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Theoretical estimation of retinal oxygenation during retinal detachment
Computers in Biology and Medicine 37 (2007) 890 – 896 www.intl.elsevierhealth.com/journals/cobm
Theoretical estimation of retinal oxygenation during retinal detachment Magnus W. Roos ∗ Department of Medical Sciences, Clinical Physiology, Uppsala University, Akademiska University Hospital, Entr. 35, 751 85 Uppsala, Sweden Received 9 May 2006; accepted 13 September 2006
Abstract The aim of the present work was to simulate the oxygenation of the whole retina during different degrees of retinal detachment. A differential equation describing the oxygenation of the whole retina, at different degrees of detachment, was set up and solved numerically. The results show that the choroid can supply the outer retina with a fairly large amount of oxygen as long as the detachment height is lower than about 1 mm. This study thus supports the view that hyperoxia may well prove to be clinically beneficial. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Detachment; Diffusion; Mathematical model; Oxygen; Retina
1. Introduction Retinal detachment may cause loss of vision. Most of the detachment-induced retinal damage appears to be directly related to the reduced supply of oxygen and to some extent also to lack of other substances, such as glucose [1–3]. The photoreceptor layer is by far the most affected area, probably because the inner segments of the photoreceptors account for almost 100% of the total oxygen consumption of the outer retina, and because the outer retina is mainly supplied with oxygen and nutrients via diffusion from the choroid. Anatomically, the retina is usually considered to consist of two parts, the outer retina (which is avascular) and the inner retina (which is supplied with blood) (Fig. 1). The outer part is mainly nourished by diffusion from the choroid, while the inner half is supplied by the retinal circulation. Although our present knowledge on retinal oxygenation is mainly based on experimental studies [4–8] mathematical modeling of the outer retina has contributed to important knowledge as well [9,10]. However, the presence of circulation in the inner retina makes
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the modeling more awkward. Consequently, this part of retina has regularly been excluded from modeling. Furthermore, in previous modeling of retinal oxygenation the oxygen consumption was set to be independent of the availability of oxygen [9–12], which seems somewhat unrealistic in conditions where the availability of oxygen is limited. However, recently a model of the oxygenation of the whole retina was introduced, which includes both retinal blood flow and oxygen consumption obeying Michaelis–Menten kinetics. This improved model allows modeling of the whole retina under different degrees of retinal ischemia as well as other conditions in which the availability of oxygen is limited [13]. However, the problem needs to be solved numerically. It has been shown experimentally that hyperoxia seems protective in retinal detachments [2,3,14]. These findings agree with results from previous modeling of the outer retina during detachment [15]. From those results it can be deduced that an increased arterial oxygenation should be beneficial, at least in detachment of lower severity (heights less than 0.5 mm). The aim of the present work was to evaluate the potential usefulness of hyperoxia in retinal detachments, by modeling the whole retinal oxygenation with intact retinal blood supply.
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where qoxmax is the maximal rate of oxygen turnover and Kox is a constant. The maximal average consumption in the outer retina during light was set to 18 mmHg s−1 and in darkness to 36 mmHg s−1 . By converting units by setting 2 , the solubility of oxygen in the retina, equal to 2.4 × 10−5 (ml O2 ml tissue−1 mmHg−1 ) [17] these consumption values correspond to 2.6 and 5.2 ml O2 per 100 g tissue per minute, respectively [17]. As region 2 occupies 20% of the outer retina and has 100% of the consumption taking place there, qoxmax in this region will be 90 mmHg s−1 in light and 180 mmHg s−1 in darkness. For the inner retina the value was set to 26 mmHg s−1 in both light and darkness, which is a value within the range reported by others [5,22] sox describes the amount of oxygen transported locally from the blood to the tissue. According to the Fick principle: sox = Fig. 1. Geometrical model. Regions 1–3 belong to the outer retina (avascular) while region 4 encompasses the whole inner retina (vascular). Region 5 is the subretinal gap (caused by the retinal detachment).
2. The model The outer retina was divided into three regions [10] (Fig. 1). The outer retina, regions 1–3, is supplied with oxygen through diffusion only, mainly from the choroid. Nearly 100% of the oxygen consumption in the outer part of the retina seems to take place in the inner segmental layer (i.e. the inner segments of the photoreceptors containing most of the mitochondria). In the present model the whole oxygen consumption by the outer retina was set to be confined to this region. This part, labeled region 2 in Fig. 1, is considered to occupy 20% of the outer retina and to have its inner and outer geometric boundaries at 75% and 85% of the total retinal thickness, respectively [10]. The location and thickness of this region has not been directly measured, but estimated by fitting a theoretical model to measured oxygen profiles [10]. The whole inner retina is considered to be one region (region 4) [11,13]. For the inner retina, the oxygen consumption was set to be uniformly distributed and the oxygen to be transported to the cells via the blood and/or by diffusion. The thickness of the retina was set to 250 m [10]. To this previously used model [13], a subretinal gap of variable size (detachment height from 0 to 2 mm) was added (Fig. 1). For the values of blood flow, oxygen consumption, etc., see Table 1. The change in local partial pressure of oxygen is given by the well-known diffusion equation: dp = ∇(Dox ∇p) − qox + sox , dt
(1)
where p is the local partial pressure of oxygen, t is time, Dox is the diffusion coefficient, qox is a consumption term and sox is a delivery term. According to the Michaelis–Menten equation, the consumption term for oxygen (qox ) is given by p × qoxmax , (2) qox = p + Kox
bf (p blood )n (pblood − ox p) + blood 60 (p )n + (Khem )n Hb · (ox p)n , − (ox p)n + (Khem )n 1
(3)
where ox is a constant, p is the local partial pressure of oxygen in the tissue, pblood is the partial pressure in arterial blood, Hb is the hemoglobin concentration in blood, 1 is the solubility of oxygen in blood and is a constant. The description of the hemoglobin saturation curve (the expression within parenthesis) is the well-known Hill equation [23]. The n is the Hill coefficient. The sox only applies to the inner retina since the outer retina does not have any blood flow. Little is known about possible changes in retinal circulation during detachment. In the present study the blood flow in the inner retina was set to 0.4 ml g−1 min−1 , which is about normal [24]. Calculations were performed for different detachment heights (0–2 mm). 2.1. Boundary conditions The partial pressure of oxygen at the choroidal border was set equal to the value in peripheral arterial blood, that is 80, 250 [5] or 405 mmHg [4]. The net diffusion at the vitreous edge was set to zero (Neumann boundary condition, dp/dx = 0). No constraints but continuity were set at the internal boundaries. 2.2. Parameters The parameter values were selected according to the available literature and are considered to apply to the cat. Selected values are given in Table 1. 2.3. Solution The system was, for convenience, set up in 2D (Fig. 1) and was solved numerically using a finite element method (FEMLAB 3.1, Comsol, Stockholm, Sweden). First, a timedependent solver was used on a coarse mesh to obtain an initial solution, which was then used as initial condition for a
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Table 1 Parameters
1 2 ox
bf Dox
Hb Khem Kox n p p blood qox qoxmax sox
The solubility of oxygen in blood, a constant set to 1.5 × 10−3 (mM mmHg−1 ) [16] The solubility of oxygen in retina, a constant set to 2.4 × 10−5 (ml O2 ml tissue−1 mmHg−1 ) [17] A constant giving the ratio of partial pressures of oxygen in retinal venous blood to retina (at steady state), set to 1 Local blood flow in the inner retina, set to 0.4 (min−1 ). In Eq. (3) this term is divided by 60 (s min−1 ) to get sox in a suitable unit (mmHg s−1 ) The diffusion coefficient of oxygen, set to 2 × 10−5 (cm2 s−1 ) [18] The oxygen carrying capacity of hemoglobin, set to 0.0616 (mmol g−1 ) [19] The hemoglobin concentration in blood, set to 140 (g l−1 ) A constant which equals the partial pressure of oxygen at which hemoglobin is 50% saturated with oxygen, set to 26 (mmHg) [20] The partial pressure of oxygen, at which the consumption runs at half maximal speed, set to 2 (mmHg) [21] Hill coefficient/hemoglobin cooperativity, set to 2.7 [20] Local partial pressure of oxygen in the retina, variable Partial pressure of oxygen in peripheral arterial blood, set to 80, 250 or 405 (mmHg) The total local consumption of oxygen, given by Eq. (2) (mmHg s−1 ) The maximal consumption rate of oxygen. For region 4, it was set to 26 independently of extent of illumination and for region 2 set to 90 in light and 180 in darkness (mmHg s−1 ) The amount of oxygen per unit time transferred locally from the blood to the retina, given by Eq. (3) (mmHg s−1 )
non-linear adaptive solver. Then, to ensure convergence, the mesh was refined and the non-linear solver was used without adaptation. 3. Results Retinal oxygen profiles during detachment in an about normoxic condition are shown in Figs. 2 (light) and 3 (dark). As can be seen in both these figures, the inner retinal profile is very little affected by the detachment. Figs. 4 and 5 exemplify the effect of different degrees of arterial oxygenation, in light and dark, respectively. Both these figures show retinal oxygen profiles where the detachment height is 1 mm. The impact of increased supply of oxygen on the total outer retinal oxygen consumption [15] in different detachment heights are shown in Figs. 6 (light) and 7 (darkness). 4. Discussion It appears that lack of oxygen is an important determinant of retinal damage caused by retinal detachment as there are results indicating that hyperoxia should be clinically useful [2,3,14]. Still, more details on the oxygenation of the detached retina are needed to guide the planning of further experimental and hopefully forthcoming clinical studies. A previous theoretical model of the oxygenation during retinal detachment has given some hints on useful treatment regimes, that is moderate retinal illumination and hyperoxia [15]. However, in their model the oxygen consumption is a constant derived from adapting a theoretical model of outer retina to experimental data. The value of the partial pressure of oxygen at the inner border is derived the same way. The obtained constants only applies for the conditions of those specific experiments (rate of retinal blood flow, partial pressure of oxygen at the choroid, etc.). One way of avoiding the need for a set of situation-dependent constant values is to add
Fig. 2. Retinal oxygen profiles in lighted conditions, where pblood was set to 80 mmHg. Partial pressure of oxygen on the y-axis and distance from the outer retinal edge on the x-axis. The profiles correspond to intact retina and to different heights of detachment; 1 and 2 mm, respectively. The oxygen consuming portion of the outer retina extends from 0.0375 to 0.0625 mm.
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Fig. 3. Retinal oxygen profiles in dark conditions, where pblood was set to 80 mmHg. Partial pressure of oxygen on the y-axis and distance from the outer retinal edge on the x-axis. The profiles correspond to intact retina and to different heights of detachment; 1 and 2 mm, respectively. The oxygen consuming portion of the outer retina extends from 0.0375 to 0.0625 mm.
Fig. 4. Retinal oxygen profiles in lighted conditions with the detachment height set to 1 mm. Partial pressure of oxygen on the y-axis and distance from the outer retinal edge on the x-axis. The profiles correspond to different values of pblood , 80 (lowest profile); 250 and 405 (highest profile) mmHg. The oxygen consuming portion of the outer retina extends from 0.0375 to 0.0625 mm.
Michaelis–Menten kinetics and include the inner retina in the modeling. These refinements should make the modeling more general, but requires that a numerical method be used to solve the problem. In the present work the retinal oxygenation at different degrees of retinal detachment was modeled. As can be seen in the Figs. 6 and 7 the consumption in the outer retina declines towards about 10% in dark and 20% in light (of base-line) as the detachment height approaches infinity. This is in good agreement with previous results [15]. Furthermore, the results of the present study suggest that in hyperoxia, the outer retinal oxygen consumption could be kept above 30% of base-line [15] in the case of a detachment height lower than about 1 mm in darkness (Fig. 7). The corresponding detachment height in the modeling by Linsenmeier and Padnick-Silver [15] was about 0.5 mm. Moreover, Linsenmeir and Padnick-Silver [15] suggested that, in light, the presence of hyperoxia could keep the oxygen consumption in the outer retina unaffected by the detachment. This seems very optimistic compared with the results from the model used in the present work. From Figs. 4 and 5 it can be deduced that at relatively high detachments, hyperoxia causes the partial pressure of oxygen in the inner retina to increase only by
some mmHg. The case of detached retina should thus not be mixed up with the case of intact retina, where hyperoxia causes the partial pressure of the inner retina to increase much more [13]. What can be further deduced from Figs. 6 and 7 is that in the present modeling, hyperoxia slightly increases the oxygen consumption above the normal value at very low detachment heights. Linsenmeier and Yancey [25] found that the oxygen consumption is not different during normoxia and hyperoxia. At a first glance, their finding seems contrary to the results shown in Figs. 6 and 7. However, the power for finding mean differences of less than 30% in that study seems to be less than 40% (according to a power analysis using a one-tailed t-test, = 0.05). Consequently, the possibility of a slight hyperoxiainduced increase in the outer retinal oxygen consumption above the normal value, as seen in Figs. 6 and 7, cannot be excluded. However, it should be mentioned that the magnitude of this effect is related to the selected value of Kox . Further, although the value of Kox will have impact on the retinal profiles of oxygen pressure, the effect on the total outer retinal consumption will be minor, e.g. at a detachment height of 1 mm (during normoxia, in light) setting Kox to 0.5 mmHg will reduce the consumption by only about 4% (data not shown). Thus, the value of Kox
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Fig. 6. Outer retinal oxygen consumption in lighted conditions. On the y-axis, the relative mean outer retinal consumption is displayed [15]. On the x-axis, the distance between the choroid and the outer retinal edge is displayed. The plots correspond to different values of pblood , 80 (lowest profile); 250 and 405 (highest profile) mmHg.
Fig. 5. Retinal oxygen profiles in dark conditions, with the detachment height set to 1 mm. Partial pressure of oxygen on the y-axis and distance from the outer retinal edge on the x-axis. The profiles correspond to different values of pblood , 80 (lowest profile); 250 and 405 (highest profile) mmHg. The oxygen consuming portion of the outer retina extends from 0.0375 to 0.0625 mm.
is not a determinant for the major conclusions of the present work. Another topic worth being mentioned is the possibility of detachment-induced changes in the inner retinal blood flow. As little is known about this issue, in the present work the retinal blood flow was left intact (bf = 0.4 ml g−1 min−1 ). However, at higher degrees of detachment, the value of blood flow in the inner retina has great impact on the outer retinal consumption of oxygen. As can be seen in Fig. 8, increasing the inner retinal blood flow may well prove to be beneficial in attempts to bring the outer retinal oxygen consumption back to normal levels. Although the results of the present study differ somewhat from previous results, this study supports the view that a combination of moderate illumination and an extra supply of oxygen [15] may be a beneficial supportive treatment for retinal detachment. 5. Summary
Fig. 7. Outer retinal oxygen consumption in dark conditions. On the y-axis the relative mean outer retinal consumption is displayed [15]. On the x-axis, the distance between the choroid and the outer retinal edge is displayed. The plots correspond to different values of p blood , 80 (lowest profile); 250 and 405 (highest profile) mmHg.
Retinal detachment may cause extensive morbidity. A better understanding of the pathophysiology is needed. It has been shown experimentally that hyperoxia seems protective in retinal detachments. These findings agree with results from previous modeling of the outer retina during detachment. In
the present work refinements (inclusion or the inner retina and Michaelis–Menten kinetics) were introduced which should make the modeling more general. In conclusion, this study supports the view that a combination of moderate illumination and
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Fig. 8. Outer retinal oxygen consumption in lighted conditions at an about normoxic condition (p blood set equal to 80 mmHg). Detachment height set to 1 mm. On the y-axis the relative mean outer retinal consumption is displayed [15]. On the x-axis, the inner retinal blood flow is displayed.
an extra supply of oxygen may be a beneficial supportive treatment for retinal detachment. Acknowledgments Thanks are extended to Professor Bertil Gustafsson, Department of Scientific Computing and Dr Göran Sperber, Department of Neuroscience, Physiology, both at Uppsala University, for comments on the manuscript. This study was financially ˚ supported by the Alzheimer Foundation, Ahlensstiftelsen, and Gun och Bertil Stohnes Stiftelse, Sweden. References [1] G. Lewis, K. Mervin, K. Valter, J. Masilm, P.J. Kappel, J. Stone, S. Fisher, Limiting the proliferation and reactivity of retinal müller cells during experimental retinal detachment: the value of oxygen supplementation, Am. J. Ophthalmol. 128 (1999) 165–172. [2] K. Mervin, K. Valter, J. Maslim, G. Lewis, S. Fisher, J. Stone, Limiting photoreceptor death and deconstruction during experimental retinal detachment: the value of oxygen supplementation, Am. J. Ophthalmol. 128 (1999) 155–164. [3] T. Sakai, G.P. Lewis, K.A. Lindberg, S.K. Fisher, The ability of hyperoxia to limit the effects of experimental detachment in cone-dominated retina, Invest. Ophthalmol. Vis. Sci. 42 (2001) 3264–3273. [4] V.A. Alder, J. Ben-Nun, S.J. Cringle, PO2 profiles and oxygen consumption in cat retina with an occluded retinal circulation, Invest. Ophthalmol. Vis. Sci. 31 (1990) 1029–1034. [5] R.D. Braun, R.A. Linsenmeier, Retinal oxygen tension and electroretinogram during arterial occlusion in the cat, Invest. Ophthalmol. Vis. Sci. 36 (1995) 523–541. [6] B.A. Berkowitz, Adult and newborn rat inner retinal oxygenation during carbogen and 100% oxygen breathing: comparison using magnetic resonance imaging PO2 mapping, Invest. Ophthalmol. Vis. Sci. 37 (1996) 2089–2098. [7] F.S. Sutherland, E. Stefansson, D.L. Hatchell, H. Reiser, Retinal oxygen consumption in vitro, the effect of diabetes mellitus, oxygen and glucose, Acta Ophthalmol. 68 (1990) 715–720.
[8] D.Y. Yu, S.J. Cringle, Oxygen distribution and consumption within the retina in vascularised and avascular retinas and in animal models of retinal disease, Prog. Retin. Eye. Res. 20 (2001) 175–208. [9] R.A. Linsenmeier, Effects of light and darkness on oxygen distribution and consumption in the cat retina, J. Gen. Physiol. 88 (1986) 521–542. [10] L.M. Haugh, R.A. Linsenmeier, T.K. Goldstick, Mathematical models of the spatial distribution of retinal oxygen tension and consumption, including changes upon illumination, Ann. Biomed. Eng. 18 (1990) 19 –36. [11] R.D. Braun, R.A. Linsenmeier, T.K. Goldstick, Oxygen consumption in the inner and outer retina of the cat, Invest. Ophthalmol. Vis. Sci. 36 (1995) 542–554. [12] S.J. Cringle, D.Y. Yu, A multi-layer model of retinal oxygen supply and consumption helps explain the muted rise in inner retinal PO2 during systemic hyperoxia, Comp. Biochem. Physiol. A 132 (2002) 61–66. [13] M.W. Roos, Theoretical estimation of retinal oxygenation during retinal artery occlusion, Physiol. Meas. 25 (2004) 1523–1532. [14] G.P. Lewis, K.C. Talaga, K.A. Lindberg, R.L. Avery, S.K. Fisher, The efficacy of delayed oxygen therapy in the treatment of experimental retinal detachment, Am. J. Ophthalmol. 137 (1999) 1085–1095. [15] R.A. Linsenmeier, L. Padnick-Silver, Metabolic dependence of photoreceptors on the choroid in the normal and detached retina, Invest. Ophthalmol. Vis. Sci. 41 (2000) 3117–3123. [16] J. Vanderkooi, M. Ereci’nska, I.A. Silver, Oxygen in mammalian tissue: methods of measurement and affinities of various reactions, Am. J. Physiol. 260C (1991) 1131–1150. [17] R.A. Linsenmeier, R.D. Braun, Oxygen distribution and consumption in the cat retina during normoxia and hypoxemia, J. Gen. Physiol. 99 (1992) 177–197. [18] B.W. Pogue, K.D. Paulsen, J.A. O’Hara, C.M. Wilmot, H.M. Schwartz, Estimation of oxygen distribution in RIF-1 tumors by diffusion model-based interpretation of pimonidazole hypoxia and Eppendorf measurements, Radiat. Res. 155 (2001) 15–25. [19] A. Despopoulos, S. Silbernagl, Color Atlas of Physiology, fifth ed., Thieme, NewYork, 2003. [20] B.D. Kavanagh, T.W. Secomb, R. Hsu, P.S. Lin, J. Venitz, M.W. Dewhirst, A theoretical model for the effects of reduced hemoglobin–oxygen affinity on tumor oxygenation, Int. J. Radiat. Oncol. Biol. Phys. 53 (2002) 172–179.
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[21] R.A. Ganfield, P. Nair, W.J. Whalen, Mass transfer storage and utilization of O2 in cat cerebral cortex, Am. J. Physiol. 219 (1970) 814–821. [22] J. Ahmed, R.D. Braun, R. Dunn, R.A. Linsenmeier, Oxygen distribution in the macaque retina, Invest. Ophthalmol. Vis Sci. 34 (1993) 516–521. [23] J. Keener, J. Sneyd, Mathematical Biology, Springer, Berlin, 2001 pp. 3–32, 482–487. [24] J. Ahmed, M.K. Pulfer, R.A. Linsenmeier, Measurement of blood flow through the retinal circulation of the cat during normoxia and hypoxemia using fluorescent microspheres, Microvasc. Res. 62 (2001) 143–153.
[25] R.A. Linsenmeier, C.M. Yancey, Effects of hyperoxia on the oxygen distribution in the intact cat retina, Invest. Ophthalmol. Vis. Sci. 30 (1989) 612–618. K. Magnus W. Roos was born in Uppsala in 1968. He received his PhD degree (Doctor of Medical Sciences) in 1997 and his MD in 2000. His main research interests are blood flow and metabolism of the central nervous system, microcirculation and diffusion in particular.
Theoretical estimation of retinal oxygenation during retinal detachment Magnus W. Roos ∗ Department of Medical Sciences, Clinical Physiology, Uppsala University, Akademiska University Hospital, Entr. 35, 751 85 Uppsala, Sweden Received 9 May 2006; accepted 13 September 2006
Abstract The aim of the present work was to simulate the oxygenation of the whole retina during different degrees of retinal detachment. A differential equation describing the oxygenation of the whole retina, at different degrees of detachment, was set up and solved numerically. The results show that the choroid can supply the outer retina with a fairly large amount of oxygen as long as the detachment height is lower than about 1 mm. This study thus supports the view that hyperoxia may well prove to be clinically beneficial. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Detachment; Diffusion; Mathematical model; Oxygen; Retina
1. Introduction Retinal detachment may cause loss of vision. Most of the detachment-induced retinal damage appears to be directly related to the reduced supply of oxygen and to some extent also to lack of other substances, such as glucose [1–3]. The photoreceptor layer is by far the most affected area, probably because the inner segments of the photoreceptors account for almost 100% of the total oxygen consumption of the outer retina, and because the outer retina is mainly supplied with oxygen and nutrients via diffusion from the choroid. Anatomically, the retina is usually considered to consist of two parts, the outer retina (which is avascular) and the inner retina (which is supplied with blood) (Fig. 1). The outer part is mainly nourished by diffusion from the choroid, while the inner half is supplied by the retinal circulation. Although our present knowledge on retinal oxygenation is mainly based on experimental studies [4–8] mathematical modeling of the outer retina has contributed to important knowledge as well [9,10]. However, the presence of circulation in the inner retina makes
∗ Tel.: +46 186114133; fax: +46 18550133.
E-mail address: [email protected]. 0010-4825/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compbiomed.2006.09.005
the modeling more awkward. Consequently, this part of retina has regularly been excluded from modeling. Furthermore, in previous modeling of retinal oxygenation the oxygen consumption was set to be independent of the availability of oxygen [9–12], which seems somewhat unrealistic in conditions where the availability of oxygen is limited. However, recently a model of the oxygenation of the whole retina was introduced, which includes both retinal blood flow and oxygen consumption obeying Michaelis–Menten kinetics. This improved model allows modeling of the whole retina under different degrees of retinal ischemia as well as other conditions in which the availability of oxygen is limited [13]. However, the problem needs to be solved numerically. It has been shown experimentally that hyperoxia seems protective in retinal detachments [2,3,14]. These findings agree with results from previous modeling of the outer retina during detachment [15]. From those results it can be deduced that an increased arterial oxygenation should be beneficial, at least in detachment of lower severity (heights less than 0.5 mm). The aim of the present work was to evaluate the potential usefulness of hyperoxia in retinal detachments, by modeling the whole retinal oxygenation with intact retinal blood supply.
M.W. Roos / Computers in Biology and Medicine 37 (2007) 890 – 896
891
where qoxmax is the maximal rate of oxygen turnover and Kox is a constant. The maximal average consumption in the outer retina during light was set to 18 mmHg s−1 and in darkness to 36 mmHg s−1 . By converting units by setting 2 , the solubility of oxygen in the retina, equal to 2.4 × 10−5 (ml O2 ml tissue−1 mmHg−1 ) [17] these consumption values correspond to 2.6 and 5.2 ml O2 per 100 g tissue per minute, respectively [17]. As region 2 occupies 20% of the outer retina and has 100% of the consumption taking place there, qoxmax in this region will be 90 mmHg s−1 in light and 180 mmHg s−1 in darkness. For the inner retina the value was set to 26 mmHg s−1 in both light and darkness, which is a value within the range reported by others [5,22] sox describes the amount of oxygen transported locally from the blood to the tissue. According to the Fick principle: sox = Fig. 1. Geometrical model. Regions 1–3 belong to the outer retina (avascular) while region 4 encompasses the whole inner retina (vascular). Region 5 is the subretinal gap (caused by the retinal detachment).
2. The model The outer retina was divided into three regions [10] (Fig. 1). The outer retina, regions 1–3, is supplied with oxygen through diffusion only, mainly from the choroid. Nearly 100% of the oxygen consumption in the outer part of the retina seems to take place in the inner segmental layer (i.e. the inner segments of the photoreceptors containing most of the mitochondria). In the present model the whole oxygen consumption by the outer retina was set to be confined to this region. This part, labeled region 2 in Fig. 1, is considered to occupy 20% of the outer retina and to have its inner and outer geometric boundaries at 75% and 85% of the total retinal thickness, respectively [10]. The location and thickness of this region has not been directly measured, but estimated by fitting a theoretical model to measured oxygen profiles [10]. The whole inner retina is considered to be one region (region 4) [11,13]. For the inner retina, the oxygen consumption was set to be uniformly distributed and the oxygen to be transported to the cells via the blood and/or by diffusion. The thickness of the retina was set to 250 m [10]. To this previously used model [13], a subretinal gap of variable size (detachment height from 0 to 2 mm) was added (Fig. 1). For the values of blood flow, oxygen consumption, etc., see Table 1. The change in local partial pressure of oxygen is given by the well-known diffusion equation: dp = ∇(Dox ∇p) − qox + sox , dt
(1)
where p is the local partial pressure of oxygen, t is time, Dox is the diffusion coefficient, qox is a consumption term and sox is a delivery term. According to the Michaelis–Menten equation, the consumption term for oxygen (qox ) is given by p × qoxmax , (2) qox = p + Kox
bf (p blood )n (pblood − ox p) + blood 60 (p )n + (Khem )n Hb · (ox p)n , − (ox p)n + (Khem )n 1
(3)
where ox is a constant, p is the local partial pressure of oxygen in the tissue, pblood is the partial pressure in arterial blood, Hb is the hemoglobin concentration in blood, 1 is the solubility of oxygen in blood and is a constant. The description of the hemoglobin saturation curve (the expression within parenthesis) is the well-known Hill equation [23]. The n is the Hill coefficient. The sox only applies to the inner retina since the outer retina does not have any blood flow. Little is known about possible changes in retinal circulation during detachment. In the present study the blood flow in the inner retina was set to 0.4 ml g−1 min−1 , which is about normal [24]. Calculations were performed for different detachment heights (0–2 mm). 2.1. Boundary conditions The partial pressure of oxygen at the choroidal border was set equal to the value in peripheral arterial blood, that is 80, 250 [5] or 405 mmHg [4]. The net diffusion at the vitreous edge was set to zero (Neumann boundary condition, dp/dx = 0). No constraints but continuity were set at the internal boundaries. 2.2. Parameters The parameter values were selected according to the available literature and are considered to apply to the cat. Selected values are given in Table 1. 2.3. Solution The system was, for convenience, set up in 2D (Fig. 1) and was solved numerically using a finite element method (FEMLAB 3.1, Comsol, Stockholm, Sweden). First, a timedependent solver was used on a coarse mesh to obtain an initial solution, which was then used as initial condition for a
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Table 1 Parameters
1 2 ox
bf Dox
Hb Khem Kox n p p blood qox qoxmax sox
The solubility of oxygen in blood, a constant set to 1.5 × 10−3 (mM mmHg−1 ) [16] The solubility of oxygen in retina, a constant set to 2.4 × 10−5 (ml O2 ml tissue−1 mmHg−1 ) [17] A constant giving the ratio of partial pressures of oxygen in retinal venous blood to retina (at steady state), set to 1 Local blood flow in the inner retina, set to 0.4 (min−1 ). In Eq. (3) this term is divided by 60 (s min−1 ) to get sox in a suitable unit (mmHg s−1 ) The diffusion coefficient of oxygen, set to 2 × 10−5 (cm2 s−1 ) [18] The oxygen carrying capacity of hemoglobin, set to 0.0616 (mmol g−1 ) [19] The hemoglobin concentration in blood, set to 140 (g l−1 ) A constant which equals the partial pressure of oxygen at which hemoglobin is 50% saturated with oxygen, set to 26 (mmHg) [20] The partial pressure of oxygen, at which the consumption runs at half maximal speed, set to 2 (mmHg) [21] Hill coefficient/hemoglobin cooperativity, set to 2.7 [20] Local partial pressure of oxygen in the retina, variable Partial pressure of oxygen in peripheral arterial blood, set to 80, 250 or 405 (mmHg) The total local consumption of oxygen, given by Eq. (2) (mmHg s−1 ) The maximal consumption rate of oxygen. For region 4, it was set to 26 independently of extent of illumination and for region 2 set to 90 in light and 180 in darkness (mmHg s−1 ) The amount of oxygen per unit time transferred locally from the blood to the retina, given by Eq. (3) (mmHg s−1 )
non-linear adaptive solver. Then, to ensure convergence, the mesh was refined and the non-linear solver was used without adaptation. 3. Results Retinal oxygen profiles during detachment in an about normoxic condition are shown in Figs. 2 (light) and 3 (dark). As can be seen in both these figures, the inner retinal profile is very little affected by the detachment. Figs. 4 and 5 exemplify the effect of different degrees of arterial oxygenation, in light and dark, respectively. Both these figures show retinal oxygen profiles where the detachment height is 1 mm. The impact of increased supply of oxygen on the total outer retinal oxygen consumption [15] in different detachment heights are shown in Figs. 6 (light) and 7 (darkness). 4. Discussion It appears that lack of oxygen is an important determinant of retinal damage caused by retinal detachment as there are results indicating that hyperoxia should be clinically useful [2,3,14]. Still, more details on the oxygenation of the detached retina are needed to guide the planning of further experimental and hopefully forthcoming clinical studies. A previous theoretical model of the oxygenation during retinal detachment has given some hints on useful treatment regimes, that is moderate retinal illumination and hyperoxia [15]. However, in their model the oxygen consumption is a constant derived from adapting a theoretical model of outer retina to experimental data. The value of the partial pressure of oxygen at the inner border is derived the same way. The obtained constants only applies for the conditions of those specific experiments (rate of retinal blood flow, partial pressure of oxygen at the choroid, etc.). One way of avoiding the need for a set of situation-dependent constant values is to add
Fig. 2. Retinal oxygen profiles in lighted conditions, where pblood was set to 80 mmHg. Partial pressure of oxygen on the y-axis and distance from the outer retinal edge on the x-axis. The profiles correspond to intact retina and to different heights of detachment; 1 and 2 mm, respectively. The oxygen consuming portion of the outer retina extends from 0.0375 to 0.0625 mm.
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Fig. 3. Retinal oxygen profiles in dark conditions, where pblood was set to 80 mmHg. Partial pressure of oxygen on the y-axis and distance from the outer retinal edge on the x-axis. The profiles correspond to intact retina and to different heights of detachment; 1 and 2 mm, respectively. The oxygen consuming portion of the outer retina extends from 0.0375 to 0.0625 mm.
Fig. 4. Retinal oxygen profiles in lighted conditions with the detachment height set to 1 mm. Partial pressure of oxygen on the y-axis and distance from the outer retinal edge on the x-axis. The profiles correspond to different values of pblood , 80 (lowest profile); 250 and 405 (highest profile) mmHg. The oxygen consuming portion of the outer retina extends from 0.0375 to 0.0625 mm.
Michaelis–Menten kinetics and include the inner retina in the modeling. These refinements should make the modeling more general, but requires that a numerical method be used to solve the problem. In the present work the retinal oxygenation at different degrees of retinal detachment was modeled. As can be seen in the Figs. 6 and 7 the consumption in the outer retina declines towards about 10% in dark and 20% in light (of base-line) as the detachment height approaches infinity. This is in good agreement with previous results [15]. Furthermore, the results of the present study suggest that in hyperoxia, the outer retinal oxygen consumption could be kept above 30% of base-line [15] in the case of a detachment height lower than about 1 mm in darkness (Fig. 7). The corresponding detachment height in the modeling by Linsenmeier and Padnick-Silver [15] was about 0.5 mm. Moreover, Linsenmeir and Padnick-Silver [15] suggested that, in light, the presence of hyperoxia could keep the oxygen consumption in the outer retina unaffected by the detachment. This seems very optimistic compared with the results from the model used in the present work. From Figs. 4 and 5 it can be deduced that at relatively high detachments, hyperoxia causes the partial pressure of oxygen in the inner retina to increase only by
some mmHg. The case of detached retina should thus not be mixed up with the case of intact retina, where hyperoxia causes the partial pressure of the inner retina to increase much more [13]. What can be further deduced from Figs. 6 and 7 is that in the present modeling, hyperoxia slightly increases the oxygen consumption above the normal value at very low detachment heights. Linsenmeier and Yancey [25] found that the oxygen consumption is not different during normoxia and hyperoxia. At a first glance, their finding seems contrary to the results shown in Figs. 6 and 7. However, the power for finding mean differences of less than 30% in that study seems to be less than 40% (according to a power analysis using a one-tailed t-test, = 0.05). Consequently, the possibility of a slight hyperoxiainduced increase in the outer retinal oxygen consumption above the normal value, as seen in Figs. 6 and 7, cannot be excluded. However, it should be mentioned that the magnitude of this effect is related to the selected value of Kox . Further, although the value of Kox will have impact on the retinal profiles of oxygen pressure, the effect on the total outer retinal consumption will be minor, e.g. at a detachment height of 1 mm (during normoxia, in light) setting Kox to 0.5 mmHg will reduce the consumption by only about 4% (data not shown). Thus, the value of Kox
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Fig. 6. Outer retinal oxygen consumption in lighted conditions. On the y-axis, the relative mean outer retinal consumption is displayed [15]. On the x-axis, the distance between the choroid and the outer retinal edge is displayed. The plots correspond to different values of pblood , 80 (lowest profile); 250 and 405 (highest profile) mmHg.
Fig. 5. Retinal oxygen profiles in dark conditions, with the detachment height set to 1 mm. Partial pressure of oxygen on the y-axis and distance from the outer retinal edge on the x-axis. The profiles correspond to different values of pblood , 80 (lowest profile); 250 and 405 (highest profile) mmHg. The oxygen consuming portion of the outer retina extends from 0.0375 to 0.0625 mm.
is not a determinant for the major conclusions of the present work. Another topic worth being mentioned is the possibility of detachment-induced changes in the inner retinal blood flow. As little is known about this issue, in the present work the retinal blood flow was left intact (bf = 0.4 ml g−1 min−1 ). However, at higher degrees of detachment, the value of blood flow in the inner retina has great impact on the outer retinal consumption of oxygen. As can be seen in Fig. 8, increasing the inner retinal blood flow may well prove to be beneficial in attempts to bring the outer retinal oxygen consumption back to normal levels. Although the results of the present study differ somewhat from previous results, this study supports the view that a combination of moderate illumination and an extra supply of oxygen [15] may be a beneficial supportive treatment for retinal detachment. 5. Summary
Fig. 7. Outer retinal oxygen consumption in dark conditions. On the y-axis the relative mean outer retinal consumption is displayed [15]. On the x-axis, the distance between the choroid and the outer retinal edge is displayed. The plots correspond to different values of p blood , 80 (lowest profile); 250 and 405 (highest profile) mmHg.
Retinal detachment may cause extensive morbidity. A better understanding of the pathophysiology is needed. It has been shown experimentally that hyperoxia seems protective in retinal detachments. These findings agree with results from previous modeling of the outer retina during detachment. In
the present work refinements (inclusion or the inner retina and Michaelis–Menten kinetics) were introduced which should make the modeling more general. In conclusion, this study supports the view that a combination of moderate illumination and
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Fig. 8. Outer retinal oxygen consumption in lighted conditions at an about normoxic condition (p blood set equal to 80 mmHg). Detachment height set to 1 mm. On the y-axis the relative mean outer retinal consumption is displayed [15]. On the x-axis, the inner retinal blood flow is displayed.
an extra supply of oxygen may be a beneficial supportive treatment for retinal detachment. Acknowledgments Thanks are extended to Professor Bertil Gustafsson, Department of Scientific Computing and Dr Göran Sperber, Department of Neuroscience, Physiology, both at Uppsala University, for comments on the manuscript. This study was financially ˚ supported by the Alzheimer Foundation, Ahlensstiftelsen, and Gun och Bertil Stohnes Stiftelse, Sweden. References [1] G. Lewis, K. Mervin, K. Valter, J. Masilm, P.J. Kappel, J. Stone, S. Fisher, Limiting the proliferation and reactivity of retinal müller cells during experimental retinal detachment: the value of oxygen supplementation, Am. J. Ophthalmol. 128 (1999) 165–172. [2] K. Mervin, K. Valter, J. Maslim, G. Lewis, S. Fisher, J. Stone, Limiting photoreceptor death and deconstruction during experimental retinal detachment: the value of oxygen supplementation, Am. J. Ophthalmol. 128 (1999) 155–164. [3] T. Sakai, G.P. Lewis, K.A. Lindberg, S.K. Fisher, The ability of hyperoxia to limit the effects of experimental detachment in cone-dominated retina, Invest. Ophthalmol. Vis. Sci. 42 (2001) 3264–3273. [4] V.A. Alder, J. Ben-Nun, S.J. Cringle, PO2 profiles and oxygen consumption in cat retina with an occluded retinal circulation, Invest. Ophthalmol. Vis. Sci. 31 (1990) 1029–1034. [5] R.D. Braun, R.A. Linsenmeier, Retinal oxygen tension and electroretinogram during arterial occlusion in the cat, Invest. Ophthalmol. Vis. Sci. 36 (1995) 523–541. [6] B.A. Berkowitz, Adult and newborn rat inner retinal oxygenation during carbogen and 100% oxygen breathing: comparison using magnetic resonance imaging PO2 mapping, Invest. Ophthalmol. Vis. Sci. 37 (1996) 2089–2098. [7] F.S. Sutherland, E. Stefansson, D.L. Hatchell, H. Reiser, Retinal oxygen consumption in vitro, the effect of diabetes mellitus, oxygen and glucose, Acta Ophthalmol. 68 (1990) 715–720.
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[21] R.A. Ganfield, P. Nair, W.J. Whalen, Mass transfer storage and utilization of O2 in cat cerebral cortex, Am. J. Physiol. 219 (1970) 814–821. [22] J. Ahmed, R.D. Braun, R. Dunn, R.A. Linsenmeier, Oxygen distribution in the macaque retina, Invest. Ophthalmol. Vis Sci. 34 (1993) 516–521. [23] J. Keener, J. Sneyd, Mathematical Biology, Springer, Berlin, 2001 pp. 3–32, 482–487. [24] J. Ahmed, M.K. Pulfer, R.A. Linsenmeier, Measurement of blood flow through the retinal circulation of the cat during normoxia and hypoxemia using fluorescent microspheres, Microvasc. Res. 62 (2001) 143–153.
[25] R.A. Linsenmeier, C.M. Yancey, Effects of hyperoxia on the oxygen distribution in the intact cat retina, Invest. Ophthalmol. Vis. Sci. 30 (1989) 612–618. K. Magnus W. Roos was born in Uppsala in 1968. He received his PhD degree (Doctor of Medical Sciences) in 1997 and his MD in 2000. His main research interests are blood flow and metabolism of the central nervous system, microcirculation and diffusion in particular.