J. BW
Vol. 24,No. IO.pp. 907422
RiolcdiaGmt
002142!40/91s3.00+.00
1991.
Q 1991 Peqamon
Britain
PIeM pk
A MICROSTRUCTURALLY-BASED FINITE ELEMENT MODEL OF THE INCISED HUMAN CORNEA PETERM. PINSKYand DEEPAKV. DAT~E Department of Civil Engineering, Stanford University, Stanford, CA 94305-4020, U.S.A. Abstract-A mechanical model of the human cornea is proposed and employed in a finite element formulation for simulating the effects of surgical procedures, such as radial keratotomy, on the cornea. The model a+mes that the structural behavior of the cornea is governed by the properties of the stroma. Argumenb based on the microstructural organization and properties of the stroma lead to the conclusion that the tiuman cornea exhibits flexural and shear rigidities which are negligible compared to its membrane rigidity. Accordingly, it is proposed that to a first approximation, the structural behavior of the cornea is that of a thick membrane shell. The tensile forces in the cornea are resisted by very fine collagen fibrils embed* in the ground substance of the stromal lamellae. When the collagen fibrils are cut, as in radial keratotorpy, it is argued that they become relaxed since there is negligible transfer ofload between adjacent fibrils dd to the low shear modulus of the ground substance. The forces in the cornea are then resisted only by the rdmaining uncut fibrils. The cutting of fib& induces an anisotropy and inhomogeneity in the membrae rigidity. By assuming a uniform angular distribution of stromal lame& through the corneal thickness, geometric arguments lead to a quantitative representation for the anisotropy and inhomogeneity. All material behavior is assumed to be in the linear elastic regime and with no time-dependency. The resulting ~constitutivemodel for the incised cornea has been employed in a geometrically non-linear 6nite element membrane shell formulation for small strains with moderate rotations. A number of numerical exampkslare pmsented to illustrate the elkctiveness of the proposed constitutive model and 6nite element formulation. The dependence of the outcome of radial keratotomy, measured in terms of the immediate postopcr$ive shift in d power, on a number of important factors is investigated. These factors include the value of the elastic module of the stromal lamellae (dependent on the patient’s age), the incision depth, the opt&one size, the number of incisions and their positions, and the intraocular pressure. Results have also b#n compared with expected surgical corrections predicted by thre.e expert surgeons and show an excellcnt correspondence.
INTRODUCMON
Surgical procedkues on the cornea which change its geometry have r&cntly assumed considerable importance for correcting &active errors of the eye. The best known is rqdial keratotomy in which a pattern of deep incisions is’made on the cornea in order to flatten it and thus reliek myopia At present, however, there is no accurate mode1 for objective estimates of refractive correction fair this and similar forms of surgery. It is believed that a structural mode1 of the cornea, based consistently on !the microstructural organization and properties of the tissue, will provide a means of improving the accuracy of predicted refractive effects. The response of the cornea to incisions will have an instantaneous component and a time-dependent component associated with a viscoelastic-like behavior having a relaxation time of the order of an hour. The subsequent hea!ing of the incisions will also cause a progressive cha&e in the comeal geometry over the healing period. The compkx process of wound healing takes about three to four years to complete and hence. the shape of the incised cornea will not generally stabilixe until a&r this period of time. The surgical correction changes by about 25% during the healing process and it L the final long-term outcome of the Received injinal fonrc 15 March 1991.
surgery which is of most importance to the patient. The comeal model proposed here does not take into account the effects of viscoelasticity or of the healing process, although consideration of these is an important research area. A number of researchers have proposed models for the incised cornea. Most are based on the assumption of isotropic elasticity. Furthermore, they have treated the cornea as a shell-like structure and thus implicitly assume it to have shear and bending stillbesses consistent with isotropic solids. Schachar et al. (198Oa and 198Ob) have attempted a comeal model based on the momentless theory of thin shells. Bryant et al. (1989) have proposed a model based on the assumption of transverse isotropy of the incised cornea and have attempted a design procedure for keratorefractive surgery based on their model. Hanna et al. (1988 and 1989) have described an isotropic model for the cornea and have attempted to obtain the stress distribution within it. Payne and Knasel(l987) have used isotropic solid brick elements in their attempt towards the modeling of the cornea. Huang et al. (1988), and Arciniegas and Amaya (1988) have proposed comeal models on similar lines. As described below, the primary load carrying component of the cornea is the stroma which is composed of a jelly-like ground substance supporting many strong but very fine collagen fibrils. When these
907
P. M. PINSKY and D. V. Dine
908
fibrils are cut in a surgical procedure they lose their ability to carry tension and cause a stress redistribution in the remaining uncut fib& This stress redistribution will be a function of the depth and pattern of incisions and will dictate the resulting deformation of the cornea. In this paper it is argued that the cornea behaves structurally, to a first approximation, as a thick membrane rather than as a shell structure, i.e. its shear and bending rigidities are several orders of magnitude smaller than its membrane rigidity. Furthermore, the primary structural effect of incisions can be modeled by the introduction of anisotropy and inhomogeneity in the membrane rigidity. A new constitutive equation for the incised cornea is developed and employed in a geometrically non-linear finite element model of the cornea which can be used to compute the instantaneous response due to incisions. This preliminary model is thought to provide a useful basis for future rational developments including the important effects of time-dependent phenomena and wound healing. MECHANICAL SMNJCTURE OF THE CORNEA
Anatomyand physiologyof the cornea The human eyeball is an imperfect spherical globe, with the cornea having a smaller radius of curvature than the remaining portions of the eyeball. The visible white opaque portion of the eyeball is called the sclera. The cornea forms the transparent outer covering of the visible colored portion of the eyeball, the color being due to the underlying iris. The interface between air and the cornea forms the main refractive component of the eye. The radius of curvature of the human cornea in its central region is 7.86 mm with a standard deviation of 0.26 mm. The horizontal diameter is in the range of 11-12 mm in 95% of the cases. The cornea1 thickness in the central region is 0.52 mm with a standard deviation of 0.04 mm. The cornea thickens towards it periphery, where its value is about 0.65 mm. The human cornea is divided into five layers lying parallel to its surface, from outside to inside they arez the epithelium, Bowman’s membrane, stroma, Descemet’smembrane and the endothelium (Maurice, 1984).The stroma makes up about 90% of the thickness of the cornea and is divided into 300-500 sheets of collagenous material the stromal lamellae, lying parallel to the comeal surface. These stromal lamellae can be distinguished clearly under electron microscopy, a typical result is shown in Fig. 1. The lamellae appear to run uninterruptedly from limbus to limbus. Each lamella is about 2-3 mm broad and about 1.5-2.5 JCII thick, and these do not appear to be interwoven. Each lamella can be se.en to be composed of long collagen fib& embedded in a ground substance. In each lamella the collagen fib& lie parallel to each other and run continuous along the length of the lamella. The diameter of the dry collagen fibrils in the human stroma is about 25 nm. This diameter does not vary
significantly across the thickness of the human cornea or over its surface. In accordance with this microstructure, the mechanical properties of a lamella are orthotropic. The transparency of the cornea requires a regular distribution of thi collagen fib& in addition to their being of equal diameter. The collagen fibrils are 20 times smaller than the average wavelength of visible light and in general would scatter light and make the cornea opaque if they were not uniformly arranged and of equal diameter (Gallagher and Mauria, 1977; and Maurice, 1987a). The cohesive forces in the surface cellular layers of the cornea are orders of magnitude smaller than the in-plane elastic stiffness of the stroma. Furthermore, the stroma along with Bowman’s membrane account for more than 90% of the thickness of the cornea. Hena it can be assumed that the in-plane elastic properties of the cornea are governed by those of the stroma. Mechanical properties of the cornea Collagen fibrils have a Young’s modulus of the order of 1.0 GPa along the fibril direction, whereas the Young’s modulus and shear modulus of the ground substana are of the order of 10” GPa. Although there are many fibrils in each lamella, they have individual diameters of only 25 nm and so have extremely low moments of inertia and bending rigidities. Due to the ‘weak’ properties of the ground substana and orientation of the fibrils, the stroma offers relatively little resistance to shear forces; in an isolated cornea the two faces slide relative to one another quite freely (Maurice, 1987a). Further, although the radius of curvature to thickness ratio of the cornea is on the order of 14-15, there is experimental evidena that in the in uivocornea, the stroma bears a constant tension at all points across the thickness (McPhee et al., 1985;
Eliason and Mauria, 1981).Based on these properties of the stroma it is reasonable to infer that the structural behavior of the cornea, to a 6rst approximation, can be idealized to that of a membrane structure, i.e. it can support in-plane (or ‘membrane’) forces but not significant transverse shear forces or bending moments. Soft biological tissues often have a ‘S-shaped stress-strain curve, which offers them the dual advantages of avoiding stress concentrations at low strains and achieving toughness at high strains by increasing their work of fracture (Fun& 1981; Lee and Tseng, 1982; and Gordon, 1988). The human cornea also shows a similar non-linear curve (Nyquist, 1968, Woo et al., 1972; Jue and Mauria, 1986). However, it has been observed that the strain regime in collagen fib& at physiological intraocular pressures corresponds to the linear elastic range (Schachar et al., 1980). Accordingly, linear elasticity is assumed at this stage of the analysis. It is possible that strain levels in the incised cornea may be in the non-linear regime. Exact characterization requires further study.
Fig. 1. El@ctron micrograph of a croswection of the corneaI stroma, showing the lame& and the arrangement ofthe collagen fibrils. The micrograph is of a ~&on of size 4.5 m x 3.3 m Individual stromal lamellae can be identified, and each is seen to contain evenly spaced collagen fibrile which have a characteristic direction associated with the lamella. The fibrils are identified aa circular dots or short oblique lines, d-ding on their angle of orientation relative to the cutting plane. (Photograph courtesy of T. Singh.)
909
Finite element model of incised cornea CONSUTWI’WE MODEL FOR THE INCISED CORNEA Properties
of thestromallamellae
The collagen f&rib in a stromal lamella lie parallel to each other asd are embedded in the ground substance which swells due to hydration and shrinks when it is dehydrated; the collagen water content is unchanged. The water content of the cornea is governed by an active mechanism located in the endothelium and the mechanical properties of the stromal lamella and in turn those of the whole cornea are a function of the water content. However, at normal physiological levels of water content, the Young’s modulus and the shear modulus of the ground substance are very small relative to those of the collagen fib@. The two-ph+ (fib&ground substance) stromal lamella is modeled as an orthotropic elastic material with principal axes de&ted by the direction of the fibrils. Rach s omal lamella is a thin membrane curved in space1 ollowing the geometry of the cornea. At every point on the mid-surface of a representative lamella, we construct an orthogonal lamella coordinate system (
Vol. 24,No. IO.pp. 907422
RiolcdiaGmt
002142!40/91s3.00+.00
1991.
Q 1991 Peqamon
Britain
PIeM pk
A MICROSTRUCTURALLY-BASED FINITE ELEMENT MODEL OF THE INCISED HUMAN CORNEA PETERM. PINSKYand DEEPAKV. DAT~E Department of Civil Engineering, Stanford University, Stanford, CA 94305-4020, U.S.A. Abstract-A mechanical model of the human cornea is proposed and employed in a finite element formulation for simulating the effects of surgical procedures, such as radial keratotomy, on the cornea. The model a+mes that the structural behavior of the cornea is governed by the properties of the stroma. Argumenb based on the microstructural organization and properties of the stroma lead to the conclusion that the tiuman cornea exhibits flexural and shear rigidities which are negligible compared to its membrane rigidity. Accordingly, it is proposed that to a first approximation, the structural behavior of the cornea is that of a thick membrane shell. The tensile forces in the cornea are resisted by very fine collagen fibrils embed* in the ground substance of the stromal lamellae. When the collagen fibrils are cut, as in radial keratotorpy, it is argued that they become relaxed since there is negligible transfer ofload between adjacent fibrils dd to the low shear modulus of the ground substance. The forces in the cornea are then resisted only by the rdmaining uncut fibrils. The cutting of fib& induces an anisotropy and inhomogeneity in the membrae rigidity. By assuming a uniform angular distribution of stromal lame& through the corneal thickness, geometric arguments lead to a quantitative representation for the anisotropy and inhomogeneity. All material behavior is assumed to be in the linear elastic regime and with no time-dependency. The resulting ~constitutivemodel for the incised cornea has been employed in a geometrically non-linear 6nite element membrane shell formulation for small strains with moderate rotations. A number of numerical exampkslare pmsented to illustrate the elkctiveness of the proposed constitutive model and 6nite element formulation. The dependence of the outcome of radial keratotomy, measured in terms of the immediate postopcr$ive shift in d power, on a number of important factors is investigated. These factors include the value of the elastic module of the stromal lamellae (dependent on the patient’s age), the incision depth, the opt&one size, the number of incisions and their positions, and the intraocular pressure. Results have also b#n compared with expected surgical corrections predicted by thre.e expert surgeons and show an excellcnt correspondence.
INTRODUCMON
Surgical procedkues on the cornea which change its geometry have r&cntly assumed considerable importance for correcting &active errors of the eye. The best known is rqdial keratotomy in which a pattern of deep incisions is’made on the cornea in order to flatten it and thus reliek myopia At present, however, there is no accurate mode1 for objective estimates of refractive correction fair this and similar forms of surgery. It is believed that a structural mode1 of the cornea, based consistently on !the microstructural organization and properties of the tissue, will provide a means of improving the accuracy of predicted refractive effects. The response of the cornea to incisions will have an instantaneous component and a time-dependent component associated with a viscoelastic-like behavior having a relaxation time of the order of an hour. The subsequent hea!ing of the incisions will also cause a progressive cha&e in the comeal geometry over the healing period. The compkx process of wound healing takes about three to four years to complete and hence. the shape of the incised cornea will not generally stabilixe until a&r this period of time. The surgical correction changes by about 25% during the healing process and it L the final long-term outcome of the Received injinal fonrc 15 March 1991.
surgery which is of most importance to the patient. The comeal model proposed here does not take into account the effects of viscoelasticity or of the healing process, although consideration of these is an important research area. A number of researchers have proposed models for the incised cornea. Most are based on the assumption of isotropic elasticity. Furthermore, they have treated the cornea as a shell-like structure and thus implicitly assume it to have shear and bending stillbesses consistent with isotropic solids. Schachar et al. (198Oa and 198Ob) have attempted a comeal model based on the momentless theory of thin shells. Bryant et al. (1989) have proposed a model based on the assumption of transverse isotropy of the incised cornea and have attempted a design procedure for keratorefractive surgery based on their model. Hanna et al. (1988 and 1989) have described an isotropic model for the cornea and have attempted to obtain the stress distribution within it. Payne and Knasel(l987) have used isotropic solid brick elements in their attempt towards the modeling of the cornea. Huang et al. (1988), and Arciniegas and Amaya (1988) have proposed comeal models on similar lines. As described below, the primary load carrying component of the cornea is the stroma which is composed of a jelly-like ground substance supporting many strong but very fine collagen fibrils. When these
907
P. M. PINSKY and D. V. Dine
908
fibrils are cut in a surgical procedure they lose their ability to carry tension and cause a stress redistribution in the remaining uncut fib& This stress redistribution will be a function of the depth and pattern of incisions and will dictate the resulting deformation of the cornea. In this paper it is argued that the cornea behaves structurally, to a first approximation, as a thick membrane rather than as a shell structure, i.e. its shear and bending rigidities are several orders of magnitude smaller than its membrane rigidity. Furthermore, the primary structural effect of incisions can be modeled by the introduction of anisotropy and inhomogeneity in the membrane rigidity. A new constitutive equation for the incised cornea is developed and employed in a geometrically non-linear finite element model of the cornea which can be used to compute the instantaneous response due to incisions. This preliminary model is thought to provide a useful basis for future rational developments including the important effects of time-dependent phenomena and wound healing. MECHANICAL SMNJCTURE OF THE CORNEA
Anatomyand physiologyof the cornea The human eyeball is an imperfect spherical globe, with the cornea having a smaller radius of curvature than the remaining portions of the eyeball. The visible white opaque portion of the eyeball is called the sclera. The cornea forms the transparent outer covering of the visible colored portion of the eyeball, the color being due to the underlying iris. The interface between air and the cornea forms the main refractive component of the eye. The radius of curvature of the human cornea in its central region is 7.86 mm with a standard deviation of 0.26 mm. The horizontal diameter is in the range of 11-12 mm in 95% of the cases. The cornea1 thickness in the central region is 0.52 mm with a standard deviation of 0.04 mm. The cornea thickens towards it periphery, where its value is about 0.65 mm. The human cornea is divided into five layers lying parallel to its surface, from outside to inside they arez the epithelium, Bowman’s membrane, stroma, Descemet’smembrane and the endothelium (Maurice, 1984).The stroma makes up about 90% of the thickness of the cornea and is divided into 300-500 sheets of collagenous material the stromal lamellae, lying parallel to the comeal surface. These stromal lamellae can be distinguished clearly under electron microscopy, a typical result is shown in Fig. 1. The lamellae appear to run uninterruptedly from limbus to limbus. Each lamella is about 2-3 mm broad and about 1.5-2.5 JCII thick, and these do not appear to be interwoven. Each lamella can be se.en to be composed of long collagen fib& embedded in a ground substance. In each lamella the collagen fib& lie parallel to each other and run continuous along the length of the lamella. The diameter of the dry collagen fibrils in the human stroma is about 25 nm. This diameter does not vary
significantly across the thickness of the human cornea or over its surface. In accordance with this microstructure, the mechanical properties of a lamella are orthotropic. The transparency of the cornea requires a regular distribution of thi collagen fib& in addition to their being of equal diameter. The collagen fibrils are 20 times smaller than the average wavelength of visible light and in general would scatter light and make the cornea opaque if they were not uniformly arranged and of equal diameter (Gallagher and Mauria, 1977; and Maurice, 1987a). The cohesive forces in the surface cellular layers of the cornea are orders of magnitude smaller than the in-plane elastic stiffness of the stroma. Furthermore, the stroma along with Bowman’s membrane account for more than 90% of the thickness of the cornea. Hena it can be assumed that the in-plane elastic properties of the cornea are governed by those of the stroma. Mechanical properties of the cornea Collagen fibrils have a Young’s modulus of the order of 1.0 GPa along the fibril direction, whereas the Young’s modulus and shear modulus of the ground substana are of the order of 10” GPa. Although there are many fibrils in each lamella, they have individual diameters of only 25 nm and so have extremely low moments of inertia and bending rigidities. Due to the ‘weak’ properties of the ground substana and orientation of the fibrils, the stroma offers relatively little resistance to shear forces; in an isolated cornea the two faces slide relative to one another quite freely (Maurice, 1987a). Further, although the radius of curvature to thickness ratio of the cornea is on the order of 14-15, there is experimental evidena that in the in uivocornea, the stroma bears a constant tension at all points across the thickness (McPhee et al., 1985;
Eliason and Mauria, 1981).Based on these properties of the stroma it is reasonable to infer that the structural behavior of the cornea, to a 6rst approximation, can be idealized to that of a membrane structure, i.e. it can support in-plane (or ‘membrane’) forces but not significant transverse shear forces or bending moments. Soft biological tissues often have a ‘S-shaped stress-strain curve, which offers them the dual advantages of avoiding stress concentrations at low strains and achieving toughness at high strains by increasing their work of fracture (Fun& 1981; Lee and Tseng, 1982; and Gordon, 1988). The human cornea also shows a similar non-linear curve (Nyquist, 1968, Woo et al., 1972; Jue and Mauria, 1986). However, it has been observed that the strain regime in collagen fib& at physiological intraocular pressures corresponds to the linear elastic range (Schachar et al., 1980). Accordingly, linear elasticity is assumed at this stage of the analysis. It is possible that strain levels in the incised cornea may be in the non-linear regime. Exact characterization requires further study.
Fig. 1. El@ctron micrograph of a croswection of the corneaI stroma, showing the lame& and the arrangement ofthe collagen fibrils. The micrograph is of a ~&on of size 4.5 m x 3.3 m Individual stromal lamellae can be identified, and each is seen to contain evenly spaced collagen fibrile which have a characteristic direction associated with the lamella. The fibrils are identified aa circular dots or short oblique lines, d-ding on their angle of orientation relative to the cutting plane. (Photograph courtesy of T. Singh.)
909
Finite element model of incised cornea CONSUTWI’WE MODEL FOR THE INCISED CORNEA Properties
of thestromallamellae
The collagen f&rib in a stromal lamella lie parallel to each other asd are embedded in the ground substance which swells due to hydration and shrinks when it is dehydrated; the collagen water content is unchanged. The water content of the cornea is governed by an active mechanism located in the endothelium and the mechanical properties of the stromal lamella and in turn those of the whole cornea are a function of the water content. However, at normal physiological levels of water content, the Young’s modulus and the shear modulus of the ground substance are very small relative to those of the collagen fib@. The two-ph+ (fib&ground substance) stromal lamella is modeled as an orthotropic elastic material with principal axes de&ted by the direction of the fibrils. Rach s omal lamella is a thin membrane curved in space1 ollowing the geometry of the cornea. At every point on the mid-surface of a representative lamella, we construct an orthogonal lamella coordinate system (
