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Four-shell ellipsoidal model employing multipole expansion in ellipsoidal coordinates

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Abstract

Although the head is more closely represented as an ellipsoid than a sphere, calculation in ellipsoidal coordinates is difficult. This paper presents a four shell ellipsoidal model, employing multipole expansion in ellipsoidal coordinates, for EEG, MEG, and evoked potential applications. Computational detail and insight into efficient calculation of the Lamé functions of the first and second kind are provided to demonstrate feasibilty. The Lamé function of the second kind, derived from the Lamé function of the first kind, can be computed at higher degrees by means of partial fraction expansion.

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References

  1. Babiloni F, Mattia D, Babiloni C, Astolfi L, Salinari S, Alessandra Basilisco, Rossini PM, Marciani MG, Cincotti F (2004) Multimodal integration of EEG, MEG, fMRI data for the solution of the neuroimage puzzle. Magn Reson Imaging 22:1471–1476

  2. Baker L (1989) C tools for scientists and engineers. McGraw-Hill, NY. ISBN 0-07-003355-2

    Google Scholar 

  3. Blimke J, Volkmer H, Spilker M, Myklebust J (1997) Comparison of evoked potential source localization and functional MRI in a four-shell ellipsoidal volume conductor. In: Proceedings of the 19th annual international conference on the IEEE engineering in medicine and biology society, pp 470–473

  4. Blimke JW (1999) Cerebral mapping: a comparison of electroencephalographic images and blood oxygenation weighted MR images. PhD thesis, Marquette University

  5. Buchner H, Waberski TD, Fuchs M, Wischmann HA, Wagner M, Drenckhahn R (1995) Comparison of realistically shaped boundary-element and spherical head models in source localization of early somatosensory evoked potentials. Brain Topogr 8(2):137–143

    Article  Google Scholar 

  6. Byerly WE (1893) An elementary treatise on Fourier’s series and spherical, cylindrical, and ellipsoidal harmonics with applications to problems in mathematical physics. Ginn and Company, Boston

    Google Scholar 

  7. Byrd PF, Friedman MD (1971) Handbook of elliptic integrals for engineers and scientists, 2nd edn. Springer, NY. ISBN 0-387-05318-2

    Google Scholar 

  8. Carlson BC (1979) Computing elliptic integrals by duplication. Numerische Mathematik 33:1–16

    Article  MATH  MathSciNet  Google Scholar 

  9. Chua LO, Lin PM (1975) Computer-aided analysis of electronic circuits, algorithms and computational techniques. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  10. Coburn B (1985) A theoretical study of epidural electrical stimulation of the spinal cord. Part II: Effects on long myelinated fibers. IEEE Trans Biomed Eng BME-32(11):978–986

    Article  Google Scholar 

  11. Coburn B, Sin WK (1985) A theoretical study of epidural electrical stimulation of the spinal cord. Part I: Finite element analysis of stimulus fields. IEEE Trans Biomed Eng BME-32(11):971–977

    Article  Google Scholar 

  12. Cohen D, Cuffin BN, Yunokuchi K, Maniewski R, Purcell C, Cosgrove GR, Ives J, Kennedy JG, Schomer DL (1990) MEG versus EEG localization test using implanted sources in the human brain. Ann Neurol 28:811–817

    Article  Google Scholar 

  13. MacKinnon CD, Webb RM, Silberstein P, Tisch S, Asselman P, Limousin P, Rothwell JC (1993) Stimulation through electrodes implanted near the subthalamic nucleus activates projections to motor areas of cerebral cortex in patients with Parkinson’s disease. Eur J Neurosci 16:1063–1065

    Google Scholar 

  14. Cuffin BN (1990) Effects of head shape on EEG’s and MEG’s. IEEE Trans Biomed Eng 37(1):44–52

    Article  Google Scholar 

  15. Cuffin BN (1995) A method for localizing EEG sources in realistic head models. IEEE Trans Biomed Eng 42(1):68–71

    Article  Google Scholar 

  16. Cuffin BN (1996) EEG localization accuracy improvements using realistically shaped head models. IEEE Trans Biomed Eng 43(3):299–303

    Article  Google Scholar 

  17. Cuffin BN, Cohen D, Yunokuchi K, Maniewski R, Purcell C, Cosgrove GR, Ives J, Kennedy J, Schomer D (1991) Tests of EEG localization accuracy using implanted sources in the human brain. Ann Neurol 29:132–138

    Article  Google Scholar 

  18. Dassios G, Kariotou F (2003) Magnetoencephalography in ellipsoidal geometry. J Math Phys 44(1):220–241

    Article  MATH  MathSciNet  Google Scholar 

  19. Dassios G, Miloh T (1999) Rayleigh scattering for the Kelvin-inverted ellipsoid. Q Appl Math 57(4):757–770

    MATH  MathSciNet  Google Scholar 

  20. de Munck JC (1988) The potential distribution in a layered anisotropic spheroidal volume conductor. J Appl Phys 64(2):464–470

    Article  Google Scholar 

  21. de Munck JC, Peters MJ (1993) A fast method to compute the potential in the multisphere model. IEEE Trans Biomed Eng 40(11):1166–1174

    Article  Google Scholar 

  22. Eshel Y, Abboud S (1997) Correlation between source asymmetry and scalp potential asymmetry in a prolate spheroid model of the head. Comput Biol Med 27(2):87–96

    Article  Google Scholar 

  23. Fender DH (1991) Models of the human brain and the surrounding media: their influence on the reliability of source localization. J Clin Neurophysiol 8(4):381–390

    Google Scholar 

  24. Fischer G, Tilg B, Wach P, Lafer G, Rucher W (1998) Analytical validation of the BEM-application of the BEM to the electrocardiographic forward and inverse problem. Comput Methods Programs Biomed 55:99–106

    Article  Google Scholar 

  25. George JS, Aine CJ, Mosher JC, Schmidt DM, Ranken DM, Schlitt HA, Wood CC, Lewine JD, Sanders JA, Belliveau JW (1995) Mapping function in the human brain with magnetoencephalography, anatomical magnetic resonance imaging, and functional magnetic resonance imaging. J Clin Neurophysiol 12(5):406–431

    Google Scholar 

  26. de Simone G, Verdencchia P, Schillaci G, Devereux RB (1998) Clinical impact of various geometric models for calculations of echocardiographic left ventricular mass. J Hypertens 16:1207–1214

    Google Scholar 

  27. He B, Li G, Zhang X (2003) Noninvasive imaging of cardiac transmembrane potentials within three-dimensional myocardium by means of a realistic geometry anisotropic heart model. IEEE Trans Biomed Eng 50(10):1190–1202

    Article  Google Scholar 

  28. Heine E (1881) Handbuch der Kugelfunctionen, vol 2. G. Reimer Verlag, Berlin

    Google Scholar 

  29. Hobson EW (1931) The theory of spherical and ellipsoidal harmonics. Cambridge University Press, Cambridge

  30. Hosek RS, Anthony Sances J, Jodat RW, Larson SJ (1978) The contributions of intracerebral currents to the EEG and evoked potentials. IEEE Trans Biomed Eng BME-25(5):405–413

    Article  Google Scholar 

  31. Iannetti GD, Niazy RK, Wise RG, Jezzard P, Brooks JC, Zambreanu L, Vennart W, Matthews PM, Tracey I (2005) Simultaneous recording of laser-evoked brain potentials and continuous, high-field functional magnetic resonance imaging in humans. Neuroimage 28:708–719

    Article  Google Scholar 

  32. Jenkins MA, Traub JF (1970) A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration. Numerische Mathematik 14:252–263

    Article  MATH  MathSciNet  Google Scholar 

  33. Johnston PR (1996) The potential for Laplacian maps to solve the inverse problem of electrocardiography. IEEE Trans Biomed Eng 43(4):384–393

    Article  Google Scholar 

  34. Kariotou F (2004) Electroencephalography in ellipsoidal geometry. J Math Anal Appl 290:324–342

    Article  MATH  MathSciNet  Google Scholar 

  35. Krings T, Chiappa KH, Cuffin BN, Buchbinder BR, Cosgrove GR (1998) Accuracy of electroencephalographic dipole localization of epileptiform activities associated with focal brain lesions. Ann Neurol 44(1):76–86

    Article  Google Scholar 

  36. Law SK, Nunez PL, Wijesinghe RS (1993) High-resolution EEG using spline generated surface Laplacians on spherical and ellipsoidal surfaces. IEEE Trans Biomed Eng 40(2):145–153

    Article  Google Scholar 

  37. Li G, He B (2001) Localization of the site of origin of cardiac activation by means of a heart-model-based electrocardiographic imaging approach. IEEE Trans Biomed Eng 48(6):660–669

    Article  Google Scholar 

  38. Miloh T (1973) Forces and moments on a tri-axial ellipsoid in potential flow. Isr J Technol 11:63–74

    MathSciNet  Google Scholar 

  39. Moon P, Spencer DE (1961) Field theory handbook, including coordinate systems, differential equations, and their solutions. Springer, Berlin

    MATH  Google Scholar 

  40. Morse PM, Feshbash H (1953) Methods of theoretical physics, vol 2. McGraw-Hill, NY

    Google Scholar 

  41. van Oosterom A (1991) History and evolution of methods for solving the inverse problem. J Clin Neurophysiol 8(4):371–380

    Article  Google Scholar 

  42. Plonsey R, Flemming DG (1969) Bioelectric phenomena. McGraw-Hill, NY

    Google Scholar 

  43. Szmurlo R, Sawicki B, Starzynski J, Wincenciak S (2006) A comparison of two models of electrodes for ECT simulations. IEEE Trans Magn 42(4):1395–1398

    Google Scholar 

  44. Roth BJ, Ko D, von Albertini-Carletti IR, Scaffidi D, Sato S (1997) Dipole localization in patients with epilepsy using the realistically shaped head model. Electroencephalogr Clin Neurophysiol 102:159–166

    Article  Google Scholar 

  45. Rush S, Driscoll DA (1969) EEG electrode sensitivity—an application of reciprocity. IEEE Trans Biomed Eng BME-16(1):15–22

    Article  Google Scholar 

  46. Srinivasan R, Nunez PL, Tucker DM, Silberstein RB, Cadusch PJ (1996) Spatial sampling and filtering of EEG with spline Laplacians to estimate cortical potentials. Brain Topogr 8(4):355–366

    Article  Google Scholar 

  47. Stancak A, Polacek H, Vrana J, Rachmanova R, Hoechstetter K, Tintera J, Scherg M (2005) EEG source analysis and fMRI reveal two electrical sources in the fronto-parietal operculum during subepidermal finger stimulation. Neuroimage 25:8–20

    Article  Google Scholar 

  48. Krings T, Chiappa KH, Cuffin BN, Cochius JI, Connolly S, Cosgrove GR (1999) Accuracy of EEG dipole source localization using implanted sources in the human brain. Clin Neurophysiol 110(1):106–114

    Google Scholar 

  49. Wilson FN, Bayley RH (1950) The electric field of an eccentric dipole in a homogeneous spherical conducting medium. Circulation 1:84–92

    Google Scholar 

  50. Yeh GCK, Martinek J (1957) The potential of a general dipole in a homogeneous conducting prolate spheroid. Ann N Y Acad Sci 65:1003–1006

    Article  Google Scholar 

  51. Yvert B, Bertrand O, Echallier JF, Pernier J (1996) Improved dipole localization using local mesh refinement of realistic head geometries: an EEG simulation study. Electroencephalogr Clin Neurophysiol 99:79–89

    Article  Google Scholar 

  52. Zanow F, Peters MJ (1995) Individually shaped volume conductor models of the head in EEG source localisation. Med Biol Eng Comput 33(4):582–588

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Max McGee National Research Center For Juvenile Diabetes, Medical College of Wisconsin, 8701 Watertown Plank Road, Milwaukee, WI, 53226, USA

    John Blimke

  2. Office of Science and Engineering Laboratories, Center for Devices and Radiologic Health, FDA, Silver Spring, MD, USA

    Joel Myklebust

  3. Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI, USA

    Hans Volkmer

  4. Department of Mathematics, Statistics, and Computer Science, Marquette University, Milwaukee, WI, USA

    Stephen Merrill

Authors
  1. John Blimke
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  2. Joel Myklebust
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  3. Hans Volkmer
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  4. Stephen Merrill
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Corresponding author

Correspondence to John Blimke.

Appendices

Appendix A: Computation of external ellipsoidal harmonics using partial fraction expansion

The external ellipsoidal harmonics require the Lamé function of the second kind derived from Eq. (5). This equation can easily be determined for the first few degrees. However, as the degree, n, increases, the elliptic integral with the squared polynomial in the denominator becomes increasingly difficult to solve. To reduce the computational difficulty, the equation can be simplified by means of partial fraction expansion, each fraction containing an elliptic integral.

The roots of the Lamé functions of the first kind are obtained (e.g. using the Jenkins-Traub method [2, 32]), and the corresponding equation is expanded. The resulting partial fractions for each class of the Lamé functions are shown below. For odd n, K m n (λ) is used to compute U m n (λ)

$$ \begin{aligned} U_n^m(\lambda)&= (2n+1)K_n^m(\lambda) \int\limits_\lambda^\infty \frac{d\tau}{[\tau(\tau^n+a_{n-3}\tau^{n-2}+a_{n-5}\tau^{n-4} +\cdots+a_1)]^{2}\sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}}\\ &= (2n+1)K_n^m(\lambda)\left[ \int\limits_\lambda^\infty \frac{Ad\tau} {\tau^{2}\sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}} + \int\limits_\lambda^\infty \frac{Bd\tau}{(\tau^{2}-\gamma_1)^{2} \sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}}\right.\\ &\quad \left.+ \int\limits_\lambda^\infty \frac{Cd\tau}{(\tau^{2}-\gamma_2) \sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}} + \cdots \right] \end{aligned} $$
(22)

where A, B, C,... are the constants of the partial fractions, and γ’s are the roots from the K m n (λ) polynomial. For n = 3, only the first three terms on the right side of Eq. (22) exist; for higher n, additional terms similar to those having B and C are present. For even n, Eq. (22) is the same except that A = 0.

Similarly, the expansion of V m n using L m n (λ) for even n is

$$ \begin{aligned} V_n^m(\lambda)&= (2n+1)L_n^m(\lambda)\int\limits_\lambda^\infty \frac{d\tau} {(\tau^{2}-b^{2}) [\tau(\tau^n+a_{n-2}\tau^{n-2}+a_{n-4}\tau^{n-4} +\cdots+a_0)]^{2}\sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}} \\& = (2n+1)L_n^m(\lambda)\left[\int\limits_\lambda^\infty \frac{Ad\tau} {\tau^{2}\sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}} + \int\limits_\lambda^\infty \frac{Bd\tau}{(\tau^{2}-\gamma_1)^{2} \sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}}\right.\\ &\quad \left.+ \int\limits_\lambda^\infty \frac{Cd\tau}{(\tau^{2}-\gamma_2) \sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}} + \cdots + \int\limits_\lambda^\infty {{Dd\tau}\over {(\tau^{2}-b^{2}) \sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}}}\right] \end{aligned} $$
(23)

The expansion of W m n using M m n (λ) is the same as Eq. (23) except that the terms involving (τ2−b 2) are replaced with those involving (τ2−c 2). For odd n, both V m n (λ) and W m n (λ) are the same for even n except that A = 0. The expansion of X m n using N m n (λ) for odd n is

$$ \begin{aligned} X_n^m(\lambda) = (2n+1)N_n^m\int\limits_\lambda^\infty \frac{d\tau} {(\tau^{2}-b^{2})(\tau^{2}-c^{2}) [\tau(\tau^n+a_{n-2}\tau^{n-2}+ \cdots+a_0)]^{2}\sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}}= (2n+1)N_n^m\left[\int\limits_\lambda^\infty \frac{Ad\tau} {\tau^{2}\sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}} + \int\limits_\lambda^\infty \frac{Bd\tau}{(\tau^{2}-\gamma_1)^{2} \sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}}\right. + \int\limits_\lambda^\infty \frac{Cd\tau}{(\tau^{2}-\gamma_2) \sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}} + \cdots + \int\limits_\lambda^\infty \frac{Dd\tau}{(\tau^{2}-b^{2}) \sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}}\left.\quad + \int\limits_\lambda^\infty \frac{Ed\tau}{(\tau^{2}-c^{2}) \sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}}\right] \end{aligned} $$
(24)

For even n, Eq. (24) can be used with A = 0.

Notice that the terms of all the classes (U m n (x), V m n (x), W m n (x), and X m n (x)) employ only five basic elliptic integrals for any degree. To simplify computations, these integrals can be replaced with the Carlson elliptic integrals [8] as follows:

$$ \begin{aligned} \int\limits_\lambda^\infty\frac{d\tau} {\tau^{2}\sqrt{\tau^{2}-b^{2}}\sqrt{\tau^{2}-c^{2}}}&= \frac{1} {3\lambda^3} R_D\left(1- \frac{u^{2}}{\lambda^{2}}, 1-\frac{v^{2}} {\lambda^{2}}, 1\right) \\ \int\limits_\lambda^\infty\frac{d\tau} {(\tau^{2}-b^{2})\sqrt{\tau^{2}-b^{2}} \sqrt{\tau^{2}-c^{2}}} &= \frac{1} {\lambda(b^{2}-c^{2})}\left[\sqrt{\frac{1-c^{2}/\lambda^{2}} {1-b^{2}/\lambda^{2}}}-R_F\left(1-\frac{u^{2}}{\lambda^{2}}, 1-\frac{v^{2}}{\lambda^{2}},1\right)\right. \\ &\quad \left.+\frac{c^{2}} {3\lambda^{2}}R_D\left(1-\frac{u^{2}}{\lambda^{2}}, 1-\frac{v^{2}} {\lambda^{2}},1\right)\right] \\ \int\limits_\lambda^\infty\frac{d\tau} {(\tau^{2}-c^{2})\sqrt{\tau^{2}-b^{2}} \sqrt{\tau^{2}-c^{2}}}= &\frac{1} {\lambda(c^{2}-b^{2})}\left[\sqrt{\frac{1-b^{2}/\lambda^{2}} {1-c^{2}/\lambda^{2}}}-R_F\left(1-\frac{u^{2}}{\lambda^{2}}, 1-\frac{v^{2}}{\lambda^{2}},1\right)\right. \\ &\quad \left.+\frac{b^{2}} {3\lambda^{2}}R_D\left(1-\frac{u^{2}}{\lambda^{2}}, 1-\frac{v^{2}} {\lambda^{2}},1\right)\right] \\ \int\limits_\lambda^\infty\frac{d\tau} {(\tau^{2}-\gamma)^{2}\sqrt{\tau^{2}-b^{2}} \sqrt{\tau^{2}-c^{2}}}= &\frac{1}{\lambda^3}\left\{\frac{1} {2(\gamma-b^{2})(\gamma-c^{2})} \left[\frac{\sqrt{\lambda^{2}-b^{2}}\sqrt{\lambda^{2}-c^{2}}} {1-\gamma/\lambda^{2}} \right.\right.\\ & \left.\quad+\frac{b^{2}c^{2}}{3\gamma} R_D\left(1-\frac{u^{2}}{\lambda^{2}},1-\frac{v^{2}} {\lambda^{2}},1\right)- \lambda^{2} R_F\left(1-\frac{u^{2}} {\lambda^{2}},1-\frac{v^{2}}{\lambda^{2}},1\right) \right] \\ &\quad \left.-\frac{1}{3\gamma}\left[1+\frac{\gamma^{2}-b^{2}c^{2}} {2(\gamma-b^{2})(\gamma-c^{2})}\right] R_J\left(1-\frac{u^{2}} {\lambda^{2}},1-\frac{v^{2}}{\lambda^{2}}, 1, 1-\frac{\gamma} {\lambda^{2}}\right)\right\} \\ \int\limits_\lambda^\infty\frac{d\tau} {(\tau^{2}-\gamma)\sqrt{\tau^{2}-b^{2}} \sqrt{\tau^{2}-c^{2}}}= &\frac{1}{3\lambda^3}R_J\left(1-\frac{u^{2}}{\lambda^{2}}, 1, 1-\frac{v^{2}}{\lambda^{2}},1-\frac{\gamma}{\lambda^{2}}\right) \end{aligned} $$

where R F , R D , and R J are the Carlson elliptic integrals of the first, second, and third kinds, respectively. u = b and v = c if b > c, or u = c and v = b if c > b. Note that these equations require that b ≠ c. The first three integrals are for degree of one and higher, the fourth one for degree of two and higher, and the last one for degree of three and higher. Also, because each Lamé function of the second kind is derived from the corresponding Lamé function of the first kind, there are 2n + 1 functions of the second kind at each degree, n. Combining all these functions indicates that there are 4n + 2 functions of both kinds. To ensure that the internal and external ellipsoidal harmonics align at the boundaries, it is necessary to normalize the Lamé functions using the normalization integral developed in Appendix B.

Appendix B: Computation of normalization integral

To ensure that Eq. (15) holds, the functions E m n (λ), E m n (μ), E m n (ν) must be normalized with the normalization integral [4, 28, 29]:

$$ c_{n,m}^4\int\limits_u^v\int\limits_0^u \frac{\left[\mu^{2}-\nu^{2}\right] \left[{\tilde{E}}_n^m(\mu){\tilde{E}}_n^m(\nu)\right]^{2} d\mu d\nu} {\sqrt{(\mu^{2}-b^{2}) (c^{2}-\mu^{2}) (b^{2}-\nu^{2}) (c^{2}-\nu^{2})}}=1 $$
(25)

where

$$ \begin{aligned} E_n^m(\lambda)&=c_{n,m}{\tilde{E}}_n^m(\lambda) \\ E_n^m(\mu)&=c_{n,m}{\tilde{E}}_n^m(\mu)\\ E_n^m(\nu)&=c_{n,m}{\tilde{E}}_n^m(\nu). \end{aligned} $$
(26)

\({\tilde{E}}_n^m(\psi)\) and E m n (ψ) are unnormalized and normalized, respectively. u = b and v = c if c > b or u = c and v = b if b > c. c n,m are the normalization constants.

The normalization integral has the products of the functions \({\tilde{E}}_n^m(\mu){\tilde{E}}_n^m(\nu),\) which are equal to \({\tilde{K}}_n^m(\mu){\tilde{K}}_n^m(\nu), {\tilde{L}}_n^m(\mu){\tilde{L}}_n^m(\nu), {\tilde{K}}_n^m(\mu){\tilde{K}}_n^m(\nu),\) or \({\tilde{K}}_n^m(\mu){\tilde{K}}_n^m(\nu).\) Starting with the product

$$ {\tilde{E}}_n^m(\mu){\tilde{E}}_n^m(\nu)= {\tilde{K}}_n^m(\mu){\tilde{K}}_n^m(\nu), $$

the integral indicates that the product is squared. Squaring it results in

$$ \left[{\tilde{K}}_n^m(\mu){\tilde{K}}_n^m(\nu)\right]^{2}= \left\{\begin{array}{ll} \sum\limits_{\begin{subarray}{l}{i=0,1,2,\ldots,n}\\{j=0,1,2,\ldots, n}\end{subarray}} a_{ij}\mu^{2i}\nu^{2j} &\quad\hbox{for\ even}\ n\\ \sum\limits_{\begin{subarray}{l}{i=0,1,2,\ldots,n}\\{j=0,1,2,\ldots, n}\end{subarray}} a_{ij}\mu^{2(i+1)}\nu^{2(j+1)} &\quad\hbox{for\ odd}\ n \end{array}\right. $$

where a ij ’s are the products of the coefficients of \({\tilde{K}}_n^m(\mu)\;\hbox{and}\;{\tilde{K}}_n^m(\nu).\) 2i, 2j, 2(i + 1), and 2(j + 1) are always even whether n is even or odd. These expressions are simplified because the squared product is intricate. Thus, each term in the normalization integral is

$$ \begin{aligned} \,&c_{n,m}^4\sum\limits_{\begin{subarray}{l}{i=0,1,2,\ldots,n}\\{j=0,1,2, \ldots,n}\end{subarray}}a_{ij}\int\limits_u^v\int\limits_0^u \frac{\mu^{2}-\nu^{2}}{\sqrt{(\mu^{2}-b^{2}) (c^{2}-\mu^{2}) (b^{2}-\nu^{2}) (c^{2}-\nu^{2})}} \mu^{2i}\nu^{2j} d\mu d\nu=1 \\ &\quad \quad \hbox{for\;even}\;n \\ &\, c_{n,m}^4 \sum\limits_{\begin{subarray}{l}{i=0,1,2,\ldots,n}\\{j=0,1,2,\ldots, n}\end{subarray}} a_{ij}\int\limits_u^v\int\limits_0^u \frac{\mu^{2}-\nu^{2}}{\sqrt{(\mu^{2}-b^{2}) (c^{2}-\mu^{2}) (b^{2}-\nu^{2}) (c^{2}-\nu^{2})}}\mu^{2(i+1)}\nu^{2(j+1)} d\mu d\nu=1 \\ &\quad \quad \hbox{for\;odd}\;n \\ \end{aligned} $$

where u = c and v = b if b > c, or u = b and v = c if c > b, and c n,m is the normalization constant. Hence, the Lamé functions necessitate two normalization integrals: one is for odd n, and the other for even n.

Arranging the normalization integral,

$$ \begin{aligned} &c_{n,m}^4\sum\limits_{\begin{subarray}{l}{i=0,1,2,\ldots,n}\\{j=0,1,2, \ldots,n}\end{subarray}} a_{ij}[I_{i+1}G_j - I_iG_{j+1}]=1 \quad\hbox{for\;even}\;n\\ &c_{n,m}^4\sum\limits_{\begin{subarray}{l}{i=0,1,2,\ldots,n}\\{j=0,1,2, \ldots,n}\end{subarray}} a_{ij}[I_{i+2}G_{j+1} - I_{i+1}G_{j+2}]=1 \quad\hbox{for\;odd}\;n \end{aligned} $$
(27)

where

$$ \begin{aligned} I_i&=\int\limits_u^v \frac{\mu^{2i} d\mu} {\sqrt{|\mu^{2}-b^{2}|}\sqrt{|\mu^{2}-c^{2}|}} \quad\hbox{and}\\ G_j&=\int\limits_0^u\frac{\nu^{2j} d\nu} {\sqrt{b^{2}-\nu^{2}}\sqrt{{c^{2}-\nu^{2}}}}. \end{aligned} $$

Solving I and G with the elliptic integrals [7] and reducing them results in

$$ \left.\begin{array}{l}P_0=K(k^\prime)\\ P_1=E(k^\prime)\\ P_{i+1}=\frac{(1-2i)k^{2}P_{i-1}+2i(1+k^{2})P_i}{2i-1}\\ I_i=v^{2i-1}P_i \end{array} \right\} \quad \hbox{for the}\;I\;\hbox{integrals and} $$
(28)
$$ \left. \begin{array}{l}P_0=K(k)\\ P_1=K(k) - E(k)\\ P_{j+1}=\frac{(1-2j)k^{2}P_{j-1}+2j(1+k^{2})P_j}{2j-1}\\ G_j=v^{2j-1}P_j\end{array} \right\} \quad \hbox{for the}\;G\;\hbox{integrals} $$
(29)

where k = c/b if b > c or k = b/c if c > b, and \(k^\prime=\sqrt{1-k^{2}}.\) K(k) and E(k) are the complete elliptic integrals of the first and second kinds, respectively. They can be replaced with the Carlson elliptic integrals [8]. Because P i+1 and P j+1 are recursive, I i and G j can easily be computed by programs at any degree n. Consequently, determining I i and G j yields c n,m that normalize \({\tilde{K}}_n^m(\psi).\)

Similarly,

$$ \left[{\tilde{L}}_n^m(\mu){\tilde{L}}_n^m(\nu)\right]^{2}= \sum\limits_{\begin{subarray}{l}{i=0,1,2,\ldots,n}\\{j=0,1,2,\ldots, n}\end{subarray}} a_{ij}|\mu^{2}-b^{2}||\nu^{2}-b^{2}|\mu^{2i}\nu^{2j} $$

where a ij ’s are the products of the coefficients of \({\tilde{L}}_n^m(\mu)\;\hbox{and}\;{\tilde{L}}_n^m(\nu).\) For odd n, separating, arranging, and simplifying the equation leads to

$$ c_{n,m}^4\sum\limits_{\begin{subarray}{l}{i=0,1,2,\ldots,n}\\{j=0,1,2, \ldots,n}\end{subarray}} a_{ij}\left|(I_{i+1}G_{j+2}-I_{i+2}G_{j+1})+b^{2}(I_{i+2}G_j- I_iG_{j+2})+b^4(I_iG_{j+1}-I_{i+1}G_j)\right|=1. $$
(30)

For even n, the subscripts of I and G are incremented by one. Equation (30) is applicable to the functions \({\tilde{M}}_n^m(\mu){\tilde{M}}_n^m(\nu)\) except that b’s are replaced with c’s. Additionally, for \({\tilde{N}}_n^m(\mu){\tilde{N}}_n^m(\nu)\) of even n,

$$ \begin{aligned} \,&c_{n,m}^4\sum\limits_{\begin{subarray}{l}{i=0,1,2,\ldots,n} \\{j=0,1,2, \ldots,n}\end{subarray}} a_{ij}\left| I_{i+3}G_{j+2}-I_{i+2}G_{j+3}+ (b^{2}+c^{2})\left[I_{i+1}G_{j+3}-I_{i+3}G_{j+1}\right]\right.\\&\quad + b^{2}c^{2}[I_{i+2}G_{j+1}-I_{i+1}G_{j+2}+ I_{i+3}G_j-I_iG_{j+3}]+ b^{2}c^{2}(b^{2}+c^{2})[I_iG_{j+2}-I_{i+2}G_j]\\&\left.\quad + (b^4+c^4)[I_{i+2}G_{j+1}-I_{i+1}G_{j+2}]+ b^4c^4[I_{i+1}G_j-I_iG_{j+1}]\right|=1. \end{aligned} $$

Again, for odd n, the subscripts are incremented by one. Thus, solving I and G by means of P i+1 and P j+1 results in c n,m that normalize the Lamé functions of the corresponding classes with Eq. (26).

In brief, the normalization integral normalizes the Lamé functions by setting their coefficients to obtain correct potentials. Arranging and simplifying the integral facilitates the computation of it by means of Eqs. (28) and (29) for computer programs. As a result, solving the normalization constant c n,m leads to normalizing the functions with Eq. (26).

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Blimke, J., Myklebust, J., Volkmer, H. et al. Four-shell ellipsoidal model employing multipole expansion in ellipsoidal coordinates. Med Biol Eng Comput 46, 859–869 (2008). https://doi.org/10.1007/s11517-008-0352-9

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