5.2 Generation of 3D shapes from sparse contours
6.1.3 Fibril-reinforced poro-visco-elastic cartilage models
σsolid=−nsolidpI+σeffect
σfluid=−nfluidpI, (6.3)
where σsolid is stress of the solid phase, nsolid is the partition of the solid phase, σfluidstress of the fluid phase,nfluidis the partition of the fluid phase, pis the fluid pressure, andIis a unit matrix. The total stressσtotis a sum of solid and fluid parts, and assuming that partitions of solid and fluid part equal one, it can be presented as [99]
σtot=σeffect−pI. (6.4)
When placed under stress, articular cartilage deforms, and its properties change.
Darcy’s law can be used to model interstitial fluid flow during deformation as [105]
q=−k∇p, (6.5)
whereqis the flux through tissue andkis the permeability of the material. During this deformation, the permeability of cartilage changes from the initial permeability k0as [106]
where the void ratioeisnfluid/nsolid,e0is the original void ratio, andMis a positive constant.
6.1.3 Fibril-reinforced poro-visco-elastic cartilage models
Biphasic models take into account the multifold structure of articular cartilage, however, they ignore the non-linearity of the behavior of the material. The fibril-reinforced poro-visco-elastic model considers the total stress of the cartilage as sum of stresses in a fibrillar and non-fibrillar matrix combined with the fluid pressure.
The non-fibrillar matrix can be modelled using Hooke’s law when deformation is relatively small. However, this is not the case in knee joint cartilage where the strains may well be rather high [107]. The neo-hookean hyperelastic model assumes that the elastic behavior of cartilage to be compressible and it can be formulated as
σnf =Kmln(J)
J I+Gm(F·FT−J23I) (6.7) whereJis the determinant of the deformation gradient tensorF[108]. Bulk modulus (Gm) and shear modulus (Km) are further defined using non-fibrillar matrix modulus (En f) and Poisson ratio (νm) as [99]
In its simplest form, fibrillar matrix can be modelled using linearly elastic springs which resist forces. It is more realistic to use nonlinear and viscoelastic springs: one
individual spring would have an initial Young’s modulus E0, a nonlinear modulus E1 = Eeef, where ef is a fibril strain andEe a strain-dependent modulus, and η a viscoelastic damping coefficient [99, 109]. Thus, the total stress of one fibril can be defined as [108]
σf =
− η
2√
(σf−E0ef)Eeσ˙f+E0ef +
η+ ηE0
2√
(σf−E0ef)Ee
˙
ef, e>0
0, e≤0.
(6.9)
Since the fibrils in the model have direction, any changes in their orientation can be modelled introducing a directional dependency on their response into compressive and tensile forces [110]. When attempting to mimic collagen crosslinking and random orientation of collagen network, the model could be improved by adding secondary collagen fibrils into the models with a defined volume fraction [99].
Hence, the total stress of the fibrils would then be a sum of all of the individual fibril stresses.
In this model, the total stress of the material is a sum of the non-fibrillar matrix stress (σnf), the combined stress of each individual collagen fibril (σf), and the negative fluid pressure (p) that is determined similarly as in the case of poroelastic biphasic models [99].
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7 Aims and hypothesis
The main aim of this thesis was to develop and implement quantitative contrast-enhanced CT methods to aid in the diagnostics of knee joint injuries and osteoarthritis. The methods included a quantitative analysis of tissue characteristics, the development of a semiautomatic segmentation method for articular cartilage, and computational modelling of the knee joint function. The specific aims of each individual study were
1. to analyse defects of articular cartilage and subchondral bone simultaneously usingin vivocontrast-enhanced CBCT images.
2. to generate an automatized and accurate method for segmentation of articular cartilages usingin vivocontrast-enhanced CT images.
3. to demonstrate the functionality of generated segmentation method for finite element modelling of knee joint function.
4. to evaluate the risk of degeneration of articular cartilage defects based onin vivocontrast-enhanced CT images using finite element modelling.
8 Materials and methods
This thesis comprises four independent studies (I-IV). The studies aimed to evaluate whether contrast-enhanced CT images are beneficial in OA and injury diagnostics and biomechanical modelling of chondral lesions. Another aim was to automate the segmentation of articular cartilages from contrast-enhanced CT images and, moreover, apply the segmentation method for finite element modelling of the knee joint. The study subjects, imaging techniques, the main aims, and the methods used in studiesI-IVare summarised in Table 8.1.
Table 8.1: The data, main purposes and applied methods of studiesI-IV.
Study Imaging Aim Method
I n= 18,in vivoCBCT Simultaneous, quantitative analysis Image analysis and of cartilage and bone manual segmentation II n= 9,in vivoCT Develop automated segmentation method Algorithm and
software development
III n= 6,in vivoCT Build computational model FEM
from automated segmentation
IV n= 5,in vivoCBCT Evaluate biomechanical characteristics FEM of lesions
n= the number of imaged knees, CBCT = cone beam computed tomography, FEM = finite element modelling
8.1 CONTRAST-ENHANCED COMPUTED TOMOGRAPHY OF THE KNEE
The material in studiesI-IV consisted of patients who either had arthroscopically determined lesions or constant pain in the knee giving a suspicion for a cartilage defect. In studies I(n = 17) and IV (n = 5), the knees of the study subjects were imaged with delayed CBCT arthrography using a peripheral CBCT scanner (Verity, Planmed Oy, Finland). The knees were imaged at 5 minutes and 45 minutes after the intra-articular administration of diluted anionic contrast agent (q = 21, M = 1269 g/mol, 160 mg iodine/ml, Hexabrix, Mallinckrodt Inc., St. Louis, MO, USA).
Patients wore a hydroxyapatite phantom belt around their tibia during imaging, which enabled BMD analysis of the bone. In studiesII(n= 9) andIII (n= 6), the knees of the patients were imaged with a full body CT scanner (Discovery, PET/CT, 690 GE Medical Systems, Waukesha, WI, USA). Similarly to studiesIand IV, ioxaglate (105 mM, Hexabrix 320, Guerbet, Roissy, France) diluted in 0.9%
saline was injected into a joint cavity prior to arthrographic (5 min) and delayed (45 min) imaging.
The study protocols were reviewed by the Ethical Committee of Kuopio University Hospital, Kuopio, Finland (No: 54/2011) (Studies I and IV) or the Ethical Committee of Northern Ostrobothnia Hospital District, Oulu (No: 33/2010) (StudiesIIandIII).