Optimal design for parameter estimation in EEG problems in a 3D multilayered domain

  • Received: 01 April 2014 Accepted: 29 June 2018 Published: 01 April 2015
  • MSC : 62F12, 62K05, 65M32.

  • The fundamental problem of collecting data in the ``best way'' in order to assure statistically efficient estimation of parameters is known as Optimal Experimental Design. Many inverse problems consist in selecting best parameter values of a given mathematical model based on fits to measured data. These are usually formulated as optimization problems and the accuracy of their solutions depends not only on the chosen optimization scheme but also on the given data.We consider an electromagnetic interrogation problem, specifically one arising in an electroencephalography (EEG) problem, of finding optimal number and locations for sensors for source identification in a 3D unit sphere from data on its boundary. In this effort we compare the use of the classical $D$-optimal criterion for observation points as opposed to that for a uniform observation mesh. We consider location and best number of sensors and report results based on statistical uncertainty analysis of the resulting estimated parameters.

    Citation: H. T. Banks, D. Rubio, N. Saintier, M. I. Troparevsky. Optimal design for parameter estimation in EEG problems in a 3D multilayered domain[J]. Mathematical Biosciences and Engineering, 2015, 12(4): 739-760. doi: 10.3934/mbe.2015.12.739

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  • The fundamental problem of collecting data in the ``best way'' in order to assure statistically efficient estimation of parameters is known as Optimal Experimental Design. Many inverse problems consist in selecting best parameter values of a given mathematical model based on fits to measured data. These are usually formulated as optimization problems and the accuracy of their solutions depends not only on the chosen optimization scheme but also on the given data.We consider an electromagnetic interrogation problem, specifically one arising in an electroencephalography (EEG) problem, of finding optimal number and locations for sensors for source identification in a 3D unit sphere from data on its boundary. In this effort we compare the use of the classical $D$-optimal criterion for observation points as opposed to that for a uniform observation mesh. We consider location and best number of sensors and report results based on statistical uncertainty analysis of the resulting estimated parameters.


    Gender hormones regulate structure and function of many tissue and organ systems [1],[2]. Sexual dimorphism is defined as “the differences in appearance between males and females of the same species, such as in colour, shape, size, and structure, that are caused by the inheritance of one or the other sexual pattern in the genetic material” [3]. Some studies have reported that gender hormones affect renal morphology and physiology, and gender differences exist in the prevalence and prognosis of renal diseases. However, there are inconsistent results across study outcomes. There are also limited data available on this issue in humans [1],[2],[4],[5].

    It is emphasizes that women have a slower rate of decline in renal function than men. This condition can be due to gender differences in kidney size and weight, biological, metabolic and hemodynamic processes [1],[4]. In a study of 13,925 Chinese adults, Xu et al. [6] reported that the rates of decline in estimated glomerular filtration rate in men in both the at-risk group and the chronic kidney disease (CKD) group were faster compared to women, after referencing to the healthy group. Fanelli et al. [7] investigated gender differences in the progression of experimental CKD induced by chronic nitric oxide inhibition in rats. Their findings have indicated that female rats developed less severe CKD compared to males. According to Fanelli et al. [7], “female renoprotection could be promoted by both the estrogen anti-inflammatory activity and/or by the lack of testosterone, related to renin-angiotensin-aldosterone system hyperactivation and fibrogenesis” [p. 1]. Other studies have also reported that CKD was slightly more common among women than men [8],[9]. In a prospective, community-based, cohort study of 5488 participants from the Netherlands, Halbesma et al. [10] investigated gender differences as predictors of the decline of renal function. They found that systolic blood pressure and plasma glucose level negatively associated with renal function decline for both genders. Interestingly, this follow-up study demonstrated that waist circumference was positively associated with renal function in men only [10]. In another community-based, cohort study of 1876 Japanese adults, a higher body mass index was also found to be an independent risk factor for the development of CKD in women only [11]. On the other hand, compared with men, women tend to initiate hemodialysis with an arteriovenous fistula less frequently, and have greater risk of arteriovenous fistula failure [8]. Carrero et al. [5] also reported that women are less likely to receive kidney transplants than men. Further research is therefore needed to better understand the effect of gender on kidney function and health outcomes.

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