- Research
- Open Access
Experimental designs for a Benign Paroxysmal Positional Vertigo model
- Santiago Campos-Barreiro^{1} and
- Jesús López-Fidalgo^{1}Email author
https://doi.org/10.1186/1742-4682-10-21
© Campos-Barreiro and López-Fidalgo; licensee BioMed Central Ltd. 2013
Received: 22 November 2012
Accepted: 23 February 2013
Published: 19 March 2013
Abstract
Background
The pathology of the Benign Paroxysmal Positional Vertigo (BPPV) is detected by a clinician through maneuvers consisting of a series of consecutive head turns that trigger the symptoms of vertigo in patient. A statistical model based on a new maneuver has been developed in order to calculate the volume of endolymph displaced after the maneuver.
Methods
A simplification of the Navier‐Stokes problem from the fluids theory has been used to construct the model. In addition, the same cubic splines that are commonly used in kinematic control of robots were used to obtain an appropriate description of the different maneuvers. Then experimental designs were computed to obtain an optimal estimate of the model.
Results
D‐optimal and c‐optimal designs of experiments have been calculated. These experiments consist of a series of specific head turns of duration Δ t and angle α that should be performed by the clinician on the patient. The experimental designs obtained indicate the duration and angle of the maneuver to be performed as well as the corresponding proportion of replicates. Thus, in the D‐optimal design for 100 experiments, the maneuver consisting of a positive 30° pitch from the upright position, followed by a positive 30° roll, both with a duration of one and a half seconds is repeated 47 times. Then the maneuver with 60° /6° pitch/roll during half a second is repeated 16 times and the maneuver 90° /90° pitch/roll during half a second is repeated 37 times. Other designs with significant differences are computed and compared.
Conclusions
A biomechanical model was derived to provide a quantitative basis for the detection of BPPV. The robustness study for the D‐optimal design, with respect to the choice of the nominal values of the parameters, shows high efficiencies for small variations and provides a guide to the researcher. Furthermore, c‐optimal designs give valuable assistance to check how efficient the D‐optimal design is for the estimation of each of the parameters. The experimental designs provided in this paper allow the physician to validate the model. The authors of the paper have held consultations with an ENT consultant in order to align the outline more closely to practical scenarios.
Keywords
Background
Introduction
First described by Bárány [1], Benign Paroxysmal Positional Vertigo (BPPV) is the most common vestibular disorder leading to vertigo. These vestibular symptoms are precipitated when the orientation of the head or body is changed relative to gravity, provoking brief periods (2‐3 minutes) of vertigo, imbalance, and nausea. These changes can occur during daily activities such as lying down in bed or reaching up to retrieve an object from a high shelf. Benign Paroxysmal Positional Vertigo is commonly called top‐shelf vertigo[2].
When does BPPV occur? Semicircular canals are identified as the origin of BPPV. There, calcium carbonate particles (C a C O_{3}) called otoliths, which are normally affixed to the canal walls, are detached by the aforementioned head or body changes. This extra mass floating in the endolymph causes an abnormal movement of the cupula, since these particles displace more volume of endolymph than usual. The brain misinterprets this displacement and sends erroneous information to the eyes, provoking a characteristic ocular nystagmus and the subsequent vertigo.
Dix and Hallpike [3] were pioneers in developing maneuvers which led to the detection of BPPV. These maneuvers consist of a series of consecutive head turns that trigger ocular responses in the patient, on the basis of which clinicians can determine whether a patient suffers from BPPV or not. Rabbit [4] developed a model which calculates the volume of endolymph displaced when the Dix and Hallpike maneuver is put into practice. In this model, a mathematical approximation consisting of a curve crossing the different angular positions was used. This is a theoretical model never validated with real data as far as the authors know. In this paper, the model is particularized to a specific real situation and an experimental plan is produced. In our case, a maneuver composed of two consecutive turns has been developed: a positive pitch (turning the head back) from the upright position, followed by a positive roll (turning the head right), as these are the most common two head movements that trigger the above‐mentioned nystagmus. In order to reflect a more realistic situation, a cubic interpolation between points has been carried out. Another advantage of carrying out the cubic interpolation with respect to Rabbit’s model, is to obtain an analytical expression of the curve. This will permit us to design statistical experiments aimed at deriving an optimal estimation of the unknown parameters of the model.
Optimal experimental design
The problem of determining which set of observations to collect is what will define the design. It is common to say that the input variables are controlled by the researcher, while the unknown parameters are determined by nature. Optimal Design of Experiments theory allows us to find the best design in the sense of obtaining an optimal estimate of the parameters of the model. Next, basic concepts of this theory will be briefly presented as well as the two main criteria to obtain this optimal estimation. Following that, we will explain how the model for the maneuver has been constructed and finally, optimal designs for this model are calculated both for discrete and continuous design space.
The collection of weights, p_{ i }=ξ(z_{ i }), provide a probability measure on χ supported on the points z_{1},…,z_{ k }. Thus, an experiment will be replicated about N p_{ i } times on value z_{ i }. Kiefer [5] pioneered this approach, and its many advantages are well documented in design monographs, see Silvey [6] for example. This approach has been applied to optimal treatment allocation [7], optimal estimation of kinetic parameters of the Michaelis‐Menten model [8] and the Arrhenius equation [9]. In what follows, the approximate design approach is adopted without loss of generality, restricting the attention to designs with a finite set of support points. For convenience, the design is described using a two‐row matrix, with the support points displayed in the first row and their corresponding proportions of observations in the second row (2).
where p_{ i }=ξ(z_{ i }) is the proportion of observations to be taken at point z_{ i } (see e.g.[10]). The covariance matrix of the least squares and maximum likelihood estimator is asymptotically proportional to the inverse of this matrix [6]. The use of this matrix is very important when it comes to designing the experiment in an optimal way.
The objective to be achieved is to find a design which gives the best estimation of the parameters (or linear functions of them), usually by using the least squares method or maximum‐likelihood estimation method. Through what it is defined as criteria Φ, we will be able to measure the accuracy of the design and to compare different designs of the same model.
The design criteria used in this work for estimating the model parameters are D‐optimality and c‐optimality [11]. The D‐optimality criterion minimizes the volume of the confidence ellipsoid of the parameters and is given by Φ_{ D }[M(ξ,θ)]=detM(ξ,θ)^{−1/m}, where m is the number of parameters in the model. The D‐optimal design will be that which minimizes the function Φ_{ D }[M(ξ,θ)]. The c‐optimality criteria is used to estimate a linear combination of the parameters, say c^{ T }θ; it is the variance of this estimate which is Φ_{ c }[M(ξ,θ)]=c^{ T }M^{−1}(ξ,θ) c. It is known that these criteria are all convex and nonincreasing functions of the designs and so, designs with small criterion values are desirable [6]. A design that minimizes one of these functions Φ over all the designs on χ is called a Φ‐optimal design, or more specifically, a D‐ or a c‐optimal design, respectively.
The function f^{ T }(z)M^{−1}(ξ^{∗},θ)f(z) is known as the generalized variance. This important theorem provides methods for constructing optimal designs [11, 12].
Derivation of a model for the maneuver
where stands for the tangential acceleration, being the angular acceleration of the head relative to the ground‐fixed inertial frame resolved into the head‐fixed frame and the vector running from the head‐fixed coordinate system’s origin to the centerline of the canal. The parameterization of is made with respect to the arc length s (also called natural parameter). The head‐fixed coordinate system was defined when the subject was in the upright position prior to movement of the head. The constant ρ stands for the density of the canal and l_{ n } is the length covered by the otolith.
In this equation, is the gravitational acceleration, is the unit normal tangent vector to the canal centerline and is the velocity of the particle. The constants A_{ s }, N, A_{ e }, a, ρ_{ s }, ρ_{ e } and μ_{ e }, stand for frontal area of the particle, number of particles which are floating inside the canal, cross‐sectional area of the canal, radius of the particle, density of the particle, density of the endolymph and endolymph viscosity, respectively.
where is the angular acceleration referred to the inertial system (for example the clinician who makes the maneuvers) and M(t) a rotation matrix. Since these maneuvers consist of a pitch (y‐axis) followed by a roll movement (x‐axis), the vector for the pitch is (0,Ω(t),0)^{ T } and for the roll it is (Ω(t),0,0)^{ T }.
where A_{ s }=Π a^{2}, A_{ e }=Π b^{2}, b stands for the radius of the cross‐sectional area of the canal and [2].
Optimal experimental designs
Physical parameters
Parameter | Value |
---|---|
A_{ s } : Frontal area of the particle | 3.14×10^{−4}c m^{2} |
ρ: density of the canal | 1.0 g c m^{−3} |
ρ_{ s }: density of the particle | 2.7 g c m^{−3} |
ρ_{ e }: density of the endolymph | 1.0 g c m^{−3} |
μ_{ e }: viscosity of the endolymph | 8.5×10^{−3}d y n s^{−1}c m^{−1} |
g: gravitational acceleration | 981 c m s^{−2} |
D‐optimal design for a discrete design space
Values of the generalized variance
Π/6 | Π/4 | Π/3 | 5Π/12 | Π/2 | |
---|---|---|---|---|---|
0.5 | 1.96 | 2 | 1.24 | 1.53 | 2 |
1 | 1.93 | 1.34 | 0.19 | 0.45 | 0.4 |
1.5 | 2 | 1.38 | 0.06 | 0.29 | 0.24 |
Sensitivity analysis of the D‐optimal design
Values of the efficiency
0.1 | 0.7 | xx‐xx 0.85 | 0.9 | 2 | |
---|---|---|---|---|---|
0.015 | 37 % | 98% | 98% | 78% | 28% |
0.2 | 35% | 97% | 100% | 80% | 31% |
0.9 | 29% | 84% | 90% | 80% | 30 |
D‐optimal design for a continuous design space
c‐optimal design for a discrete design space
If we are interested in estimating a linear combination of the parameters, say c^{ T }θ, then we use the c‐optimality criterion. An elegant way for finding an optimal design that estimates a linear combination of the parameters was given by Elfving [18]. This method is nicely illustrated and explained, e.g. by Chernoff [19], Kitsos [20] or Wiley[10].
For a given regression problem with regression function f(Δ t,α,θ), the method first defines the Elfving’s set given by set G, the convex hull of {f(Δ t,α,θ^{(0)})∪−f(Δ t,α,θ^{(0)})}, θ^{(0)} being a nominal value for the parameter θ. This means that the set G is the smallest convex set containing {f(Δ t,α,θ^{(0)})∪−f(Δ t,α,θ^{(0)})}. The point of intersection of the straight line defined by the vector c with the boundary of the Elfving’s set determines the c‐optimal design, ξ^{∗}, as a convex combination of the vertices of G. These vertices provide the support points of the optimal design. The weights in the convex combinations are the weights of the optimal design. Furthermore, Φ_{ c }(ξ^{∗})=c^{ T }M^{−1}(ξ,θ) c=(∥ c ∥ / ∥ c^{∗} ∥)^{2}, where c^{∗} is the vector defined by the cut point of the straight line defined by c with the boundary of G.
provides a way to see how good the D‐optimal design is at estimating each of the parameters. In the case of θ_{1}, the D‐optimal design and c‐optimal design are compared and the efficiency is around 75%, while for θ_{2}, and are compared, having an efficiency of around 85%. These results show the D‐optimal design is more efficient for estimating θ_{2} than for estimating θ_{1}, that is, with this design, the test power for testing {H_{0}:θ_{2}=0} will be greater that the test power for {H_{0}:θ_{1}=0}.
c‐optimal design for a continuous design space
As we can observe, the results obtained are quite similar to the case concerning the discrete design space, except for the estimation of θ_{1}+θ_{2}, where for the continuous case the c‐optimal design is only supported at one point, although in both cases the angle of turn is similar. If the number of support points in a c‐optimal design is less than the number of parameters, this design allows the computation of the maximum likelihood estimate of this linear combination. But, in this case, not all the parameters are identifiable individually, that is, some of them cannot be estimated.
Discussion
The present biomechanical model was derived to provide a quantitative basis for the detection of BPPV. This model is based on a maneuver consisting of two consecutive head turns. These are the most common head movements leading to vertigo symptoms. We would like to remark that although the model can only be applied for this specific maneuver, it could be extended to other types. The experiment, that is, the duration and angle of the head movements to be applied to the patients should be based on the design provided by the D‐optimal design since it helps us estimate the parameters simultaneously, minimizing the confidence ellipsoid. The c‐optimal design is used either for estimating linear combinations of the parameters, or for estimating the parameters separately. But in this case, it also provides valuable assistance to check how efficient the D‐optimal design is for the estimation of each of the parameters. This is an interesting check of the sensitivity since a D‐optimal design could be quite efficient for estimating a particular parameter but quite inefficient for estimating another one.
will be minimized. Symbol ∝ stands for “asymptotically proportional” in this case.
Finally, we would like to point out that, as far as the authors know, the models found in the published works describing this sort of maneuvers have not been validated with data yet. The clinicians hold that the extra volume of endolymph displaced by the otoliths are directly related to the eye movements provoked in the patient under vertigo symptoms. Therefore, in some way, to validate the model, response y should be measured through some variable related to eye movement.
Authors’ information
S. Campos is pursuing a PhD under the supervision of Prof. López‐Fidalgo who is Professor of Statistics and Dean of the Industrial Engineering School of the University of Castilla‐La Mancha (Spain). He is an ISI Elected Member and holds Visiting Positions at the University of Manchester, Institute of Sciences and Technology (UMIST), the University of California, Los Angeles (UCLA) and the University of California, Riverside (UCR). Prof. López‐Fidalgo has written numerous academic publications; he is the former Editor of the Bulletin of the Spanish Statistical Society, Editor of the proceedings MODA8 (Springer), Associate Editor of Test and Sankhya B. He has published more than 70 papers, in such publications as the Journal of the American Statistical Association, Journal of the Royal Statistical Society, series B, Bioinformatics or Pharmaceutical Statistics, amongst others. Prof. López‐Fidalgo has also been representative of the Spanish Agency of Research for Mathematics.
Declarations
Acknowledgements
The authors have been sponsored by Ministerio de Ciencia e Innovación and fondos FEDER MTM2010‐20774‐C03‐01 and ‐03, Junta de Comunidades de Castilla‐La Mancha PEII10‐0291‐1850, Fondo Social Europeo FSE2007‐2013 and Junta de Castilla y León SA071A09. They would like to thank Mr. Ruíz for the help he provided with the mathematical model and ENT consultant Dr. Martín‐Sanz for his invaluable advice with regard to the clinical aspects. The authors would further like to express their gratitude to the referees for their comments and suggestions which helped improve the quality of the paper.
Authors’ Affiliations
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