Retinal Venous Vulnerability in Primary Open Angle Glaucoma: The Combined Effects of Intraocular Pressure and Blood Pressure with Application to the Thessaloniki Eye Study

Abstract

Primary open angle glaucoma (POAG) is a leading cause of irreversible blindness with risk factors including elevated intraocular pressure (IOP), and both high and low blood pressure (BP). This study investigates the joint influence of IOP and BP on retinal hemodynamics, emphasizing venous circulation. A synthetic dataset comprising 2500 eyes with varied IOP [5–45] mmHg, systolic BP (SBP) [90–200] mmHg and diastolic BP (DBP) [40–120] mmHg was created. Mean pressure $(\bar{P})$, mean flow $(\bar{Q})$, and mean resistance $(\bar{R})$, were estimated using a validated mathematical model. The values of these hemodynamic output variables were then analyzed in relation to different values of IOP and mean arterial pressure (MAP; MAP = 1/3 SBP + 2/3 DBP). Clinical data from a population-based Greek study were similarly analyzed. Differences in the simulated hemodynamic output variables and clinical markers between healthy and POAG eyes were then measured. Synthetic dataset analysis revealed that $\bar{R}$ and $\bar{P}$ vary significantly depending on different IOP-MAP combinations. Notably, eyes with low MAP and high IOP demonstrated a drastic increase in $\bar{R}$ in the venules accompanied with a dramatic decrease in $\bar{P}$ in the central retinal vein (CRV). Clinical data indicated that venules in POAG eyes had significantly higher $\bar{R}$ than healthy eyes (p < 0.01), along with decreased $\bar{P}$ in the CRV of POAG eyes compared to healthy eyes (p = 0.01). The study highlights the increased susceptibility to venous collapse in POAG eyes and the importance of considering the venous side of retinal circulation in the combined impact of risk factors in POAG.

1. Introduction

Primary Open Angle Glaucoma (POAG) is a multifactorial progressive optic neuropathy, and in 2024 remains a leading cause of irreversible blindness worldwide. Global prevalence of POAG is estimated to be approximately 3% in those aged 40 to 80, and the disease is expected to afflict over 100 million individuals by 2040 [1]. Patients with glaucoma experience apoptosis of retinal ganglion cells, damage to the optic nerve, thinning of the retinal nerve fiber layer (RNFL), and corresponding visual field (VF) defects. The pathogenesis of POAG is poorly understood in terms of assessing individual risk and the disease is influenced by multiple risk factors including the biomechanical stress and physiological consequences caused by imbalances of intraocular pressure (IOP) and blood pressure (BP).

While IOP is the only currently approved modifiable risk factor, no specific threshold of IOP has been established for diagnostic purposes [2]. Additionally, many patients with POAG do not have elevated IOP, while those with medically lowered IOP often still experience disease progression [3–5]. Biomarkers of both ocular and systemic hemodynamics, including both high and low BP, have previously been identified as potential contributory factors for POAG [6]. While numerous studies have examined the individual correlations of IOP and BP with POAG status, the combined and weighted effects of these two factors and their impact on retinal hemodynamics and POAG disease remain poorly understood [7]. Ocular perfusion pressure, calculated from BP and IOP, represents the passive driving force of circulation within retinal tissues. Thus, understanding the complex interplay between IOP and BP, and their effects on retinal hemodynamics in healthy and POAG eyes, is critical for improved risk modeling and identifying novel approaches for disease management.

The current understanding of combined effects of IOP and BP risk factors is limited by the available clinical data, with substantial variation in IOP and BP values. A major limiting factor is the relatively low prevalence of POAG [8]. Consequently, even in large population-based studies consisting of several thousand eyes, those diagnosed with POAG may only represent a small fraction of the total dataset. Additionally, most clinical analyses focus solely on arterial hemodynamic biomarkers, leaving susceptibility to venous collapse unassessed.

To address these challenges, we introduce an approach that combines both synthetic and real-world datasets. Additionally, we use a validated mathematical model [9] to simulate hemodynamic output variables along the retinal vasculature. Based on the findings on the synthetic dataset, we analyze the clinical dataset using statistical methods. Our analysis focuses particularly on the venous side of the circulation, providing, for the first time, an assessment of susceptibility to venous collapse in patients with POAG within the Greek population.

2. Methods

2.1. Physiology-Based Hemodynamic Modeling

Several physics-based models have been proposed to study ocular hemodynamics, as reviewed in [10]. Here, we use the mathematical model proposed and validated by Guidoboni et al. in [9], which simulates retinal circulation leveraging an electric analogy to fluid flow [11]. This model was selected for three main reasons: (i) it is the simpler model that accounts for the combined effects of BP, IOP, and venous collapsibility; (ii) It allows BP and IOP to be used as individualized inputs; (iii) the results obtained from the model have been substantiated a posteriori by the findings of the Singapore Epidemiology of Eye Disease Study [12], which included nearly 20,000 eyes.

The model consists of the following four nonlinear differential equations describing retinal hemodynamics using the electric analogy to fluid flow:

$$C_1\frac{d(P_1-RL{T}_p)}{dt}=\frac{P_{in}-P_1}{R_{in}+R_{1a}}-\frac{P_1-P_2}{R_{1b}+R_{1c}+R_{1d}+R_{2a}}$$ (1)

$$C_2\frac{d(P_2-IOP)}{dt}=\frac{P_1-P_2}{R_{1b}+R_{1c}+R_{1d}+R_{2a}}-\frac{P_2-P_4}{R_{2b}+R_{3a}+R_{3b}+R_{4a}}$$ (2)

$$C_4\frac{d(P_4-IOP)}{dt}=\frac{P_2-P_4}{R_{2b}+R_{3a}+R_{3b}+R_{4a}}-\frac{P_4-P_5}{R_{4b}+R_{5a}+R_{5b}+R_{5c}}$$ (3)

$$C_5\frac{d(P_5-RL{T}_p)}{dt}=\frac{P_4-P_5}{R_{4b}+R_{5a}+R_{5b}+R_{5c}}-\frac{P_5-P_{\mathit{out}}}{R_{5d}+R_{\mathit{out}}}$$ (4)

Here, $P,C$, and $R$ represent pressures, capacitances, and resistances across the different vascular compartments within the eye. The system results from applying balance equations at each pressure node in Fig. 1, details of which are in the Appendix A.

Fig. 1.

The Retina Model overlayed on the different regions of the retinal vasculature represented by the model

The system is driven by the inlet and outlet pressures $P_{\mathit{in}}$ and $P_{\mathit{out}}$ respectively. $P_{\mathit{in}}$ represents the arterial pressure upstream of the retina and is computed as a function of the SBP and the DBP using the relationship B4 in the Appendix. It is worth noting that SBP and DBP can be measured clinically and, therefore, B4 allows to directly translate clinical data into model inputs. $P_{\mathit{out}}$ represents the venous pressure downstream of the retina. Unlike its arterial counterpart, systemic venous pressure is not typically measured in clinical settings; thus, we maintain it at a fixed value of 14 mmHg as in [9].

The pressures $P_1(t),P_2(t),P_4(t)$, and $P_5(t)$ represent the blood pressure within the CRA, arterioles, venules and CRV respectively. The values of the capacitances ($C_1$, $C_2,C_4,C_5$) and some of the resistances ($R_{1a},R_{1b},R_{2a},R_{2b},R_{3a},R_{3b},R_{5c}$) are set as constants, consistent with what was used in [9]. Other resistances ($R_{1c}(t),R_{1d}(t),R_{4a}(t),R_{4b}(t),R_{5a}(t),R_{5b}(t)$ depend nonlinearly on the specific transmural pressure at each vascular compartment. Specifically, for the arterial resistances we have:

$$R_x\left(t\right)=\frac{k_{r,x}\rho L_x}{A_{ref,x}^2}{\left(1+\frac{P_x\left(t\right)-\mathrm{I}\mathrm{O}\mathrm{P}}{k_{p,x}{k}_{L,x}}\right)}^{-4},\text{with}x=1c,1d$$ (5)

while for the venous resistances, we have the following:

$$R_x\left(t\right)=\left\{\begin{array}{ll}\frac{k_{r,x}\rho L_x}{A_{ref,x}^2}{\left(1+\frac{P_x\left(t\right)-\mathrm{I}\mathrm{O}\mathrm{P}}{k_{p,x}{k}_{L,x}}\right)}^{-4},& P_x\left(t\right)\ge \mathrm{I}\mathrm{O}\mathrm{P}\\ \frac{k_{r,x}\rho L_x}{A_{ref,x}^2}{\left(1+\frac{P_x\left(t\right)-\mathrm{I}\mathrm{O}\mathrm{P}}{k_{p,x}}\right)}^{\frac{4}{3}},& P_x\left(t\right)<\mathrm{I}\mathrm{O}\mathrm{P}\end{array},\right.\text{with}x=4a,4b,5a,5b.$$ (6)

Here we have

$$k_{r,x}=\frac{8\pi {\mu }_x}{\rho }$$ (7)

where $\rho$ and ${\mu }_x$ represents the density and viscosity of blood, respectively. We note that the blood viscosity may be different depending on the vessel segment $x$. In addition, we have that

$$k_{p,x}=\frac{E_x{h}_x^3}{\sqrt{1-v_x^2}}\frac{\pi }{A_{ref,x}^{3/2}},k_{L,x}=\frac{12A_{ref,x}\pi h_x^2}{L_x^3},$$ (8)

and $E_x,h_x$, and $v_x$ representing Young’s modulus, Poisson’s ratio, and thickness of the vessel wall of the segment $x$, respectively. The cross-sectional area of the vessel x is denoted by $A_{\mathit{ref},x}$. A complete list of the parameter values used in this work is reported in Table C1 in the Appendix C. It is worth noting that IOP can be measured clinically and explicitly appears in the functional relationships (5) and (6), describing the nonlinear resistances. Thus, (5) and (6) allow to directly translate clinical data into model inputs.

In this work, we utilize the mean value of hemodynamic output variables, defined as an integral average over the cardiac cycle. Specifically, mean resistances in the CRA, venules, and CRV- denoted as $\overline{R_1},\overline{R_4},\overline{R_5}$, respectively- are computed as integral averages over a cardiac cycle of the sum of resistances within each specific vascular compartment. This yields:

$$\overline{R_1}=\frac{1}{t_n-t_1}{\int }_{t_1}^{t_n}\left[R_{1a}+R_{1b}+R_{1c}\left(t\right)+R_{1d}\left(t\right)\right]dt$$ (9)

$$\overline{R_4}=\frac{1}{t_n-t_1}{\int }_{t_1}^{t_n}\left[R_{4a}(t)+R_{4b}(t)\right]dt$$ (10)

$$\overline{R_5}=\frac{1}{t_n-t_1}{\int }_{t_1}^{t_n}\left[R_{5a}\left(t\right)+R_{5b}\left(t\right)+R_{5c}+R_{5d}\right]dt$$ (11)

Where $t_1$ and $t_n$ represent the time instances determining the beginning and the end of one cardiac cycle, while the subscripts $a,b,c$ and $d$ represent different regions across which the CRA and the CRV lie, as shown in Fig. 1.

Similarly, mean flow rates $\overline{Q_{12}},\overline{Q_{24}},\overline{Q_{45}}$ are computed as

$$\overline{Q_{12}}=\frac{1}{t_n-t_1}{\int }_{t_1}^{t_n}\frac{P_1\left(t\right)-P_2\left(t\right)}{R_{1b}+R_{1c}\left(t\right)+R_{1d}\left(t\right)+R_{2a}}dt$$ (12)

$$\overline{Q_{24}}=\frac{1}{t_n-t_1}{\int }_{t_1}^{t_n}\frac{P_2\left(t\right)-P_4\left(t\right)}{R_{2b}+R_{3a}+R_{3b}+R_{4a}\left(t\right)}dt$$ (13)

$$\overline{Q_{45}}=\frac{1}{t_n-t_1}{\int }_{t_1}^{t_n}\frac{P_4\left(t\right)-P_5\left(t\right)}{R_{4b}\left(t\right)+R_{5a}\left(t\right)+R_{5b}\left(t\right)+R_{5c}}dt$$ (14)

where $\overline{Q_{12}}$ is the mean flow rate between the CRA and the arterioles, $\overline{Q_{24}}$ is the mean flow rate between the arterioles and venules, and $\overline{Q_{45}}$ is the mean flow rate between the venules and CRV. The mean retinal blood flow $\bar{Q}$ is computed as the arithmetic average of these quantities, namely:

$$\bar{Q}=\frac{\left(\overline{Q_{12}}+\overline{Q_{24}}+\overline{Q_{45}}\right)}{3}$$ (15)

Overall, the model takes SBP, DBP, and IOP as inputs and provides simulated values of mean blood flow, vascular pressures, and vascular resistances as outputs along the retinal vasculature. These simulated outputs, which represents properties of retinal hemodynamics, are referred to as hemodynamic output variables throughout the paper. Different segments of the blood vessels represented by the model include the central retinal artery (CRA), central retinal vein (CRV), and the arterioles, capillaries, and venules in between. A summary of the input and hemodynamic output variables throughout the different segments are summarized in Table 1.

Table 1.

Variables used in the study. Input variables are constant in time, while the hemodynamic output variables are time-dependent and averaged over a cardiac cycle

Input Variables
Notation	Variable	Notation	Variable
IOP	Intraocular pressure	MAP	Mean arterial pressure
SBP	Systolic blood pressure	DBP	Diastolic blood pressure
Hemodynamic output variables
Notation	Variable	Notation	Variable
$\overline{P_1}$	Mean blood pressure in the CRA	$\overline{R_1}$	Mean resistance in the CRA
$\overline{P_2}$	Mean blood pressure in the arterioles	$\overline{R_4}$	Mean resistance in the venules
$\overline{P_4}$	Mean blood pressure in the venules	$\overline{R_5}$	Mean resistance in the CRV
$\overline{P_5}$	Mean blood pressure in the CRV	$\bar{Q}$	Mean retinal blood flow

The system of ODEs represented by Eqs. (1–4) above is solved using a numerical solver that implements an implicit method. Given the strong nonlinearity of the system, it is important to choose appropriate initial conditions and error tolerances to obtain accurate solutions. Furthermore, as the synthetic dataset includes IOP ranges that are quite high, it is also crucial to prevent issues with convergence. To address this, we use the solution of the stationary problem as the initial conditions for the time dependent system, and set both the relative and absolute tolerance parameters to a value of 10⁻⁸.

Using this model allows us to estimate the hemodynamic properties of eyes using input variables that are almost universally available across POAG datasets, including the dataset being used in this analysis, discussed in Sect. 2.3. Furthermore, as the properties of the model such as the diameter of the blood vessels and capillary density were based on that of healthy eyes, this would allow us to the analyze the hemodynamic properties of POAG eyes in relation to healthy eyes of different IOP-MAP combinations when used with the synthetic dataset described in Sect. 2.2 below.

2.2. Synthetic Dataset and Physiology-Based Hemodynamic Enhancement

A dataset of simulated eyes was created to fully explore the relationship between IOP and BP. The rationale for using a synthetic dataset alongside a real dataset lies in the numbers and the features. Even in large population-based studies including thousands of eyes, POAG eyes are typically in the low hundreds. For example, the Thessaloniki Eye Study (TES) includes a total of 3136 eyes, but only 88 are diagnosed with POAG (see Sect. 2.3). In the quest to study how IOP and BP levels affect POAG, real datasets may only provide a glimpse of the few specific IOP-BP combinations that characterize the limited number of eyes in that specific dataset. In contrast, a synthetic dataset encompasses a broader range of IOP and BP values than what would typically be found in real datasets; Therefore, the synthetic dataset serves as a master canvas that can help contextualize the findings within specific datasets, such as TES.

A synthetic dataset can also complement real datasets in terms of features. Even in advanced technologies for vascular and hemodynamic imaging, such as Optical Coherence Tomography Angiography (OCTA), the measured features do not encompass all aspects of the circulation. For instance, blood pressure within the ocular blood vessels is not easily accessible in a clinical setting; Moreover, most clinical imaging focuses primarily on the arterial side of the vasculature, leaving the venous side largely unrepresented in clinical datasets. From this perspective, the synthetic dataset can be used to estimate what cannot be measured directly.

A range of values was considered for SBP [90–260] mmHg, DBP [40–150] mmHg and IOP [5–45] mmHg. Next, 50 evenly spaced points were created within these ranges for each of the three input variables. Subsequently, two separate 2500 × 2500 meshgrids were generated: one for IOP-SBP and another for IOP-DBP. Then, using the relationship MAP = 2/3 DBP + 1/3 SBP, an IOP-MAP meshgrid was created as shown in Fig. 2. The ranges for SBP, DBP, and IOP were taken based on existing clinical datasets from POAG studies [13–15]. The values of the simulated hemodynamic output variables in the retinal vasculature for these eyes in the IOP-MAP plane were obtained by passing the IOP, SBP, and DBP for each of the eyes as inputs to the mathematical model discussed above in Sect. 2.1.

Fig. 2.

Synthetic eyes in the simulated IOP-MAP plane, serving as a canvas to analyze POAG-eyes in the real world

The behaviors of the model-estimated hemodynamic output variables were then visualized by generating 3-D surface plots. The IOP-MAP were resampled onto 2-D (50 × 50) arrays, and the hemodynamic output variables were plotted on the Z-axis, as illustrated by figures in Sect. 3.1. The trends of these hemodynamic output variables were later used, alongside results from the statistical analysis of the clinical dataset, to contextualize the behaviors of said variables in POAG eyes.

As the focus of the study was on the role of the CRV and venules, arterial autoregulation was not considered, and the vascular resistance of the retinal arterioles of the eyes in the synthetic dataset was assumed to be constant.

2.3. Clinical Datasets and Physiology-Based Hemodynamic Enhancement

Following the analysis on the hemodynamic output variables of the synthetic dataset, patient data from the Thessaloniki Eye Study (TES) conducted in Thessaloniki, Greece, [13], was utilized for clinical dataset analysis. The TES was a population-based study consisting of 3,136 eyes, among which 88 were diagnosed with POAG. For each participant, IOP, SBP, DBP, and ocular structure and hemodynamics were recorded. Specifically, Heidelberg Retinal Tomography (HRT) was used to measure vertical and horizontal cup-to-disk (CD) ratios, while Heidelberg Retinal Flowmetry (HRF) was used to measure the percentage of avascular space (zero blood flow detection via pixel flow analysis) and the mean flow in the upper temporal (UT) and lower temporal (LT) regions of the optic nerve head (ONH) [16].

Following the steps for enhancing the synthetic dataset discussed in Sect. 2.2, the values of SBP, DBP and IOP in TES were used as eye-specific inputs for the mathematical model of retinal circulation to calculate the associated mean values of vascular pressures, resistances, and flow rates. Comparing the differences in clinically measured markers and model-estimated hemodynamic output variables between healthy and POAG-diagnosed eyes in the TES would allow us to understand both structural and hemodynamic differences between them. The non-parametric Mann-Whitney U test was chosen for this comparison, as not all hemodynamic output variables and clinical markers being compared met the assumptions required by parametric tests. The differences between the model-estimated hemodynamic output variables and clinical markers were then reviewed in relation to the behavior of the hemodynamic output variables in the synthetic eyes, as discussed in Sect. 2.2.

3. Results

3.1. Findings on the Physiology-Enhanced Synthetic Dataset

3.1.1. IOP-MAP and Blood Flow

First, a 3-D plot illustrating the relationship between IOP, MAP, and $\bar{Q}$ was generated (Fig. 3a) as discussed in Sect. 2.2. The plot revealed that overall, lower values of $\bar{Q}$ were estimated for synthetic eyes with lower MAP values and higher IOP. However, the slope of the decrease in $\bar{Q}$ varied depending on specific IOP-MAP combinations. When IOP was relatively low, the decrease of flow with decreasing MAP appeared to be insensitive to the specific IOP level. Conversely, as IOP increased, the flow rapidly decreased to values very close to zero when MAP was sufficiently low. Between these two extremes, a region of rapid flow decrease was observed.

Fig. 3.

a Relationship between $\overline{Q}$, IOP and MAP; b peaks in the magnitude of the gradients of $\overline{Q}$ with respect to IOP and MAP; c separation of regions based on the peaks

To quantify the results observed above, the gradient of $\bar{Q}$ with respect to MAP and IOP was computed, and its magnitude was plotted along with IOP and MAP in Fig. 3b. The plot also included peaks of the magnitudes (indicated as red dots), which were calculated as the local maxima. Using these peaks, curves were first generated in 2-D in the MAP-IOP plane, and were then interpolated into the 3-D space with the MAP, IOP and $\bar{Q}$. The interpolation was performed using MATLAB functions that utilizes Delaunay Triangulation for linear interpolation, after which a curve was fit into the 3-D space. Through this analysis, three regions of interest emerged (Fig. 3c): the region to the left of the blue curve (Region I), the region between the two curves (Region II), and the region to the right of the red curve (Region III). After dividing these regions based on $\bar{Q}$ in the synthetic eyes, we proceeded to investigate how the pressures and resistances differed among them. The results of this exploration are discussed below in Sects. 3.1.2 and 3.1.3. Ultimately, this analysis would enable us to explore and understand hemodynamic differences between healthy and POAG eyes in a population-based dataset, as discussed in Sect. 3.2.

It is to be noted that these curves and the resulting regions are not intended to serve as hard cut-offs for eyes in the real world, but more as a conceptual framework to illustrate the combined effects of MAP and IOP on retinal hemodynamics, as well as to contextualize alterations in hemodynamic properties that may arise in POAG eyes. Following this, other hemodynamic output variables mentioned in Table 1 were examined in relation to the IOP-BP plane through similar 3-D plots, with the hemodynamic output variables plotted on the Z-axis. To assess the influence of $\bar{Q}$ on these variables, specifically, $\bar{P}$ and $\bar{R}$ along the retinal vasculature- the two curves obtained based on analysis of $\bar{Q}$ above were projected onto the corresponding 3D-plots.

3.1.2. IOP-MAP and Pressures

The projected curves on the plots of the mean pressure exerted on the CRA $\left(\overline{P_1}\right)$, arterioles $\left(\overline{P_2}\right)$, venules $\left(\overline{P_4}\right)$ and CRV $\left(\overline{P_5}\right)$ are shown in Fig. 4a–d respectively.

Fig. 4.

Separation of regions in the 3-D plot of mean pressures ($\overline{P}$), MAP and IOP throughout the a CRA, b arterioles, c venules, d and CRV based on peaks obtained from $\overline{Q}$

The level of pressure varied based on the region: the gradual decrease of $\bar{Q}$ observed from Region I to Region II was accompanied by gradual increase of mean pressure in the vascular compartments upstream of the CRV, namely in the CRA through the venules (Fig. 4a–d), and a gradual decrease of $\overline{P_5}$ in the CRV (Fig. 4c). This behavior of $\overline{P_5}$ continued throughout Region II.

Upon reaching Region III, the decrease in $\overline{P_5}$ became much more pronounced. $\overline{P_1}$, $\overline{P_2}$, and $\overline{P_4}$ increased significantly across this region, while $\overline{P_5}$ drastically decreased to values very close to zero. When considered alongside the substantial increase of the resistance in the venules $\left(\overline{R_4}\right)$ discussed in Sect. 3.1.3, the decrease in $\overline{P_5}$ could suggest a collapse of the CRV of eyes in Region III. This is because reduced blood flow due to a collapse would cause a decrease in the pressure downstream and an increase in pressure upstream of the constriction, as blood flows from the CRA to the CRV.

3.1.3. IOP-MAP and Resistances

The curves were then projected on the plots of the mean resistances in the CRA (Fig. 5a), venules (Fig. 5b) and CRV (Fig. 5c).

Fig. 5.

Separation of regions in the 3-D plot of $\overline{R}$, MAP and IOP based on peaks obtained from $\overline{Q}$ in the CRA (a), CRV (b), and the venules (c)

Using the same colors for the three regions as defined in the previous sections, we examined the behavior of the mean resistances. Expected behavior of resistances in different parts of the vasculature were hypothesized based on that of the pressures observed above in Sect. 3.1.2 and the properties of fluid flow across a collapsible hollow tube: increasing the resistance (restriction) in an internal segment of a tube with fluid flowing through it leads to an increase of pressure upstream of the restriction, a decrease of pressure downstream of the restriction, and an overall decrease in flow rate. Consequently, eyes in Region III were anticipated to exhibit particularly high $\overline{R_4}$ as referred from the behavior of $\overline{P_5}$, as discussed in Sect. 3.1.2. This was found to be true, as illustrated in Fig. 5c, which features a zoomed-in area, showing that $\overline{P_4}$ in Region III was several orders of magnitude higher than the rest other regions. This significant increase in mean resistance in the venules also corresponded with a decrease in $\overline{Q}$ for synthetic eyes in Region III, as discussed in Sect. 3.1.1.

3.2. Findings on the Physiology-Enhanced Population-Based Dataset

Using the findings regarding the susceptibility of venous collapse in the synthetic dataset as motivation, eyes in TES were analyzed to understand the combined effects of MAP and IOP in the context of POAG. The eyes in TES were categorized based on whether they were healthy or diagnosed with POAG and then plotted over the canvas of synthetic eyes (Fig. 6). Following this, a Mann–Whitney U test was performed on both clinical markers measured in the TES, and the model-estimated hemodynamic output variables to see if they are significantly different between the two groups. The clinically measured markers included those representing structural damage to the ONH (Horizontal and Vertical CD ratios) as well as markers that represent flow (zero flow pixels and mean flow in the UT and LT of the ONH. The results of the test were as reported in Table 2.

Fig. 6.

Position of the eyes in the TES in relation to the synthetic eyes in the simulated IOP-MAP plane

Table 2.

Differences in model-estimated hemodynamic output variables and clinically-measured markers between Healthy eyes and POAG eyes from TES

Variable	Medians Clinical markers
	Healthy eyes	POAG eyes	p-value
IOP (mmHg)	14.5	18	<0.01
MAP (mmHg)	103.7	104.7	.32
Horizontal CD ratio	.43	.67	<0.01
Vertical CD ratio	.37	.72	<0.01
Zero flow pixels -UT	18.8	20.6	.09
Zero flow pixels -LT	17.9	19.2	.06
Mean flow - UT	324.9	350.2	<0.01
Mean flow - LT	325.7	358.5	<0.01
	Hemodynamic output variables
	Healthy eyes	POAG eyes	p-value
$\bar{Q}$ (cm3 s−1)	0.0009	0.0008	0.02
${\bar{P}}_1$ (mmHg)	50	50.86	.30
${\bar{P}}_2$ (mmHg)	40.2	41	0.06
${\bar{P}}_4$ (mmHg)	23.5	24.1	<0.01
${\bar{P}}_4$ (mmHg)	20.1	19.9	.01
${\bar{R}}_1$ (mmHg s cm−3)	7420	7429	<0.01
${\bar{R}}_4$ (mmHg s cm−3)	4283	4890	<0.01
${\bar{R}}_5$ (mmHg s cm−3)	2232	2255	<0.01

Values that are significantly different are in bold

Clinical markers that exhibited a statistically significant difference in the medians included the mean flow in the upper and lower temporal regions. This was accompanied by the statistically significant difference in the simulated $\overline{Q}$ between the two groups of eyes. Additionally, the differences in CD ratios, mean pressures in the venules and the CRV, and the mean resistances in the CRA, CRV and venules were also found to be statistically significant.

The MAP between the two groups of eyes were not significantly different. However, the POAG eyes exhibited a higher IOP compared to healthy eyes, consistent with previous findings [17, 18]. Additionally, higher CD ratios were observed in the POAG eyes, as expected. Furthermore, POAG eyes had higher mean flows in both the UT and LT regions of the ONH. A possible explanation for this could be a reduced number of capillaries in the POAG eyes for blood to flow through, resulting in a higher mean flow along the remaining capillaries via shunting. Our results provide indication of this as seen in the higher amount of retinal avascular space, or capillary loss, in eyes with POAG. However, the differences in the avascular space biomarkers were not found to be statistically significant by the Mann–Whitney U test.

Among the model simulated values, it was observed that POAG eyes exhibited higher $\overline{R_4}$ and $\overline{R_5}$. Additionally, they displayed lower $\overline{P_5}$. Interestingly, the observed behaviors of these hemodynamic output variables were similar to that displayed by the synthetic eyes transitioning from Region I to Region II and III in Sect. 3.1. This similarity suggests that POAG eyes could be susceptible to venous collapse akin to the synthetic eyes in Regions II and III. While the behaviors of the POAG eyes and the synthetic eyes susceptible to venous collapse were similar, the values of the model-simulated hemodynamic output variables are evidently different. Several factors could account for this disparity, including the design of the mathematical model itself. Notably, the model does not account for variability among patients as it assumes identical ocular dimensions for all eyes. Furthermore, it does not incorporate vascular regulation. Additionally, given the POAG is a multifactorial disease, numerous other risk factors associated with POAG are not considered in this study which could also potentially be contributing to these observations [19–22].

4. Conclusions and Perspectives

In this study, we first investigated the retinal venous circulation in synthetic eyes by employing a hybrid clinical and mathematical modeling approach. Our analysis indicated that eyes with specific combinations of IOP and MAP increase susceptibility to venous collapse, thereby elevating risk for POAG. We then translated these mathematical modeling results to clinical data, using a population-based dataset from the TES, identifying specific similarities between eyes diagnosed with POAG and the synthetic eyes at risk of venous collapse. These findings underscore the potential importance of assessing venous circulation in eyes at risk for POAG. These observations are consistent with the findings of the Singapore Epidemiology of Eye Diseases study, which included nearly 20,000 eyes, and found that eyes exhibiting low blood pressure and high IOP were more likely to have POAG [23]. However, it is important to note that the mathematical model used in the study did not account for variability in vascular regulatory ability among patients, nor their specific vascular structure. The heterogeneity of available data on BP, IOP, and POAG also presents a challenge. Future investigations employing more sophisticated theoretical models could help address these limitations. Examples include models that incorporate regulatory mechanisms as proposed by Arciero et al [24] or those that integrate patient-specific OCTA vascular markers as explored by Chiaravalli et al [25]. Ultimately, using modeling to account for both the arterial and venous side of the ocular circulation, alongside IOP and BP, may better inform clinicians in assessing individual risk for POAG.

Appendix A. Functional Relationships Between P, C, Q, and R

The relationships between P, C, Q, and R, along with Kirchoff’s Current Law (KCL) applied to each pressure node in Fig. 1, to obtain (1) – (4) are given below:

KCL states that the algebraic sum of all currents in a node is equal to 0. For the Retinal Model, this means that the current flowing in ($Q_{\mathit{in}}$) is equal to the current exiting the same node ($Q_{\mathit{out}}$):

$$Q_{\mathit{in}}=Q_{\mathit{out}}$$ (A1)

Ohm’s Law for fluid flow, assuming the fluid incompressible, Newtonian is given by:

$$\mathrm{\Delta }P=QR$$ (A2)

where $\mathrm{\Delta }P$ is the pressure difference between two points in the system; Q is the flow rate of the fluid; R is the hydraulic resistance.

Finally, the flow through a capacitor is expressed as:

$$C\frac{d\mathrm{\Delta }P}{dt}=Q$$ (A3)

where C represents the compliance of the vessel fluid is flowing through.

Appendix B. Calculation of Inlet Pressure ($P_{in}$)

The inlet Pressure $P_{\mathit{in}}(t)$ is calculated based on the following:

$$P_{in}(t)=\left\{\begin{array}{ll}0.65\cdot \mathrm{S}\mathrm{B}\mathrm{P}-0.475\cdot \mathrm{D}\mathrm{B}\mathrm{P}& \\ \cdot \mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{2\pi (Tm)}{4\cdot 0.082\cdot (Thr)}+\frac{2\pi \cdot 0.082\cdot (Thr)}{0.328\cdot (Thr)}\right),& (Tm)\le 0.082\cdot (Thr)\\ 0.65\cdot \mathrm{S}\mathrm{B}\mathrm{P}+0.9\cdot \mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{2\pi (Tm)}{0.03\cdot (Thr)}-\frac{2\pi \cdot 0.082\cdot (Thr)}{0.03\cdot (Thr)}\right),& 0.082\cdot (Thr)<(Tm)\le 0.112\cdot (Thr)\\ 0.65\cdot \mathrm{S}\mathrm{B}\mathrm{P}+0.118\cdot \mathrm{S}\mathrm{B}\mathrm{P}& \\ \cdot \mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{2\pi (Tm)}{0.572\cdot (Thr)}-\frac{2\pi \cdot 0.112\cdot (Thr)}{0.572\cdot (Thr)}\right),& 0.112\cdot (Thr)<(Tm)\le 0.398\cdot (Thr)\\ -\frac{0.13\cdot SBP\cdot (Tm)}{0.034\cdot (Thr)}+0.65\cdot \mathrm{S}\mathrm{B}\mathrm{P}+\frac{0.13\cdot SBP\cdot 0.398\cdot (Thr)}{0.034\cdot (Thr)},& 0.398\cdot (Thr)<(Tm)\le 0.432\cdot (Thr)\\ 0.52\cdot \mathrm{S}\mathrm{B}\mathrm{P}-0.8\cdot \mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{2\pi (Tm)}{0.05\cdot (Thr)}+\frac{2\pi \cdot 0.322\cdot (Thr)}{0.05\cdot (Thr)}\right),& 0.432\cdot \frac{60}{\mathrm{H}\mathrm{R}}<(Tm)\le 0.482\cdot \frac{60}{\mathrm{H}\mathrm{R}}\\ 0.52\cdot \mathrm{S}\mathrm{B}\mathrm{P}+(0.52\cdot \mathrm{S}\mathrm{B}\mathrm{P}-0.5\cdot \mathrm{D}\mathrm{B}\mathrm{P})& \\ \cdot \mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{2\pi (Tm)}{2.072\cdot (Thr)}+\frac{2\pi \cdot 0.554\cdot (Thr)}{2.072\cdot (Thr)}\right),& \text{otherwise}\end{array}\right.$$ (B4)

where HR represents the heart rate of the patient, which was taken to be 80 beats per minute for this study; T hr represents the time interval of one heart beat, calculated as $Thr=\frac{60}{\mathrm{H}\mathrm{R}}$; T m represents the phase within each interval, calculated as $Tm=t\mathit{mod}\mathit{Thr}$.

Appendix C. Parameters in the Model

See Table 3.

Table 3.

Parameters in the model and their respective values

Parameter	Value
Constant Capacitances
C 1	7.22 × 10−7 ml/mmHg
C 2	7.53 × 10−7 ml/mmHg
C 4	1.67 × 10−5 ml/mmHg
C 5	1.07 × 10−5 ml/mmHg
Constant Resistances
$R_{1a}$	4.30 × 103 mmHg s/ml
$R_{1b}$	4.30 × 103 mmHg s/ml
$R_{2a}$	6 × 103 mmHg s/ml
$R_{2b}$	6 × 103 mmHg s/ml
$R_{3a}$	5.68 × 103 mmHg s/ml
$R_{3b}$	5.68 × 103 mmHg s/ml
$R_{5c}$	1.35 × 103 mmHg s/ml
$R_{5d}$	1.35 × 103 mmHg s/ml
Blood Viscosity
${\mu }_{1c},{\mu }_{1d}$	3 × 10−3 × 0.00750062 mmHg s
${\mu }_{4a},{\mu }_{4b},{\mu }_{5a},{\mu }_{5b}$	3.24 × 10−3 × 0.00750062 mmHg s
Poisson’s Ratio
$h_{1c},h_{1d},h_{4a},h_{4b},h_{5a},h_{5b}$	0.49
Vessel wall thickness
${\nu }_{1c},{\nu }_{1d}$	39.7239 × 10−4 cm
${\nu }_{4a},{\nu }_{4b}$	10.7 × 10−4/20 cm
${\nu }_{5a},{\nu }_{5b}$	10.7 × 10−4 cm
Cross-sectional Area
$A_{\mathit{ref},1a},A_{\mathit{ref},1b}$	$\pi$(87.5 × 10−4)2 cm2
$A_{\mathit{ref},4a},A_{\mathit{ref},4b}$	$\pi$(0.007738342862) cm2
$A_{\mathit{ref},5a},A_{\mathit{ref},5b}$	$\pi$(119 × 10−4)2 cm2

Data Availability

The synthetic dataset generated during the current study are available from the corresponding author on reasonable request. Restrictions apply to the availability of the population-based dataset, a de-identified version of which was used for the current study, and are not publicly available.