Mechanical Properties of the Rabbit Iris Smooth Muscles

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Vision Research 43 (2003) 479–487 www.elsevier.com/locate/visres

Mechanical properties of the rabbit iris smooth muscles

Kazutsuna Yamaji a,*, Takeshi Yoshitomi b, Shiro Usui a, Yoshitaka Ohnishi b a Laboratory for Neuroinformatics, RIKEN Brain Science Institute, Saitama 351-0198, Japan b Department of Ophthalmology, Wakayama Medical University, Wakayama 641-8509, Japan Received 11 March 2002; received in revised form 15 July 2002

Abstract The study focuses on obtaining the visco-elastic properties of the iris sphincter and dilator muscles. Two kinds of experiments were performed: the isometric contraction experiment and the isotonic quick release experiment. The length–tension relationship was obtained from the former experiment. This relationship clariﬁed the contribution of each muscle in determining the statics of the pupil. The viscous and serial elastic properties were obtained from the latter experiment. The viscosity could be expressed by the expanded HillÕs equation as a function of velocity and contractile tension. We argue that serial elasticity is independent of contractile tension. These properties provide insights into the pupillary mechanism. 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Pupil; Smooth muscle; Isometric contraction; Isotonic contraction; Mathematical model

1. Introduction 2. Materials and methods

After the first work done by Stark and Sherman All experiments were performed according to the (1957), several mathematical models of pupillary light guide for care and use of laboratory animals (DHEW reflex have been developed. The pupillary light reflex arc publication, NIH 80-23). Male albino rabbits weighting can be regarded as consisting of three components: 2–3 kg were sacrificed with an overdose of intravenous retina, central nervous system and iris muscle. In each pentobarbital sodium (Abbott, North Chicago, IL, component, iris muscle has been modeled in detail using USA). The eyes were immediately enucleated and placed viscous and elastic components by adopting the results in oxygenated (95%O2 þ 5%CO2) Krebs solution (NaCl: of skeletal muscle modeling (Usui & Hirata, 1995). 94.8, KCl: 4.7, MgSO4: 1.2, CaCl2: 2.5, KH2PO4: 1.2, However, physiological data for determining the model NaHCO3: 25.0, Glucose: 11.7, unit: mM). After removal structure or function of the iris muscle have not yet been of cornea, ring-shaped iris sphincter specimen or radial- sufficiently observed. shaped dilator muscle specimen were prepared accord- Development of physiologically plausible iris muscle ing to the method previously reported (Kern, 1970). The model makes it possible not only to understand its sphincter muscle: two places were tied by silk thread as control mechanism but also to develop a new method shown in Fig. 1(left). Then, 1 mm width muscle was for monitoring the autonomic nervous activity (Yamaji, separated from the iris (Fig. 1(center)). The dilator Hirata, & Usui, 2000, 2001). Mechanical properties of muscle: it was cut in the sector of 22.5 or 45 degrees iris muscle have to be examined in this context. This depending on the experiment, and then pupillary edge study aims at obtaining the visco-elastic properties of and sclera side were tied as shown in Fig. 1(right). iris sphincter and dilator muscles from the isometric contraction experiment and the isotonic quick release 2.1. Isometric contraction experiment experiment. In this experiment, length–tension relationship of the iris muscle was obtained. The active and passive tensions were recorded at several steps of muscle length. One end * Corresponding author. of the specimen was connected to the isometric tension E-mail address: [email protected] (K. Yamaji). transducer (Nihon Kohden Co., TB-612T), and another

Fig. 1. Preparation for the sphincter (left and center) and dilator (right) muscle specimens. Photographs show an eye of which cornea was removed. See the text for details of the preparation. end was secured to the hook at the bottom of an problems indicated by these results, active tension has organ bath (1.5 ml). The organ bath was perfused con- been obtained by using exogenous drugs. Acetylcholine tinuously (0.17 ml/s) with Krebs solution warmed to 10 mM (Yoshitomi, Sakamoto, & Ohnishi, 2001) and 37 C. phenylephrine 0.1mM (Ishikawa, Yoshitomi, Harada, In the skeletal muscle, tetanic contraction caused by Katori, & Ishikawa, 1993) (Wako Chemical Inc., Osaka, electric stimuli is usually regarded as the active tension Japan) were adopted to produce the maximum tension (Mashima & Tsuchiya, 1968). Fig. 2 shows data which of sphincter and dilator, respectively. was recorded to define the optimum condition of the Experiments were performed after steadying the ini- electric stimuli. The rabbit sphincter response to electric tial muscle tone. It required more than 120 min after stimuli composes of fast cholinergic response and slow mounting the muscle. In each muscle, experiments for substance P-ergic response (Yoshitomi, Ishikawa, Ha- lengthening and shortening the muscle were done using runo, & Ishikawa, 1995). The former originates from the same number of specimens (each n ¼ 4). Selected parasympathetic nerve which mainly mediates the light muscle lengths were 2, 3, 4, 6, 8, 10, 12, 14 mm in the reflex. The latter originates from trigeminal nerve and is sphincter, and 2–7 mm in the dilator. After producing related to the sensory response. Responses in Fig. 2(left) the maximum contraction, acetylcholine or phenyleph- have already eliminated the substance P-ergic response rine was washed out (more than 30 min). Then, the by capsaicin (Ueda, Muramatsu, Sakakibara, & Fujiw- muscle length was readjusted by a micrometer attached ara, 1981) (SIGMA Chemical Co., St. Louis, USA) to a transducer stage for the next session. because this study discusses the muscle properties to the In the sphincter, the tension recorded from prepared light response. The figure shows that the sphincter specimen corresponds to the whole sphincter tension. In cannot produce a tetanic response in spite of stimulation the dilator, as shown in Fig. 1, only a part of the muscle frequency. Although the maximum tension can be found was used because of the difficulty in bundling the whole in Fig. 2(left), there is possibility that transient response muscle. 45 degrees arc was selected for preparation. is attenuated by viscosity (Mashima, Akazawa, Ku- However, it is unsure whether the specimen can be shima, & Fujii, 1973). Electric stimulation should not be stretched uniformly in the experiment. There is a pos- used to obtain the active tension in the sphincter. In sibility that 8 times of experimental data does not cor- contrast, the dilator tension is sustained at certain level respond to the whole dilator behavior that should be after relatively long settling time (Fig. 2(right)). A compared with the sphincter behavior. Two kinds of tetanic response is obtained in the dilator. However, specimens of which sector area was 22.5 and 45 degrees long time stimulation causes electrolysis. Thus compo- were prepared in order to compare the passive and ac- sition of Krebs solution may change. Because of the tive tension between the two (n ¼ 4). 400 200 300 150 200 100 Tension [mg] Tension [mg] 100 0 50 0 2 4 6 8 10 0 10 20 30 40 Time [s] Time [s]

Fig. 2. Sphincter (left) and dilator (right) responses to electric ﬁeld stimulation. Interval of pulse stimuli changed in each response. In the sphincter response, thick black: 10 ms, thin black: 100 ms, dashed: 1000 ms. In the dilator response, thick black: 5 ms, thin black: 10 ms, dashed: 100 ms. K. Yamaji et al. / Vision Research 43 (2003) 479–487 481

This experiment provides data concerning the fol- • Load–displacement behavior of the sphincter and di- lowing: lator.

• Relationship between the extracted sector area, and Viscous behavior and load–displacement behavior passive and active tensions of the dilator. are in accordance with the viscous component and the • Length–tension relationship of the sphincter and dila- serial elastic component of the mathematical model. tor.

It has been observed that passive length–tension re- 3. Results lationship is in accordance with the parallel elastic component of the mathematical model. 3.1. Isometric contraction experiment

2.2. Isotonic quick release experiment In order to obtain the length–tension relationship, the active, the passive and the total tensions were calculated In this experiment, viscous behavior and load–dis- from the experimental data. The passive tension was placement behavior were observed. These were obtained defined as an average tension over 10 s before contrac- from the experiment which recorded tension and dis- tion. The total tension was calculated as an average placement simultaneously under the condition that one tension over 10 s, after the contraction reached steady end of the isometrically contracted specimen was state. The active tension was simply calculated by sub- quickly released with various load weights. The muscle tracting the passive tension from the total tension. was mounted in a 5 ml organ bath which was perfused Fig. 3 shows the results for extracted sector area of the with Krebs solution as in the previous experiment. One dilator. In each passive and active tension, the tension of end was connected to an isometric tension transducer 45 degrees specimen (light gray) is almost double of the and another end was connected to a load. Measurement tension of 22.5 degreesÕ one (dark gray). This suggests of muscle displacement was realized by image process- that 8 times of the data from 45 degrees specimen cor- ing. Muscle image was captured using CCD camera responds to all dilator tension. Since the sphincter and (SONY TRV20) and displacement was detected from dilator are arranged circularly and radially, respectively, binarized image by off-line processing. direction of tension development is different. These dif- Experiments were performed more than 120 min after ferences can be compensated by the following equation mounting the muscle for the same reason as in Section (Hansmann, 1972). Eq. (1) is derived by simplifying two 2.1. At first, the load was fixed on an electromagnet. dimensional structure of each muscle into one dimension Initial muscle length was set to a length which produces using mechanically equivalent transformation. That is, maximum active tension by referring to the result in Eq. (1) describes static equilibrium of push–pull struc- Section 2.1(sphincter: 10mm, dilator: 5 mm). Next, ture between the sphincter and dilator. isometric contraction was produced by the drugs des- Td ¼ 2pTs ð1Þ cribed previously. Finally, the electromagnet was cut off at the time when the tension reached the maximum. If the load weight was lower than muscle contraction,

muscle was shortened; otherwise muscle was lengthened. 150 After a session, acetylcholine or phenylephrine was washed out by Krebs solution (more than 30 min). The viscous behavior is known as a function of not only velocity but also contractile tension (Mashima, 100 Akazawa, Kushima, & Fujii, 1972). To examine this dependency, contractile tension before quick release was changed by attenuating drug concentration. Number of specimens used for the shortening and the lengthening Tension [%] 50 experiment were 3 and 4 respectively and drug concentration varied in each specimen. Eﬀect of extracted sector area on the dilator was also examined as well as in Section 2.1( n ¼ 4). 0 Data obtained in this experiment were the following: Passive Active

Fig. 3. Eﬀect of section area on the dilator response. Dark gray: 22.5 • Relationship between extracted sector area, and vis- degree (n ¼ 4), light gray: 45 degree (n ¼ 4), bar: standard deviation. cous and load–displacement behavior of the dilator. Tension of 22.5 degrees specimens were normalized by 45 degrees • Viscous behavior of the sphincter and dilator. specimens as 100%. 482 K. Yamaji et al. / Vision Research 43 (2003) 479–487

Ts and Td are all tension developed by the sphincter and 11 dilator. Therefore, each muscle tension can be compared 400 fairly from 8 times of dilatorÕs tension and from 2p times 10 of sphincterÕs tension. 200 9 Fig. 4 shows the length–relationship properties of the Length [mm] Tension [mg] 8 0 sphincter and dilator. Each data is average over eight -0.5 0.0 0.5 1.0 1.5 specimens. The active tension curve indicates that Time [s] maximum tension of the sphincter and dilator is 2 and 0.68 g, respectively. In the previous study (Loewenfeld, Fig. 5. An example of experimental data from the isotonic quick release method. Thick solid line: muscle displacement, thick dashed line: 1993), it is qualitatively stated that the sphincter is more tension change. One end of the muscle was released at 0 s. At first, powerful than the dilator. The difference between the muscle is quickly shortened, and then continues to shorten at constant two muscles is about 3 times in the case of rabbit. The speed. shape of active tension curve of iris muscles is flattened as compared with that of skeletal muscle (Csapo, 1962). viscous behavior as a function of velocity requires three Due to the limitation of experimental equipment, the parameters: the tension before and after release (dashed sphincter could not be stretched more than 14 mm. This line), and the displacement velocity. The tensions before makes it difficult to define the shape of active tension and after release were each replaced by an average ten- curve. However, there is a significant difference between sion over 1s. The tension after release corresponds to the data in 12 and 14 mm by WilcoxenÕs rank sum test the load weight. The viscosity was obtained from the (P < 0:05). This result suggests that the data at 14 mm subtraction of these two tensions as explained in the indicates the downtrend of the curve. next section (Eqs. (2) and (3)). The latter velocity was Passive tension of the sphincter starts to increase extracted from the first derivative curve of muscle dis- where its active tension starts to decrease. On the other placement as an average velocity over 0.2 s after release. hand, passive tension of the dilator starts to increase at Determination of the load–displacement behavior relatively short muscle length in terms of active tension requires the muscle length before and just after release development. Moreover, passive tension of the dilator is (broken line), and the tension after release (load weight). higher than that of the sphincter. These differences may The length before release was extracted as an average result from the fact that the dilator specimen contains over 1s before release. The length after release was ex- larger amount of connective tissue than the sphincter pressed in terms of the velocity change. The quick dis- (Gordon & Siegman, 1971). placement was calculated from the subtraction of these two lengths. 3.2. Isotonic quick release experiment 3.2.1. Viscous behavior Fig. 5 shows tension change (thin line) and muscle In the previous study on skeletal muscle, muscle vis- displacement (thick line) of the sphincter under the cosity is described by the following equations (Mashima condition that one end of the muscle was released at 0 s et al., 1972), which expands the HillÕs equation (Hill, and was shortened isotonically. Determination of the 1938). 2.5 2.5 300 2.0 2.0 300 200 1.5 1.5

* 200 1.0 1.0 100 100 Converted Tension [g] Converted Tension [g] 0.5 0.5 Measured Tension [mg] Measured Tension [mg] 0 0 0.0 0.0 2 4 6 8 10 12 14 2 4 6 8 Length [mm] Length [mm]

Fig. 4. Length–tension properties of the sphincter (left, n ¼ 8) and dilator (right, n ¼ 8). Circle: total tension, triangle: active tension, square: passive tension, bar: standard deviation. Active tension of dilator is shifted to 0.1mm rightward for distinguishing the error bars. ‘‘Converted tension’’ of sphincter was calculated by 2p times the recorded tension refering to Eq. (1). Dilator tension was calculated by 8 times the recorded tension based on the results in Fig. 3. (Ã) P < 0:05 by WilcoxenÕs runk sum test. K. Yamaji et al. / Vision Research 43 (2003) 479–487 483

F v Fvþðv; F Þ¼F À P ¼ ðP0 þ aÞ ð2Þ 200 P0 v þ b F v F v; F P F P a0 3 150 vÀð Þ¼ À ¼ ð 0 þ Þ 0 ð Þ P0 v À b Eqs. (2) and (3) refer to the condition at which muscle is 100 contracted and extended. P0, F and P denote maximum tension, contractile tension and load weight, respectively. v is muscle velocity, and a and b are constants 50 Viscous-like Force [mg] called the heat constant and the rate constant of energy liberation. F =P0 was added to the original HillÕs equa- 0 tion. This term was introduced empirically in order to 0.0 0.2 0.4 0.6 represent the experimental result that viscous-like force Contraction Velocity [mm/s] of skeletal muscle was changed depending on contractile Fig. 6. Effect of the extracted sector area on the dilator viscosity. tension (Mashima et al., 1972). In the above equations, Symbols are experimental data. Filled symbols: 45 degrees (n ¼ 4), if the contractile tension is equal to the maximum ten- outlined symbols: 22.5 degrees (n ¼ 4). Different symbols (circle, tri- sion (F ¼ P0), the equations are equivalent to the origi- angle, square and diamond) refer to the data recorded from different nal HillÕs equation which represents the hyperbolic specimen. Solid and dashed curves were calculated by fitting Eq. (2) to relationship between load and velocity. the data from 45 to 22.5 degrees specimens, respectively. Broken curve is double of the dashed curve (22.5 degrees). Standard deviations of Accordingly, viscosity becomes 0 mg when the con- residue from estimated curve to experimental data are as follows, 45 tractile tension is equal to 0 mg. However, since the degrees: 10.6, 22.5 degrees: 6.78 [mg]. muscle can be stretched even when its contractile tension is 0 mg, the model of muscle extension (Eq. (3)) should have a function for producing the viscosity at 0 mg. One was used for estimating the constant c in Eq. (4) accu- of the solutions to this problem is that the model in- rately. Each curve in the figure is determined by fitting corporates a kind of intercept of linear function into Eq. Eqs. (2) and (4) to each experimental data using a (3) by factor c P a0 v=v b0 at the point F 0 and ð 0 þ Þð À Þ ¼ nonlinear optimization method. The estimated para- thus Eq. (3) was expanded empirically as follows. meters are shown in Table 1. Vmax is the maximum F ðv; F Þ¼P À F contractile velocity calculated from a and b (Stephens, vÀ F v Kroeger, & Mehta, 1969). ¼ ð1 À cÞþc ðP þ a0Þ ð4Þ Results show that Eqs. (2) and (4) express all data for P 0 v À b0 0 which the contractile tensions are different. That is, Here, c is a constant describing viscosity when the active HillÕs equation can also be adapted to iris smooth tension is 0 mg. If F ¼ P0, Eq. (4) is equivalent to Eq. muscle and viscous behavior can be appropriately (3), then if F ¼ 0, c Â 100% of the viscosity at F ¼ P0 modeled by using Eqs. (2) and (4). still remains. That is, Eq. (4) provides an intercept in viscosity, and compresses Eq. (3) into a range from the 3.2.2. Load–displacement behavior intercept to F ¼ P0. Relationship between extracted sector area and serial Fig. 6 shows the relationship between extracted sector elastic property of the dilator is shown in Fig. 8. Sym- area and viscosity of the dilator. Different symbol refers bols in the figure denote load weight as a function of to the data recorded from different specimens. Solid and quick displacement. Tension–extension curve of serial dashed curves were calculated by fitting Eq. (2) to the elasticity (curves raised to rightward) is calculated as experimental data. Constants a and b in Eq. (2) for each follows (Jewell & Wilkie, 1958). curve are estimated by the least square method at F ¼ P0. The figure indicates that the double of 22.5 de- 1. Since the experimental data varies exponentially grees curve (broken curve) almost corresponds to 45 (Sugi & Tsuchiya, 1979), Eq. (5) was incorporated degrees curve (solid curve). Therefore, the viscosity to empirically express the load–displacement behav- generated by the whole dilator muscle is calculated by ior. This model was necessary for estimation of multiplying 8 times the data from 45 degrees specimen. the tension–extension relationship from the load– Fig. 7 shows the viscous behavior of the sphincter and displacement behavior. Quick displacement in Fig. 8 dilator. The figure takes shortening velocity as positive vanishes when the muscle tension equals to the load velocity. The circle symbol corresponds to the data at weight. a À c corresponds to the point of contact with the maximum contraction P0 and other symbols denote Y axis, that is, the muscle tension before release. b data for which contractile tension was less than P0. The which corresponds to the time constant of the first- open lozenge symbol of which active tension was 0 mg order lag system expresses the attenuation factor of 484 K. Yamaji et al. / Vision Research 43 (2003) 479–487 3 3 400 400 2 2 200 200 1 1 0 0 0 0 -1 -1 -200 -200 -2 -2 Converted Viscous-like Force [g] -400 Measured Viscous-like Force [mg] Converted Viscous-like Force [g] Measured Viscous-like Force [mg] -3 -3 -400 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -0.5 0.0 0.5 1.0 Contraction Velocity [mm/s] Contraction Velocity [mm/s]

Fig. 7. Viscosity–velocity properties of the sphincter (left, n ¼ 7) and dilator (right, n ¼ 7). In diﬀerent specimens (indicated by diﬀerent symbols), contractile tension was changed in order to verify applicability of Eqs. (2) and (3). Conversion of the axis was done as in Fig. 4 according to the result in Fig. 6. Contractile tension before quick release of each data are as follows: (d) 383.8/218.3 (sphincter/dilator), (N) 295.4/150.8, (j) 206.5/105.6, () 385.2/223.3, (Ã) 281.0/182.1, (M) 153.3/110.0, (}) 26.79/61.11 [mg]. The tension denoted by } corresponds to passive tension (active tension is zero). Standard deviations of residue from estimated curve to experimental data are as follows: (d) 4.76/4.62 (sphincter/dilator), (N) 8.11/3.38, (j) 8.85/5.52, () 3.26/5.22, (Ã) 4.60/13.13, (M) 5.79/6.66, (}) 4.33/2.96 [mg].

Table 1 Parameter values of viscous-like force. The cont and ext indicate the parameters of the model for muscle contraction (Eq. (2)) and for muscle extension (Eq. (4)), respectively. The ext=cont is the ratio of the values in the extension vs. contraction domains Sphincter Dilator cont ext ext=cont cont ext ext=cont

a=P0 0.11 0.90 8.33 0.28 2.52 9.15 b=l0 0.035 0.054 1.54 0.035 0.077 2.17 c – 0.22 – – 0.22 –

Vmax=l0 0.32 0.13

Extension [mm] exponential curve. The parameters are estimated us- 0.0 0.25 0.5 0.75 1.0 1.25 ing a nonlinear optimization method.

200 200 2. Quick displacement at 0 mg load weight means a displacement of the serial elasticity at which the muscle produces maximum tension. This displacement is cal- 150 150 culated by Eq. (5). 3. Load–displacement curve is shifted so that the maximum displacement point which is calculated in (2) 100 100 becomes the origin. This curve is mirrored against Load [mg]

Tension [mg] Y axis in order to obtain the tension–extension rela-

50 50 tionship of serial elasticity.

Serial elastic properties from 45 and 22.5 degrees 0 0 0.0 0.25 0.5 0.75 1.0 1.25 specimens are shown in the same manner as in Fig. 6. It Quick Displacement [mm] can be seen that the double of 22.5 degrees curve (broken curve) almost corresponds to the 45 degrees Fig. 8. Eﬀect of extracted sector area on dilator serial elasticity. Data curve (solid curve). That is, serial elasticity of the whole rise leftward are the load–quick displacement properties. Symbols and curves are shown in the same manner as in Fig. 6. The original ex- dilator is also estimated by 8 times multiplying experimental data from 45 degrees specimen. perimental data are the same as in Fig. 6. Curves which indicate the load–quick displacement property are calculated by adopting Eq. (5). x From these curves, data rises rightward which corresponds to tension– P ¼ a Á exp À À c ð5Þ extension properties (serial elastic properties) are estimated. Broken b curve is double of the dashed curve (22.5 degrees). Detailed estimation procedure is written in the text. Standard deviations of residue from Fig. 9 indicates the serial elastic property of the sphincter estimated curve to experimental data are as follows: 45 degrees; 5.04, (left) and dilator (right). The experimental data to ex- 22.5 degrees; 7.90 [mg]. tract load weight and quick displacement are the same K. Yamaji et al. / Vision Research 43 (2003) 479–487 485

Extension [mm] Extension [mm] 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 2.5 2.5 2.5 2.5 300 2.0 2.0 2.0 2.0 300 200 1.5 1.5 1.5 1.5 200 1.0 1.0 1.0 1.0 100 Converted Load [g] Converted Load [g] Measured Load [mg] Measured Load [mg] Converted Tension [g] Converted Tension [g] 100 0.5 0.5 0.5 0.5 0 0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.0 0.0 0.5 1.0 0.0 Quick Displacement [mm] Quick Displacement [mm]

Fig. 9. Load–quick displacement properties of the sphincter (left, n ¼ 3) and dilator (right, n ¼ 3). The original experimental data are the same as for positive velocity (solid symbols) in Fig. 7. Contractile tension of each data, therefore, are also the same. Conversion of the axis was done as in Fig. 4 according to the results in Fig. 8. Standard deviations of residue from estimated curve to experimental data are as follows: (d) 6.11/0.49 (sphincter/ dilator), (N) 3.69/5.34, (j) 1.53/0.54 [mg].

Table 2 PDmin PDmax Parameter values of serial elasticity based on Eq. (5) Sphincter Dilator 2.5 abcabc

d 378 0.43 3.09 211 0.19 1.24 2.0 N 261 0.44 3.18 138 0.19 0.96 j 192 0.412.82 98.10.170.90 1.5 1.0 as in Section 3.2.1. Therefore, there are three kinds of Tension [g] diﬀerent data and in each of which contractile tension is diﬀerent. The estimated parameters of Eq. (5) are shown 0.5 in Table 2. In each contractile tension, load–displace-

ment curves diﬀer. However, tension–extension curves 0.0 almost overlap. It is demonstrated that, in contrast to 0 2 4 6 8 10 viscosity, serial elasticity is independent of contractile Pupil Diameter [mm] tension. Fig. 10. Tension properties as function of pupil diameter. (––) sphincter, (– –) dilator. (d) total tension, (N) active tension, (j) passive tension. Data for dilator at 7 mm in Fig. 4 is not indicated in this ﬁgure because it becomes a negative value in pupil diameter. 4. Discussion

4.1. Length–tension relationship the physical movable range of the pupil. This prediction was compared with the pupil diameter of enucleated eye In order to compare the length–tension relationship which was immersed in acetylcholine 10 mM or between the sphincter and dilator more directly, abscissa phenylephrine 0.1mM. Experimentally obtained maxi- of Fig. 4 was converted into pupil diameter (Fig. 10). A mum and minimum pupil diameters were: 1:38 0:17 pupil diameter at which passive tension of each muscle mm (n ¼ 8) and 7:98 0:24 mm (n ¼ 8). They almost crosses, denotes a pupil diameter at death since active correspond to the prediction. It has been confirmed that tension is no more produced in muscle. Fig. 10 shows length–tension relationship measured from the isolated that diameter is about 6 mm. This diameter corresponds muscle accurately reflects the pupil property in vivo. to the pupil diameter (5:55 0:86 mm, n ¼ 8) of enu- Furthermore, pupil specific characteristics can be cleated eyes just before preparing specimens. Moreover, found in Fig. 10. In the skeletal muscle, in general, the pupil diameters indicated by PDmax (PDmin) refer to the muscle length which produces the maximum contractile diameter at which passive tension of the sphincter (di- tension (optimum length: l0) corresponds to the muscle lator) is greater than the total tension of the dilator length in vivo (Murphy, 1983). In the iris muscle, there is (sphincter). The range from PDmin to PDmax have to be an optimum length at the maximum pupil diameter in 486 K. Yamaji et al. / Vision Research 43 (2003) 479–487 the sphincter and at the minimum pupil diameter in the model of which serial elastic component is independent dilator. This discloses the optimized pupillary mecha- of other components. nism that the highest constriction force is produced The stretched length of serial elasticity at optimum at the maximum pupil diameter, while the highest di- length (l0) is about 3%–10% in the skeletal muscle latation force is produced at the minimum pupil dia- (Cavagna, 1970; Hill, 1949) and about 3% in the guinea meter. pig teanea coli (Mashima et al., 1979). The extension of serial elasticity calculated from Table 2 is 20% in the sphincter and 18% in the dilator. Difference between the 4.2. Viscous behavior iris muscle and the other muscles originated from difference in amount of tissue contained in specimens. The To compare the viscosity between two muscles, vis- result shown in this study not only indicates the prop- cosity at normalized contractile tension (1g) was simu- erty of muscle clearly but also includes the property of lated. As a result, the dilator viscosity was 1.35 times iris tissue. However, the mixed property is required higher than that of the sphincter viscosity. This differ- when the dynamic pupillary response is discussed. This ence comes from the fact that the dilator specimen will be the next step of our study. contains larger amount of tissue than the sphincter dose. In each muscle, viscosity in negative velocity is 1.52 and 2.13 time higher than that of the positive velocity in the References sphincter and dilator. In the skeletal muscle, viscosity in negative velocity is also higher (1.4 times) (Mashima Cavagna, G. A. (1970). The series elastic component of frog et al., 1972) than that of positive case. This fact under- gastrocnemius. Journal of Physiology, 206, 257–269. lines general characteristics in muscle viscosity. Csapo, A. I. (1962). Smooth muscle as a contractile unit. Physiological The parameter values shown in Table 1determine the Reviews, 42, 7–33. viscous behavior. Parameter a=P differs between the Gordon, A. R., & Siegman, M. J. (1971). Mechanical properties of 0 smooth muscle. I. Length–tension and force–velocity relations. two muscles, however, its ratio expressed as ext=cont is American Journal of Physiology, 221, 1243–1249. almost the same. According to the physical meaning of a Hansmann, D. (1972). Human pupillary mechanics: physiology and explained in Section 3.2.1, it can be said that heat loss of control. Ph.D. thesis, University of California, Berkeley. the dilator is 2.5 times higher than that of the sphincter. Hill, A. (1938). The heat of shortening and the dynamic constants of However, heat loss ratio of lengthening to shortening is muscle. Proceedings of the Royal Society of London B, 126, 136– 195. almost the same. On the other hand, there is little dif- Hill, A. (1949). The abrupt transition from rest to activity in muscle. ference in b=l0 and its ratio ext=cont between the two Proceedings of the Royal Society of London B, 136, 399–420. muscles. The comparison of Vmax=l0 shows that maxi- Ishikawa, H., Yoshitomi, T., Harada, Y., Katori, M., & Ishikawa, S. mum velocity of the sphincter is about 2.5 times higher (1993). The presence of two sites of action of endothelins in the than that of the dilator. Table 3 shows the list for vis- isolated rabbit iris sphincter and dilator muscles. Current Eye Research, 12, 1049–1055. cous parameter in each organ and species. The viscous Jewell, B., & Wilkie, D. (1958). An analysis of the mechanical parameters of the rabbit iris sphincter and dilator are components in frogÕs striated muscle. Journal of Physiology, 143, not much different from the others. 515–540. Kern, R. (1970). Die adrenergischen Receptoren der intraaoculaen€ Muskeln der menschen. Graefes Archive for Clinical and Experi- 4.3. Load–displacement behavior mental Ophthalmology, 180, 231–248. Loewenfeld, I. (1993). The pupil: anatomy, physiology and clinical applications. Boston: Butterworth Heinemann. 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