G. Tenti

Mathematical pressure volume models of the cerebrospinal fluid

Numerous mathematical models have emerged in the medical literature over the past two decades attempting to characterize the pressure and volume dynamics of the central nervous system compartment. These models have been used to study the behavior of this compartment under such pathological clinical conditions as hydrocephalus, head injury and brain edema. The number of different approaches has led to considerable confusion regarding the validity, accuracy or appropriateness of the various models. In this paper we review the mathematical basis for these models in a simplified fashion, leaving the mathematical details to appendices. We show that most previous models are in fact particular cases of a single basic differential equation describing the evolution in time of the cerebrospinal fluid pressure (CFS). Central to this approach is the hypothesis that the rate of change of CSF volume with respect to pressure is a measure of the compliance of the brain tissue which as a consequence leads to particular models depending on the form of the compliance function. All such models in fact give essentially no information on the behavior of the brain itself. More recent models (solved numerically using the Finite Element Method) have begun to address this issue but have difficulties due to the lack of information about the mechanical properties of the brain. Suggestions are made on how development of models which account for these mechanical properties might be developed.

Estimation of the quasi-linear viscoelastic parameters using a genetic algorithm

The quasi-linear viscoelastic (QLV) theory of Fung has been widely used for the modeling of viscoelastic properties of soft tissues. The essence of Fungs approach is that the stress relaxation can be expressed in terms of the instantaneous elastic response and the reduced relaxation function. Using the Boltzmann superposition principle, the constitutive equation can be written as a convolution integral of the strain history and reduced relaxation function. In the appropriate models, QLV theory usually consists of five material parameters (two for the elastic response and three for the reduced relaxation function), which must be determined experimentally. However, to be consistent with the assumptions of QLV theory, the material functions should be obtained based on a step change in strain which is not possible to be performed experimentally. It is known that this may result in regression algorithms that converge poorly and yield non-unique solutions with highly variable constants, especially for long ramp times. In this paper, we use the genetic algorithm approach, which is an adaptive heuristic search algorithm premised on the evolutionary ideas of natural selection and genetics, and simultaneously fit the ramping and relaxation experimental data (on ligaments) to the QLV constitutive equation to obtain the material parameters.

Brain biomechanics: Mathematical modeling of hydrocephalus

Abstract The considerable amount of literature on mathematical models of hydrocephalus and other brain abnormalities is critically reviewed. These models have various degrees of mathematical sophistication, and have influenced not only the diagnosis of hydrocephalus, but also its treatment with CSF shunts. The mathematical models are classified into two classes, pressure-volume models, and consolidation models. Advantages and disadvantages of both types are pointed out with a view to removing the confusion frequently generated by the technical aspects of the subject. The conclusion is reached that, while none of the current models are good enough to be of immediate use to the neurosurgeon, mathematical models are likely in the future to be a powerful tool for the understanding and the treatment of hydrocephalus, as well as other conditions related to brain biomechanics. The amount of mathematics has been kept to the absolute minimum, but it is cited and appended for those who would like to dig further into this fascinating area of research. [Neural Res 2000; 22: 19-24]

A viscoelastic model of the brain parenchyma with pulsatile ventricular pressure

In this paper, we present an extension of the model developed by Sivaloganathan et al. [Appl. Math. and Comput., to appear], which is of more physical relevance. We obtain explicit solutions for the displacement and stresses, and show how the mechanical parameters, that appear in the constitutive equation for the viscoelastic solid, can be calculated from data obtained from dynamic load experiments. Finally, we solve the boundary value problems corresponding to the case of adult hydrocephalus, as well as the more general case where both dilatational and deviatoric responses are assumed to be viscoelastic.

The synchrony of arterial and CSF pulsations is not due to resonance.

Accessible online at: www.karger.com/journals/pne Dear Sir, The recently published paper ‘A model of intracranial pulsations’ by Egnor et al. [1] proposes to explain the synchrony of arterial and CSF pulsations by means of the phenomenon of resonance. We would like to point out that, as a matter of fact, resonance has nothing to do with the synchrony of the arterial and CSF pulsations. There is a much more prosaic explanation of the phenomenon. When the arterial pulse arrives at the wall of the CSF space, it produces a pressure wave in the fluid. As CSF is basically incompressible, the pressure wave propagates through the CSF space at the speed of sound in water. And since the latter is quite high, the pulse is felt almost instantaneously throughout the fluid. The technical part of the argument presented in the paper by Egnor et al. [1] proceeds as follows. A quantity of CSF oscillates back and forth inside the cranial cavity in response to the arterial pulsations. These oscillations may be described mathematically by a standard, one-dimensional, damped, forced oscillator. The corresponding differential equation is easily solved for the steady state velocity of the CSF pulsations, and the result is given in equation 31 of [1], which we reproduce for convenience: vCSF(t) = Real{∑p } = F0 sin(ˆt – £ ) R2 + ˆmCSF – kE ˆ 2 (1)

Mathematical modeling of the brain: principles and challenges.

OBJECTIVE The use of mathematics in the study of phenomena and systems of interest to medicine has become quite popular in recent years, but not much progress has been made as a result of these efforts. The aim of this article is to identify the reasons for this failure and to suggest procedures for more successful outcomes. METHODS We review and assess a variety of mathematical modeling procedures, from microscopic (at the level of molecular behavior) to macroscopic standpoints, from lumped-parameters to distributed-parameters approaches. Using examples that are as simple as possible, we elucidate the difference between the predictive and the explanatory powers of mathematical models, as well as the uses (and abuses) of analogy in their construction. RESULTS Mathematical medicine is a truly interdisciplinary area that brings together medical researchers, engineers, and applied mathematicians whose vast differences in expertise and background make collaboration difficult. CONCLUSION The lack of a common language and a common way of understanding what a mathematical model is, and what it can do, is identified as the main source of the slow progress to date, and constructive suggestions are made to improve the situation.

Dynamical Morphology of the Brain's Ventricular Cavities in Hydrocephalus

Although interest in the biomechanics of the brain goes back over centuries, mathematical models of hydrocephalus and other brain abnormalities are still in their infancy and a much more recent phenomenon. This is rather surprising, since hydrocephalus is still an endemic condition in the pediatric population with an incidence of approximately 1 – 3 per 1000 births. Treatment has dramatically improved over the last three decades, thanks to the introduction of cerebrospinal fluid (CSF) shunts. Their use, however, is not without problems and the shunt failure at two years remains unacceptably high at 50%. The most common factor causing shunt failure is obstruction, especially of the proximal catheters. There is currently no agreement among neurosurgeons as to the optimal catheter tip position; however, common sense suggests that the lowest risk location is the place that remains larger after ventricular decompression drainage. Thus, success in this direction will depend on the development of a quantitative theory capable of predicting the ultimate shape of the ventricular wall. In this paper, we report on some recent progress towards the solution to this problem.

A Poroelasticity Theory Approach to Study the Mechanisms Leading to Elevated Interstitial Fluid Pressure in Solid Tumours

Although the mechanisms responsible for elevated interstitial fluid pressure (IFP) in tumours remain obscure, it seems clear that high IFP represents a barrier to drug delivery (since the resulting adverse pressure gradient implies a reduction in the driving force for transvascular exchange of both fluid and macromolecules). R. Jain and co-workers studied this problem, and although the conclusions drawn from their idealized mathematical models offered useful insights into the causes of elevated IFP, they by no means gave a definitive explanation for this phenomenon. In this paper, we use poroelasticity theory to also develop a macroscopic mathematical model to describe the time evolution of a solid tumour, but focus our attention on the mechanisms responsible for the rise of the IFP, from that for a healthy interstitium to that measured in malignant tumours. In particular, we discuss a number of possible time scales suggested by our mathematical model and propose a tumour-dependent time scale that leads to results in agreement with experimental observations. We apply our mathematical model to simulate the effect of “vascular normalization” (as proposed by Jain in Nat Med 7:987–989, 2001) on the IFP profile and discuss and contrast our conclusions with those of previous work in the literature.

MATHEMATICAL MODELING OF THE BRAINPRINCIPLES AND CHALLENGES

OBJECTIVEThe use of mathematics in the study of phenomena and systems of interest to medicine has become quite popular in recent years, but not much progress has been made as a result of these efforts. The aim of this article is to identify the reasons for this failure and to suggest procedures for more successful outcomes. METHODSWe review and assess a variety of mathematical modeling procedures, from microscopic (at the level of molecular behavior) to macroscopic standpoints, from lumped-parameters to distributed-parameters approaches. Using examples that are as simple as possible, we elucidate the difference between the predictive and the explanatory powers of mathematical models, as well as the uses (and abuses) of analogy in their construction. RESULTSMathematical medicine is a truly interdisciplinary area that brings together medical researchers, engineers, and applied mathematicians whose vast differences in expertise and background make collaboration difficult. CONCLUSIONThe lack of a common language and a common way of understanding what a mathematical model is, and what it can do, is identified as the main source of the slow progress to date, and constructive suggestions are made to improve the situation.