Accommodative microfluctuations and iris contour

Abstract

Mechanical interaction between aqueous humor, iris, and intraocular structures can alter the iris profile from its normal curvature. In particular, significant changes to the iris profile occur during accommodation as the anterior lens movement forces the iris into greater posterior bowing. We extended a previous mathematical model of the anterior segment and investigated the response of this coupled fluid–solid system due to accommodative microfluctuations. The results showed that the system response exhibited the same waveform as the stimulus for small-amplitude microfluctuations generally associated with the high-frequency component. Low-frequency microfluctuations with relatively larger amplitudes elicited a response different from the stimulus, indicating that the forces generated by the lens movement significantly affected the aqueous–iris mechanical interaction.

Introduction

As the lens moves forward during accommodation, the mechanical interaction between the lens, the iris, and the aqueous humor has ramifications for the iris contour, especially in pigment dispersion syndrome. For example, Campbell (1979) proposed that mechanical friction between the iris and the zonules leads to the release of iris pigment particles in pigmentary dispersion syndrome and pigmentary glaucoma. Karickhoff (1992) further suggested that the posterior iris bowing is due to reverse pupillary block, in which fluid pressure in the anterior chamber is greater than that in the posterior. Using ultrasound biomicroscopy, Pavlin, Harasiewicz, and Foster (1994) observed increased posterior iris bowing as the lens moves during accommodation. These studies demonstrate the importance of mechanical interaction during the relatively large-scale motion of accommodation. 

Given a steady visual stimulus, minute variations occur in human lens position and curvature. These accommodative microfluctuations have been measured by infrared optometry to have frequencies ranging between 0 and 6 Hz, with power spectra showing two dominant frequency bands (Campbell, Robson, & Westheimer, 1959): a low-frequency component (LFC) composed of frequencies less than 0.5 Hz and a high-frequency component (HFC) typically centered near 2.0 Hz. The definition of the HFC varies between reported studies. Campbell et al. (1959) reported the HFC to be found between 1.3 and 2.2 Hz, whereas Kotulak and Schor (1986b) reported it to be found between 1.5 and 2.5 Hz. Winn, Pugh, Gilmartin, and Owens (1990) showed a correlation between arterial pulse and the HFC, defined as the frequency band between 1.0 and 2.3 Hz. 

In general, microfluctuations increase with target vergence and accommodative response (Charman & Heron, 1988). For example, the mean-to-peak HFC amplitude is 0.02–0.04 D for a 1-D stimulus but increases to 0.06–0.10 D for a 4-D stimulus (Kotulak & Schor, 1986b). HFC activity decreases, however, as the stimulus approaches either the near or far points of focus, exhibiting a maximum near the center of accommodative range (Miege & Denieul, 1988). 

The amplitude of LFC microfluctuations is typically 0.5 D or less (Campbell et al., 1959; Charman & Heron, 1988). Given a 2-D steady stimulus, Mordi and Ciuffreda (2004) recorded an amplitude of approximately 0.4 D. 

It has been proposed that the purpose of microfluctuations is to serve as a feedback mechanism for accommodative control (Kotulak & Schor, 1986a), potentially explaining how the accommodative system can respond perfectly, with regard to initial accommodative direction, to a blurred image (Smithline, 1974). 

Given the importance of accommodation in determining iris contour, the objective of this study was to test the hypothesis that accommodative microfluctuations also play a significant role. Because experimental evaluation of the hypothesis would be difficult, a theoretical analysis was performed, extending our previous finite-element models of the anterior segment (Heys & Barocas, 2002; Huang & Barocas, 2004). 

Methods

Model equations

The computer model is a modified version of those used previously (Heys & Barocas, 2002; Huang & Barocas, 2004). We present here a summary, including any necessary changes to incorporate microfluctuations for the current study. 

Aqueous humor was modeled as an incompressible Newtonian fluid:  

$\nabla ·\mathbf{v}=0$

(1)

 

$\nabla ·\mathbf{T}=\rho \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}$

(2)

 

$\mathbf{T}=-P\mathbf{I}+\mu (\nabla \mathbf{v}+(\nabla \mathbf{v}{)}^{\mathrm{T}}),$

(3)

where v is the velocity, T is the stress tensor, ρ is the density, P is the pressure, I is the identity matrix, and μ is the viscosity. The density and viscosity of aqueous humor are similar to those of water (Beswick & McCulloch, 1956; Scott, 1988; Vass et al., 2004). 

The iris was modeled as an incompressible neo-Hookean solid:  

$\det \mathbf{F}=1$

(4)

 

$\nabla ·\sigma =0$

(5)

 

$\sigma =-P\mathbf{I}+G(\nabla \mathbf{u}+(\nabla \mathbf{u}{)}^{\mathrm{T}}-\left(\nabla \mathbf{u}\right)\left(\nabla \mathbf{u}{)}^{\mathrm{T}}\right),$

(6)

where F is the deformation tensor, σ is the Cauchy stress tensor, G is the shear modulus, and u is the displacement. A shear modulus of 9 kPa was used in the model based on bovine iris tissue (Heys & Barocas, 1999). 

Figure 1 shows the model domain and the boundaries that represent different ocular structures. The lens was modeled as a rigid solid. Lens movement due to microfluctuations and accommodation is described below. The cornea was modeled as an elastic shell with a modulus of 10.3 MPa (Hjortdal, 1994; Smolek, 1994) and a thickness of 0.5 mm (Pepose & Ubels, 1992). Aqueous humor inflow occurred at the ciliary processes at a rate of 2.5 μl/min (Caprioli, 1992). Pressure-dependent outflow was modeled as: 

$Q_{\mathrm{PD}}=k(\mathrm{IOP}-P_{\mathrm{vein}}),$

(7)

where Q_PD is the pressure-dependent outflow, k is the fluid conductivity through the trabecular meshwork, IOP is the intraocular pressure, and P_vein is the episcleral venous pressure. In the simulation, values of 0.3 μl·min⁻¹·mmHg⁻¹ and 9 mmHg were used for k and P_vein, respectively (Kaufman, 1996). Pressure-independent outflow through the uveoscleral pathway was also included at the same boundary at a rate of 0.4 μl/min (Kaufman, 1996). 

Figure 1

 

Model domain depicting various boundaries. Aqueous humor flows from the ciliary body, through the iris–lens gap, and exits through the trabecular meshwork.

For steady conditions, fluid outflow into the posterior segment was set to zero (Araie, Sugiura, Sakurai, & Oshika, 1991; Epstein, Hashimoto, Anderson, & Grant, 1979). For a steady inflow of 2.5 μl/min, this model results in an IOP of 16 mmHg. Coupling of fluid and solid mechanics was accomplished by imposing no-slip and stress balance conditions at the aqueous–iris interface. 

Accommodative microfluctuations

During accommodation, the anterior lens surface can be described accurately by a parabolic function (Koretz, Handelman, & Brown, 1984): 

$z=a{r}^2+c,$

(8)

where r and z are radial and axial coordinates of the anterior lens surface and a and c are experimentally determined parameters. Equation 8 can be recast as: 

$z=a(r^2-r_0^2)+z_0,$

(9)

where (r₀,z₀) are the coordinates of a point on the peripheral anterior lens that is assumed to remain fixed during accommodation. This assumption was justified by the fact that the human eye accommodates primarily by varying lens curvature and not position. Based on measurements of Cook and Koretz (1991), we chose this point to be approximately 4 mm from the corneal axis. Under steady conditions, the initial position of the lens was set such that the calculated anterior chamber depth was approximately 3 mm, consistent with data for a normal eye (Fontana & Brubaker, 1980). 

Lens movement was incorporated in the model by implementing a and c as time-dependent functions. The function c( t) was determined in the model according to how anterior chamber depth varied with time. The function a( t) was subsequently calculated by combining Equations 8 and 9 to yield:  

$a=(z_0-c)/r_0^2.$

(10)

 

Microfluctuations were incorporated into the model by oscillating the lens boundary sinusoidally. Because the model included only the anterior segment, volume conservation of the lens was enforced by requiring that any change in anterior chamber volume (due to the movement of the anterior lens surface) be matched by an equal volume of fluid flow through the vitreous boundary (Heys & Barocas, 2002). 

Based on the aforementioned studies, we considered LFC microfluctuations with a frequency of 0.1–0.5 Hz and an amplitude of 0.1–0.5 D and HFC microfluctuations with a frequency of 1.0–6.0 Hz and an amplitude of 0.01–0.1 D. For conversion from the focal dioptric unit to length, we assumed that the anterior chamber depth decreased by 0.037 mm/D, as measured by Koretz, Cook, and Kaufman (1997). van der Heijde, Beers, and Dubbelman (1996) reported that the lens thickness increases by about 0.056 mm/D, while Strenk et al. (1999) measured a maximum change in lens thickness of 0.63 mm, given a stimulus change from 0.1 to 8 D (equivalent to 0.080 mm/D). If lens thickness changes are taken to be symmetric with respect to the lens equator, then the change in anterior chamber depth is half of the change in lens thickness. Thus, the van der Heijde and Strenk measurements would correspond to 0.028 and 0.040 mm/D, respectively, comparing favorably with the Koretz value. 

The initial state of the system was the calculated steady solution without any microfluctuations. Calculations were performed for 200 s of simulated time. 

Accommodation

In addition to examining the effects of microfluctuations on an otherwise steady system, we also performed simulations in which microfluctuations were coupled to an initial accommodative step of 5 D. Kasthurirangan, Vilupuru, and Glasser (2003) reported that accommodative response has an exponential form. The associated time constant was approximately 0.46 s for a step of 5 D. 

Numerical solution

Spatial discretization of the model equations was accomplished using a Galerkin finite element code developed internally. Aqueous humor velocity, iris displacement, and fluid domain pseudosolid displacement (Sackinger, Schunk, & Rao, 1996) were discretized by biquadratic basis functions, and pressure was discretized by bilinear basis functions. The finite element mesh was generated using the FIGEN module of the FIDAP software package (Fluent, Inc., Lebanon, NH) and contained approximately 5,000 elements—mesh refinement that has previously been shown to yield accurate results (Heys, Barocas, & Taravella, 2001). A solution to the differential–algebraic equation system was accomplished using DASPK (Brown, Hindmarsh, & Petzold, 1998). 

Analysis

Iris–lens contact distance was estimated by calculating the distance over which the iris and lens were separated by less than 50 μm, the approximate resolution of 50 MHz ultrasound imaging (Liebmann & Ritch, 1996). Curvature of the iris profile was defined as the maximum distance between the posterior surface of the iris and the line connecting the iris root and pupil margin (Liebmann, Tello, Chew, Cohen, & Ritch, 1995). The pressure difference between the anterior and posterior chambers (ΔP_AP) was determined by noting pressure values representative of the respective regions, justified because pressure variation within each chamber was much less than ΔP_AP. 

Results

Movie 1 shows the effect of LFC microfluctuations at the iris–lens gap. As the lens moved anteriorly, local pressurization occurred in the iris–lens “pinch” region, driving aqueous humor into both the anterior as well as the posterior chamber. The pressures in the respective chambers are relatively uniform compared with the pressure change occurring at the iris–lens gap, which accounts for the bulk of Δ P _AP. 

 

Movie 1

 

Mechanical response at the iris–lens gap during the initial 6 s of LFC microfluctuation (0.5 Hz, 0.5 D).

Figure 2 shows the waveforms of anterior chamber depth (corresponding to lens position), iris–lens contact, Δ P _AP, and iris curvature during the first 20 s of LFC microfluctuations relative to the steady case (i.e., without any microfluctuation). Three different cases are shown to illustrate the effects of varying amplitude and frequency within the defined ranges for LFC microfluctuations. For all LFC cases tested, the system reached steady oscillatory response after 20 s. 

Figure 2

 

(A) Anterior chamber depth, (B) iris–lens contact, (C) Δ P _AP, and (D) iris curvature during the first 20 s of LFC microfluctuations.

For small microfluctuation amplitudes (≤0.2 D), iris–lens contact, Δ P _AP, and iris curvature all exhibited sinusoidal waveforms similar to that of the microfluctuation. At larger amplitudes, however, the response waveform became asymmetric. This is demonstrated further in Figure 3, which plots the maximum and minimum deviations in iris curvature from the steady case as a function of LFC microfluctuation amplitude and frequency. 

Figure 3

 

Maximum (blue) and minimum (red) deviation in iris curvature from the steady case as a function of LFC microfluctuation (A) amplitude and (B) frequency. Frequency is 0.5 Hz for data shown in (A) and amplitude is 0.5 D in (B).

Figure 4 shows the iris curvature response during the first 3 s of HFC microfluctuations. Because the HFC amplitudes were less than 0.1 D, the resultant waveforms were also sinusoidal, similar to the results for small-amplitude LFC microfluctuations. Iris–lens contact and Δ P _AP (not shown) exhibited the same sinusoidal response as iris curvature. Variations in amplitude or frequency within the HFC range did not affect the general waveform of iris–lens contact, Δ P _AP, and iris curvature. An increase in either amplitude or frequency magnified the mechanical response due to a faster lens velocity. 

Figure 4

 

Iris curvature response during the first 3 s of HFC microfluctuations.

Results for simulations with an initial accommodative step prior to onset of microfluctuations showed a large initial pressure change in the posterior chamber, accompanied by the associated posterior bowing of the iris ( Movie 2). Figure 5 shows the waveform for microfluctuations at 0.5 Hz and 0.5 D coupled to an initial accommodative step of 5 D. During the initial time period (approximately between 2 and 2.5 min) when there was a negative Δ P _AP, the system exhibited a sinusoidal response. After sufficient time, the system recovered and exhibited the same oscillatory response as the cases without the initial accommodation. 

 

Movie 2A

 

Movie 2B

 

Movie 2. Mechanical response of (A) the iris–lens gap and (B) the entire model domain during the initial 6 s of LFC microfluctuation (0.5 Hz, 0.5 D) coupled to an initial accommodative step of 5 D. The accommodation caused a relatively large change in pressure and iris curvature.

Figure 5

 

(a) Anterior chamber depth, (b) iris–lens contact, (c) Δ P _AP, and (d) iris curvature for microfluctuations at 0.5 Hz and 0.5 D coupled to an initial accommodative step of 5 D.

Discussion

Microfluctuations have attracted interest as various investigators studied its relationship to accommodative feedback control (Charman & Heron, 1988; Winn & Gilmartin, 1992). To our knowledge, our study is the first to examine how microfluctuations affect the fluid–solid mechanical interaction in the anterior segment and how the mechanics affect iris profile. 

As the lens moved anteriorly, there were several mechanical effects that contributed to the iris profile response. First, physical proximity to the lens forced iris curvature change because there was increased iris–lens contact during anterior lens movement. Second, Δ P _AP directly affected the iris profile through pressure forces. During steady conditions, Δ P _AP was determined by the flow resistance in the iris–lens gap and bulk aqueous flow (Silver & Quigley, 2004). During anterior lens movement, ΔP_AP was further affected by the associated fluid outflow to the posterior segment (as necessitated by volume conservation of the lens), which caused a depressurization in the posterior chamber (Figure 5). Finally, there was also localized pressurization as fluid was squeezed out of the iris–lens gap (Movie 1). The complicated interplay between iris and aqueous humor mechanics during lens movement dictated the overall ΔP_AP. In all cases, changes in iris curvature were reflective of the respective changes in ΔP_AP. 

At larger amplitudes, the pressure changes due to squeeze flow in the iris–lens gap intensified due to higher lens velocities. The pressure forces generated were sufficient to push the iris away from the lens, lowering the flow resistance through the pupil margin. Therefore, excessively high values of Δ P _AP, and consequently, iris curvature, were effectively “blunted” ( Figure 2). The overall waveform at larger amplitudes became asymmetric, with high Δ P _AP and iris curvature values being attenuated. Decreasing the frequency further dampened high Δ P _AP and iris curvature values, as the lens velocity during microfluctuations was lower. 

The asymmetric waveform response showed that large-amplitude (≥0.2 D) LFC microfluctuations produced pressure forces sufficient in magnitude to affect the fluid–solid mechanical interaction between the aqueous humor and iris. Conversely, HFC and small-amplitude LFC microfluctuations produced mechanical responses similar to that of the microfluctuation itself, indicating that these microfluctuations did not affect Δ P _AP sufficiently to complicate the aqueous–iris mechanical interaction. 

Results from simulations including an initial accommodative step showed that the mechanical response waveforms were different from the microfluctuation waveform only if the posterior chamber pressure was greater than the anterior chamber pressure. Because the posterior chamber pressure remained low for more than 2 min following accommodation, we would expect the irregular response waveform only in prolonged near viewing. 

It is evident that the iris curvature is directly affected by the pressure difference between the anterior and posterior chambers. This finding is identical to that reported in a previous simulation study on accommodation (Heys & Barocas, 2002). These findings are consistent with ultrasound visualization studies that show an increase in posterior iris bowing when the lens moves anteriorly during accommodation (Adam, Pavlin, & Ulanski, 2004; Pavlin et al., 1994). Compared with accommodation, the magnitude of the changes in iris curvature due to HFC and small-amplitude LFC microfluctuations was small. Large-amplitude LFC microfluctuations, however, caused oscillations in iris curvature with amplitude roughly 10% of the change due to a 5-D accommodative step. Whether this effect significantly contributes to pigmentary dispersion syndrome and pigmentary glaucoma is unclear. 

An additional consideration is iris–zonule contact, which is important in pigmentary dispersion syndrome, but it is not addressed by the current model. One could certainly speculate that any effect that increases iris bowing in the current model would likely aggravate iris–zonule contact, but no direct conclusion can be made. Improvements in ultrasound imaging of the zonules (Inazumi, Takahashi, Taniguchi, & Yamamoto, 2002; Ludwig, Wegscheider, Hoops, & Kampik, 1999; McWhae, Crichton, & Rinke, 2003) may provide further insight into this question in the future. 

Model sensitivity to various model parameters has been published in a previous study (Heys et al., 2001). It was found that increasing iris stiffness by a factor of 4 resulted in a similar change in ΔP_AP. Iris–lens contact, however, changed only by a factor of 2, and the total contact area changed by less than 10%. Because it is likely that the iris modulus varies with pupil dilation, the aqueous–iris mechanical response could be further complicated during accommodation. Efforts are currently underway to investigate the aqueous–iris response to multiple stimuli, such as the “near triad”—simultaneous positive accommodation and pupillary constriction during viewing of a near object. 

Conclusion

Our computer simulations showed that the forces generated by the LFC of accommodative microfluctuations were sufficient to affect the fluid–solid mechanical interaction between the aqueous humor and the iris. Waveforms of the aqueous–iris mechanical response were significantly different from waveforms of the lens movement for microfluctuations with amplitudes greater than approximately 0.2 D. 

Acknowledgments

We thank Nicha Chitphakdithai for her efforts in background research and preliminary test trials during simulation development. This work was supported by the University of Minnesota Biomedical Engineering Institute and by the National Institutes of Health (1 R01 EY15795-01). Computations were made possible by a supercomputing resources grant from the University of Minnesota Supercomputing Institute for Digital Simulation and Advanced Computation. 

Commercial relationships: none. 

Corresponding author: Victor H. Barocas. 

Email: [e-mail]. 

Address: Department of Biomedical Engineering, University of Minnesota, 7-105 Nils Hasselmo Hall, 312 Church Street S.E., Minneapolis, MN 55455.