A stochastic differential equation analysis of cerebrospinal fluid dynamics
© Raman; licensee BioMed Central Ltd. 2011
Received: 1 September 2010
Accepted: 18 January 2011
Published: 18 January 2011
Clinical measurements of intracranial pressure (ICP) over time show fluctuations around the deterministic time path predicted by a classic mathematical model in hydrocephalus research. Thus an important issue in mathematical research on hydrocephalus remains unaddressed--modeling the effect of noise on CSF dynamics. Our objective is to mathematically model the noise in the data.
The classic model relating the temporal evolution of ICP in pressure-volume studies to infusions is a nonlinear differential equation based on natural physical analogies between CSF dynamics and an electrical circuit. Brownian motion was incorporated into the differential equation describing CSF dynamics to obtain a nonlinear stochastic differential equation (SDE) that accommodates the fluctuations in ICP.
The SDE is explicitly solved and the dynamic probabilities of exceeding critical levels of ICP under different clinical conditions are computed. A key finding is that the probabilities display strong threshold effects with respect to noise. Above the noise threshold, the probabilities are significantly influenced by the resistance to CSF outflow and the intensity of the noise.
Fluctuations in the CSF formation rate increase fluctuations in the ICP and they should be minimized to lower the patient's risk. The nonlinear SDE provides a scientific methodology for dynamic risk management of patients. The dynamic output of the SDE matches the noisy ICP data generated by the actual intracranial dynamics of patients better than the classic model used in prior research.
Intracranial dynamics play a central role in healthy brain function because disturbances in the internal fluid environment of the skull can lead to multiple complications such as, among other things, hydrocephalus . Intracranial dynamics, driven by the circulation of CSF, are important because CSF protects the brain from injury, contains nutrients enabling normal functioning of the brain and, transports waste products away from the surrounding tissues. Much more is involved in hydrocephalus than a simple disorder of CSF circulation ; it is considered a complex spectrum of diseases, primarily defined by perturbation of the cranial contents--operationalized as CSF volume--and the intracranial pressure . Given the complex nature of hydrocephalus, we define hydrocephalus as a disease associated with disturbances in the CSF dynamics, as in .
The importance of modeling noise in CSF dynamics
Generalizing the CSF dynamics to incorporate noisy flow
Visual examination of the time-series of ICP recordings shows that the fluctuations are smooth (unlike electrons in a wire which generate shot noise, characterized by jumps ), and therefore continuous state space Markov processes are appropriate to capture the noisy dynamics of CSF flow. A large class of Markov processes can be represented by SDEs, and here a methodological choice must be made--noisy dynamic processes can be represented by stochastic differential equations of the Ito type or the Stratonovich type which correspond to two different ways of introducing noise into a dynamic system. A central difference between the two is that the Stratonovich SDE uses the usual deterministic calculus whereas the Ito SDE requires a completely new stochastic calculus. Extensive conceptual, empirical and philosophical discussions of this issue exist in the literature on mathematical models of electrical, biological and physical phenomena [19–21]. The overwhelming majority of these discussions conclude that Ito processes, generated by stochastic differential equations of the Ito type, are superior to Stratonovich processes, generated by stochastic differential equations of the Stratonovich type [22, 23]. Ito  extended standard deterministic calculus to a "stochastic calculus" applicable to functions of a wide class of continuous-time random processes, known as Ito processes. Given the SDE for the process under consideration, a result called Ito's Lemma yields the SDE driving the dynamics of a general transformation of the original process . This utilitarian result allows deducing the stochastic properties of considerably complex models driven by Ito processes . An essential property of Ito processes is that nonlinear functions of Ito processes remain Ito processes--a property called closure under nonlinear transformations, indispensable for practical reasons. From an empirical standpoint, a compelling advantage of Ito processes is that they often yield very precise statistical specifications for estimation . An attractive property of Ito processes--on theoretical, mathematical, practical and computational grounds--is that they are Markov processes. Finally, the Ito calculus has been extended to embrace general martingale processes --a development that permits joint consideration of both smooth noise and noise that occurs in jumps. Thus our modeling framework can accommodate neurological phenomena requiring noise that encompasses both smooth and jumpy variations in the state of the system, such as the firing of neurons .
Modeling the CSF dynamics as an Ito process to incorporate noisy flow
Note that in order for equation (9) to be dimensionally consistent, the unit of σ is mL/min. Because the 'input' I(t) is the infusion rate which is under direct experimental control, therefore, in the language of control theory, I(t) is a 'control' variable. In the infusion studies conducted at Addenbrookes's Hospital in Cambridge, UK, I(t) is maintained at a constant rate of 1.5 mL/min. However, factors not within the experimenter's control also influence the input flow rate. In addition to the infusion rate of the experimenter which influences CSF formation, CSF is produced inside the brain, but much about its production remains unknown at the present time. Currently, there are no direct methods to measure the CSF production rate over short periods of time. Globally, the average secretion rate--used as a proxy for the production--is 0.35 mL/min with a 95% confidence range of 0.27 mL/min to 0.45 mL/min . The lack of precise knowledge about the CSF production rate and the unmeasured factors that influence it are sources of noise in the total CSF formation rate. Consequently the stochastic Marmarou model may be conceptualized as the classic Marmarou model with a noisy input flow rate that reflects uncertainty about CSF formation.
The clinical significance of the stochastic Marmarou model
By building the fluctuations right into the dynamics of the model structure, the stochastic model makes full use of the information in the variations of the ICP waveform. From this additional information, the time-varying probability distributions of the ICP waveform can be extracted, and it is these latter quantities that enable computation of the probabilities of clinically relevant events. It is the knowledge of these probabilities of clinically relevant events that facilitate dynamic risk management of the patient. Conceptually, the average value of p(t) at any given time 't' is the average ICP at that time in an ensemble of patients with a similar CSF flow profile, as reflected in the values of the CSF flow parameters.
Analyzing the stochastic Marmarou model
In the Results section, we will display the exact analytical solution to the stochastic Marmarou model and derive insights from the solution into the influence of noise on the ICP at each point in time, and on average. Under the normal conditions described in , biological processes will settle down to a steady state after the transients have died out. In the deterministic Marmarou model , the steady state (equilibrium) is found by setting the time rate of change of the ICP equal to zero. What is the corresponding steady-state concept for a stochastic process? The stochastic counterpart to the time-independent steady-state level of the ICP is the time-independent probability distribution of the ICP, and the equilibrium probability distribution is to the stochastic environment as the stable equilibrium point is to the deterministic one . We derive the equilibrium probability distribution for the ICP, and from it, draw conclusions for the influence of CSF flow parameters and noise intensity upon the average steady-state ICP level. We compute a measure relevant to the treatment and control of hydrocephalus: given the current value of the patient's ICP, what is the probability that it will exceed a critical high level? And how is that probability influenced by neurological characteristics of the patient such as their resistance to CSF flow and the noise intensity of the fluctuations in CSF formation rate which in turn drives the fluctuations in their ICP?
Computing probabilities of clinically relevant events
The mathematical formulation of the problem posed in the previous paragraph is: given that a patient's ICP is currently x mmHg, where x is an arbitrary value, what is the probability that the ICP will exceed a critical threshold 'b' (mmHg) at a future time? Mathematically stated: given that p(s) = x, find the following transition probability-- P[p(t) > b | p(s) = x], t > s. Simple though the question seems, finding the answer requires computing the conditional probability distributions of the CSF process. Since the conditional probability distributions follow the Fokker-Planck partial differential equation, the problem is non-trivial, but Karlin and Taylor  circumvent the difficulty by solving a boundary-value problem associated with this dynamically changing probability. They show that the required probability satisfies a nonlinear ordinary differential equation which must be solved subject to two conditions on the probability that are natural consequences of the current ICP level when it is at one of the two extreme points of the range of ICP values under consideration. It is these conditions that give rise to the term 'boundary value problem.'
Results and Discussion
We state and discuss the significance of the mathematical results, deferring their proofs to the Appendices (Additional file 1, Additional file 2 and Additional file 3) in the interest of maintaining clarity of exposition. Our first result is the exact analytical solution to the stochastic Marmarou model.
Solution to stochastic Marmarou model with constant rate infusion
The proof is provided in additional file 1: Solving the stochastic Marmarou model. Note that the solution to the stochastic Marmarou model is found through an "integrating factor" which involves an integration constant, the evaluation of which necessitates a unit of 1/min unit for the 2 inside the exponent of the exponential function. The noise intensity parameter σ and the Brownian Motion process W(t) in the solution show the explicit influence of noise on the evolution of the ICP, underscoring the importance of modeling the noise in the clinical ICP data. In addition to the practical utility of offering a closed-form analytical solution, this result has value for another reason: it shows explicitly that noise cannot be averaged away when the process is nonlinear. If the Brownian motion process W(t) entered the solution for p(t) in an additive linear way, its effect would disappear on average. But the Brownian motion process enters the solution in a highly nonlinear fashion, making it impossible to average out its effect to zero. Finally, the solution depends upon the noise intensity parameter σ in a mathematically continuous way, a fact that is meaningful because the result shows that the solution to the deterministic Marmarou model  emerges as the special case corresponding to σ = 0 mL/min, and so, it is natural to ask if the simpler deterministic model would suffice when the noise intensity is small. Should the influence of noise be negligible in a particular case, the value of σ will be very small, and, because of the mathematical continuity in its dependence upon σ, the stochastic solution will be very close to the deterministic solution in such a case, and we may use the simpler deterministic model with confidence. However, the stochastic model is preferable in general for two reasons: it captures the dynamics of the ICP data better than the deterministic model when the noise intensity is larger, and furthermore, the stochastic model characterizes the risk profile of the patient probabilistically. Almost tautologically, the deterministic model cannot evaluate the risks due to the errors that are an inseparable part of medical data because deterministic modeling philosophy sees the future as completely predictable from the present situation. These considerations suggest that, from a conservative modeling perspective, incorporating the influence of noise into the dynamics is conceptually more defensible.
In principle, the solution contains all the transient probability distributions of the ICP process that characterize it on its way to equilibrium. In practice, mathematical difficulties may make these transient distributions hard to extract from the solution. But we can still compute the probability of the critical events by using a methodology that does not depend on that knowledge. And we can still draw useful information about the nature of the process at steady-state. Next, we find the steady-state probability distribution of the ICP process.
Steady-state probability distribution of ICP
The steady-state probability distribution of the ICP is gamma with the parameters shown p.149 in , and will exist provided that the noise intensity parameter σ satisfies the condition: .
Our next three results are motivated by the following considerations. A larger cerebrospinal fluid resistance R tends to increase ICP by increasing the pressure due to the circulatory CSF component. This is a direct consequence of Davson's equation : ICPCSF = (resistance to CSF outflow) × (CSF formation) + (pressure in sagittal sinus). This naturally leads to the following questions. How will the intensity of the fluctuations influence the relationship between resistance and ICP? The same relationship may hold on average, but, as anticipated in the solution to the stochastic Marmarou model, it may be moderated by the noise intensity parameter because of the nonlinearity of the ICP process. How will the intensity of fluctuations affect the average steady-state ICP--is the average steady-state ICP smaller or larger when the intensity of fluctuations increases? Finally, will the intensity of fluctuations attenuate or amplify the effect of resistance to CSF flow on the average steady-state ICP?
Relationship between average steady-state ICP and cerebrospinal fluid resistance
The average steady-state ICP, denoted by μ, increases with the cerebrospinal fluid resistance R--thus the relationship between R and ICP holds on average.
The steady-state probability distribution of ICP is gamma with the parameters shown in the previous subsection. From well-known properties of the gamma distribution, it follows that the steady-state mean ICP level μ is given by: . Therefore, . From the expression for , it is clear that the average ICP level does indeed increase with R, provided that . This condition is satisfied, using the values of the parameters in the previous subsection. Thus, the increasing relationship between the actual ICP level and the cerebrospinal fluid resistance, predicted by Davson's equation when ICP is conceptualized as a deterministic process, also holds on average at steady-state when ICP is modeled as a stochastic process.
Relationship between average steady-state ICP and noise intensity
The average steady-state ICP level, decreases with the intensity of fluctuations, measured by the infinitesimal variance parameter σ2.
From the relationship derived in the previous subsection, , it is clear that μ decreases as σ2 increases. A larger noise intensity corresponds to greater variation in the CSF input flow rate which translates into greater variation in ICP, and these larger fluctuations could cause the average ICP level to increase, decrease or remain unaffected. The nonlinear influence of the parameters of CSF flow dynamics on ICP level turns out to reduce the average ICP value when the fluctuations in ICP are greater. This is an outcome that one would expect to find when steady-state has been achieved--when the transition probabilities have settled down to constant levels so that the probability distribution of ICP is no longer changing over time. This mathematical finding could be tested by separating a random sample of patients into two groups, such that one group has more variability in its ICP levels (due to higher variability in its CSF input flow rate) than the other group, and then conducting a statistical test of significance--such as a t-test--on the difference in mean ICP levels in these two groups at steady-state.
Effect of noise intensity on the relationship between average steady-state ICP and cerebrospinal fluid resistance
The resistance increases the ICP on average by a smaller amount when the intensity of fluctuations is higher.
From , it is clear that a higher σ2 will dampen the effect of the cerebrospinal resistance on the average steady-state ICP level. This is an outcome that one would expect to find at steady-state. The mathematical finding could be tested by separating a random sample of patients into two groups, such that one group has more variability in its ICP levels than the other group (due to higher variability in its CSF input flow rate), and then correlating the mean ICP level with the cerebrospinal resistance in each group at steady-state. According to the mathematics, the correlation should be smaller in the group with more variable ICP. Given the linear relationship between the steady-state mean and the cerebrospinal resistance, a simple correlation coefficient such as the Pearson product moment should suffice.
Next we turn our attention to dynamic management of the patient's risk. Risk may be quantified in terms of the probability of the onset of some critical event, say the ICP exceeding a dangerously high level. Given the current value of the patient's ICP, what is the probability that it will exceed a high level? Such a probability is intrinsically dynamic because it depends upon the patient's current condition (their current ICP), the dynamics of the patient's CSF flow and the noise intensity σ2. We want to understand how the probability is influenced by important clinical characteristics of the patient such as their resistance to CSF flow, and by the noise intensity.
Computing clinically relevant dynamic probabilities
The integrals defining s(x) and S(x) are indefinite at the lower end because the final answer is unaffected by its choice. For our clinical applications, it is natural to take the lower end point to be zero.
The stochastic generalization of the Marmarou model offers a tractable analytical description of the noisy ICP dynamics and yields insights into the impact of noise. The SDE offers a rigorous analytical framework to study issues of clinical interest and neurological significance such as the patient's risk. A key clinical implication is that fluctuations in the CSF formation rate--which increase the fluctuations in ICP-- should be minimized to lower the patient's risk. Future work could extend the framework developed in this research to accommodate the non-zero reference pressure case. Finally, the stochastic differential equation framework, in conjunction with nonlinear control theory, can be used to develop a nonlinear automatic controller to regulate shunts to facilitate continuous CSF drainage.
The author is grateful to Dr. Marek Czosnyka, Neuroscience Group, University of Cambridge, UK, for introducing him to the key issues in CSF dynamics and hydrocephalus, for making data available from infusion studies conducted at Addenbrooke's Hospital, UK, and to the late Professor Anthony Marmarou for suggesting the probabilistic computations for critical events.
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