Mathematical Modeling of Human Eye the Eye: a Window on the Body

Mathematical modeling of human eye The eye: a window on the body

Diseases of the Eye:

Glaucoma Retinopathies Age-related Macular ? Degeneration (AMD)

Diseases of the Body:

Diabetes Need of quantitative methods to Hypertension detect and grade vascular abnormalities in the Neurodegenerative eyes and identify underlying pathogenic Disorders (NDD) mechanisms Modeling of Ocular Blood Flow

Math PoliMi Math IUPUI Glick Eye Institute Riccardo Sacco Giovanna Guidoboni IU Medicine Francesca Malgaroli Lucia Carichino Simone Cassani Alon Harris Brent Siesky

Math WSU Sergey Lapin Tyler Campbell Motivation

• Alterations in retinal hemodynamics are associated with – Ocular diseases (e.g. glaucoma, age-related macula degeneration - AMD) – And more (e.g. hypertension, diabetes, multiple sclerosis, Alzheimer, Parkinson) • The Retinal circulation can be assessed non-invasively – Visualization (e.g. fundus camera) – Hemodynamic measurements in macro- and microcirculation

Diabetic Normal AMD Retinopathy

Fundus Camera http://www.medicine.uiowa.edu/eye/Ocular-Fundus-Photograhy/ http://en.wikipedia.org/wiki/Macular_degeneration The Human Eye: Physiology of the Retina

• Central Retinal Artery (CRA) Delivers blood to the retina in the form of capillaries which are one cell thick blood vessels. • Central Retinal Vein (CRV) Retrieves used blood and waste back to the heart for more oxygen. • Retinal Arterioles Small branches of the artery that distribute blood throughout the retina. • Retinal Venules Small branches of the vein that collect blood throughout the retina. Ocular Blood Flow

Is driven by: Is impeded by: Difference in Arterial and Intraocular pressure (IOP) Venous Blood pressure Cerebrospinal Fluid pressure (CSFp) Intracranial pressure (ICP) Intraocular Is modulated by: choroid Pressure ret ina Int raocular Vascular regulation Vascular Pressure Regulation Opht halmic Opht halmic Vein Vein

lamina Cerebrospinal cribrosa Cent ral Cent ral Ret inal Ret inal Intracranial Fluid Pressure A rt ery Vein

Nasal Temporal Pressure Post erior Post erior Ciliary Ciliary Art ery Art ery Arterial Blood from Int ernal Opht halmic A rt ery t o ant erior part of eye, Carot id face and nose Pressure A rt ery

Cavernous Venous Blood Sinus t o Int ernal Pressure Jugular Vein

1 Intraocular Pressure and Auto Regulation

• Intraocular pressure (IOP) is the overall pressure within the eye. Intraocular choroid ret ina PressureInt raocular • During each cardiac cycle, the Pressure

velocity of the blood-flow Opht halmic Opht halmic within the retina is changed Vein Vein dramatically. lamina cribrosa Cent ral Cent ral Ret inal Ret inal • Because of this change, the eye A rt ery Vein

has built in autoregulation Nasal Temporal Post erior Post erior Ciliary Ciliary mechanisms that attempt to Art ery Art ery from Int ernal Opht halmic A rt ery t o ant erior maintain constant blood part of eye, Carot id face and nose pressure within the retina. A rt ery

Cavernous • These mechanisms can fail and Sinus t o Int ernal without proper autoregulation, Jugular Vein some eye diseases can develop.

1 Diseases related to increased IOP

• Glaucoma is a term used to describe a group of diseases that affect the optic nerve. • This occurs in patients with excess IOP caused by poor drainage of the Aqueous Humor fluid. • Performing a Trabeculectomy can alleviate some of this pressure but the damage to the optic nerve is not reversible. • Macular Degeneration occurs when the central portion of the retina deteriorates and causes vision loss in more than 10 million Americans. • Both Glaucoma and Macular Degeneration are incurable diseases and there no reliable methods to predict their development. Why modeling?

• Interpretation of clinical data is challenging! Understand physiology in health and disease • Pressurized ambient (intraocular pressure - IOP) • Fluid-structure interactions • Complex vascular system • Sub-systems • Flow regulation • Traditionally: IOP • Animal studies • Clinical or population-based studies • Our Approach CSF • Mathematical models • Clinical data Why modeling?

• In a disease state, some of the vascular regulation mechanisms might be impaired, compromising the oxygenation in the retina. • There is inconsistency in the scientific literature regarding the vascular response to changes in oxygen demand. • Inconsistent clinical observations are due to the numerous factors, including arterial blood pressure and vascular regulation, that influence the relationship between IOP and ocular hemodynamics. • Mathematical modeling can be used to investigate the complex relationship among these factors and to interpret the outcomes of clinical studies. Mathematical Model

• The retinal circulation is described using the analogy between the flow of a fluid in a hydraulic network and the flow of current in an electric circuit • The vasculature supplying the retina is divided into five main compartments the CRA, arterioles, capillaries, venules, and the CRV. • Using the analogy between hydraulic and electrical circuits, blood flow is modeled as current flowing through a network of resistors (R), representing the resistance to flow offered by blood vessels, and capacitors (C), representing the ability of blood vessels to deform and store blood volume. Mathematical Model

IOP Mathematical Model

• Intraocular segments are exposed to the IOP. • Retrobulbar segments are exposed to the retro laminar tissue pressure. • Translaminar segments are exposed to an external pressure based on stress within the lamina cribosa. • Diameters of the venules vary passively with IOP whereas the arterioles are assumed to be affected by the blood pressure. Mathematical Model

Flow Q through a resistor is directly proportional to the pressure drop P across the resistor.

Flow Q through a capacitor is directly proportional to the time derivative of the product between the pressure drop across the capacitor and the capacitance.

Kirchoff’s law applied to the retinal vascular network volume change = flow in - flow out Mathematical Model

Obtain system of ordinary differential equations for the nodal pressures 푃1, 푃2, 푃4, 푃5:

The inlet and outlet pressures 푃푖푛and 푃표푢푡 vary with time along a cardiac cycle and, consequently, the calculated pressures are time dependent. Control state for the system - conditions of a healthy eye

• Control state for flow - Poiseuille’s law applied to the CRA • The control states of arteriolar, capillary, and venular resistances use the dichotomous network (DN) model for the retinal microcirculation • Input pressure is two-thirds of the MAP control pressures at all the other nodes of the network are computed using Ohm’s law and the control values of the resistances • The time profile of input pressure and output pressure at the control state are determined through an inverse problem based on color Doppler imaging measurements of blood velocity in the CRA and CRV Passive variable resistances

Start with Navier–Stokes equations in a straight cylinder. Passive variable resistances

Assume:

• Body forces and mass sources are absent • 푝 is constant on each Σ • 푢푧 = 푢 푧 푓 Σ , where 푢 푧 is average axial velocity and 푓 Σ is appropriate shape function • Axial motion is predominant

Obtain reduced equations:

Where 퐾푟 depends on 푓(Σ), 푄 푧 is volumetric flow and 퐴 푧 is cross-sectional area Variable passive resistances

• Arterial walls are thicker than venous walls: – Arteries described as compressible tubes – Veins described as collapsible tubes – Sterling resistor

The cross-section changes as transmural pressure difference 2 푟푟푒푓 decreases. 푘퐿 = 12 ℎ Active variable resistances

The resistances for arterioles are modeled through phenomenological description of blood flow autoregulation:

• Without autoregulation the resistances kept constant equal to their control value. • With autoregulation the resistance are computed using:

푐퐿 + 푐푈exp(퐾 푄푛표퐴푅 − 푄 − 푐 ) 푅2푎 = R2b = 1 + exp(퐾 푄푛표퐴푅 − 푄 − 푐 ) Mathematical Model

CRA

푐퐿 + 푐푈exp(퐾 푄푛표퐴푅 − 푄 − 푐 ) arterioles 푅2푎 = R2b = 1 + exp(퐾 푄푛표퐴푅 − 푄 − 푐 ) venules

CRV Results

Comparison of model predicted values with measured data. Results

Model predicted values of total retinal blood flow; peak systolic velocity in the CRA; end diastolic velocity in the CRA; and resistivity index ((PSV- EDV)/PSV) in the CRA as IOP varies between 15 and 45 mmHg for theoretical patients with low, normal or high blood pressure (LBP-, NBP-, HBP-) and functional or absent blood flow autoregulation. Computer-aided identification of novel ophthalmic artery waveform parameters in healthy subjects and glaucoma patients.

L. Carichino, G. Guidoboni, A.C. Verticchio Vercellin, G. Milano, C.A. Cutolo, C. Tinelli, A. De Silvestri, S. Lapin, J.C. Gross, B.A. Siesky, A. Harris. CDI and waveform parameters

• Significant blood velocity derangements in the OA, CRA, and PCAs are associated with diabetic retinopathy and glaucoma • CDI is a consolidated noninvasive technique to measure blood velocity profile in some of the major ocular vessels • Typical waveform parameters utilized in ophthalmology are peak systolic velocity (PSV), end diastolic velocity (EDV) and resistive index (RI). • CDI is commonly used in the fields of radiology, cardiology, and obstetrics, and various waveform parameters have been proposed in the scientific literature. • Recently, waveform parameters commonly used in renal and hepatic arteries have been used to characterize OA velocity waveform in glaucoma patients.

We propose a computer-aided manipulation process of OA-CDI images that enables the extraction of a novel set of waveform parameters that might help better characterize the disease status in glaucoma. Baseline characteristics of the study group

CDI images: • Pavia: 50 images acquired by 4 different operators on 9 healthy individuals (Siemens Antares Stellar Plus™, probe VFX 9-4 MHz vascular linear array) • Indianapolis: 38 glaucoma patients (Philips HDI 5000 SonoCT Ultrasound System, 7.5 MHz linear probe)

The PSV, EDV and RI raw data are obtained directly from the ultrasound machine as an average over at least three cardiac cycles. Computer-aided image manipulation process

Waveform parameters: • peak systolic velocity (PSV) • dicrotic notch velocity (DNV) • end diastolic velocity (EDV) • resistive index RI = (PSV-EDV)/PSV • period of a cardiac cycle (T) • first systolic ascending time (PSVtime) • difference between PSV time and DNV time (Dt) • subendocardial viability ratio between the diastolic time interval (DTI) and the systolic time interval (STI) • area under the wave (A) • area ratio f = Aw/Abox = Aw/(PSV Dt) • normalized distance between ascending and descending limb of the wave at two thirds of the difference between PSV and EDV (DAD/T) Computer-aided image manipulation process

• OAG patients had a statistically significant higher DAD/T than healthy subjects (p<0.001) • female OAG patients had a statistically significant higher DAD/T than male OAG patients (p=0.002)

The correlation between DAD/T, vascular status, and OAG could prove to enhance the screening of OAG, and potentially serve as a marker for progression.