT eq eq Cm eq eq
E M T d M C d T C d m C ρ ρ ρ+ + = + + (2) 0
PT eq T ρ = (3) eq eq M eq eq m P m P μ ρ= (4) ( / ) M eq eq M eq eq eq m P d M M ρ η= (5) ( / ) ( / ) in eq eq eq M eq eq eq M M L d M ρ η η η= (6) 0
T eq d T = (7) ( / )[ ] 0 M eq eq eq MM eq T TT eq d M d M d d T η η + + = (8) All variable and parameter definitions are as in Little et al. (2009). Assuming 0 PT ρ ≠ then by (3): 0 eq T = (9) If M μ ρ ≠ then by (4): 0 eq P = or (10) 0 eq m = If ( / ) 0
M eq eq d M η ≠ and M μ ρ ≠ then by (4) and (5): 0 eq M = (11) If ( / ) 0 M eq eq d M η ≠ and M μ ρ = (so that eq M is not necessarily 0) by (6) we have that: / ( / ) / ( / eq eq in eq eq M eq eq M M L d ) M η ρ η η = (12) Under the assumption that equation (12) can be iteratively solved (using the parameter values of Little et al. (2009)) to yield: 0 eq M ≠
17 1 / 4.63 x 10 M cell eq eq M η − − = (13) and at this value the functions ( / ), ( / ) in eq eq M eq eq M d M ρ η η can be evaluated (using the parameter values of Little et al. (2009)) to give: ( )
10 1 1, ,0 ( / ) 5.34 x 10 cell ml s in eq in eq eq in M ρ ρ η ρ − − − = = ≈ (14) ( / ) 3.55 x 10 s M eq M eq eq d d M η − − = = (15) If we perform the obvious linearisations in equation (1), and make use of the parametric forms for the macrophage bound-lipid ingestion rate: ( ) , ,0 , 3 / [ ]exp[ / ] in in high in in high M R M ρ η ρ ρ ρ η= + − − (16) and the macrophage mortality rate: ( ) ,0 2 / M M d M d R η = + / M η (17) assumed by Little et al. (2009) we obtain: , 00 , 3 3 2 2, 2 2 2 2 ( )exp / [ / ][ / / ](1/ ) ( / ) | | ( / ) M eq in eqin high eq eq eq eqM eq eq eq eq eqM eq eq eq eq eq
C C C M LtL R R M M Md R M M MD M M M M M M η χ η η ρρ η η ηη η η ηη η η
Δ Δ Δ Δ ΔΔ ΔΔ Δ ΔΔ Δ Δ Δ ∂ ⎡ ⎤+ ∇ ⋅ ∇ + ∇ =⎣ ⎦∂ ⎡ ⎤+ − −⎣ ⎦− − −⎡ ⎤+ ∇ ⋅ ∇ − ∇ + ∇⎣ ⎦ (18) If we neglect second and higher order terms in , ,
M C η Δ Δ Δ : ( ) exp / [ / ][ / / ] ( / ) M eq in eq in high eq eq eq eqM eq eq eq eq eq M eq eq
C C M L L R R M M Mtd R M M M D M M η χ η ρ ρ η η ηη η η η η
Δ Δ Δ Δ ΔΔ Δ Δ Δ ∂ ⎡ ⎤+ ∇ ≈ + − −⎣ ⎦∂− − − + ∇ (19) [Parenthetically, we notice that in the limit assumed by Little et al. (2009), of , lim 0 n eq n η →∞ = , , , lim 0 n eq n M →∞ = , , lim / 0 n eq n eq n M η →∞ = , with each set of n , , , , ( , , , n n eq n eq n M M ) η η Δ Δ satisfying: , 2, , 0 0 , 3 3 , , , , ,2 2 2, 2 , , , , , , , , , ( ) exp / [ / ][ / / ] ( / ) n , M eq n in n in high eq n eq n n eq n n eq nM n eq n n eq n eq n n eq n M eq n eq n n
C C M L L R R M M Mtd R M M M D M M η χ η ρ ρ η η ηη η η η η
Δ Δ Δ Δ ΔΔ Δ Δ Δ ∂ ⎡ ⎤+ ∇ = + − −⎣ ⎦∂− − − + ∇ (19’) then by the Arzelà-Ascoli theorem (Kelley 1975, chapter 7), and by considering a subsequence if necessary, , lim nn η η Δ →∞ ≡ Δ n and , lim n M M
Δ →∞ ≡ Δ exist and satisfy expression (19) in which we replace eq η , eq M and / eq eq M η by 0 (in particular , , , M eq in eq d ρ by (0), (0) M in d ρ ). Therefore, in this limit expression (19) reduces to: (0) [ (0) ] in M in high M L d L Rt η ρ ρ Δ Δ ∂ ≈ − −∂ η Δ R (20) For the values of the parameters (0), , , M in high d L ρ given in Little et al. (2009) we have that whereas . Therefore, to a good approximation (20) reduces to: (0) 9.3 x 10 s M d − − =
17 1 s (0) M d − − << in high L R ρ = (0) (0) in M M L dt η ρ η
Δ Δ ∂ ≈ −∂ Δ (21) in other words, equation (37) of Little et al. (2009).] Integrating over the intima, and using Green’s first identity and the fact that macrophage and chemo-attractant flux over the boundary is generally zero: , 00 , 3 3 2, 2 2 exp / ( / )( / ) ( / ) I I I II I in eqin high eq eq eq eqM eq eq eq eq eq d dx L M dxdtL R R M dx M M dxd dx R M dx R M M η ρρ η η ηη η η η
Δ ΔΩ Ω ΔΩ ΩΔ ΔΩ Ω = ⎡ ⎤⎡ ⎤+ − −⎢⎣ ⎦ ⎢ ⎥⎣ ⎦− − + ∫ ∫ ∫ ∫∫ ∫ I dx ΔΔΩ ⎥ ∫ (22) , 0 0 , 3 3220 , 3 3 , 2 ( / ) exp /( / )exp / ( / ) I I in eq in high eq eq eq eqeq eqin high eq eq M eq eq eq
L L R M R M M dxR ML R R M d R M ρ ρ η ηηρ η η
ΔΩ ΔΩ ⎡ ⎤⎡ ⎤− −⎣ ⎦⎢ ⎥= ⎢ ⎥+⎣ ⎦⎡ ⎤⎡ ⎤+ − − −⎣ ⎦⎣ ⎦ ∫ dx η ∫ et al. (2009) although the constants are slightly different. As can be seen from the form of equations (42)-(48) of Little et al. (2009) this change makes no difference to the numerical estimates of averaged MCP-1 etc. given in the paper. Acknowledgments
This work has been partially funded by the European Commission under contract FP6-036465 (NOTE).
References
Keller EF, Segel LA (1971a) Model for chemotaxis.
J. Theoret. Biol. : 225–234. Keller EF, Segel LA (1971b) Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theoret. Biol. : 235–248. Kelley JL (1975) General Topology.
Springer-Verlag, 1975. Little MP, Gola A, Tzoulaki I (2009) A model of cardiovascular disease giving a plausible mechanism for the effect of fractionated low-dose ionizing radiation exposure.
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