Knowledge about geometric properties such as shape and volume and Poisson's ratio of the nucleus can be used in the mechanical and optical modeling of the accommodation process. Therefore, Scheimpflug imaging was used to determine the shape of the human lens nucleus during accommodation in five subjects. To describe the shape of the nucleus, we fitted a parametric model of the cross-sectional geometry to the gradient of the Scheimpflug images using the Hough transform. The geometric model made it possible to estimate the anterior and the posterior central radius, central thickness, equatorial diameter, and cross-sectional area of the nucleus. Assuming that the nucleus is rotationally symmetric, the volume of the nucleus can be estimated by integrating around the circumference. For all five subjects, the results show that during accommodation the nucleus became more convex and that the central thickness increased whereas the equatorial diameter decreased. This decrease in equatorial diameter of the nucleus with accommodation is in accordance with the Helmholtz accommodation theory. Finally, the volume of the nucleus (on average 35 mm ^{3}) showed no significant change during accommodation in any of the subjects, presumably due to the fact that the human nucleus consists of incompressible material with a Poisson's ratio that is near .5.

*nucleus*has not yet been measured in vivo as a function of accommodation.

*G,*with

*G*

_{x}and

*G*

_{y}the components in

*x*- and

*y*-directions (radial and axial directions, respectively). The Canny edge filter was composed of the directional derivatives of a Gaussian filter (standard deviation

*σ*= 1 pixel) and was convoluted with the original image. Cook and Koretz (1998) showed experimentally that for the detection of internal lens boundaries, the inverse magnitude of the gradient

*I*

_{grad}provides good contrast:

*I*

_{grad}to values between zero (black) and one (white). Figure 2 gives an example of the result of this procedure for the Scheimpflug image as shown in Figure 1.

*x*

_{0},

*y*

_{0}) as the apex position and

*R*as the central radius of curvature at an aperture of 3.5 mm. The positions of left (

*x*

_{EQL},

*y*

_{EQL}) and right (

*x*

_{EQR},

*y*

_{EQR}) equators were used to form a nucleus shape, using four curves that closed the periphery of the nucleus. It was assumed that the curve describing the outline of the nucleus was continuous, that the derivative of this curve was continuous, and that the

*y*derivative was zero at the equators. We chose to describe the parametric curve with as few parameters as possible. Therefore, nucleus geometry was closed with four polynomials of the third order. The four curves in the periphery were described with the following equation:

*x*

_{EQ},

*y*

_{EQ}) as the position of left or right equator and (

*x*

_{perifery},

*y*

_{perifery}) as the coordinates of curves in the periphery.

*y*derivative was zero at the equator (

*y*

_{perifery}=

*y*

_{EQ}). For every curve in the periphery, we constrained the parameters

*a*and

*b*by the necessary continuity of function and derivative at the interception point with the central parabolic:

*x*

_{ i},

*y*

_{ i}) as the position of the interception point between the central parabolic (3.5 mm aperture) and the specific curve in the periphery.

*I*

_{grad}were selected manually by mouse to obtain this initial estimate. Subsequently, two parabolics were fitted through these selected points with the Levenberg–Marquardt method. Furthermore, the initial parabolics were plotted, and mouse dragging of the equator positions was possible with a real-time update of the curves closing the periphery. The left and right equator positions were determined manually, based on the darkest transition in equatorial direction of the gradient image. It was possible to make an extra visual check of equator position because the curves in the periphery should lie on the darkest pad between equator and interception point.

*x*

_{0 start},

*y*

_{0 start}, and

*R*

_{start}:

*x*

_{0},

*y*

_{0}, and

*R*formed (9 × 9 × 21 =) 1701 parabolics around the first estimate. The value for the Hough transform of a specific set (

*x*

_{0},

*y*

_{0},

*R*) was composed by evaluating the line integral of the gradient image

*I*

_{grad}along that specific corresponding parabolic. To fit not only the central parabolic, but also the complete anterior and posterior geometry of the nucleus, we included the anterior and the posterior curves in the periphery in the evaluation of the line integral. For every combination of (

*x*

_{0},

*y*

_{0},

*R*), we carried out the Hough transform, resulting in a three-dimensional Hough transform matrix. The global minimum in the Hough transform matrix corresponds to the parameters that describe an edge. If the initial estimate was close enough, the global minimum of the Hough transforms corresponded to the best fit of parametric geometry of the nucleus.

*f*

_{ant}(

*x*) and

*f*

_{post}(

*x*) as the anterior and the posterior curves, respectively, as a function of the radial position (

*x*). The volume of a solid of revolution with respect to an axis of rotation can be determined according to the following equation (Arfken & Weber, 2001):

*f*(

*y*) as the axisymmetric cross-section curve as a function of the axial position (

*y*) with respect to the axis of rotation. To calculate the volume of the nucleus, we choose to rotate along the vertical axis (

*x*= 0 in Figure 3) that was defined by the mean of anterior and posterior apex locations. Because the nucleus is not symmetric around this rotation axis, we calculated the volume for both the left and the right half of the nucleus using Equation 7. Finally, the mean of these two volumes was taken to obtain the volume of the nucleus.

*p*< .0001,

*r*= .71) with accommodation, but that there was no significant change (

*p*= .61,

*r*= −.08) in volume of the nucleus. Moreover, none of the other subjects showed any significant change in lens volume (

*p*> .05), and the mean volume was 35 mm

^{3}. If no volume transport takes place in or out of the nucleus, the constant volume implies that the material inside the lens nucleus can be assumed to be incompressible.

*p*> .05). However, for the other three subjects there was a significant change in the ratio of anterior and posterior thickness during accommodation. On average, the mean value for this ratio at 0 D was .95.

Age (y) | Max. stimulus (D) | Anterior radius (mm) | Posterior radius (mm) | Anterior thickness (mm) | Posterior thickness (mm) | ANT/POST thickness ratio | Equatorial diameter (mm) | CSA
(mm ^{2}) | Volume
(mm ^{3}) | |
---|---|---|---|---|---|---|---|---|---|---|

16 | 10.4 | Intercept | 4.62 (± 0.05) | 4.02 (± 0.05) | 1.28 (± 0.01) | 1.20 (± 0.01) | 1.07 (± 0.01) | 5.87 (± 0.02) | 10.52 (± 0.07) | 39.2 (± 0.4) |

Slope | −0.147 (± 0.009) | −0.094 (± 0.008) | 0.018 (± 0.002) | 0.017 (± 0.001) | 0.0001 (± 0.002) | −0.027 (± 0.004) | 0.07 (± 0.011) | 0.03 (± 0.05) | ||

p | 0.98 | 0.61 | ||||||||

r | −0.94 | −0.89 | 0.79 | 0.92 | 0.00 | −0.76 | 0.71 | 0.08 | ||

19 | 8.2 | Intercept | 4.14 (± 0.08) | 3.09 (± 0.04) | 1.10 (± 0.02) | 1.26 (± 0.01) | 0.88 (± 0.02) | 6.12 (± 0.04) | 9.55 (± 0.06) | 35.0 (± 0.3) |

Slope | −0.149 (± 0.017) | −0.041 (± 0.009) | 0.028 (± 0.004) | 0.012 (± 0.003) | 0.014 (± 0.005) | −0.065 (± 0.009) | 0.079 (± 0.013) | −0.06 (± 0.07) | ||

p | 0.005 | 0.38 | ||||||||

r | −0.87 | −0.65 | 0.82 | 0.61 | 0.53 | −0.81 | 0.79 | −0.18 | ||

24 | 7.4 | Intercept | 3.77 (± 0.04) | 3.35 (± 0.05) | 1.25 (± 0.02) | 1.24 (± 0.01) | 1.00 (± 0.02) | 5.99 (± 0.01) | 10.18 (± 0.07) | 37.2 (± 0.3) |

Slope | −0.161 (± 0.010) | −0.088 (± 0.010) | 0.021 (± 0.005) | 0.029 (± 0.003) | −0.005 (± 0.004) | −0.080 (± 0.008) | 0.085 (± 0.016) | −0.13 (± 0.08) | ||

p | 0.32 | 0.11 | ||||||||

r | −0.96 | −0.85 | 0.66 | 0.87 | −0.19 | −0.88 | 0.71 | −0.31 | ||

28 | 7.4 | Intercept | 3.67 (± 0.04) | 2.59 (± 0.03) | 1.14 (± 0.01) | 1.22 (± 0.01) | 0.93 (± 0.01) | 5.59 (± 0.02) | 8.81 (± 0.04) | 29.5 (± 0.2) |

Slope | −0.145 (± 0.010) | −0.045 (± 0.006) | 0.029 (± 0.001) | 0.009 (± 0.002) | 0.016 (± 0.002) | −0.040 (± 0.004) | 0.059 (± 0.008) | −0.06 (± 0.04) | ||

p | 0.003 | 0.12 | ||||||||

r | −0.94 | −0.79 | 0.98 | 0.54 | 0.77 | −0.88 | 0.80 | −0.30 | ||

32 | 7.4 | Intercept | 3.73 (± 0.04) | 3.03 (± 0.06) | 1.16 (± 0.01) | 1.34 (± 0.01) | 0.87 (± 0.01) | 5.69 (± 0.02) | 9.84 (± 0.06) | 34.3 (± 0.3) |

Slope | −0.113 (± 0.009) | −0.070 (± 0.013) | 0.026 (± 0.002) | 0.015 (± 0.003) | 0.010 (± 0.002) | −0.038 (± 0.004) | 0.081 (± 0.013) | 0.03 (± 0.06) | ||

p | 0.69 | |||||||||

r | −0.93 | −0.74 | 0.95 | 0.75 | 0.64 | −0.87 | 0.78 | 0.08 |

Anterior radius (mm) | Posterior radius (mm) | Anterior thickness (mm) | Posterior thickness (mm) | ANT/POST thickness ratio | Equatorial diameter (mm) | CSA
(mm ^{2}) | Volume (mm ^{3}) | |
---|---|---|---|---|---|---|---|---|

Uncorrected | 5.30 | 4.49 | 1.20 | 1.16 | 1.03 | 6.92 | 10.95 | 45.5 |

Corrected | 3.67 | 2.59 | 1.14 | 1.22 | 0.93 | 5.59 | 8.81 | 29.5 |

^{3}). The CSA of the nucleus increased with accommodation but no significant change of volume of the nucleus was found. Therefore, the nucleus can be assumed to be incompressible with a Poisson's ratio near .5 if no volume transport occurs in or out of the nucleus. This finding seems to be in contrast with Strenk et al. (2004), in which it was concluded that the volume of the lens should increase with accommodation because of the increase of CSA that was measured. Nevertheless, an increase of CSA does not necessarily imply a change of volume (Judge & Burd, 2004) as also has been shown for the nucleus in the present study.

*R*

_{ant}= 3.6 mm,

*R*

_{post}= 3.4 mm) and the results of our study (on average

*R*

_{ant}= 4.0 mm,

*R*

_{post}= 3.2 mm) were of the same order. Furthermore, the central anterior and posterior thickness measurements (on average 1.2 and 1.3 mm, respectively) were in agreement with those reported by Brown (on average 1.3 and 1.4 mm, respectively), which result in a ratio of .93 between anterior and posterior thickness. Although no Type II correction has been applied in these former studies only the estimated equatorial diameter (on average 5.8 mm) was approximately 15% lower than the Scheimpflug measurements reported by Brown (on average 7.0 mm). Table 2 shows that the determination of the thickness of the nucleus was not influenced by distortion. This could be explained by the image formation of the central part of the lens. A chief ray that originates from the central lens area crosses the cornea at an almost perpendicular angle and is only slightly refracted. The chief ray continues to the image plane of the camera and forms the image. A chief ray in the peripheral part of the lens crosses the cornea under a more oblique angle, resulting in more image distortion. As a result, the difference found in the equatorial diameters can be explained by the fact that the peripheral equators were more distorted by the refraction of cornea and lens (Type II).

^{3}. This slightly higher result can probably be explained by the image distortion (Type II), as already indicated in the above section, and the simplification of nucleus geometry.

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