Morphology and Morphometry of Human Chronic Spinal Cord Injury Using Diffusion Tensor Imaging and Fuzzy Logic

Abstract

Diffusion tensor imaging (DTI) was performed on regions rostral to the injury site in four human subjects with chronic spinal cord injury (SCI) and equivalent regions in four neurologically intact subjects. Apparent diffusion coefficients were measured and compared between subjects. A fuzzy logic tissue classification algorithm was used to segment gray and white matter regions for morphometric analysis, including comparisons of cross-sectional areas of gray and white matter along with frontal and sagittal diameters. Results indicated a general decrease in both longitudinal and transverse diffusivity in the upper cervical segments of subjects with chronic SCI. Further, a decrease in the cross-sectional area of the entire spinal cord was observed in subjects with SCI, consistent with severe atrophy of the spinal cord. These observations have implications in tracking the progression of SCI from the acute to the chronic stages. We conclude that DTI with fuzzy logic tissue classification has potential for monitoring morphological changes in the spinal cord in people with SCI.

Appendix A: Mathematics of FIS Used for Spinal Cord Tissue Classification

The three anisotropy indices defined in Eqs. (1)–(3) were stored in a separate input matrix defined as $\varvec{\upchi} = {\left[ {\begin{array}{*{20}c} {{\psi _{1} }} & {{\psi _{2} }} & {{\psi _{3} }} \\ \end{array} } \right]}$. For each index, Gaussian membership functions were created to reflect the distributions of each region of interest (GM, WM, and CSF) for use in a Mamdani-type FIS.34 The membership functions represent the degree of membership a particular index value has in a specific region of interest (tissue type). The process of calculating the degree of membership a particular index value has in a particular region of interest is termed fuzzification and consists of normalized values from 0 to 1. The fuzzification matrix is defined as

$$ \varvec{\upmu}{\left( \varvec{\upchi} \right)} = {\left[ {\begin{array}{*{20}c} {{\varvec{\upmu}_{{{\text{CSF}}}} {\left( \varvec{\upchi} \right)}}} & {{\varvec{\upmu}_{{{\text{GM}}}} {\left( \varvec{\upchi} \right)}}} & {{\varvec{\upmu}_{{{\text{WM}}}} {\left( \varvec{\upchi} \right)}}} \\ \end{array} } \right]} = {\left[ {\begin{array}{*{20}c} {{\mu _{{{\text{CSF}}}} {\left( {\psi _{1} } \right)}}} & {{\mu _{{{\text{GM}}}} {\left( {\psi _{1} } \right)}}} & {{\mu _{{{\text{WM}}}} {\left( {\psi _{1} } \right)}}} \\ {{\mu _{{{\text{CSF}}}} {\left( {\psi _{2} } \right)}}} & {{\mu _{{{\text{GM}}}} {\left( {\psi _{2} } \right)}}} & {{\mu _{{{\text{WM}}}} {\left( {\psi _{2} } \right)}}} \\ {{\mu _{{{\text{CSF}}}} {\left( {\psi _{3} } \right)}}} & {{\mu _{{{\text{GM}}}} {\left( {\psi _{3} } \right)}}} & {{\mu _{{{\text{WM}}}} {\left( {\psi _{3} } \right)}}} \\ \end{array} } \right]}, $$

(A.1)

where

$$ \mu _{{{\text{tissue}}}} {\left( {{\text{index}}} \right)} = \frac{1} {{{\sqrt {2\pi s_{{{\text{index,tissue}}}} } }}}\;{\text{exp}}\left ( \frac{{ -({\text{index}} - m_{{{\text{index,tissue}}}} )^{2} }} {{2 \cdot s_{{{\text{index,tissue}}}} }} \right)$$

is the Gaussian membership function for a specific anisotropy index and tissue type, m _index,tissue is the sampled mean value for a particular index and tissue type obtained with the histological template, and s _index,tissue is the sampled standard deviation for a particular index and tissue type obtained with the histological template.

Once the membership of a particular index value (ψ₁, ψ₂ or ψ₃), for a particular tissue type (CSF, GM or WM), was calculated, we determined the tissue type physically represented in the particular voxel of interest. Since all three anisotropy indices must agree on the type of tissue represented in the voxel, a logical ‘and’ operation for each column in the fuzzification matrix was performed by finding the product of the degree of membership for all three anisotropy indices. Note that if the value of membership for a particular tissue type was high for all three anisotropy indices, the result after the product was high, thus providing confidence about the classification of this voxel as that particular tissue type. The degree of membership as a particular tissue type, for a specific voxel, was defined as

$$ \varvec{\upbeta} = {\text{and}}{\left( {\varvec{\upmu}{\left( \varvec{\upchi} \right)}} \right)} = {\left[ {\begin{array}{*{20}c} {{{\text{and}}{\left( {\varvec{\upmu}_{{{\text{CSF}}}} {\left( \varvec{\upchi} \right)}} \right)}}} & {{{\text{and}}{\left( {\varvec{\upmu}_{{{\text{GM}}}} {\left( \varvec{\upchi} \right)}} \right)}}} & {{{\text{and}}{\left( {\varvec{\upmu}_{{{\text{WM}}}} {\left( \varvec{\upchi} \right)}} \right)}}} \\ \end{array} } \right]}, $$

(A.2)

where

$$ {\text{and}}{\left( {\varvec{\upmu}_{{{\text{tissue}}}} {\left( \varvec{\upchi} \right)}} \right)} = {\prod\limits_{ \in {\text{ index}}} {\mu _{{{\text{tissue}}}} {\left( {{\text{index}}} \right)}} }. $$

(A.3)

The degree of membership of a specific voxel as a particular tissue type was then mapped to the new variable space termed the fuzzy anisotropy index (FAI) by first defining a set of output membership functions. To allow for a specific voxel being a combination of tissue types, as occurs with partial volume effects, we defined an output membership function matrix as

$$ \varvec{\upxi}(y) = {\left[ {\begin{array}{*{20}c} {{\xi _{{{\text{CSF}}}} (y)}} & {{\xi _{{{\text{GM}}}} (y)}} & {{\xi _{{{\text{WM}}}} (y)}} \\ \end{array} } \right]}, $$

(A.4)

where

$$ \xi _{{{\text{CSF}}}} (y) = \left\{ {\begin{array}{*{20}c} {{ - 2y + 1}} \\ {{\text{0}}} \\ \end{array} } \right.{\text{ }}\begin{array}{*{20}c} {{{\text{for 0}} \le y{\text{ }} \le {\text{0}}{\text{.5}}}} \\ {{{\text{for 0}}{\text{.5}} \le y{\text{ }} \le {\text{1}}}} \\ \end{array} , $$

(A.5)

$$ \xi _{{{\text{GM}}}} (y) = \left\{ {\begin{array}{*{20}c} {{2y}} \\ {{ - 2y - 2}} \\ \end{array} } \right.{\text{ }}\begin{array}{*{20}c} {{{\text{for 0}} \le y{\text{ }} \le {\text{0}}{\text{.5}}}} \\ {{{\text{for 0}}{\text{.5}} \le y{\text{ }} \le {\text{1}}}} \\ \end{array} , $$

(A.6)

$$ \xi _{{{\text{WM}}}} (y) = \left\{ {\begin{array}{*{20}c} {0} \\ {{2y - 1}} \\ \end{array} } \right.{\text{ }}\begin{array}{*{20}c} {{{\text{for 0}} \le y{\text{ }} \le {\text{0}}{\text{.5}}}} \\ {{{\text{for 0}}{\text{.5}} \le y{\text{ }} \le {\text{1}}}} \\ \end{array} , $$

(A.7)

and y is the output variable termed the fuzzy anisotropy. To perform the mapping operation, the minimum between the membership grade for each tissue type, $\varvec{\upbeta}$, and the output membership function for that particular tissue type, $\varvec{\upxi}{\left( y \right)}$, must be calculated. The membership grade for each tissue type was assumed constant for all values of the fuzzy anisotropy, y, thus $\varvec{\upbeta}{\left( y \right)} = \varvec{\upbeta}$. The mapped distributions were defined as

$$ \varvec{\upeta}{\left( y \right)} = {\left[ {\begin{array}{*{20}c} {{\eta _{{{\text{CSF}}}} {\left( y \right)}}} & {{\eta _{{{\text{GM}}}} {\left( y \right)}}} & {{\eta _{{{\text{WM}}}} {\left( y \right)}}} \\ \end{array} } \right]} = \min {\left( {\varvec{\upbeta},\varvec{\upxi}{\left( y \right)}} \right)}, $$

(A.8)

where

$$ \eta _{{{\text{tissue}}}} {\left( y \right)} = \min {\left( {{\text{and}}{\left( {\varvec{\upmu}_{{{\text{tissue}}}} {\left( \varvec{\upchi} \right)}} \right)},\xi _{{{\text{tissue}}}} (y)} \right)}. $$

(A.9)

The result of the mapping operation was a row vector, $\varvec{\upeta}{\left( y \right)}$, containing three functions of the fuzzy anisotropy for each tissue type, $\eta _{{{\text{tissue}}}} {\left( y \right)}$. A single value for the FAI was computed by first summing the three distributions in $\varvec{\upeta}{\left( y \right)}$ and then finding the centroid location along y. The FAI was defined as

$$ {\text{FAI}} = {\text{centroid}}{\left( {\varvec{\upeta}{\left( y \right)}{\varvec{\upvarphi}}} \right)}, $$

(A.10)

where ${\varvec{\upvarphi }} = {\left[ {\begin{array}{*{20}c} {1} & {1} & {1} \\ \end{array} } \right]}^{T}$ is used as the matrix sum. The resulting FAI was used to classify gray matter regions and intact white matter tracts. The Fuzzy Logic Toolbox in MATLAB (MathWorks, Inc., Natick, MA) was used for implementation of the FIS.