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Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy Jeffrey L. Olson a,b, Masoud Asadi-Zeydabadi c, Randall Tagg c a
University of Colorado Denver, Rocky Mountain Lions Eye Institute, Denver, CO, USA University of Colorado Denver, Department of Ophthalmology, Denver, CO, USA c University of Colorado Denver, Department of Physics, Denver, CO, USA b
art ic l e i nf o
a b s t r a c t
Article history: Received 18 April 2014 Accepted 24 December 2014
This paper uses computer modeling to estimate the progressive decline in oxygenation that occurs in the human diabetic retina after years of slowly progressive ischemic insult. An established model combines diffusion, saturable consumption, and blood capillary sources to determine the oxygen distribution across the retina. Incorporating long-term degradation of blood supply from the retinal capillaries into the model yields insight into the effects of progressive ischemia associated with prolonged hyperglycemia, suggesting time-scales over which therapeutic mitigation could have beneficial effect. A new extension of the model for oxygen distribution introduces a feedback mechanism for vasodilation and its potential to prolong healthy retinal function. & 2015 Published by Elsevier Ltd.
Keywords: Diabetes mellitus Retinopathy Computer model
1. Introduction Diabetic retinopathy is a leading cause of blindness in the industrialized world and results in a progressive loss of the capillary bed in the retina. This loss of blood supply in turn causes a shortage of oxygen critically needed by metabolically active retinal photoreceptors. In the face of this chronic anoxic challenge, retinal cells die and vision is progressively lost. Previous theoretical modeling of the retina has dealt with acute events, such as ophthalmic artery occlusion and retinal detachment. The aim of the present work is to develop a mathematical model of the retina under progressive and chronic ischemic conditions to study various aspects of retinal oxygenation. Diabetes is a rapidly growing problem in the industrialized world – there are 18 million known diabetics in the United States, and 170 million worldwide [1,2]. For millions of diabetics, each day brings a further diminution of their vision. Despite recent technological and pharmacologic advances in medicine, the number of people losing sight continues to increase. In fact, diabetic retinopathy is the leading cause of blindness in adults aged 25–55 years old, and will blind over 25,000 Americans this year alone [3]. Often the vision loss is irreversible. Vision loss carries profound impact for both the individual and society at large, and currently affects more people than ever. It has been noted that the population of those with visual impairment in the United States is expected to double over the next thirty years [4]. Diabetic retinopathy is characterized by progressive closure and loss of native blood vessels, usually on the anatomic level of the
capillary, thought to be secondary to pericyte loss caused by chronic hyperglycemia. Over time, the loss of individual blood vessels results in decreased blood flow to the retinal tissue as a whole. This chronic, relentless loss of the capillary bed and tissue blood supply is fundamentally the same process that occurs in the kidney, heart, and peripheral nerves of the diabetic resulting in renal failure, cardiac disease, and neuropathy. The major difference between these diseases and retinopathy, is that the loss of capillaries can be visualized and photographed through the lens of the eye either non-invasively or minimally invasively via dye injection (Fig. 1). The loss of blood supply that characterizes diabetes leads to the fundamental underlying problem associated with diabetic retinopathy: a chronic shortage of oxygen, along with other nutrients such as glucose. This decreased oxygen supply adversely affects retinal cells, which are some of the most metabolically active cells in the human body, with a concurrent high demand for oxygen [5]. With prolonged ischemia, these cells in the retina progressively lose their ability to function, and the patient suffers a progressive loss of vision [6]. While the current state of treatment using either lasers or injectable medication provides a means of slowing the rate of vision loss, it does not treat the underlying problem, i.e., a lack of tissue oxygen [7,8]. Conventionally, the retina is divided into two vascular zones: the inner vascularized retina and the outer avascular retina. The outer retina derives its oxygen by diffusion from the underlying choroid, which is a high flow network of interconnected channels not under metabolic regulation. The inner retina is nourished by an intrinsic vascular system under metabolic control. The watershed area between
http://dx.doi.org/10.1016/j.compbiomed.2014.12.021 0010-4825/& 2015 Published by Elsevier Ltd.
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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Fig. 1. Human retinal appearance, as photographed during fluorescein angiography. After injection of a fluorescent dye, blood flow can be imaged in the major retinal arterial (arrows) and venous (arrowheads) circulations. Further, capillary perfusion can be assessed. The areas where the capillaries have closed secondary to diabetic retinopathy are non-perfused. These areas of ischemia are treated with laser surgery (asterisks).
the two circulations typically occurs at the junction of the inner and outer retinal layers [9]. The rates of oxygen consumption in various levels of the retina have been determined by experimentation [10–12] and described by mathematical modeling [13,14]. Building upon these previous computer models, other researchers have used this dual domain blood flow concept to predict oxygen levels in different acute disease states such as retinal artery occlusion [15] and retinal detachment [16]. To date, there have been no computer models of chronic or progressive anoxic states in the retina to simulate diseases such as diabetic retinopathy, which may run their course over many decades. The prior models also did not include vasodilation as a possible feedback mechanism for compensating capillary loss in the inner retina. It has been demonstrated that hyperoxia may provide some benefit in ameliorating the progressive of diabetic retinopathy by administering oxygen by nasal cannula [17]. Further, there is the possibility that moderate illumination may prevent progression of diabetic retinopathy by inhibiting the increase in photoreceptor oxygen demand that occurs during the dark cycle [9]. Attempts have been made to restore oxygen supply to the retina by means of oxygenated irrigating fluids [18], intraocular implants [19], and systemic oxygen delivery [17]. In theory, higher levels of oxygen delivered to the lungs results in higher oxygen levels carried by the blood to end organs. However, there are two factors which counteract this effect. First, increasing systemic arterial oxygen levels has differing effects on the two blood supplies of the eye: the choroidal supply increases dramatically with systemic hyperoxia, but the effect on the inner retinal is blunted dramatically secondary to autoregulation of retinal blood flow [20]. The second major obstacle in this treatment paradigm is the upper limit on oxygenation that occurs in the blood once hemoglobin is saturated, as defined by the oxyhemoglobin dissociation curve. For example, at an arterial partial pressure of oxygen of 90 mmHg, hemoglobin is about 97% saturated. Increasing the arterial pressure of oxygen to 100 mmHg increases the saturation of hemoglobin to 98% [21]. In short, there is not much room for physiologic improvement once the hemoglobin is saturated. High levels of inhaled oxygen treatment have some serious drawbacks and side effects as well. The first and most obvious is the need to wear a nasal cannula and carry a continuous supply of oxygen. Second, high oxygen levels have been shown to induce hyperoxic lung damage through receptor mediated and mitochondrial cell death pathways [22]. In order to better frame the problem of maintaining adequate oxygen over many years of progressive decline of retinal capillary
function, a model-based approach is explored. We first describe the mathematical representation of oxygen delivery, transport, and consumption, including numerical procedures used to calculate oxygen distributions from the model. Then we describe results both in terms of spatial distribution of oxygen in the retina and in terms of long-term effects of decreasing blood flow in the retina. The model is one-dimensional, capturing the dominant features of oxygen transport due to the thinness of the retina layer with the assumption that tissue parameters do not vary rapidly in the lateral directions at the 100 μm scale. While significant variation of parameters governing blood supply, diffusion and consumption can occur across the full spatial extent of the retinal layer and some highly localized regions of different structure exist (for example, the fovea centralis), we believe this 1-dimensional model permits primary understanding of long-term effects for a given location once local conditions are defined through the governing parameters. After exploring long-term effects within the established model, we add a new two-parameter extension of the model incorporating vasodilation to explore its potential mitigating effects over long times. Finally, we interpret the results to arrive at some conclusions to help guide future treatment.
2. Methods 2.1. The model Histological and anatomical depictions of the retina are shown in Fig. 2. The total thickness of the retina has been measured histologically and in vivo by optical coherence tomography to be about 0.250 mm. To describe the distribution of oxygen within this tissue, we take as a starting point the model used by Roos [15,16]. In this one-dimensional model, the retinal structure is idealized as four distinct regions of oxygen shown schematically in Fig. 3. The outer retina, which is avascular and receives its oxygen supply from choroidal blood flow, is divided to regions 1–3. The delivery of oxygen in the outer retina depends on diffusion from choroid into the retina. Oxygen transport is modeled as pure diffusion across region 1, which includes Bruch’s membrane, the pigmented epithelium, and the outer segmental layers of the rods and cones where light-transduction principally occurs. The majority of oxygen consumption in the outer retina occurs in region 2, the inner segmental layer of the retinal photoreceptors, which contain numerous mitochondria and are highly metabolically active with active transport of ions across membranes. Oxygen that is not consumed in this layer may further diffuse with assumed negligible consumption across region 3, consisting of the external limiting membrane and the outer nuclear layer. The thicknesses of layers 1–3 are taken to be 0.0375, 0.025, and 0.0625 mm, respectively, for a total of 0.125 mm which is half of the retinal thickness.
Fig. 2. Histological appearance of human retina. The cell layers are shown adjacent to the regions of oxygen consumption used in the mathematical model. (Histology microphotograph courtesy of Dr. Olson’s lab).
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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Fig. 3. Schematic of the model of retina depicting the different regions of oxygen consumption and their corresponding depth in millimeters.
The remaining layers of the retina are a complex structure of sources and sinks of oxygen. Metabolic activity is assumed high in the outer and inner plexiform layers. Derived from the central retinal artery, capillaries infiltrate the inner retina, providing the predominant source of oxygen for this region. It is the closure of these capillaries that leads to ischemia and neovascularization associated with diabetic retinopathy. This complex region, extending through the ganglion layer to the inner limiting membrane that borders the vitreous humor is modeled by uniformly distributed sources and sinks of oxygen. The strength of the oxygen supply depends on the blood flow rate per unit tissue volume of the inner retina capillaries that are fed by the ophthalmic artery and this blood flow rate is a key parameter in the current investigation. The thickness of this region of distributed consumption and supply of oxygen is taken to be 0.125 mm, the innermost half of the retinal thickness. The partial pressure of oxygen po ðt; rÞ depends on time t and spatial position r time according to a diffusion equation that contains, in general, a time and space dependent consumption term qo ðt; rÞ and source term so ðt; r Þ. The diffusion equation is: ∂po ðt; rÞ ¼ ∇ Do ðrÞ∇po ðt; rÞ qo ðt; rÞ þ so ðt; rÞ; ∂t
ð1Þ
where Do ðrÞ is the diffusion coefficient of oxygen in the medium. In this problem we assume the diffusion coefficient is constant in the entire region. While different values of diffusion constant might be appropriate in different layers, the gradients are strongest in regions 1 and 2 containing photoreceptors (see Section 3). Thus diffusion has the greatest effect in these regions where a common value of diffusion constant is presumed reasonable. As noted in the introduction, the problem is considered as onedimensional due to the thinness of the retinal layer, with x representing the distance across the retina from the choroid. Therefore ∂po ðt; xÞ ∂2 po ðt; xÞ ¼ Do qo ðt; xÞ þ so ðt; xÞ: ∂t ∂x2
ð2Þ
The consumption term is nonzero only in regions 2 and 4. Following Roos [15,16] we allow a nonlinear saturation of the oxygen consumption using a Michaelis–Menten expression: qo ðt; xÞ ¼
po ðt; xÞ q ; po ðt; xÞ þ K o oMax
ð3Þ
where K o is the Michaelis constant (the partial pressure where the consumption is half of its maximum) and qoMax is the maximum rate of consumption. The maximum rates are assumed different for the two consumption regions (the inner segments of the photoreceptors in the outer retina and the entire inner retina); moreover, the inner segmental region (region 2) has maximum rates that differ between light and dark conditions (the dark rate is nearly double). Both consumption regions are modeled with the same Michaelis constant that has a value small relative to normal homeostatic oxygen partial pressures: in other words, under normal (non-ischemic) conditions, the consumption rate approaches saturation. The actual values chosen are given below. The source term is nonzero only in inner retina (region 4) and is based on Fick’s principle that oxygen transported into the tissue from a capillary source is given by the difference between oxygen
flow into the capillary from the arterial end and oxygen flow out of the capillary at the venous end. Oxygen is present both as dissolved oxygen and as oxygen bound to hemoglobin. The binding of oxygen to hemoglobin is assumed to be saturable according to a Hill-type expression. Thus the overall transport per unit volume of tissue in the inner retina is given by [15,16] so ðt; xÞ ¼
bf Fðp ; p ðt; xÞÞ; 60 b o
ð4Þ
where bf is local blood flow rate in milliliters per gram of tissue in the retina per minute, the factor 60 converts this to a rate per second, and the final term is a nonlinear function of arterial oxygen partial pressure pb and tissue oxygen partial pressure po ðt; xÞ given by " Fðpb ; po ðt; xÞÞ ¼
pb βo po ðt; xÞ þ
! n # pnb βo po ðt; xÞ δ:cHb : n α1 pnb þ ðK hem Þn βo po ðt; xÞ þ ðK hem Þn
ð5Þ Here F has units of partial pressure (mmHg), βo is the ratio of the partial pressures of oxygen in venous blood and retina at steady state which is constant, K hem is the partial pressure of oxygen at 50% hemoglobin saturation with oxygen, n is the Hill exponent, δ is the oxygen carrying capacity of hemoglobin, cHb is the concentration of hemoglobin in blood, and α1 is the solubility of oxygen in blood. As will be seen below, the two factors in the blood supply term (the blood flow and the nonlinear function of partial pressures) interact through the diffusion process in a way that is, for normal physiological conditions, somewhat offsetting so as to create a feedback process that partially stabilizes oxygen supply in response to blood flow changes. This feedback effect is found to diminish as blood flow drops to levels low enough where tissue oxygen is strongly depleted (see below). (Note: Later in this paper we introduce a more explicit feedback process into the model in terms of vasodilation, choosing first to examine here the purely kinetic and diffusive effects by themselves.)
2.2. Parameters The parameters used for the model are taken from Roos [15,16] and are considered to apply to human physiology. The parameters are shown in Table 1, including the initial level of blood flow used by Roos that we progressively decrease to assess chronic impact of capillary degradation in the inner retina. Here are some interpretations of the chosen parameters. Given the thickness of the retina of h¼0.25 mm, a diffusive time scale for transport across regions 1–3, where there is no internal blood 2 supply, is h=2 =D0 ¼ 8 s. Next, the combination δ:cHb =α1 has the value 5750 mmHg, indicating that hemoglobin transport should dominate over dissolved oxygen transport in the blood. Finally, one key assumption, in setting βo ¼ 1 (a constant), is that the venous blood oxygen partial pressure is always close to the tissue blood oxygen partial pressure even as the latter falls off for low blood flow rates.
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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Table 1 Parameter values and definitions. Parameter with value Do ¼ 2 10 5 cm2 =s pb ¼ 80 mmHg βo ¼ 1 K hem ¼ 26 mmHg n ¼ 2:7 δ ¼ 0:0616 mmol=g cHb ¼ 140 g=l a1 =1.5 * 10-3 mM/ mmHg K o ¼ 2 mmHg qoMax ¼ 26 mmHg=s qoMax ¼ 90 and 170 mmHg/s bf (max) ¼ 0.4 ml/ (g min)
Definition Diffusion coefficient of oxygen Arterial oxygen partial pressure The ratio of the partial pressures of oxygen in venous blood and retina at steady state The partial pressure of oxygen at 50% hemoglobin saturation with oxygen The Hill exponent for Hemoglobin oxygen uptake The oxygen carrying capacity of hemoglobin The solubility of oxygen in blood The Michaelis constant for oxygen consumption The maximum rate of consumption in inner retina (region 4) The maximum rate of consumption in region 2 in light and darkness, respectively. For this paper we compare long term effects of decreasing blood flow under lighted conditions, qoMax ¼ 90 mmHg=s The starting value of blood flow in the capillaries of region 2 which is progressively decreased as degradation occurs
2.3. Boundary conditions The partial pressure of oxygen at the outer retinal border (choroidal side) was set at 80 mmHg, equivalent to peripheral arterial blood. At boundaries between the distinct regions of the retina, oxygen partial pressure and its derivative (and hence the transport rate) are assumed continuous. This is done on physical grounds that the tissue structure itself transforms smoothly from region to region and that there are no highly localized sinks or sources of oxygen. At the innermost boundary of the retina, where the retinal tissue contacts the vitreous humor, a zero-flux condition dp=dx ¼ 0 is used. Physiologically it might be more realistic to model the vitreous humor as a spatially extended reservoir capable of storing dissolved oxygen. Transport would be characterized by internal diffusion, negligible internal consumption and zero flux at the far (anterior) side of the vitreous humor. To reach steady state, time-dependent relaxation of the oxygen distribution in the retina would need to run for a longer time until net transport into the reservoir is zero. If the size of the vitreous humor is H ¼20 mm then a diffusive time scale is H 2 =D0 ¼ 2 105 s, or roughly two days. Thus the choice of zero flux boundary condition at the retinal edge is seen as a tool to accelerate the arrival to steady state conditions. Since we are examining the response to retinal capillary blood flow changes over a much longer time scale (years), this artificial acceleration of arrival to steady state for a given rate of blood flow is considered acceptable. 2.4. Numerical solution Eq. (2) is a time dependent partial differential equation. We implemented an explicit finite difference method in MATLAB to solve this problem numerically. The initial partial pressure across the retina was assumed to be equal zero (po ðt ¼ 0; xÞ ¼ 0) subject to the boundary conditions that were mentioned before (po ðt; x ¼ 0Þ ¼ 80 mm Hg and ð∂p=∂xÞðt; x ¼ hÞ ¼ 0). We have run the program until the results reach a steady-state condition. We have used Δt ¼ 0:0002 s and Δx ¼ 0:001 mm as time and spatial step sizes, respectively. Notice that Δt o ðΔx2 =Do Þ has been chosen to insure the numerical solution is stable.
3. Results Fig. 4 shows partial pressure of oxygen for different final times of 0.1 s, 1.0 s, 2.0 s, 4.0 s, 6.0 s, 8.0 s, 10.0 s and 20.0 s. This figure shows the partial pressure reaches steady-state equilibrium around 6 s. Note that this is comparable to the diffusive time scale
Fig. 4. Partial pressure of oxygen in retina: pb ¼ 80 mmHg, bf ¼0.4 ml/(g min), for different final times tf ¼ 0.1 s, 1 s, 2 s, 4 s, 6 s, 8 s, 10 s and 20 s.
for transport across regions 1–3 of the retina. All subsequent results are given for the steady sate condition obtained by running the time-dependent numerical solver for longer than 6 s. The partial pressure of oxygen across the retina for different values of blood flow for the case of lighted conditions is given in Fig. 5. As bf decreases from 0.4 ml/(g min) to 0.3 ml/(g min), the partial pressure at boundary between the retina and vitreous (prv ) drops 30% from 20.32 mmHg to 14.24 mmHg. As blood flow decreases from 0.3 ml/(g min) to 0.2 ml/(g min) and 0.2 ml/(g min) to 0.1 ml/(g min) prv decreases from 14.24 mmHg to 5.16 mmHg (64% drop), and from 5.16 mmHg to 1.13 mmHg (78% drop), respectively. The oxygen consumption in the retina is given in Fig. 6. In this model, there is oxygen consumption only in regions 2 and 4. As we can observe from Fig. 6 for the case of high blood flow the oxygen consumption is almost constant across those regions. It is nearly independent of retinal blood flow in region 2, implying that most of the oxygen consumed in this region is supplied by the choroid. Moreover, the oxygen partial pressure remains well above the Michaelis constant K o ¼ 2 mmHg so that the consumption is almost equal the maximum consumption, (qo ðt; xÞ ffi qoMax ¼ 90 mmHg=s). In region 4, the consumption remains saturated near the maximum level (qo ðt; xÞ ffiqoMax ¼ 26 mmHg=s) for initial steps downward in blood flow. However, as the region 4 oxygen partial pressure falls towards low values near the Michaelis constant of 2 mmHg (refer back to Fig. 5), the consumption term falls more markedly with the
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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Fig. 5. Steady state partial pressure of oxygen versus position across the retina in lighted conditions with choroidal blood pressure fixed at pb ¼ 80 mmHg. Separate curves correspond to the different blood flow rates bf¼ 0.1, 0.2, 0.3 and 0.4 ml/(g min).
Fig. 6. Oxygen consumption in the retina in lighted conditions with choroidal blood pressure fixed at pb ¼ 80 mmHg. Separate curves correspond to the different blood flow rates bf¼ 0.1, 0.2, 0.3 and 0.4 ml/(g min). Increasing sensitivity to blood flow is seen in the inner retina as diminished oxygen partial pressure removes the consumption kinetics from saturated levels.
same downward step in blood flow. This suggests that the retinal tissue becomes more acutely sensitive to blood flow changes when the nonlinear consumption kinetics falls away from the saturated level that is maintained for well-oxygenated tissue. The supply of oxygen from retinal capillaries in region 4 is shown in Fig. 7. As we see, the supply varies only moderately (within 10%) across this region and it decreases as blood flow decreases. Initially the effect of lower blood flow is compensated somewhat by a greater differential between arterial and venous oxygen partial pressures (where the venous partial pressure is assumed to be close to the tissue partial pressure). According to Eq. (4), the source term is a product of blood flow and a nonlinear function of the arterial and tissue oxygen partial pressures. As seen in Fig. 8, the latter factor has a steep increase with decrease in tissue partial pressure near the normal steady value of around 20 mmHg. However, as the tissue partial pressure falls significantly below the parameter K hem ¼ 26 mmHg this compensating effect levels off and the oxygen supply falls in greater jumps. Again, the nonlinearity built into the assumed transport
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Fig. 7. The oxygen transfer rate from blood to the retina in lighted conditions with choroidal blood pressure fixed at pb ¼ 80 mmHg. Separate curves correspond to the different blood flow rates bf¼ 0.1, 0.2, 0.3 and 0.4 ml/(g min).
Fig. 8. Nonlinear oxygen partial pressure function (in mmHg) (Eq. (5)), as used in the oxygen source equation (Eq. (4)). As diminished blood flow results in lower tissue oxygen partial pressure, there is a steep increase in this factor for the normal inner retina partial pressure of 20 mmHg, tending to compensate for the loss of blood flow. However, at low tissue oxygen partial pressures, the Hill kinetics for hemoglobin binding of oxygen levels off and the compensating effect of a greater differential between arterial and tissue oxygen partial pressures diminishes. Thus the supply of oxygen is much more sensitive to blood flow rate under ischemic conditions.
kinetics leads to increased sensitivity to drops in blood flow as the inner retina oxygen falls to low levels. Having considered the effects of several levels of blood flow below normal, we now consider how the oxygen levels in the retina respond over a long-term chronic decrease in blood flow associated with diabetes-induced hyperglycemia. As can be seen in Fig. 5, the four-region model gives a location of smallest oxygen partial pressure at the boundary between the inner retina and the vitreous humor. Using this partial pressure as an indicator for the critical oxygen concentration, we now examine in Fig. 9 show this oxygen partial pressure behaves as a function of blood flow over a continuous range of blood flow values. The pressure (prv ) is shown in Fig. 9 as a function of the blood flow. This pressure is about 20 mmHg for the normal blood flow (bf ¼0.4 ml/(g min)). It decreases to half (10 mmHg) for a blood flow of bf ¼ 0.25 ml/ (g min). This is close to an inflection point in the curve where the slope is steepest and hence the sensitivity to blood flow changes might be considered the greatest. Partial pressure has fallen to 25% (5 mmHg) at about bf ¼ 0.18 ml/(g min), somewhat less than half of normal blood flow. For clinical response, it is helpful to consider how such a decrease in retinal oxygen might develop chronically over time.
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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Fig. 11. Partial pressure at the boundary between retina and vitreous (prv) as function of time (year), assuming 1% and 2% loss per year.
Fig. 9. The partial pressure of oxygen at the boundary between retina and vitreous as a function of the blood flow in the lighted condition and pb ¼ 80 mmHg.
arterioles that supply parallel bundles of capillaries whose conductance is degrading. The arterioles respond to consequent loss in tissue oxygen partial pressure po by dilating in order to maintain the normal healthy pressure pn. The amount of dilation however is limited. We capture this behavior with the following modified expression for the oxygen source term originally specified in Eq. (4): so ðt; xÞ ¼
bf 0 1
Fðpb ; po ðt; xÞÞ; 60 0:51 þα tanhβðp ðt; xÞ=pn Þ 1 þ0:5 ð2=f N Þ 1 o b
ð7Þ
Fig. 10. Blood flow (bf) as function of time (year) assuming 1% and 2% loss per year.
Suppose the initial (normal) blood flow is bf 0 and it decreases by fraction of f b per year, then blood flow after N year (bf ðNÞ) is: N ð6Þ bf ðNÞ ¼ bf 0 f b : The average decline of the retinal blood flow for purposes of this model in diabetic people was estimated to be an average of 2% per year (f b ¼ 0:98). The blood flow decreases to 75%, 50% and 25% after about 14, 34 and 68.5 years, respectively. The blood flow (bf) versus time is given in Fig. 10. The partial pressure at the boundary between retina and vitreous (prv ) as a function of time is shown in Fig. 11. This pressure declines to half (10 mmHg) after about 23 years and drops to 25% (5 mmHg) after about 35 years of disease, at which point the blood flow is less than half its normal value. These time scales correspond to the periods when blindness is found to increase in aging diabetic adults. If the blood flow decrement is reduced to 1% per year, either by better maintenance of blood glucose or by other therapeutic intervention, the time intervals for progressive ischemia are considerably lengthened. Oxygen partial pressure at the retinal boundary with the vitreous humor declines to half (10 mmHg) after about 60 years instead of 23 years and drops to 25% (5 mmHg) after about 75 years instead of 35 years of disease.
4. Vasodilation We now extend the model to include feedback regulation of blood flow via vasodilation in region 4. In this region, we envision
where the transport function F(pb,po) was defined in Eq. (5). The first term in the denominator represents the fraction of series resistance due to each arteriole and the second term represents the fraction of series resistance due to the capillaries fed by the arteriole. The latter is N influenced by a conductance degradation given by f b where N is the number of years and, as before, we examine the value f b ¼ 0:98 (2% loss per year). Under healthy conditions when p¼pn and there has been no degradation (N¼0) we assume the two components of resistance are “impedance matched” to contribute equal fractions 0.5 to the total flow resistance. In healthy conditions, the denominator has value unity and the original source term is recovered. If there is capillary degradation, we have constructed the modified expression so that when there is no vasodilation (α¼0), the model reverts to that used in previous sections. The new element is the feedback response represented by the hyperbolic tangent term, which gives initial linear response to small perturbations but which saturates since the value of the hyperbolic tangent function ranges from 1 to 1. The parameter α represents the saturation level of change in arteriole resistance due to the maximum possible expansion or contraction. O’Halloran et al. [23] demonstrated a 9.7% constriction in arteriole diameter in response to patient hyperoxia (breathing 100 oxygen). This would translate to a 34% reduction in laminar blood flow due to the fourth-power dependence of flow on diameter. This leads us to an upper limit choice α¼0.4. We also examine results for a more conservative value of α ¼0.2. The parameter β represents the sensitivity of the feedback to deviations in pressure from the “set point” pn. The values of response sensitivity β that we explore are 1, 10, and 100. We will show below how the combination αβ is related to the rate of system response to a small perturbation away from the set point pressure pn. We seek to interpret the effect of vasodilation described by this model in region 4 by exploring the change in the way the partial pressure of oxygen decays at the boundary of the vitreous humor, originally depicted in Fig. 11 without vasodilation. Near the vitreous humor, and over much of region 4, the oxygen partial pressure is nearly constant with position (see Fig. 5), so gradients are weak and diffusion might be neglected. We exploit this to simplify the analysis by eliminating the diffusion term from Eq. (2). This reduces
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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the problem to a one-variable nonlinear ordinary differential equation with a fixed point given by the steady state pressure we are detemining at the boundary of the vitreous humor. We find that the computed fixed point for the parameters given in Table 1 is 20.55 mmHg, very close to the value 20.72 mmHg found when diffusion is included, justifying our simplification. We have also checked full results such as selected curves in Fig. 12 with diffusion included and find correspondingly small differences. As noted above, we can interpret the values of sensitivity parameter β by looking at the system response. For small perturbations of oxygen partial pressure p about the fixed point, we linearize the ordinary differential equation obtained by eliminating diffusion term from Eq. (2). The sink term is still given by Eq. (3) and the new source term is given by Eq. (7). Using the parameters of Table 1 and the fixed point value p ¼20.55 mmHg, we obtain (with details omitted)
dð∂pÞ ¼ ð1:25 þ 0:593αβÞð∂pÞ mmHg=s: dt
ð8Þ
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This leads to an exponential relaxation time given by τ ¼ ð1:25 þ 0:593αβ Þ
1
s:
ð9Þ
For α ¼0.4, the choices β¼0, 1, 10, and 100 give τ¼ 0.8 s, 0.675 s, 0.276 s, and 0.040 s, respectively. Note that the time 0.8 s is the rate at which the system relaxes to equilibrium in the absence of vasodilation and is mostly determined by the linearized Hill kinetics of the hemoglobin transport of oxygen. Fig. 13 shows the relaxation to the healthy (N ¼0) equilibrium fixed point 20.55 mmHg of the full nonlinear equation for a small perturbation reducing the initial value of pressure to 20.45 mmHg. We use α¼ 0.4 and β¼0, 1, 10, and 100. The 1/e point for decay of the 0.1 mmHg perturbation is 20.513 mmHg and Fig. 13 shows that the times at which this point is reached agree with the expected relaxation times. Parametrically, the second term in Eq. (9) can be identified as a “vasodilation relaxation time” τvd ¼
120po n 1 : bf 0 Fðpb ; po nÞ αβ
ð10Þ
With arteriole pressure pb ¼80 mmHg, healthy steady state oxygen partial pressure computed to be pon ¼20.55 mmHg, and the parameter values of Table 1, the transport factor F(pb, pon) defined in Eq. (5) takes the value 3650 mmHg. Using nominal blood flow bfo ¼0.4 ml/(g min), this gives τvd ¼
1 s: 0:593αβ
ð11Þ
For α¼0.4, the choices β¼ 1, 10, and 100 give τvd ¼ 4.2 s, 0.42 s, and 0.042 s, respectively. Nagel and Vilser [24] used imaging to measure the effect of changes in intra-ocular pressure on retinal vessel diameter, observing time scales of several seconds for the response. Since the perturbation was indirect (physical pressure applied with a suction cup) and was not locally-induced changes of the oxygenation of the retinal tissue, we infer that a reasonable choice for local response would be of order of a second or less. Further experiments are suggested to narrow the choice. Now we examine the response of the system to progressive N degradation given by fractional loss f b where N is the number of
Fig. 12. (a) Effect of vasodilation modeled by Eq. (7) with saturation parameter α ¼0.2 and sensitivity factor β ¼ 1, 10, and 100. Capillary conductance degradation occurs by a factor of 0.98 per year. (b) Effect of vasodilation modeled by Eq. (7) with saturation parameter α ¼0.4 and sensitivity factor β ¼1, 10, and 100. Capillary conductance degradation occurs by a factor of 0.98 per year.
Fig. 13. Relaxation of a 0.1 mmHg oxygen partial pressure deficit to the healthy (N ¼ 0) fixed point 20.55 mmHg. For α ¼0 there is no vasodilation and response is dominated by hemoglobin transport of more oxygen when tissue partial pressure is reduced. The effects of vasodilation are shown for saturation parameter α ¼0.4 matched with values of the sensitivity factor β ¼ 1, 10, and 100. The plots confirm the relaxation times given by Eq. (9).
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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Fig. 14. Comparisons of the effect of an increase in arterial partial pressure from 80 mmHg to 100 mmHg to the effect of the vasodilation model with saturation parameter α¼ 0.4 and sensitivity factor β ¼10. Capillary conductance degradation occurs by a factor of 0.98 per year.
years and, as before, f b ¼ 0:98 (2% loss per year). We find the fixed point for each value of N by time integration for 20 s from an arbitrary initial condition p ¼19.0 mmHg. The purpose is to find the extent to which vasodilation can prolong healthy tissue oxygen levels. Fig. 13a and b shows the model behavior for values of saturation parameter α ¼0.2 and α¼ 0.4, respectively. The figures show that sustained vasodilation would be capable of maintaining healthy partial pressure for a time largely determined by the saturation parameter α, but eventually is overwhelmed by the continued fall-off in blood supply. Nonetheless, the “half-life” of the oxygen level is extended, for about 3 years in the case of α ¼0.2 and for about 6 years in the case of α¼ 0.4. If the sensitivity factor β is too small (such as β ¼1), saturation at full dilation cannot be achieved and the effect of vasodilation as a feedback mechanism is diminished. Finally, in Fig. 14 we compare this model of vasodilation response to the effect of a possible systemic or induced increase of arterial blood oxygen partial pressure pb from 80 mmHg to 100 mmHg. We show for comparison the vasodilation response for α¼ 0.4 and β¼10. It is clearly seen that the effect of increase in pb is not very strong: a 25% increase in arterial oxygen partial pressure only yields a roughly 2.5% increase in the initial tissue partial pressure and the delay of decay is a year or two. This picture is consistent with our introductory discussion stating that, for normal or elevated arterial oxygen partial pressures, the blood hemoglobin is largely saturated and only small increases in transport can be obtained. Vasodilation would appear to be potentially more effective in maintaining oxygen levels in the inner retina.
5. Conclusion The most striking feature of the model is the relentless loss of capillary perfusion that results from sustained hyperglycemia. Assuming a modest 2% loss of capillary perfusion per year, the blood flow decreases by 25% after 14 years, and is cut in half after 34 years of disease. This correlates with clinical findings in diabetic patients, who typically have a more rapid progression of retinopathy and other diabetic complication with persistently elevated glycosylated hemoglobin levels [25]. The most important intervention at this point is tighter glycemic control. In our model, decreasing the HgA1c by 2 points was assumed to decrease the annual loss of blood flow to 1%,
which results in a 50% loss of tissue oxygen after 60 years instead of 23 years. A novel addition to the model for retina oxygenation is a twoparameter model for vasodilation. The saturation parameter α sets the maximum change in resistance to blood flow due to arteriole dilation. The tissue oxygen sensitivity factor β is like a feedback gain in response to deviation of tissue oxygen pressure from the healthy homeostatic level. The product determines α β the response time. If dilation can achieve 40% reduction in arterial resistance (α¼0.4) and sensitivity β¼10 is chosen upon the assumption that the response time is a fraction of second (0.42 s), then vasodilation can achieve a notable prolongation of healthy oxygenation in the inner retina, extending the time of decay to 50% of initial oxygenation by six years in spite of a 2% per year fall-off in capillary conductance. Some of the limitations of our model include an assumption of a constantly elevated blood sugar, whereas in reality many patients have an extremely variable levels over the course of their disease—our goal was not to model this variability, but to model the average. The vasodilation model has support in the literature for parameter values but further determination of actual response times is needed. Also, we have assumed that vasodilation can be sustained in response to chronic conditions. Also, the model neglects a possible more active role of the vitreous humor in regulating oxygen [26]. Finally, we again recognize that lateral variation of parameters and tissue structure occurs across the retina and that the effects of ischemia can and will occur at different times, in accord with the locally adjusted application of the model presented here. In conclusion, this study demonstrates a mathematic model of chronic retinal ischemia associated with diabetic retinopathy that appears to correspond to time scales over which clinical aspects of the disease become acute. Interventions that diminish annual decrements in retinal capillary blood delivery can greatly increase these time scales, offering hope for lower incidence of blindness during the lifetimes of people suffering from diabetes.
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[17] Q.D. Nguyen, et al., Supplemental oxygen improves diabetic macular edema: a pilot study, Invest. Ophthalmol. Vis. Sci. 45 (2) (2004) 617–624. [18] R.N. Khurana, et al., A novel method to oxygenate intraocular irrigation fluids with an in-line oxygenator, Retina 27 (1) (2007) 83–86. [19] E. Greenbaum, et al., Metabolic prosthesis for oxygenation of ischemic tissue, IEEE Trans. Biomed. Eng. 56 (2) (2009) 528–531. [20] S.J. Cringle, D.Y. Yu, A multi-layer model of retinal oxygen supply and consumption helps explain the muted rise in inner retinal PO(2) during systemic hyperoxia, Comp. Biochem. Physiol. A: Mol. Integr. Physiol. 132 (1) (2002) 61–66. [21] S.G. Bell, An introduction to hemoglobin physiology, Neonatal Netw. 18 (2) (1999) 9–15. [22] W.A. Altemeier, S.E. Sinclair, Hyperoxia in the intensive care unit: why more is not always better, Curr. Opin. Crit. Care 13 (1) (2007) 73–78.
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[23] M O’ Halloran, E. O’Donoghue, C. Dainty, Measurement of the retinal arteriolar response to a hyperoxic provocation in nonsmokers and smokers, using a high-resolution confocal scanning laser ophthalmoscope, J. Biomed. Opt. 19 (7) (2014) (076012–1–14). [24] E. Nagel, W. Vilser, Autoregulative behavior of retinal arteries and veins during changes of perfusion pressure: a clinical study, Graefes Arch. Clin. Exp. Ophthalmol. 242 (1) (2004) 13–17. [25] The effect of intensive treatment of diabetes on the development and progression of long-term complications in insulin-dependent diabetes mellitus. The Diabetes Control and Complications Trial Research Group, N. Engl. J. Med. 329 (14) (1993) 977–986. [26] N.M. Holekamp, The vitreous gel: more than meets the eye, Am. J. Ophthalmol. 149 (1) (2010) 32–36.
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Computers in Biology and Medicine journal homepage: www.elsevier.com/locate/cbm
Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy Jeffrey L. Olson a,b, Masoud Asadi-Zeydabadi c, Randall Tagg c a
University of Colorado Denver, Rocky Mountain Lions Eye Institute, Denver, CO, USA University of Colorado Denver, Department of Ophthalmology, Denver, CO, USA c University of Colorado Denver, Department of Physics, Denver, CO, USA b
art ic l e i nf o
a b s t r a c t
Article history: Received 18 April 2014 Accepted 24 December 2014
This paper uses computer modeling to estimate the progressive decline in oxygenation that occurs in the human diabetic retina after years of slowly progressive ischemic insult. An established model combines diffusion, saturable consumption, and blood capillary sources to determine the oxygen distribution across the retina. Incorporating long-term degradation of blood supply from the retinal capillaries into the model yields insight into the effects of progressive ischemia associated with prolonged hyperglycemia, suggesting time-scales over which therapeutic mitigation could have beneficial effect. A new extension of the model for oxygen distribution introduces a feedback mechanism for vasodilation and its potential to prolong healthy retinal function. & 2015 Published by Elsevier Ltd.
Keywords: Diabetes mellitus Retinopathy Computer model
1. Introduction Diabetic retinopathy is a leading cause of blindness in the industrialized world and results in a progressive loss of the capillary bed in the retina. This loss of blood supply in turn causes a shortage of oxygen critically needed by metabolically active retinal photoreceptors. In the face of this chronic anoxic challenge, retinal cells die and vision is progressively lost. Previous theoretical modeling of the retina has dealt with acute events, such as ophthalmic artery occlusion and retinal detachment. The aim of the present work is to develop a mathematical model of the retina under progressive and chronic ischemic conditions to study various aspects of retinal oxygenation. Diabetes is a rapidly growing problem in the industrialized world – there are 18 million known diabetics in the United States, and 170 million worldwide [1,2]. For millions of diabetics, each day brings a further diminution of their vision. Despite recent technological and pharmacologic advances in medicine, the number of people losing sight continues to increase. In fact, diabetic retinopathy is the leading cause of blindness in adults aged 25–55 years old, and will blind over 25,000 Americans this year alone [3]. Often the vision loss is irreversible. Vision loss carries profound impact for both the individual and society at large, and currently affects more people than ever. It has been noted that the population of those with visual impairment in the United States is expected to double over the next thirty years [4]. Diabetic retinopathy is characterized by progressive closure and loss of native blood vessels, usually on the anatomic level of the
capillary, thought to be secondary to pericyte loss caused by chronic hyperglycemia. Over time, the loss of individual blood vessels results in decreased blood flow to the retinal tissue as a whole. This chronic, relentless loss of the capillary bed and tissue blood supply is fundamentally the same process that occurs in the kidney, heart, and peripheral nerves of the diabetic resulting in renal failure, cardiac disease, and neuropathy. The major difference between these diseases and retinopathy, is that the loss of capillaries can be visualized and photographed through the lens of the eye either non-invasively or minimally invasively via dye injection (Fig. 1). The loss of blood supply that characterizes diabetes leads to the fundamental underlying problem associated with diabetic retinopathy: a chronic shortage of oxygen, along with other nutrients such as glucose. This decreased oxygen supply adversely affects retinal cells, which are some of the most metabolically active cells in the human body, with a concurrent high demand for oxygen [5]. With prolonged ischemia, these cells in the retina progressively lose their ability to function, and the patient suffers a progressive loss of vision [6]. While the current state of treatment using either lasers or injectable medication provides a means of slowing the rate of vision loss, it does not treat the underlying problem, i.e., a lack of tissue oxygen [7,8]. Conventionally, the retina is divided into two vascular zones: the inner vascularized retina and the outer avascular retina. The outer retina derives its oxygen by diffusion from the underlying choroid, which is a high flow network of interconnected channels not under metabolic regulation. The inner retina is nourished by an intrinsic vascular system under metabolic control. The watershed area between
http://dx.doi.org/10.1016/j.compbiomed.2014.12.021 0010-4825/& 2015 Published by Elsevier Ltd.
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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Fig. 1. Human retinal appearance, as photographed during fluorescein angiography. After injection of a fluorescent dye, blood flow can be imaged in the major retinal arterial (arrows) and venous (arrowheads) circulations. Further, capillary perfusion can be assessed. The areas where the capillaries have closed secondary to diabetic retinopathy are non-perfused. These areas of ischemia are treated with laser surgery (asterisks).
the two circulations typically occurs at the junction of the inner and outer retinal layers [9]. The rates of oxygen consumption in various levels of the retina have been determined by experimentation [10–12] and described by mathematical modeling [13,14]. Building upon these previous computer models, other researchers have used this dual domain blood flow concept to predict oxygen levels in different acute disease states such as retinal artery occlusion [15] and retinal detachment [16]. To date, there have been no computer models of chronic or progressive anoxic states in the retina to simulate diseases such as diabetic retinopathy, which may run their course over many decades. The prior models also did not include vasodilation as a possible feedback mechanism for compensating capillary loss in the inner retina. It has been demonstrated that hyperoxia may provide some benefit in ameliorating the progressive of diabetic retinopathy by administering oxygen by nasal cannula [17]. Further, there is the possibility that moderate illumination may prevent progression of diabetic retinopathy by inhibiting the increase in photoreceptor oxygen demand that occurs during the dark cycle [9]. Attempts have been made to restore oxygen supply to the retina by means of oxygenated irrigating fluids [18], intraocular implants [19], and systemic oxygen delivery [17]. In theory, higher levels of oxygen delivered to the lungs results in higher oxygen levels carried by the blood to end organs. However, there are two factors which counteract this effect. First, increasing systemic arterial oxygen levels has differing effects on the two blood supplies of the eye: the choroidal supply increases dramatically with systemic hyperoxia, but the effect on the inner retinal is blunted dramatically secondary to autoregulation of retinal blood flow [20]. The second major obstacle in this treatment paradigm is the upper limit on oxygenation that occurs in the blood once hemoglobin is saturated, as defined by the oxyhemoglobin dissociation curve. For example, at an arterial partial pressure of oxygen of 90 mmHg, hemoglobin is about 97% saturated. Increasing the arterial pressure of oxygen to 100 mmHg increases the saturation of hemoglobin to 98% [21]. In short, there is not much room for physiologic improvement once the hemoglobin is saturated. High levels of inhaled oxygen treatment have some serious drawbacks and side effects as well. The first and most obvious is the need to wear a nasal cannula and carry a continuous supply of oxygen. Second, high oxygen levels have been shown to induce hyperoxic lung damage through receptor mediated and mitochondrial cell death pathways [22]. In order to better frame the problem of maintaining adequate oxygen over many years of progressive decline of retinal capillary
function, a model-based approach is explored. We first describe the mathematical representation of oxygen delivery, transport, and consumption, including numerical procedures used to calculate oxygen distributions from the model. Then we describe results both in terms of spatial distribution of oxygen in the retina and in terms of long-term effects of decreasing blood flow in the retina. The model is one-dimensional, capturing the dominant features of oxygen transport due to the thinness of the retina layer with the assumption that tissue parameters do not vary rapidly in the lateral directions at the 100 μm scale. While significant variation of parameters governing blood supply, diffusion and consumption can occur across the full spatial extent of the retinal layer and some highly localized regions of different structure exist (for example, the fovea centralis), we believe this 1-dimensional model permits primary understanding of long-term effects for a given location once local conditions are defined through the governing parameters. After exploring long-term effects within the established model, we add a new two-parameter extension of the model incorporating vasodilation to explore its potential mitigating effects over long times. Finally, we interpret the results to arrive at some conclusions to help guide future treatment.
2. Methods 2.1. The model Histological and anatomical depictions of the retina are shown in Fig. 2. The total thickness of the retina has been measured histologically and in vivo by optical coherence tomography to be about 0.250 mm. To describe the distribution of oxygen within this tissue, we take as a starting point the model used by Roos [15,16]. In this one-dimensional model, the retinal structure is idealized as four distinct regions of oxygen shown schematically in Fig. 3. The outer retina, which is avascular and receives its oxygen supply from choroidal blood flow, is divided to regions 1–3. The delivery of oxygen in the outer retina depends on diffusion from choroid into the retina. Oxygen transport is modeled as pure diffusion across region 1, which includes Bruch’s membrane, the pigmented epithelium, and the outer segmental layers of the rods and cones where light-transduction principally occurs. The majority of oxygen consumption in the outer retina occurs in region 2, the inner segmental layer of the retinal photoreceptors, which contain numerous mitochondria and are highly metabolically active with active transport of ions across membranes. Oxygen that is not consumed in this layer may further diffuse with assumed negligible consumption across region 3, consisting of the external limiting membrane and the outer nuclear layer. The thicknesses of layers 1–3 are taken to be 0.0375, 0.025, and 0.0625 mm, respectively, for a total of 0.125 mm which is half of the retinal thickness.
Fig. 2. Histological appearance of human retina. The cell layers are shown adjacent to the regions of oxygen consumption used in the mathematical model. (Histology microphotograph courtesy of Dr. Olson’s lab).
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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Fig. 3. Schematic of the model of retina depicting the different regions of oxygen consumption and their corresponding depth in millimeters.
The remaining layers of the retina are a complex structure of sources and sinks of oxygen. Metabolic activity is assumed high in the outer and inner plexiform layers. Derived from the central retinal artery, capillaries infiltrate the inner retina, providing the predominant source of oxygen for this region. It is the closure of these capillaries that leads to ischemia and neovascularization associated with diabetic retinopathy. This complex region, extending through the ganglion layer to the inner limiting membrane that borders the vitreous humor is modeled by uniformly distributed sources and sinks of oxygen. The strength of the oxygen supply depends on the blood flow rate per unit tissue volume of the inner retina capillaries that are fed by the ophthalmic artery and this blood flow rate is a key parameter in the current investigation. The thickness of this region of distributed consumption and supply of oxygen is taken to be 0.125 mm, the innermost half of the retinal thickness. The partial pressure of oxygen po ðt; rÞ depends on time t and spatial position r time according to a diffusion equation that contains, in general, a time and space dependent consumption term qo ðt; rÞ and source term so ðt; r Þ. The diffusion equation is: ∂po ðt; rÞ ¼ ∇ Do ðrÞ∇po ðt; rÞ qo ðt; rÞ þ so ðt; rÞ; ∂t
ð1Þ
where Do ðrÞ is the diffusion coefficient of oxygen in the medium. In this problem we assume the diffusion coefficient is constant in the entire region. While different values of diffusion constant might be appropriate in different layers, the gradients are strongest in regions 1 and 2 containing photoreceptors (see Section 3). Thus diffusion has the greatest effect in these regions where a common value of diffusion constant is presumed reasonable. As noted in the introduction, the problem is considered as onedimensional due to the thinness of the retinal layer, with x representing the distance across the retina from the choroid. Therefore ∂po ðt; xÞ ∂2 po ðt; xÞ ¼ Do qo ðt; xÞ þ so ðt; xÞ: ∂t ∂x2
ð2Þ
The consumption term is nonzero only in regions 2 and 4. Following Roos [15,16] we allow a nonlinear saturation of the oxygen consumption using a Michaelis–Menten expression: qo ðt; xÞ ¼
po ðt; xÞ q ; po ðt; xÞ þ K o oMax
ð3Þ
where K o is the Michaelis constant (the partial pressure where the consumption is half of its maximum) and qoMax is the maximum rate of consumption. The maximum rates are assumed different for the two consumption regions (the inner segments of the photoreceptors in the outer retina and the entire inner retina); moreover, the inner segmental region (region 2) has maximum rates that differ between light and dark conditions (the dark rate is nearly double). Both consumption regions are modeled with the same Michaelis constant that has a value small relative to normal homeostatic oxygen partial pressures: in other words, under normal (non-ischemic) conditions, the consumption rate approaches saturation. The actual values chosen are given below. The source term is nonzero only in inner retina (region 4) and is based on Fick’s principle that oxygen transported into the tissue from a capillary source is given by the difference between oxygen
flow into the capillary from the arterial end and oxygen flow out of the capillary at the venous end. Oxygen is present both as dissolved oxygen and as oxygen bound to hemoglobin. The binding of oxygen to hemoglobin is assumed to be saturable according to a Hill-type expression. Thus the overall transport per unit volume of tissue in the inner retina is given by [15,16] so ðt; xÞ ¼
bf Fðp ; p ðt; xÞÞ; 60 b o
ð4Þ
where bf is local blood flow rate in milliliters per gram of tissue in the retina per minute, the factor 60 converts this to a rate per second, and the final term is a nonlinear function of arterial oxygen partial pressure pb and tissue oxygen partial pressure po ðt; xÞ given by " Fðpb ; po ðt; xÞÞ ¼
pb βo po ðt; xÞ þ
! n # pnb βo po ðt; xÞ δ:cHb : n α1 pnb þ ðK hem Þn βo po ðt; xÞ þ ðK hem Þn
ð5Þ Here F has units of partial pressure (mmHg), βo is the ratio of the partial pressures of oxygen in venous blood and retina at steady state which is constant, K hem is the partial pressure of oxygen at 50% hemoglobin saturation with oxygen, n is the Hill exponent, δ is the oxygen carrying capacity of hemoglobin, cHb is the concentration of hemoglobin in blood, and α1 is the solubility of oxygen in blood. As will be seen below, the two factors in the blood supply term (the blood flow and the nonlinear function of partial pressures) interact through the diffusion process in a way that is, for normal physiological conditions, somewhat offsetting so as to create a feedback process that partially stabilizes oxygen supply in response to blood flow changes. This feedback effect is found to diminish as blood flow drops to levels low enough where tissue oxygen is strongly depleted (see below). (Note: Later in this paper we introduce a more explicit feedback process into the model in terms of vasodilation, choosing first to examine here the purely kinetic and diffusive effects by themselves.)
2.2. Parameters The parameters used for the model are taken from Roos [15,16] and are considered to apply to human physiology. The parameters are shown in Table 1, including the initial level of blood flow used by Roos that we progressively decrease to assess chronic impact of capillary degradation in the inner retina. Here are some interpretations of the chosen parameters. Given the thickness of the retina of h¼0.25 mm, a diffusive time scale for transport across regions 1–3, where there is no internal blood 2 supply, is h=2 =D0 ¼ 8 s. Next, the combination δ:cHb =α1 has the value 5750 mmHg, indicating that hemoglobin transport should dominate over dissolved oxygen transport in the blood. Finally, one key assumption, in setting βo ¼ 1 (a constant), is that the venous blood oxygen partial pressure is always close to the tissue blood oxygen partial pressure even as the latter falls off for low blood flow rates.
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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Table 1 Parameter values and definitions. Parameter with value Do ¼ 2 10 5 cm2 =s pb ¼ 80 mmHg βo ¼ 1 K hem ¼ 26 mmHg n ¼ 2:7 δ ¼ 0:0616 mmol=g cHb ¼ 140 g=l a1 =1.5 * 10-3 mM/ mmHg K o ¼ 2 mmHg qoMax ¼ 26 mmHg=s qoMax ¼ 90 and 170 mmHg/s bf (max) ¼ 0.4 ml/ (g min)
Definition Diffusion coefficient of oxygen Arterial oxygen partial pressure The ratio of the partial pressures of oxygen in venous blood and retina at steady state The partial pressure of oxygen at 50% hemoglobin saturation with oxygen The Hill exponent for Hemoglobin oxygen uptake The oxygen carrying capacity of hemoglobin The solubility of oxygen in blood The Michaelis constant for oxygen consumption The maximum rate of consumption in inner retina (region 4) The maximum rate of consumption in region 2 in light and darkness, respectively. For this paper we compare long term effects of decreasing blood flow under lighted conditions, qoMax ¼ 90 mmHg=s The starting value of blood flow in the capillaries of region 2 which is progressively decreased as degradation occurs
2.3. Boundary conditions The partial pressure of oxygen at the outer retinal border (choroidal side) was set at 80 mmHg, equivalent to peripheral arterial blood. At boundaries between the distinct regions of the retina, oxygen partial pressure and its derivative (and hence the transport rate) are assumed continuous. This is done on physical grounds that the tissue structure itself transforms smoothly from region to region and that there are no highly localized sinks or sources of oxygen. At the innermost boundary of the retina, where the retinal tissue contacts the vitreous humor, a zero-flux condition dp=dx ¼ 0 is used. Physiologically it might be more realistic to model the vitreous humor as a spatially extended reservoir capable of storing dissolved oxygen. Transport would be characterized by internal diffusion, negligible internal consumption and zero flux at the far (anterior) side of the vitreous humor. To reach steady state, time-dependent relaxation of the oxygen distribution in the retina would need to run for a longer time until net transport into the reservoir is zero. If the size of the vitreous humor is H ¼20 mm then a diffusive time scale is H 2 =D0 ¼ 2 105 s, or roughly two days. Thus the choice of zero flux boundary condition at the retinal edge is seen as a tool to accelerate the arrival to steady state conditions. Since we are examining the response to retinal capillary blood flow changes over a much longer time scale (years), this artificial acceleration of arrival to steady state for a given rate of blood flow is considered acceptable. 2.4. Numerical solution Eq. (2) is a time dependent partial differential equation. We implemented an explicit finite difference method in MATLAB to solve this problem numerically. The initial partial pressure across the retina was assumed to be equal zero (po ðt ¼ 0; xÞ ¼ 0) subject to the boundary conditions that were mentioned before (po ðt; x ¼ 0Þ ¼ 80 mm Hg and ð∂p=∂xÞðt; x ¼ hÞ ¼ 0). We have run the program until the results reach a steady-state condition. We have used Δt ¼ 0:0002 s and Δx ¼ 0:001 mm as time and spatial step sizes, respectively. Notice that Δt o ðΔx2 =Do Þ has been chosen to insure the numerical solution is stable.
3. Results Fig. 4 shows partial pressure of oxygen for different final times of 0.1 s, 1.0 s, 2.0 s, 4.0 s, 6.0 s, 8.0 s, 10.0 s and 20.0 s. This figure shows the partial pressure reaches steady-state equilibrium around 6 s. Note that this is comparable to the diffusive time scale
Fig. 4. Partial pressure of oxygen in retina: pb ¼ 80 mmHg, bf ¼0.4 ml/(g min), for different final times tf ¼ 0.1 s, 1 s, 2 s, 4 s, 6 s, 8 s, 10 s and 20 s.
for transport across regions 1–3 of the retina. All subsequent results are given for the steady sate condition obtained by running the time-dependent numerical solver for longer than 6 s. The partial pressure of oxygen across the retina for different values of blood flow for the case of lighted conditions is given in Fig. 5. As bf decreases from 0.4 ml/(g min) to 0.3 ml/(g min), the partial pressure at boundary between the retina and vitreous (prv ) drops 30% from 20.32 mmHg to 14.24 mmHg. As blood flow decreases from 0.3 ml/(g min) to 0.2 ml/(g min) and 0.2 ml/(g min) to 0.1 ml/(g min) prv decreases from 14.24 mmHg to 5.16 mmHg (64% drop), and from 5.16 mmHg to 1.13 mmHg (78% drop), respectively. The oxygen consumption in the retina is given in Fig. 6. In this model, there is oxygen consumption only in regions 2 and 4. As we can observe from Fig. 6 for the case of high blood flow the oxygen consumption is almost constant across those regions. It is nearly independent of retinal blood flow in region 2, implying that most of the oxygen consumed in this region is supplied by the choroid. Moreover, the oxygen partial pressure remains well above the Michaelis constant K o ¼ 2 mmHg so that the consumption is almost equal the maximum consumption, (qo ðt; xÞ ffi qoMax ¼ 90 mmHg=s). In region 4, the consumption remains saturated near the maximum level (qo ðt; xÞ ffiqoMax ¼ 26 mmHg=s) for initial steps downward in blood flow. However, as the region 4 oxygen partial pressure falls towards low values near the Michaelis constant of 2 mmHg (refer back to Fig. 5), the consumption term falls more markedly with the
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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Fig. 5. Steady state partial pressure of oxygen versus position across the retina in lighted conditions with choroidal blood pressure fixed at pb ¼ 80 mmHg. Separate curves correspond to the different blood flow rates bf¼ 0.1, 0.2, 0.3 and 0.4 ml/(g min).
Fig. 6. Oxygen consumption in the retina in lighted conditions with choroidal blood pressure fixed at pb ¼ 80 mmHg. Separate curves correspond to the different blood flow rates bf¼ 0.1, 0.2, 0.3 and 0.4 ml/(g min). Increasing sensitivity to blood flow is seen in the inner retina as diminished oxygen partial pressure removes the consumption kinetics from saturated levels.
same downward step in blood flow. This suggests that the retinal tissue becomes more acutely sensitive to blood flow changes when the nonlinear consumption kinetics falls away from the saturated level that is maintained for well-oxygenated tissue. The supply of oxygen from retinal capillaries in region 4 is shown in Fig. 7. As we see, the supply varies only moderately (within 10%) across this region and it decreases as blood flow decreases. Initially the effect of lower blood flow is compensated somewhat by a greater differential between arterial and venous oxygen partial pressures (where the venous partial pressure is assumed to be close to the tissue partial pressure). According to Eq. (4), the source term is a product of blood flow and a nonlinear function of the arterial and tissue oxygen partial pressures. As seen in Fig. 8, the latter factor has a steep increase with decrease in tissue partial pressure near the normal steady value of around 20 mmHg. However, as the tissue partial pressure falls significantly below the parameter K hem ¼ 26 mmHg this compensating effect levels off and the oxygen supply falls in greater jumps. Again, the nonlinearity built into the assumed transport
5
Fig. 7. The oxygen transfer rate from blood to the retina in lighted conditions with choroidal blood pressure fixed at pb ¼ 80 mmHg. Separate curves correspond to the different blood flow rates bf¼ 0.1, 0.2, 0.3 and 0.4 ml/(g min).
Fig. 8. Nonlinear oxygen partial pressure function (in mmHg) (Eq. (5)), as used in the oxygen source equation (Eq. (4)). As diminished blood flow results in lower tissue oxygen partial pressure, there is a steep increase in this factor for the normal inner retina partial pressure of 20 mmHg, tending to compensate for the loss of blood flow. However, at low tissue oxygen partial pressures, the Hill kinetics for hemoglobin binding of oxygen levels off and the compensating effect of a greater differential between arterial and tissue oxygen partial pressures diminishes. Thus the supply of oxygen is much more sensitive to blood flow rate under ischemic conditions.
kinetics leads to increased sensitivity to drops in blood flow as the inner retina oxygen falls to low levels. Having considered the effects of several levels of blood flow below normal, we now consider how the oxygen levels in the retina respond over a long-term chronic decrease in blood flow associated with diabetes-induced hyperglycemia. As can be seen in Fig. 5, the four-region model gives a location of smallest oxygen partial pressure at the boundary between the inner retina and the vitreous humor. Using this partial pressure as an indicator for the critical oxygen concentration, we now examine in Fig. 9 show this oxygen partial pressure behaves as a function of blood flow over a continuous range of blood flow values. The pressure (prv ) is shown in Fig. 9 as a function of the blood flow. This pressure is about 20 mmHg for the normal blood flow (bf ¼0.4 ml/(g min)). It decreases to half (10 mmHg) for a blood flow of bf ¼ 0.25 ml/ (g min). This is close to an inflection point in the curve where the slope is steepest and hence the sensitivity to blood flow changes might be considered the greatest. Partial pressure has fallen to 25% (5 mmHg) at about bf ¼ 0.18 ml/(g min), somewhat less than half of normal blood flow. For clinical response, it is helpful to consider how such a decrease in retinal oxygen might develop chronically over time.
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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Fig. 11. Partial pressure at the boundary between retina and vitreous (prv) as function of time (year), assuming 1% and 2% loss per year.
Fig. 9. The partial pressure of oxygen at the boundary between retina and vitreous as a function of the blood flow in the lighted condition and pb ¼ 80 mmHg.
arterioles that supply parallel bundles of capillaries whose conductance is degrading. The arterioles respond to consequent loss in tissue oxygen partial pressure po by dilating in order to maintain the normal healthy pressure pn. The amount of dilation however is limited. We capture this behavior with the following modified expression for the oxygen source term originally specified in Eq. (4): so ðt; xÞ ¼
bf 0 1
Fðpb ; po ðt; xÞÞ; 60 0:51 þα tanhβðp ðt; xÞ=pn Þ 1 þ0:5 ð2=f N Þ 1 o b
ð7Þ
Fig. 10. Blood flow (bf) as function of time (year) assuming 1% and 2% loss per year.
Suppose the initial (normal) blood flow is bf 0 and it decreases by fraction of f b per year, then blood flow after N year (bf ðNÞ) is: N ð6Þ bf ðNÞ ¼ bf 0 f b : The average decline of the retinal blood flow for purposes of this model in diabetic people was estimated to be an average of 2% per year (f b ¼ 0:98). The blood flow decreases to 75%, 50% and 25% after about 14, 34 and 68.5 years, respectively. The blood flow (bf) versus time is given in Fig. 10. The partial pressure at the boundary between retina and vitreous (prv ) as a function of time is shown in Fig. 11. This pressure declines to half (10 mmHg) after about 23 years and drops to 25% (5 mmHg) after about 35 years of disease, at which point the blood flow is less than half its normal value. These time scales correspond to the periods when blindness is found to increase in aging diabetic adults. If the blood flow decrement is reduced to 1% per year, either by better maintenance of blood glucose or by other therapeutic intervention, the time intervals for progressive ischemia are considerably lengthened. Oxygen partial pressure at the retinal boundary with the vitreous humor declines to half (10 mmHg) after about 60 years instead of 23 years and drops to 25% (5 mmHg) after about 75 years instead of 35 years of disease.
4. Vasodilation We now extend the model to include feedback regulation of blood flow via vasodilation in region 4. In this region, we envision
where the transport function F(pb,po) was defined in Eq. (5). The first term in the denominator represents the fraction of series resistance due to each arteriole and the second term represents the fraction of series resistance due to the capillaries fed by the arteriole. The latter is N influenced by a conductance degradation given by f b where N is the number of years and, as before, we examine the value f b ¼ 0:98 (2% loss per year). Under healthy conditions when p¼pn and there has been no degradation (N¼0) we assume the two components of resistance are “impedance matched” to contribute equal fractions 0.5 to the total flow resistance. In healthy conditions, the denominator has value unity and the original source term is recovered. If there is capillary degradation, we have constructed the modified expression so that when there is no vasodilation (α¼0), the model reverts to that used in previous sections. The new element is the feedback response represented by the hyperbolic tangent term, which gives initial linear response to small perturbations but which saturates since the value of the hyperbolic tangent function ranges from 1 to 1. The parameter α represents the saturation level of change in arteriole resistance due to the maximum possible expansion or contraction. O’Halloran et al. [23] demonstrated a 9.7% constriction in arteriole diameter in response to patient hyperoxia (breathing 100 oxygen). This would translate to a 34% reduction in laminar blood flow due to the fourth-power dependence of flow on diameter. This leads us to an upper limit choice α¼0.4. We also examine results for a more conservative value of α ¼0.2. The parameter β represents the sensitivity of the feedback to deviations in pressure from the “set point” pn. The values of response sensitivity β that we explore are 1, 10, and 100. We will show below how the combination αβ is related to the rate of system response to a small perturbation away from the set point pressure pn. We seek to interpret the effect of vasodilation described by this model in region 4 by exploring the change in the way the partial pressure of oxygen decays at the boundary of the vitreous humor, originally depicted in Fig. 11 without vasodilation. Near the vitreous humor, and over much of region 4, the oxygen partial pressure is nearly constant with position (see Fig. 5), so gradients are weak and diffusion might be neglected. We exploit this to simplify the analysis by eliminating the diffusion term from Eq. (2). This reduces
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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the problem to a one-variable nonlinear ordinary differential equation with a fixed point given by the steady state pressure we are detemining at the boundary of the vitreous humor. We find that the computed fixed point for the parameters given in Table 1 is 20.55 mmHg, very close to the value 20.72 mmHg found when diffusion is included, justifying our simplification. We have also checked full results such as selected curves in Fig. 12 with diffusion included and find correspondingly small differences. As noted above, we can interpret the values of sensitivity parameter β by looking at the system response. For small perturbations of oxygen partial pressure p about the fixed point, we linearize the ordinary differential equation obtained by eliminating diffusion term from Eq. (2). The sink term is still given by Eq. (3) and the new source term is given by Eq. (7). Using the parameters of Table 1 and the fixed point value p ¼20.55 mmHg, we obtain (with details omitted)
dð∂pÞ ¼ ð1:25 þ 0:593αβÞð∂pÞ mmHg=s: dt
ð8Þ
7
This leads to an exponential relaxation time given by τ ¼ ð1:25 þ 0:593αβ Þ
1
s:
ð9Þ
For α ¼0.4, the choices β¼0, 1, 10, and 100 give τ¼ 0.8 s, 0.675 s, 0.276 s, and 0.040 s, respectively. Note that the time 0.8 s is the rate at which the system relaxes to equilibrium in the absence of vasodilation and is mostly determined by the linearized Hill kinetics of the hemoglobin transport of oxygen. Fig. 13 shows the relaxation to the healthy (N ¼0) equilibrium fixed point 20.55 mmHg of the full nonlinear equation for a small perturbation reducing the initial value of pressure to 20.45 mmHg. We use α¼ 0.4 and β¼0, 1, 10, and 100. The 1/e point for decay of the 0.1 mmHg perturbation is 20.513 mmHg and Fig. 13 shows that the times at which this point is reached agree with the expected relaxation times. Parametrically, the second term in Eq. (9) can be identified as a “vasodilation relaxation time” τvd ¼
120po n 1 : bf 0 Fðpb ; po nÞ αβ
ð10Þ
With arteriole pressure pb ¼80 mmHg, healthy steady state oxygen partial pressure computed to be pon ¼20.55 mmHg, and the parameter values of Table 1, the transport factor F(pb, pon) defined in Eq. (5) takes the value 3650 mmHg. Using nominal blood flow bfo ¼0.4 ml/(g min), this gives τvd ¼
1 s: 0:593αβ
ð11Þ
For α¼0.4, the choices β¼ 1, 10, and 100 give τvd ¼ 4.2 s, 0.42 s, and 0.042 s, respectively. Nagel and Vilser [24] used imaging to measure the effect of changes in intra-ocular pressure on retinal vessel diameter, observing time scales of several seconds for the response. Since the perturbation was indirect (physical pressure applied with a suction cup) and was not locally-induced changes of the oxygenation of the retinal tissue, we infer that a reasonable choice for local response would be of order of a second or less. Further experiments are suggested to narrow the choice. Now we examine the response of the system to progressive N degradation given by fractional loss f b where N is the number of
Fig. 12. (a) Effect of vasodilation modeled by Eq. (7) with saturation parameter α ¼0.2 and sensitivity factor β ¼ 1, 10, and 100. Capillary conductance degradation occurs by a factor of 0.98 per year. (b) Effect of vasodilation modeled by Eq. (7) with saturation parameter α ¼0.4 and sensitivity factor β ¼1, 10, and 100. Capillary conductance degradation occurs by a factor of 0.98 per year.
Fig. 13. Relaxation of a 0.1 mmHg oxygen partial pressure deficit to the healthy (N ¼ 0) fixed point 20.55 mmHg. For α ¼0 there is no vasodilation and response is dominated by hemoglobin transport of more oxygen when tissue partial pressure is reduced. The effects of vasodilation are shown for saturation parameter α ¼0.4 matched with values of the sensitivity factor β ¼ 1, 10, and 100. The plots confirm the relaxation times given by Eq. (9).
Please cite this article as: J.L. Olson, et al., Theoretical estimation of retinal oxygenation in chronic diabetic retinopathy, Comput. Biol. Med. (2015), http://dx.doi.org/10.1016/j.compbiomed.2014.12.021i
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Fig. 14. Comparisons of the effect of an increase in arterial partial pressure from 80 mmHg to 100 mmHg to the effect of the vasodilation model with saturation parameter α¼ 0.4 and sensitivity factor β ¼10. Capillary conductance degradation occurs by a factor of 0.98 per year.
years and, as before, f b ¼ 0:98 (2% loss per year). We find the fixed point for each value of N by time integration for 20 s from an arbitrary initial condition p ¼19.0 mmHg. The purpose is to find the extent to which vasodilation can prolong healthy tissue oxygen levels. Fig. 13a and b shows the model behavior for values of saturation parameter α ¼0.2 and α¼ 0.4, respectively. The figures show that sustained vasodilation would be capable of maintaining healthy partial pressure for a time largely determined by the saturation parameter α, but eventually is overwhelmed by the continued fall-off in blood supply. Nonetheless, the “half-life” of the oxygen level is extended, for about 3 years in the case of α ¼0.2 and for about 6 years in the case of α¼ 0.4. If the sensitivity factor β is too small (such as β ¼1), saturation at full dilation cannot be achieved and the effect of vasodilation as a feedback mechanism is diminished. Finally, in Fig. 14 we compare this model of vasodilation response to the effect of a possible systemic or induced increase of arterial blood oxygen partial pressure pb from 80 mmHg to 100 mmHg. We show for comparison the vasodilation response for α¼ 0.4 and β¼10. It is clearly seen that the effect of increase in pb is not very strong: a 25% increase in arterial oxygen partial pressure only yields a roughly 2.5% increase in the initial tissue partial pressure and the delay of decay is a year or two. This picture is consistent with our introductory discussion stating that, for normal or elevated arterial oxygen partial pressures, the blood hemoglobin is largely saturated and only small increases in transport can be obtained. Vasodilation would appear to be potentially more effective in maintaining oxygen levels in the inner retina.
5. Conclusion The most striking feature of the model is the relentless loss of capillary perfusion that results from sustained hyperglycemia. Assuming a modest 2% loss of capillary perfusion per year, the blood flow decreases by 25% after 14 years, and is cut in half after 34 years of disease. This correlates with clinical findings in diabetic patients, who typically have a more rapid progression of retinopathy and other diabetic complication with persistently elevated glycosylated hemoglobin levels [25]. The most important intervention at this point is tighter glycemic control. In our model, decreasing the HgA1c by 2 points was assumed to decrease the annual loss of blood flow to 1%,
which results in a 50% loss of tissue oxygen after 60 years instead of 23 years. A novel addition to the model for retina oxygenation is a twoparameter model for vasodilation. The saturation parameter α sets the maximum change in resistance to blood flow due to arteriole dilation. The tissue oxygen sensitivity factor β is like a feedback gain in response to deviation of tissue oxygen pressure from the healthy homeostatic level. The product determines α β the response time. If dilation can achieve 40% reduction in arterial resistance (α¼0.4) and sensitivity β¼10 is chosen upon the assumption that the response time is a fraction of second (0.42 s), then vasodilation can achieve a notable prolongation of healthy oxygenation in the inner retina, extending the time of decay to 50% of initial oxygenation by six years in spite of a 2% per year fall-off in capillary conductance. Some of the limitations of our model include an assumption of a constantly elevated blood sugar, whereas in reality many patients have an extremely variable levels over the course of their disease—our goal was not to model this variability, but to model the average. The vasodilation model has support in the literature for parameter values but further determination of actual response times is needed. Also, we have assumed that vasodilation can be sustained in response to chronic conditions. Also, the model neglects a possible more active role of the vitreous humor in regulating oxygen [26]. Finally, we again recognize that lateral variation of parameters and tissue structure occurs across the retina and that the effects of ischemia can and will occur at different times, in accord with the locally adjusted application of the model presented here. In conclusion, this study demonstrates a mathematic model of chronic retinal ischemia associated with diabetic retinopathy that appears to correspond to time scales over which clinical aspects of the disease become acute. Interventions that diminish annual decrements in retinal capillary blood delivery can greatly increase these time scales, offering hope for lower incidence of blindness during the lifetimes of people suffering from diabetes.
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