l.r. ritter, akif ibragimov , jay walton # -texas tech university, #-texas a & m university...

L.R. Ritter, Akif Ibragimov*, Jay Walton#

*-Texas Tech University, #-Texas A & M University

April 18, 2011

Mathematical Modeling of Atherogenesis

http://www.spsu.edu/

Contents

The disease processBuilding the mathematical model of

atherosclerosis (Keller-Segel model of chemo-taxis)

Atherogenesis viewed as an instabilitySimple stability analysis (i) Stability analysis (ii): apoptosis/differing

roles of macrophagesStability analysis (iii): boundary

transport and anti-oxidant

AtherosclerosisAtherosclerosis is a vascular disease characterized by the build up of lipid laden cells in the walls of large muscular arteries. Low density lipoproteins can become trapped in the artery wall where they become oxidatively modified. This triggers an immune response that can become corrupted as immune cells become lipid laden. An inflammatory feedback loop mediated by chemical signaling may be instigated leading to runaway inflammation and disease progression.

Cardiovascular disease continues to be the primary cause of adult mortality through out the United States, Europe, and much of Asia.

Large Muscular Artery

Arteries are thick walled, multilayered tubes. Atherosclerosis occurs in the thin innermost layer—the intima.

Atherosclerosis

Heart Attack

http://www2.spsu.edu/math/lritter/HeartAttack.avi

Inflammatory Spiral

injury or leakiness of the endothelial layer

Macrophage

Endothelial cells or immune cells send chemical signal

Macrophage “reads” signal and migrates to intervene

Ox-LDL

Oxidized LDL attracts macrophage that attempt to consume the particles: foam cells form

Foam cell

Macrophage

Inflammatory spiral may result

Mathematical Model

The mathematical model is constructed by mass balance and consideration of the primary disease features.

• Motility/diffusion

• cell-chemical interaction—e.g. chemo-taxis

• cell-cell interactions—e.g. foam cell formation or

healthy immune function

• chemical reactions—e.g. oxidation of LDL

Mathematical Model

We consider several generalized species:

I : Immune cells, primarily macrophagesD : Debris—i.e. the bulk of the lesionC : chemo-taxis inducing agents (chemical signals)L : Low density lipoproteins in a native or partially oxygenated stateLox: LDL molecules after full peroxidation of the lipid coreR : Reactive oxygen species (free-radicals)

In addition to many parameters including:

Aox : Anti-oxidant speciesLB : Serum LDL concentration(and several reaction rates and additional parameters)

Chemo-taxis

che·mo·tax·is (kē'mōtāk'sĭs,kěm'ō) n. The characteristic movement or orientation of an organism or cell along a chemical concentration gradient either toward or away from the chemical stimulus.

The American Heritage® Dictionary of the English Language, Fourth Edition.

https://secure.reference.com/premium/login.html?rd=2&u=http://dictionary.reference.com/browse/chemotaxis

Chemo-taxis

Keller-Segel Model of Chemo-taxis:

For motile species U influenced by the chemical species V:

The flux field for U = flux due to diffusion (reaction to U) +

flux due to chemo-taxis (reaction to V)

measure of motility chemo-tactic sensitivity function

Chemo-taxis

The point-wise equation for U:

where Q is an appropriately defined reaction term.

Mathematical Model

In general, each variable I, D, C, and L, is vector of several components representing different phenotypes, chemical agents, or in the case of LDL, levels of oxidation. So that the system consists of a large number of distinct equations.

Several studies indicate for example that macrophages play many different roles in the disease process: see

The role of macrophages in atherosclerosisLibby, Peter; Clinton, Steven K. (1993),

Multifunctional roles of macrophages in the development and progression of atherosclerosis in humans and experimental animalsK. Takahashi, Motohiro Takeya, Naomi Sakashita (2002)

Pivotal Advance: Macrophages become resistant tocholesterol-induced death after phagocytosis ofapoptotic cellsCui et al (2007)

Equation for Immune Cells I

Self interaction (diffusion)

Ii is the ith immune cell type, 1≤ i ≤ NI


Response to chemical (chemo-taxis)


Binding to debris (immune response)


Subspecies interaction


Binding to oxLDL (foam cell formation)


Normal degradation

Mathematical Model

We introduce a coupled nonlinear system of reaction-diffusion-convection equations characterized by chemo-taxis,

Immune cells

Lesion bulk

Chemo-taxis agents

NI immune species, ND debris species, NC chemical species

Mathematical Model

and linear oxidation of LDL with reverse oxidation* and diffusion.

* This is a model adapted from Cobbold, Sherratt and Maxwell, 2002

Number of native LDL subspecies

Bio-physical interpretation of parameters

Stability analysis

The disease is characterized by local accumulation a cells.

We propose that atherogenesis—the onset of lesion growth—can be viewed as due to an instability in a uniform configuration of cells and chemical species.

Stability analysis (i)

A greatly simplified analysis of a single immune cell type, single debris type and single chemical agent after full oxidation of present LDL:

In the interior of an annular domain (or annular

cylinder) with homogeneous Neumann

B.C.

Introduction of small perturbations u, v, and w.


Put

where

On an annular domain Ω with smooth boundary ∂Ω.


Substitution of eigenfunctions for u, v, and w produces an algebraic system for the coefficient . s

We seek nontrivial solutions u0, v0, w0.


There are three eigenvalues s; two are necessarily real and negative, and one is real and can be positive or negative depending on parameter values.

For a given level of hostility (i.e. concentration of oxLDL) Y is a measure of the marginal ability of immune cells to respond to an increase in lesion debris.

If Y < 0, an increase in lesion drives healthy immune function.

If Y > 0 an increase in lesion results in unhealthy inflammation.

The role of apoptosis

Apoptotic macrophages are regularly found in atherosclerotic plaques


Apoptotic macrophages are regularly found in atherosclerotic plaques

During phagocytosis, macrophages take in membrane derived as opposed to lipoprotein derived cholesterol.


Apoptotic macrophages are regularly found in atherosclerotic plaques.


This has been seen to induce a “survival response” in macrophages (Cui et al 2007).


Apoptotic macrophages are regularly found in atherosclerotic plaques.


This has been seen to induce a “survival response” in macrophages (Cui et al 2007).

Apoptotic macrophages may also lead to plaque rupture in later stages of the disease (Tabas 2004).

Stability analysis (ii)We consider two “competing” macrophage species, one healthy and one corrupt. A similar eigenvalue perturbation is not feasible.

Again imposing a zero flux condition on the boundary of the domain.

Stability analysis (ii)

We again introduce perturbation variables u, v, w, and z,

but consider another approach to the stability analysis.

Stability analysis (ii)

We can state a stability theorem giving a specific set of requirements on the relative size of the system parameters. The significant requirements include

Certain domination of diffusive motility of immune cells over chemo-taxis,

An inequality relating healthy immune response to foam cell formation,

Certain domination of diffusion of the chemo-attractant over its production in response to the lesion.

Stability (ii) details

The equations governing perturbation variables are

(Classical solutions are considered.)

In an annular domain

Ω

On ∂Ω

Stability (ii) detailsWith the various parameters defined by

Stabilizing: Large E, G Strong healthy immune response

Small B, C Minimal apoptosis and foam cell formation

Stabilizing: Large mu, mv, mz, Small cu cv diffusion dominates over chemotaxis

Some key terms:o E, G—rates of debris removalo B—rate of apoptosis of macrophageso C—rate of foam cell formation


With some effort, the following inequality is derived, and the energy functional F is defined accordingly.


Transition matrix for species interaction.

Net mobility—diffusive motility –vs- chemo-taxis

The role of anti-oxidants and boundary transport

High concentrations of serum LDL is associated with high incidents of atherosclerotic lesions (Kronenberg et al. 1999).



Transport of LDL molecules and immune cells across the endothelial layer are key to atherogenesis (Ross 1999).




Anti-oxidant levels can mitigate the disease onset and progression (Howard and Kritchevsky 1997).




Anti-oxidant levels can mitigate the disease onset and progression (Howard and Kritchevsky (1997).

The present energy-estimate approach can be extended to include additional species and boundary transport.

Stability analysis (iii)

We seek to capture the full role of LDL oxidation, anti-oxidant presence and boundary transport. To include these, we begin with a system of one of each species


and allow for movement of

immune cells,

chemo-attractant,

and LDL molecule across the endothelial layer.

subendothelial intima (domain of equations)

Endothelial layer

internal elastic lamina

forward & reverse transport from blood


We again consider a perturbation off of a constant equilibrium solution

The linearized equations are considered with third type boundary conditions for those variables representing immune cells u, chemo-attractant w, and native LDL z.

forward transport (from blood to tissue)

reverse transport (from tissue to blood)


Of particular interest: (stabilizing processes)

o M & N—decay and removal rates of chemo-attractant

o μ1 & μ3—motility/diffusion rates of immune cells and chemo-

attractant (within the intima)

o P1—a decay rate of LDL within the intima

o Q3 & R3—reverse oxidation rates (due to anti-oxidant)


Construction of appropriate functionals requires treatment of the boundary terms. The following inequalities are useful and result in explicit inclusion of the geometry of the domain.


The stability criteria (sufficiency) consist of several inqualities:

Decay and uptake of chemo-attractant by macrophages

must dominate chemo-taxis both within the intima ( )c

and across the endothelial layer (a1 & a3).


Motility of macrophages and diffusion of chemo-attractant

within the intima (m1 & m3) must be significant when

compared to the influx of immune cells across the endothelium

(a1 & a3) and chemo-taxis within the intima (c).


If P1 is the rate of LDL decay within the intima

Then in the case of reverse transport of LDL (from the intima back to the blood), stability requires

P1 > 0.

In the case of forward transport, stability requires

P1 > |a4|k + oxidation rates + |Ω|

where k and Ω depend on the size of the domain.


Anti-oxidant reactions must occur at rates superior to the rates of

the LDL—free radical reaction (R1) and the peroxidation rates of

the lipid core (Q2 & Q5).

Stability analysis (iii) (result)


We impose a pair of conditions:


And define a pair of functionals:

Generalization

A complete generalization for an unspecified number of each species is currently under construction. Such analysis will give a base line set of inequalities relating various parameters that can eventually be compared with bio-medical data.

Selected References

R. Ross, Cell biology of atherosclerosis, Annu. Rev. Physiol., 57 (1995) 791—804

R. Ross, Atherosclerosis: An inflammatory disease, N. Eng. J. of Med., 340(2) (1999) 115—126

C.A. Cobbold, J.A. Sherratt, and S.J.R. Maxwell, Lipoprotein oxidation and its significance for atherosclerosis: a mathematical approach. Bull. Math. Biol., 64 (2002), 65—95

E.F.Keller and L.A. Segel, Model for chemotaxis. J. Theor. Biol., 30 (1971), 235—248

Selected ReferencesStability Analysis of a Reaction-Diffusion System Modeling Atherogenesis, With A.I. Ibragimov, and J.R. Walton, SIAM J. Appl. Math. 70 (2010), pp. 2150-2185

Stability analysis of a model of atherogenesis: An energy estimate approach II, With A.I. Ibragimov, C.J. McNeal, and J.R. Walton, J. of Comp. and Math. Meth. in Med. , Vol.11(1), (2010) pp. 67-88

Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis, With A.I. Ibragimov, C.J. McNeal, and J.R. Walton, Discrete and Continuous Dynamical Systems, Supplement (2009), pp. 630-639

Stability analysis of a model of atherogenesis: An energy estimate approach With A.I. Ibragimov, C.J. McNeal, and J.R. Walton, J. of Comp. and Math. Meth. in Med. , Vol.9(2),(2008) pp. 121-142

A dynamic model of atherogenesis as an inflammatory response, With A.I. Ibragimov, C.J. McNeal, and J.R. Walton, DCDIS A Supplement, Advances in Dynamical Systems, Vol.14(S2) (2007), pp. 185--189

A Mathematical Model of Atherogenesis as an Inflammatory Response, With A.I. Ibragimov, C.J. McNeal, and J.R. Walton, Math. Med. Biol. , 22 (2005), pp. 305--333

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