Programa Emprendedor Global (EG)

For a task as complex as a biomechanical model of syringomyelia all things cannot be resolved in one attempt. Analytical models such as that of Carpenter and colleagues are desirable as they give a deeper insight into the underlying mechanics. There is also a need for some systematic assumptions and approximations in order to make progress. The review of physiological parameters presented here establishes some guidelines for model design. In a one-dimensional model of the SSS the inviscid assumption is a rea- sonable one; as for the SC itself, a more realistic model is required taking into account the fluid-saturated poroelastic nature of the tissue. Transpial flow induced by pressure

3. The elastic jump hypothesis and 1-d continuum modelling 79

wave propagation is a possible source for attenuation, which may be exacerbated by disruptions to the SC blood supply due to a spinal disease such as syringomyelia. The analysis presented here does not support the elastic-jump hypothesis and suggests that there must be some other localising factor more critical to providing the necessary con- ditions for syrinx formation than the magnitude of the transpial pressure differential. In the next chapter attention turns to the theory of poroelasticity.

Nomenclature

Latin symbols:

A Cross-sectional area of tube;

D Distensibility of the flexible tube;

E Elastic modulus of the flexible tube;

L Length;

P The initial pressure generated by a cough (note that ∆P =PB−PA);

Q Volumetric flow rate;

R The hydraulic radius of a tube, taken to be twice the cross-sectional area divided

by the wetted perimeter;

T Nondimensional number describing ratio of viscous-to-inertial forces in a pulsatile

flow (=pπ/2·1/Wo);

Wo Womersley number, describes the ratio of viscous-to-inertial forces in a pulsatile flow;

n The number of (perivascular) spaces over the length of the spinal cord;

p A pressure pertaining to a wave travelling in the coaxial tube system (note that

r A tube radius (also denotes the radial coordinate of a cylindrical coordinate system);

t Thickness of flexible tube;

x Axial coordinate of the cylindrical coordinate system.

Greek symbols:

α Cross-sectional area ratio (=AA/AT);

β Damping coefficient;

∆r Pressure amplification ratio (= ∆pr/∆pi);

ε Small parameter used in weakly nonlinear theory (=D∆P);

κ Permeability;

µ Dynamic viscosity;

ν Kinematic viscosity;

ρ Density, both for fluids and solid tissues within the spinal canal;

τ Pressure pulse rise time, which is the time taken for pressure to reach its peak

value from the beginning of a given pulse (for a cough this corresponds to a steep

slope on thep-ttrace);

ω Angular oscillation frequency.

Subscripts:

0 Refers to quantity at timet= 0 (quiescent state);

A Refers to quantity in annular cylindrical space A; B Refers to quantity in circular cylindrical space B; pia Refers to a property of the pial membrane;

3. The elastic jump hypothesis and 1-d continuum modelling 81

PVS Refers to a property of the perivascular space; SC Refers to a property of the spinal cord;

T Refers to quantity in total of space A and space B; i Refers to a pulse incident on a stenosis;

r Refers to a pulse reflected from a stenosis;

x Refers to quantity varying in xdirection;

Poroelasticity in spinal cord tissue

“Every living membrane, of necessity, is permeable to fluid. The difference between a perforated and a permeable membrane is merely the size and number of its holes.” — Gardner (1965; p.251).

4.1

Introduction

A complete model of the cerebrospinal system, taking into account the viscous and inertial effects of both the fluid and solid components of the porous SC tissue, is a sub- stantial leap in complexity from what has been attempted to date. In the construction of any such model it is important to first understand the behaviour of each of the com- ponents before assembling them into the whole. Continuum models of the cerebrospinal system have been able to identify the dominant wave modes for approximations of the SC as either an inviscid fluid or an elastic solid but the simple Darcy’s law approach to the porous problem is unsatisfactory. Therefore, in this chapter the wave-bearing properties are investigated for an infinite fluid-saturated poroelastic continuum having material properties representative of the human SC. A series of parametric sensitivity studies are reported which indicate the behaviour specific to the porous nature of the material.

For the study of wave propagation in a porous material there are two approaches

4. Poroelasticity in spinal cord tissue 83

that are in present use: theTheory of Poroelasticity and the Theory of Porous Media.

Both model a two-phase continuum consisting of a porous solid skeleton saturated with an interstitial pore fluid. Their main difference lies in the way the solid-fluid interaction is handled.

The framework for the Theory of Poroelasticity was first presented in 1941 by Biot, based on the earlier work of von Terzaghi (1923). This was a quasi-steady-state theory that was derived in an intuitive fashion and used to study the settlement of soils under load, a phenomenon called consolidation. Later Biot extended this theory to include anisotropy (Biot, 1955), viscoelasticity (Biot, 1956a), and eventually the full unsteady formulation for low-frequency (Biot, 1956b) and high-frequency waves (Biot, 1956c). One of the major findings in these latter two papers was the identification of three different plane wave types for a fluid-saturated porous continuum: a shear wave and two kinds of dilatational wave. The shear wave, transmitted by the solid, and the faster of the two dilatational waves, transmitted by the fluid and solid moving in phase, are analogous to the two wave types in classical linear elasticity theory. The additional dilatational wave, also known as the slow wave, corresponds to the solid and fluid moving out of phase with one another and has been confirmed experimentally (Plona, 1980).

In contrast to Biot’s Theory of Poroelasticity, the Theory of Porous Media has been the incremental effort of a number of researchers and so has taken longer to develop. Inspired by the work of Fillunger (1913), this theory is based on the axioms of con- tinuum theories of mixtures (Truesdell & Toupin, 1960; Bowen, 1976) extended by the concept of volume fractions (Bowen, 1980; 1982; Ehlers, 1993). Part of the motivation in developing this theory has been to rigorously define the pore-level micromechan- ics through tensor transformations, rather than keep to the more phenomenological approach of Biot. The same three wave types are predicted as described above.

In the case of incompressible constituents, the two theories are equivalent pro- viding that the apparent mass density term is set to zero. On the other hand, if the constituents are compressible then each theory may predict a different outcome (Schanz

& Diebels, 2003). The different sets of governing equations and material parameters has made comparison of the two theories problematic. However, Biot’s theory has been around longer and it has stood up to experimental validation, and so is the approach that is adopted here. For more details on the Theory of Porous Media, the inter- ested reader is referred to the book by de Boer (2000). Next, the governing equations are given for the Theory of Poroelasticity, along with a description of the material parameters.