Mathematical modeling of human eye

The eye: a window on the body

Diseases of the Body:

Diabetes

Hypertension

Neurodegenerative

Disorders (NDD)

Diseases of the Eye:

Glaucoma

Retinopathies

Age-related Macular

Degeneration (AMD)

Need of quantitative methods to

detect and grade vascular abnormalities in the

eyes and identify underlying pathogenic

mechanisms

?

Modeling of Ocular Blood Flow

Glick Eye Institute

IU Medicine

Alon Harris

Brent Siesky

Math IUPUIGiovanna Guidoboni

Lucia Carichino

Simone Cassani

Math PoliMi

Riccardo Sacco

Francesca Malgaroli

Math WSUSergey 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 micro-

circulation

Fundus Camera

http://www.medicine.uiowa.edu/eye/Ocular-Fundus-Photograhy/

http://en.wikipedia.org/wiki/Macular_degeneration

Normal AMDDiabetic

Retinopathy

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

Ophthalmic Arteryfrom

Internal

Carot idArtery

to anter iorpart of eye,

face and nose

Temporal

Poster iorCiliaryArtery

NasalPoster ior

CiliaryArtery

Ophthalmic

Vein

Ophthalmic

Vein

Cavernous

Sinus

to InternalJugular Vein

Int raocularPressure

ret ina

choroid

laminacr ibrosa

Cent ralRet inalA rtery

Cent ralRet inalVein

1

Arterial Blood

Pressure

Venous Blood

Pressure

Intraocular

Pressure

Cerebrospinal

Fluid Pressure Intracranial

Pressure

Is driven by:Difference in Arterial and

Venous Blood pressure

Is impeded by:Intraocular pressure (IOP)

Cerebrospinal Fluid pressure (CSFp)

Intracranial pressure (ICP)

Is modulated by:Vascular regulation

Vascular

Regulation

• Intraocular pressure (IOP) is

the overall pressure within the

eye.

• During each cardiac cycle, the

velocity of the blood-flow

within the retina is changed

dramatically.

• Because of this change, the eye

has built in autoregulation

mechanisms that attempt to

maintain constant blood

pressure within the retina.

• These mechanisms can fail and

without proper autoregulation,

some eye diseases can

develop.

Ophthalmic Arteryfrom

Internal

Carot idArtery

to anter iorpart of eye,

face and nose

Temporal

Poster iorCiliaryArtery

NasalPoster ior

CiliaryArtery

Ophthalmic

Vein

Ophthalmic

Vein

Cavernous

Sinus

to InternalJugular Vein

Int raocularPressure

ret ina

choroid

laminacr ibrosa

Cent ralRet inalA rtery

Cent ralRet inalVein

1

Intraocular

Pressure

Intraocular Pressure and Auto Regulation

• 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.

Diseases related to increased IOP

Why modeling?

• Interpretation of clinical data is

challenging! Understand physiology

in health and disease

IOP

CSF

• Pressurized ambient (intraocular pressure - IOP)

• Fluid-structure interactions

• Complex vascular system

• Sub-systems

• Flow regulation

• Traditionally:

• Animal studies

• Clinical or

population-based studies

• Our Approach

• Mathematical models

• Clinical data

• 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.

Why modeling?

• 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

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.

volume change = flow in - flow outKirchoff’s law applied to the retinal vascular network

Mathematical Model

Obtain system of ordinary differential equations for the nodal pressures

𝑃1, 𝑃2, 𝑃4, 𝑃5:

The inlet and outlet pressures 𝑃𝑖𝑛and 𝑃𝑜𝑢𝑡 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

• 𝑝 is constant on each Σ• 𝑢𝑧 = 𝑢 𝑧 𝑓 Σ , where 𝑢 𝑧 is average axial velocity and 𝑓 Σ is appropriate

shape function

• Axial motion is predominant

Obtain reduced equations:

Where 𝐾𝑟 depends on 𝑓(Σ), 𝑄 𝑧 is volumetric flow and 𝐴 𝑧 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

decreases. 𝑘𝐿 = 12𝑟𝑟𝑒𝑓

ℎ

2

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:

𝑅2𝑎 = R2b =𝑐𝐿 + 𝑐𝑈exp(𝐾 𝑄𝑛𝑜𝐴𝑅 − 𝑄 − 𝑐)

1 + exp(𝐾 𝑄𝑛𝑜𝐴𝑅 − 𝑄 − 𝑐)

Mathematical Model

CRA

CRV

venules

arterioles 𝑅2𝑎 = R2b =𝑐𝐿 + 𝑐𝑈exp(𝐾 𝑄𝑛𝑜𝐴𝑅 − 𝑄 − 𝑐)

1 + exp(𝐾 𝑄𝑛𝑜𝐴𝑅 − 𝑄 − 𝑐)

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

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)

Healthy and Glaucoma

• 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.