Mathematical modeling

Introduction

Quantification is central to scientific understanding of phenomena, and performing experiments is clearly the fundamental approach in doing that. By ensuring that only one factor is varied at a time and all others remain unchanged or stable, one can quantify the effects of the manipulated factor on the matter experimented.

Despite the existence of numerous reports of experimental studies on the mechanical behavior of various fasciae (Iatridis et al. 2003; Zeng et al. 2003), specific mathematical models proposed for fascia are rare, conceivably due to complex geometry and material properties. Taking a finite deformation approach, analytical models of human fasciae were developed recently in order to quantify deformations occurring in manual therapy (Chaudhry et al. 2008). In such models, the tissue is considered to be continuously distributed in space (i.e., as a continuum). This disregards the very fine structure and benefits from a systematic treatment of forces and deformations that continuum mechanics offers.

Using analytical models, nonlinear material properties and large deformations can be handled; and solutions of the equilibrium equations are possible, but typically within a well defined material domain only. Therefore, complex geometries and mechanical interactions between adjacent structures cannot be addressed. Because fascial structures are architecturally highly complex and are continuous with other tissues (e.g., muscle, bone) this should be considered as a sizable limitation.

In contrast, with at least similar capabilities, the finite element method has the great advantage that geometrically highly complex structures can be modeled. This involves a general discretization procedure of continuum problems posed by mathematical statements. In this context, “discrete” represents constructing an adequate model using a finite number of well defined components (elements), the behavior of which is well understood, whereas “continuous” represents indefinite subdivisions implying differential equations and boundary conditions that characterize the mechanical equilibrium mathematically. The general procedure includes: (1) determination of the geometry of the system, (2) subdivision of the volume into a finite number of elements, (3) solution of the equilibrium equations for each element, and (4) assembly of the element solutions to obtain the solution for the complete system.

Modeling fascia and muscle tissue using the finite element method

The value of the finite element model may be substantial, provided that it is developed with a specific and well defined purpose. Such well established modeling goals determine: (1) the extent to which the actual problem can be simplified; (2) the modeling assumptions to be made; and (3) the relevant model output parameters and their interpretation. Recently, this approach has been applied successfully to modeling plantar fascia with a specific emphasis on surgical treatment of plantar fasciitis. Gefen (2002) developed a model for analysis of structural characteristics of the human foot during standing in order to investigate the biomechanical effects of surgical release of the plantar fascia. His results showed that a total fascia release can cause an extensive arc deformation compromising the foot’s load-bearing ability. With similar intentions, Cheung et al. (2006) constructed a finite element model of the ankle–foot complex aiming at quantifying the effects of different plantar fascia stiffness on foot geometry, with zero stiffness representing a fascia release. With decreasing fascia stiffness, their results also indicate a decreased foot arc height, as well as midfoot pronation. These studies indicate that the extent of plantar fascia release must be planned carefully and that models may provide a basis for such planning.

Finite element models developed to study the biomechanics of other fascial structures include, for example, anterior vaginal wall (Chen et al. 2008) and inguinal transversalis fascia (Fortuny et al. 2009). Also, muscle mechanics has been studied using finite element modeling (Gielen 1998; van der Linden 1998; Blemker et al. 2005). Despite the diversity of their modeling goals, the authors regarded muscle implicitly as a tissue that can be active and thereby change its properties. However, they did not model the muscle as operating within the context of a fascial integrity: (1) elements were used in which both active and passive properties of muscle tissue are lumped together, hence, the role of intramuscular connective tissues and their interaction with the contractile apparatus were not accounted for explicitly; (2) muscle was considered as an isolated entity, hence the continuity of intramuscular fascia (e.g., epimysium, perimysium, endomysium) and epimuscular fascia such as collagen reinforced neurovascular tracts or compartmental boundaries were not accounted for.

In contrast, the linked fiber-matrix mesh (LFMM) model developed by the authors during previous appointments at the University of Twente (Yucesoy et al. 2002) was designed specifically to study muscular mechanics within the context of fascial integrity. The mechanical roles played by such muscle-related fascia have been referred to as intra- and epimuscular myofascial force transmission (Huijing 1999; Yucesoy et al. 2005). Several publications focusing on the major effects of such force transmission (Maas et al. 2001; Yucesoy et al. 2003a; Meijer et al. 2007; Smeulders & Kreulen 2007; Yucesoy & Huijing 2007) typically report differences in muscle forces exerted by muscle at its origin and insertion, as well as the length range of muscle active force exertion being dependent on the actual mechanical conditions imposed (e.g., muscular relative positions).

The following concepts characterizing important phenomena also determined the modeling approach of the LFMM model:

1. Multi-molecular connections between the myofiber (i.e., muscle fiber) and the extracellular matrix (ECM) found along the full periphery (Berthier & Blaineau 1997) of myofibers are capable of transmitting force (Street 1983; Huijing et al. 1998; Yucesoy et al. 2002). Therefore, the force balance determining the length of a sarcomere is much more complex than just the interaction between two sarcomeres arranged in series within the same myofiber; it also includes intramuscular myofascial loads: forces exerted by (i) the ECM and (ii) sarcomeres located in neighboring myofibers.