Deformations Experienced in the Human Skin, Adipose Tissue, and Fascia in Osteopathic Manipulative Medicine

Development of Mathematical Model

Because the skin, adipose tissue, and fascia are known to experience large strains when subjected to longitudinal forces,¹¹ a large deformation theory¹² was used to determine the magnitude of the mechanical forces applied to the surface of the skin that results in a specified deformation. We followed the development of the large deformation theory equations from our previous study⁸ and used a 3-dimensional element comprising the skin, adipose tissue, and fascia (Figure 1). We assumed that these tissues are composed of incompressible material¹¹ (ie, the volume before and after deformation remains the same). We also included the basic kinematics and kinetics equations of Green and Zerna¹² for evaluating the stresses under specified deformations for any such tissue.

Figure 1.

Three-dimensional model of skin, adipose, and fascial layers subjected to normal and tangential forces in the undeformed state. The axes (x¹, x², x³) in the undeformed state coincide with the axes (y¹, y², y³) in the deformed state. The symbols represent the coordinate values of the faces along the x¹, x², and x³ axes.

The stresses are required to satisfy differential equations of equilibrium and boundary conditions to determine the magnitude of the mechanical forces needed to produce the specified deformations.

The metric tensors g_ij, g^ij (needed to evaluate stresses) in the Cartesian coordinates xⁱ in the undeformed state are determined by:

The repeated index means summation over r. Similarly, the metric tensors G_ij, G^ij in the deformed Cartesian coordinates y^r (r=1,2,3) are determined by

From equation (1) we find that

Where |g_ij| is the determinant of the matrix g_ij. Thus, g=1. The metric tensors defined above are the measures of deformations in 3 dimensions. Their physical meaning can be understood by their relationship to the strains E^ij through the equation:

Deformation

We assumed that the deformation produced by the manual therapy techniques of shear along the x¹ axis, and compression along the negative x³ axis (Figure 1), would be determined by the following equation:

where the yⁱ axes in the deformed state coincide with the xⁱ axes in the undeformed state.

Here, k_i denotes the shear ratio from the application of the tangential force. The maximum shear occurs at the surface of the fascia where the thickness is maximum, and is 0 at the bottom of the fascia where the thickness is 0. In equation (5), k₃ denotes the compression ratio from the applied normal pressure, and k₂ is the extension ratio resulting from compression on the surface of the fascia. Often in manual therapy, compression and shear are applied simultaneously. Values of k₃ less than 1 indicate compression. For example, k₃=0.90 indicates 10% compression. In this equation, k₂ is determined in terms of k₃ using the incompressibility condition as described below.

Using equations (2) and (5) we get

The strain invariants I₁, I₂, I₃ (needed to evaluate stresses) are determined by

where G is the determinant of the matrix G_ij, and G from (6) is k₂² [k₃²]. The strain invariants in equation (8) do not vary with the coordinate system.

Using (3) and (6) through (8), we find

Because the fasciae are assumed to be incompressible,

To determine the tensor B^ij used to evaluate stresses, we used the following equation:

Therefore, using (3), (7), and (10), we obtained

Evaluation of Stress

The stresses can be evaluated from formula (11) as follows:

where W is the strain energy function.

We used the form of the strain energy function for soft tissues described by Demiray.¹³ This function is given by

where C₁ and C₂ are mechanical parameters to be determined. These parameters are analogous to the elastic parameters in the small deformation theory of elasticity. Their values are determined by curve fitting as explained in the Evaluation of Mechanical Parameters section.

After applying the equilibrium equations and the boundary conditions, we arrived at the following equations for evaluating the normal and tangential stresses, N and T, respectively. The details are provided in Chaudhry et al.⁸

Using equations (13) and (14), we have

Then the normal pressure (N) is given by

The tangential stress (T) along CD in Figure 1 becomes

We assumed that the forces applied to the surface of the skin are transmitted entirely to the underlying fat and fascia, independent of the thickness of these tissues. We tested this hypothesis by placing various materials as proxies for skin, fat, and fascia on a pressure mat and applying forces of 10 kg, which was within the range of forces applied in manual therapies.⁸ The forces were applied through these layers to the pressure mat in several configurations, and in all cases the forces were transmitted entirely through the layers, with measurements varying from 5% to 7%, which is within the sensitivity range of this device. This hypothesis does not conflict with the results in authoritative articles by others who have studied the transmission of forces through tissue.^14-16 Bereznick et al¹⁴ studied frictional properties at the thoracic skin-fascia interface during spinal manipulative therapy. In their introduction, the authors stated that in the absence of friction, the normal component of the applied forces is directly transmitted to the underlying vertebrae, which is what we assumed. Kawchuk and Perle¹⁵ studied the relationship between the application angle of spinal manipulative therapy and the resultant vertical acceleration produced. According to their findings (presented in Table 1 of their article), the experimentally measured transmitted force can be slightly more or less than the predicted force, and the average difference between the predicted force (based on the applied force) and the transmitted force to the underlying vertebral layer is −0.59%, with an SD of more than 6%. This finding indicates that any difference between the applied force and the measured transmitted force is well within the measurement capabilities of the experiment. That is, within experimental measurement error, the applied force is totally transmitted to the underlying layer. It should be noted that Perle and Kawchuk¹⁶ studied the distribution of pressure generated on a single rigid surface during hand manipulation and its relationship with the location and magnitude of hand configurations; thus, their findings are not directly relevant to the work in the current article.

Evaluation of Mechanical Parameters

To estimate the values of C₁ and C₂ in the formulas for fascia, skin, and adipose tissue, we needed experimental data for the stress, σ, vs the extension. Then we found the values for C₁ and C₂ that minimize (in the least squares sense^17,18) the difference between the experimental values and the form given in equation (18). The values of C₁ and C₂ for superficial nasal fascia were estimated in our 2008 study.⁸

We now needed to estimate C₁ and C₂ for skin and adipose tissue. To compute the theoretical stress-strain relationship, we used the longitudinal stress-stretch relationship for incompressible material^12(p80) for our setting. This relationship is shown by

where λ is the stretch ratio.

For experimental values along the curve, we used the plot described in Lapeer et al⁹ for human skin and the plot described in Finocchietti et al¹⁰ for human adipose tissue. We assumed, as in our previous model,⁸ that the strain energy function constants, C₁ and C₂, derived from experimental data from stretch experiments, were valid for compression.

As in our previous model,⁸ we used the least squares method to find the values of C₁ and C₂ that minimize the difference between the experimental values given in Lapeer et al⁹ and in Finocchietti et al¹⁰ and the theoretical values from the form in equation (18) (Figure 2, Figure 3, and Figure 4). The values of mechanical constants C₁ and C₂ are presented in Table 1 for the 3 types of tissue considered.

Figure 2.

Experimental (measured) and theoretical (calculated) stress-stretch ratio curves for superficial nasal fascia.⁸

Figure 3.

Experimental (measured) and theoretical (calculated) stress-stretch ratio for skin.

Figure 4.

Experimental (measured) and theoretical (calculated) stress-stretch ratio for adipose tissue.