Image made from pieces.

Math and Migraines�Modeling Pressure Dynamics in the Brain��Nonlinear differential equations shed light on the pathological pressure dynamics associated with Intracranial Hypertension and Migraine

Scott Stevens

Senior Lecturer in Mathematics

Timeline:

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• 1996 – 2000: Stevens works with his Ph.D. Advisor Dr. William Lakin and Dr. Paul Penar (UVM Neurosurgery) on modeling intracranial fluid dynamics with lumped parameter models mostly as it relates to intracranial pressures in Microgravity through NASA grants. �
• 2000 – 2002: This work is expanded to include whole-body circulatory dynamics that resulted in a US Patent No. US 7,182,602 B2, Date: 02/27/2007. Title: Whole-Body Mathematical Model for Simulating Intracranial Pressure Dynamics. Inventors: William D. Lakin, Scott A. Stevens, Paul L. Penar, and Bruce I Tranmer.�
• 2002 – 2007: Stevens is joined by other collaborators to develop various models of intracranial pressure dynamics under multiple circumstances such as CSF withdrawals/infusions, collapsible sinuses, shunting, stenting, head-down tilt, and others. �
• 2008: Stevens, Stimpson, Lakin, Thakore, Penar publish “A Model for Idiopathic Intracranial Hypertension and Associated Pathological ICP Wave-Forms” in the IEEE Transactions on Biomedical Engineering that got some attention along with a couple of articles in Neurosurgery/Neurology journals. This presentation describes this model and our results. �
• 2008 – 2025: Scott leaves Penn State, moves to Vermont, stops research, writes a couple of books and becomes the Dean of the ITS Division from 2016 – 2025. Meanwhile, this body of work becomes well-cited in the literature exploring the relationships between IIH, Migraine, venous sinus stenosis, stenting, and starling resistors.�
• Now, I’m back at UVM as a Senior Lecturer in Mathematics and giving this talk today.

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Today

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• I will give the presentation from 2008 describing the model and the results from the IEEE publication. �
• I will follow that up with some research conducted by others since then which utilize, cite, and apply our model/theory. �
• I will present some current topics of interest that I have been playing around with in the last couple of years – suitable for more serious investigations.

Idiopathic Intracranial Hypertension (IIH)

• High pressure (hypertension)
• In the head (intracranial)
• Unknown cause (idiopathic)
• Symptoms: headache, nausea, papilledema (swollen optic nerve), visual obscurations possibly leading to blindness
• Often concurrent with intracranial venous-sinus stenosis

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J N P Higgins, C Cousins, B K Owler, N Sarkies and J D Pickard

Idiopathic intracranial hypertension: 12 cases treated by venous sinus stenting

Journal of Neurology Neurosurgery and Psychiatry 2003;74:1662-1666

Sinus Stenosis: Blockage or compression?

Normal

Stenosed Sinus

Prevalence

• IIH prevalence < 1%�
• IIH without papilledema (IIHWOP) unknown. �
• 6.7% of 724 migraine patients – sinus stenosis.�
• 67.8% of these - IIHWOP.�
• Possibly 1.3 million in United States�
• 9/10 chronic daily headache patients reveal IIHWOP with Pathological ICP waveforms (what is that?)

Bono 2006, Torbey 2004.

B-waves (spikes)

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Clinically Observed Pathological ICP Waveforms

A-waves (plateaus)

Risberg, Lundberg 1969

Torbey 2004

The “Lumped Parameter Model”.

Model Assumptions

Model Assumptions

Governing Equations: CSF/Brain Compartment�(Flow In) – (Flow Out) = Volume Change

Governing Equations: �Cerebral Veins and Saggital Sinus

Downstream Starling Resistor�Greater Transmural Pressure -> Less Resistance

Data: Heil (1997)

Model

Keep your eye on “m”:

the initial collapsibility parameter.

Governing Differential Equations

The compliance matrix is non-singular so you can evaluate the steady states on the right.

Steady-State Equations

Options - Based on the collapsibility parameter m.

One Stable Steady State

Crazy Stuff Happens

One Unstable Steady State

Two Stable Steady States

Bifurcation Diagram for P_F in terms of the collapsibility parameter m.�Weird stuff happens between m1 and mH. There is a Hopf bifurcation nested in a supercritical pitchfork bifurcation.

As the collapsibility parameter (m) increases, the situation gets worse.

Simulations in the Hopf Bifurcation Region

m1 < m < mH

self-excited oscillations

Simulations after the Pitchfork Bifurcation (m > mH)

Temporary Perturbations Cause Permanent Transitions between Steady States

Cerebral Blood Flow Spike (Apnea)

CSF Withdrawal

Simulations with a Periodic Forcing Term

Model Simulations with a Collapsible Sinus

Clinically Observed Pathological ICP Waves

A-Waves

B-Waves

Our results suggested that the cause was an overly collapsible venous sinus, a pitchfork bifurcation, and two stable states. We proposed that the situation should be resolved with a sinus stent. It eventually caught on and sinus stenting became more common.

Some Citations Since 2008�Supporting the theory and sinus stenting

• Of note, a mathematical modeling study [Stevens et al.] has demonstrated that placing a mathematical SR at the transverse sinus level replicates the wide spontaneous ICP fluctuations documented in IIH/IIHWOP patients by ICP monitoring studies. Neurological Sciences 2019

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• …the procedure (stenting) can be effective in preventing the positive feedback loop and breaking the starling-like resistor effect [Stevens et al.] Brain Sciences 2021.

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• If stenting the venous sinuses can allow for outflow and correct stenosis, it may thereby end the cycle that underlies the development of IIH [Stevens et al.] World Neurosurg 2024. �
• Yet, the efficacy of the procedure could stem from the prevention of the positive feedback loop and break in the starling-like resistor effect [Stevens et al.] Cureas 2019.

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• A Starling-like resistor… has important consequences… [the model] can exist in 1 of 2 stable states—a normal… and a high-pressure state.” AJNR 2011�
• “Stevens et al. suggest that transient increases in ICP can compress the transverse sinus (Starling-like resistor effect) and be the stimulus for IIH.” Neurological Science 2024

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• … increasing intracranial pressure and further exacerbating IIH by compressing the already stenosed segment acting as a Starling resistor [Stevens et al.] AJNR 2016.

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• The most accepted theory is that  intracranial hypertension causes extrinsic compression of the dural venous sinuses. Brazilian Neurosurgery 2019. �

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Current Work

• Valsava Simulations on a slightly modified version of the IEEE Model. Introduce a differentiable Pv forcing term to simulate Valsalva. �
• A simplified model that produces a single differential equation with a super-critical pitchfork bifurcation (no limit cycles)

Valsalva Simulations

Brain & CSF (F)

Brain & CSF (F)

Arterioles & Capillaries ( C )

Cerebral Veins & Sagittal Sinus (S)

Transverse Sinus & Jugular Veins (V)

Arteries �( A )

QAC

QCS

QSV

QCF

QFS

QSV

CAF

CFV

Single Differential Equation Model

Single Differential Equation

Steady-State Analysis

Classic Pitchfork Bifurcation

Our Pitchfork Bifurcation

STUDENTS: If this was interesting to you - �Take Advanced ODE’s (MATH 5230).

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Thanks

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Sometimes it’s the tragedies that don’t happen that make a great day.

Questions�Comments�Good Stories

Unused Slides

Nullclines in the transformed variables.

A fantastic, web-based direction field / phase portrait utility�Rice University

http://math.rice.edu/~dfield/dfpp.html

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• Previous Models��IIH characteristics ��1) Sinus Stenosis�2) intermittent symptoms�3) long term relief�4) fast transitions between states�5) treatment methods���Stevens, Previte, Lakin, Thakore, Penar, and Hamschin: "Idiopathic Intracranial Hypertension and Transverse Sinus Stenosis: A Modeling Study". Mathematical Medicine and Biology 2007��

• Current Model��IIHWOP characteristics��1) Retains previous results for IIH�2) Demonstrates Pathological ICP wave-forms in IIHWOP����Stevens, Stimpson, Lakin, Thakore, and Penar “A model for idiopathic intracranial hypertension and associated pathological ICP wave-forms. Accepted by IEEE Transaction on Biomedical Engineering.

Dynamics after the Pitchfork Bifurcation (m > mH)

Collapsible sinus simulation

Rigid sinus simulation

Possible Diagnostic – CSF Withdrawal

Cerebral Blood Flow Perturbations.

Spikes and plateaus together.

Post saddle-node bifurcation: Similar to our previous results.

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Two stable states: Normal and Elevated��Temporary perturbations cause fast, permanent transitions. ��Cerebral blood flow perturbation - Sleep apnea.