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July 27, 2007
Workshop on Brain Biomechanics: Mathematical Modelling of Hydrocephalus and Syringomyelia

Speaker Abstracts

Noam Alperin, University of Illinois at Chicago
"MRI based noninvasive measurement of intracranial compliance: direct vs.
modeling based approach."

Intracranial compliance (ICC) is a bio-mechanical parameter which plays an
important role in many neurological problems and therefore could
potentially become an important diagnostic marker.

Currently intracranial compliance is measured only invasively by
manipulation of the intracranial volume and recording the corresponding
pressure change. The invasiveness of the test and the risk involved limits
the clinical utility of ICC. A considerable effort has been ongoing for
over a decade to derive the bio-mechanical properties of the intracranial
compartment noninvasively using MRI measurements of blood and CSF flow. The
presentation will compare a modeling based vs. direct approach for
noninvasive derivation of the ICC. The direct approach is the noninvasive
analog of the invasive volume-pressure response test developed by Marmarou
et al, where the injected bolus is the net systolic blood volume entering
the brain with every cardiac cycle. The methodology, validation studies,
and early clinical application of the MRI based ICC measurement (MR-ICP)
will be reviewed.

Benjamin Cohen, Rensselaer Polytechnic Institute
A Mechanics-Based Framework Leading to the Diagnosis and Treatment of Hydrocephalus
with Timothy Wei, (Rensselaer Polytechnic Institute)
Michael Egnor, Mark Wagshul (State University of New York at Stony Brook)

To date hydrocephalus researchers acknowledge the need for rigorous but utilitarian fluid mechanics understanding and methodologies in studying normal and hydrocephalic intracranial dynamics. We seek to address these issues by applying a systems level fluid dynamics approach to hydrocephalus. The goal is to provide a first principles framework to integrate a broad spectrum of sometimes disparate investigations into a highly complex, multidisciplinary problem. The paradigm entails defining regions in the brain referred to as ’control volumes’. In effect, this is a budgeting procedure using fundamental conservation laws (mass and momentum) to keep track of and account for, from direct measurements, important parameters in the cranium. At a minimum we would require at least one control
volume for the CSF and at least one for the blood. However, any desired number and arrangement is possible at the investigators’ discretion.

The power of control volume analysis lies in the fact that only terms accounting
for net changes within the system and property flow across the boundaries are required. Thus, the use of control volumes preclude the need for velocity and pressure field information everywhere in the cranium, greatly simplifying the analysis. Measurements are tied directly to the physical meaning of terms in the conservation equations and can be obtained using a blend of modern techniques such as MR imaging and digital particle image velocimetry. Once clinical data are collected, the budgets are used to relate spatially separated measurements throughout the cranium. Similarly, interactions between control volumes can be studied and are potential locations for changes leading to hydrocephalus. Most importantly, the use of the conservation equations with control volumes is an effective
way to comprehensively couple diverse clinical data sets. This then is not truly mathematical modeling; because a modeler will actually attempt to solve these or similar equations. It is instead a framework to guide experimental method using the most basic fluid dynamics analysis.

Nikos P. Chrisochoides, College of William and Mary
Near Real-Time Non-Rigid Registration of pre- and intra-operative MRI for Image Guided Neurosurgery.

In this talk we present our experience and preliminary results from near real-time non-rigid registration of intra-operative Magnetic Resonance Imaging (IMRI) with the preoperative MRI, fMRI, and DT-MRI data. Previously, non-rigid registration algorithms which use landmark tracking across the entire brain volume were considered not practical because of the high computational demands. We show that we can compute and present enhanced MR images to neurosurgeons during the tumor resection within minutes after IMRI acquisition. During the last year, this software system was used routinely (on average once a month) for clinical studies at Brigham and Women's Hospital in Boston, MA.

This is join work with my students A. Fedorov and A. Kot from William and Mary and N. Archip, P. Black, O. Clatz, A. Golby, R. Kikinis, S. Warfield from Harvard Medical School.

Corina S. Drapaca, University of Waterloo
Application of Potential Theory to MR Elastography

In recent years, a lot of work has been done in the area of inverse problems of elastodynamics, especially due to its important role in the newly emerging biomedical engineering-related techniques like, for example, magnetic resonance elastography. Given that tissue stiffness can be used as an indicator of pathology and that phase-contrast MRI can help measuring displacements, the interest of researchers in accurate biomechanical models of tissues and correct techniques for determination of mechanical parameters of a chosen model increased significantly. Once the biomechanical model of the tissue is specified, one way of obtaining stiffness from measured displacement data is based on the direct local inversion of the equations of motion. In our work we model the biological tissues as almost incompressible linear viscoelastic materials and perform a simple algebraic inversion of the Navier equations in displacements to obtain the Lame coefficients. However, the higher order differentiation present in the Navier equations affects the result of the inversion. By introducing the Newtonian potential field corresponding to the displacement field, we can lower the order of numerical differentiation in the Navier equations and thus make the algebraic inversion more stable. Results of the proposed method will be shown and new mathematical concepts related to the problem of elastography will also be discussed.

Michael Egnor, M.D., Stoney Brook
Modeling of Phase, Amplitude and Transfer Function of Carotid Arterial Pulse Pressure and Intracranial Pulse Pressure in Dogs
with Mark Wagshul, PhD. , Erin McCormack, PhD. , Pat McAllister, PhD.

In recent dog experiments we have made three observations that pose new challenges to the modeling of intracranial dynamics. In our experiments, we placed pressure transducers in the carotid artery and in the brain parenchyma of 12 dogs, and recorded the changes in timing and amplitude relationships between the arterial and ICP waveforms. We also calculated the amplitude transfer functions between the arterial and the ICP pulses. We varied the mean ICP by infusing or withdrawing CSF from the spinal subarachnoid space. We found:

1) At normal mean ICP, the ICP waveform leads the carotid waveform by roughly 50 msec. With variation in mean ICP, there is a roughly linear lag of ICP with raising mean ICP, and an increasing lead of ICP with lowering mean ICP.

2) The amplitude of the ICP pulse is at a minimum at normal ICP for each animal. With raising or lowering ICP, the amplitude increases (the results are less clear at low ICP than they are at high ICP).

3) The transfer function between the arterial pulse (input) and the ICP pulse (output) tends to show a notch at the heart rate. The frequency of the notch varies with the heart rate, and the notch is ablated with substantial increases in mean ICP.

We have modeled the cranium as a single degree of freedom forced oscillator. The reference phasor is the flow variable, which we take to be the cerebral blood flow in systole/diastole, and the effort variable is the ICP. We model the carotid pulse as simply a marker for the cardiac cycle of cerebral blood flow. In our model, the ICP pulse in the cranium is a standing wave. The leading and lagging of the ICP pulse with changes in mean ICP then is simply the phase changes in the effort variable with respect to the flow variable that occur with changes in compliance in the system (high ICP is low compliance, low ICP is high compliance).

The amplitude and notch results (#2 and #3) we model as the behavior of a band stop filter tuned to the heart rate. We propose that the cranium is a cavity resonator, and that the cerebral windkessel effect is accomplished by this band stop filter effect, and is disrupted by changes in ICP and cerebral compliance.

Richard L. Ehman, Department of Radiology, Mayo Clinic
Magnetic Resonance Elastography by Direct MR Visualization of Propagating Shear Waves

The goal of our research is to develop MRI-based methods for assessing the mechanical properties of tissues in vivo. We have focused on a novel MRI technique for visualizing propagating acoustic shear waves [Science, 269, 1854-57; 1995].

Suitable dynamic shear stress for Magnetic Resonance Elastography (MRE) can be generated by surface drivers, inertial effects, acoustic radiation pressure, or endogenous physiologic mechanisms. The MRE acquisition sequence is capable of visualizing cyclic tissue motion of less than 1 micron in displacement amplitude, with imaging times ranging from 100ms to several minutes. Inversion algorithms based on continuum mechanics are used to process the acquired data to generate maps of mechanical properties such as depict stiffness, viscosity, attenuation, and anisotropic behavior.

We have applied MRE to assess specimens of a variety of tissues, ranging in stiffness from lung to cartilage. In applications to brain imaging human studies have demonstrated that it is feasible to apply MRE to quantitatively image the mechanical properties of gray and white matter, providing a new tool for tissue characterization, diagnosis, and mechanical modeling in brain pathologies such as hydrocephalus. A variation of the technique, using impulse mechanical waveforms, allows dynamic visualization of micron level strains as mechanical transients propagate through the brain, providing a direct way to study the biomechanics of traumatic brain injury.

Miles Johnston, University of Toronto
In search of mechanisms to explain ventricular expansion in communicating hydrocephalus.

David N. Levine, NYU School of Medicine
Intracranial Pressure in Hydrocephalus

Ventricular enlargement in hydrocephalus has been thought to result from elevated intraventricular pressure building behind an obstruction in the flow path of CSF from ventricles to subarachnoid space. This view led to the expectation that a large pressure difference would exist between the ventricles proximal to the obstruction and the convexity subarachnoid space distal to the obstruction. Yet measurements in both experimental and clinical settings show that such transmantle pressure differences are small or non-existent. A theory is proposed that reconciles the view that hydrocephalus is caused by an obstruction in the flow path of CSF with the observed absence of large pressure gradients across the cerebral mantle. Obstruction to CSF flow produces only a small pressure gradient that is sufficient to overcome the added resistance to CSF flow and to initiate ventricular enlargement. In the presence of a rigid skull the small increment of intraventricular pressure is transmitted to the periphery as a result of the poroelastic properties of the brain parenchyma. The efficiency of transmission is greatly influenced by the poroelastic compressibility of the brain and by ventricular size and can be calculated if these properties are known. Intraventricular and subarachnoid CSF pressure rise to levels that satisfy two physical requirements: first, maintenance of a small transmantle gradient that will overcome the resistance to CSF flow caused by the obstruction and will thus match CSF absorption to CSF production; and second, transmission of that degree of pressure from ventricles to the convexity subarachnoid space that is consistent with the poroelastic state of the brain..

David N. Levine, NYU School of Medicine
The Syrinx as a Biological Pothole

Syringomyelia is commonly associated with an extramedullary lesion at the foramen magnum, usually a Chiari I malformation. Many theories of pathogenesis have been proposed, but critical review of these theories shows that they either fail to account for the clinical or neuropathological features of syringomyelia or are biophysically implausible. I have proposed a new theory of pathogenesis (J Neurol Sci 220:3-21, 2004) in which syringomyelia is the outcome of repetitive mechanical stress of the spinal cord. This stress occurs when, as a result of a block of the subarachnoid space, activities such as standing, coughing, straining and even the rhythmic cardiac cycle, induce transiently higher CSF pressure above the block than below it. The corresponding changes in transmural venous pressure favor venous collapse above the block and venous dilation below it. In the region of the block the non-linear volume change of the cord is a source of mechanical stress, which can disrupt the neuropil and damage the blood-cord barrier. The latter allows leakage of intravascular fluid that may eventually coalesce to form a syrinx. There is a strong analogy with thermal stress, in which spatially uneven heating causes different degrees of expansion and mechanically disruptive stress. Such stresses are a major cause of potholes on roads. The equations of thermal stress can be used to model the distribution of shear stress in the presence of blockage of the subarachnoid space. The proposed theory is consistent with the main clinical and neuropathological features of syringomyelia

Joseph R. Madsen, Boston University
Intracranial pulsations and their characteristics in hydrocephalus: inferences from the frequency domain

A system linking input signal to output signal can be characterized using system identification in frequency domain. A transfer function - a mathematical representation of the relationship between input and output signals - can describe how the system responds to a certain frequency (or frequency range). In the cranial system, the intracranial pressure (ICP) waveform may be considered a signal which is derived in part from the input of other signals, particularly the arterial blood pressure (ABP). The gain (amplitude) of the transfer function is a measure in frequency domain describing how much pressure is transmitted through the system. The conventional method to calculate transfer function based on Fourier transform is unable to detect non-stationary characteristics of the biological signal. As a result of the necessity for time varying analysis, we developed a new method to calculate time varying transfer function (TVTF) by taking advantage of both autoregressive and moving-average (ARMA) modeling and short-time Fourier transform.
Recently, the TVTF analysis (applied to canine data) has revealed an important role of pulsatile intracranial cerebrospinal fluid (CSF) movement in attenuating strong arterial pulsations transmitted to the brain. The role is analogous to a notch filter centered at the cardiac frequency. The efficacy of the filter depends upon how well the center frequency of the notch filter remains tuned at the target frequency particularly. How does the notch filter respond to normal physiological conditions and to pathological conditions? To approach this question, we will show how the TVTF technique can be applied to ABP and ICP obtained from dogs with normal, hydrocephalic, and shunt treated conditions, and to human patients with clinical hydrocephalus. Correlation of dynamic analysis, biochemical markers, and structural studies in hydrocephalus and syringomyelia will improve understanding and treatment of these conditions.
With: E.-H. Park, R. Zou, M. Egnor, M. Wagshul, E. Kelly, S. Dombrowski, M. Luciano, P. Eide, T. Anor, J. Shim

Susan Margulies, University of Pennsylvania
Mechanical Properties of Brain Tissue

James P. (Pat) McAllister, Wayne State University School of Medicine
Probable Cellular Influences on the Biomechanical Properties of the Brain: The "Sponge" Changes during Hydrocephalus

Mathematical modeling in hydrocephalus often uses the physical properties of a viscoelastic "sponge" to represent the brain. While many details on the cellular pathophysiology of the hydrocephalic brain are still lacking, it is well-known that cytology and cytoarchitecture change considerably during the progression of ventriculomegaly and after treatment with CSF diversion. The purpose of this presentation is to review some of the cellular changes that most likely play major roles in altering the properties of the "sponge" during untreated and treated hydrocephalus.
Many neurons, especially those near the cerebral ventricles, become smaller, disoriented, and severely compacted. The increased density of cells and the reduced amount of neuropil may have a major effect on the properties of the "sponge". While degenerating and apoptotic neurons have been observed throughout the cerebral cortex and in subcortical nuclei, it is still not clear if neuronal loss is statistically significant. Glial alterations are numerous and can occur rapidly. Reactive astrocytosis and microgliosis occur within hours of ventricular expansion, and do not seem to depend on increased intracranial pressure. Often glial "scars" persist long after ventricular shunting; these can be found in the cortical gray and white matter, as well as in distant locations such as the thalamic relay nuclei and the myelinated corticofugal pathways. We speculate that such scarring could have a profound effect on the viscoelastic properties of the hydrocephalic brain, as well as prevent neuronal regeneration and alter the blood brain barrier. In contrast to the hypertrophy and proliferation exhibited by astrocytes and microglia, oligodendrocytes die and demyelination occurs. In addition, oligodendrocytes do not appear to regenerate, which clearly has a direct effect on remyelination after treatment. Correlated with the findings of decreased cerebral blood flow, the cerebral vasculature is disoriented and the size and number of microvessels is reduced. Furthermore, recent studies have shown that important clearance mechanisms associated with cerebral microvessels are impaired. Finally, it is well known that both interstitial and cytological edema accompany ventriculomegaly; in the cerebral cortex edema is generally restricted to the periventricular white matter but in severe ventricular enlargement can extend to layer IV.
Although the re-expansion of the cortical mantle after decompression has fostered the simplistic notion that these cellular deficits can be completely reversed, experimental studies have shown that not all cytopathology can be reversed. In general, relatively early decompression seems to prevent cell loss, but residual deficits still occur in fine structures such as dendritic spines and synaptic terminals, and residual gliosis can also be observed in "successfully" shunted brains.
It is very important to recognize that cellular plasticity is ongoing, even in the untreated hydrocephalic brain. While the exact features of this plasticity have not been revealed clearly, they most certainly contribute to the transient nature of the brains we attempt to model.
While much experimental data needs to be collected before the cytopathology of hydrocephalus can be clearly defined, the following features should be included in attempts to model this disorder: (1) cellular changes are multifaceted, transient and not always associated with intracranial pressure, (2) cell numbers vary with time and type; neurons and oligodendrocytes usually degenerate but astrocytes and microglia proliferate, (3) microvessels decrease and the amount of neovascularization is not known, (4) considerable plasticity occurs in untreated hydrocephalus and the extent of regeneration after shunting is not clear, and (5) pathology is not limited to the periventricular white matter.

Richard Penn, University of Chicago
Physics of Hydrocephalus

Harold L. Rekate, University of Arizona School of Medicine
All Forms Of Hydrocephalus Can Be Explained By Derangements of Bulk Flow
30 Years of Research Driven By A Bulk Flow Model

Purpose : To describe the findings of physiologic experiments and clinical observations arising from a mathematical model of CSF dynamics developed in collaboration with Case Institute of Technology.

The Model: The model that we derived was based on hydraulic principles and is analogous to a study of DC circuits as defined by Ohms Law. It was our intention to move on to an AC Pulsatility model once we had reached the limits of the model to describe the enigmas of hydrocephalus.

The Process: Using a multicompartment model relating the pressure, volume and flow data experiments were designed using large animal experimental models of hydrocephalus to test various hypotheses, measure resistance elements and attempt to study the enigmas of hydrocephalus. Concurrently we examined the relevance to clinical situations of the model by making use of data related to ICP monitoring in the clinical setting.

Conclusions: While the current model fails to explain the results of the experiments of Bering with unilateral hydrocephalus or those of Di Rocco related to augmentation of the pulse pressure, we have so far found no clinical setting that is not easily explained by the concepts inherent in our bulk flow model. Experimental evidence from experiments designed to understand the dynamics of CSF as a circuit would suggest that all hydrocephalus is obstructive and explainable by bulk flow of CSF.

M. E. Wagshul, Stony Brook University
Joint work with M.R. Egnor, E.J. McCormack, S. Rashid,Stony Brook University
Intracranial flow and pressure: What can we measure?

A number of groups have attempted to model intracranial dynamics, based either on first-principle fluid-dynamics or on phenomenological assumptions about the mechanisms of intracranial pressure and flow. In either case, the development of such models, and more importantly the testing of these models, must come through the predictions which they produce of expected flow and/or pressure waveforms throughout the cranium. Thus, the ability to measure flow or pressure at multiple locations inside the brain is of critical importance for the validation of any mathematical model attempting to reproduce intracranial flow o pressure. In this paper we will review 1) the basic techniques for non-invasive measurements of flow, 2) the technical challenges involved in extracting accurate quantitative measures of flow, 3) where in the cranium one can measure flow, 4) improved techniques we have developed for efficiently measuring flow in multiple regions, 5) potential applications for extracting pressure information, and finally 6) how this information might be used to drive mathematical model development and testing.

Kathleen Wilkie, University of Waterloo
A Pulsatile Cerebrospinal Fluid Model for Hydrocephalus

Cerebrospinal fluid (CSF) pulses in the cranium with the frequency of the
heartbeat. Recently, these pulsations have been proposed as a possibly important mechanism in the pathogenesis of hydrocephalus. In this talk, I will discuss a simple model which may help to determine if the CSF pulsations are mechanically relevant to the development of hydrocephalus. The brain is modelled as a thick-walled poroelastic cylinder with periodic forcing at both the ventricular and subarachnoid space boundaries. Analytic solutions and numerical simulations provide a time- and space-dependent analysis of the effects of the pulsations.

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