# The ImpedanceFitter is a package to fit impedance spectra to equivalent-circuit models using open-source software.
#
# Copyright (C) 2018, 2019 Leonard Thiele, leonard.thiele[AT]uni-rostock.de
# Copyright (C) 2018, 2019 Julius Zimmermann, julius.zimmermann[AT]uni-rostock.de
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <https://www.gnu.org/licenses/>.
from scipy.constants import epsilon_0 as e0
from .suspensionmodels import eps_sus_MW, bhcubic_eps_model
[docs]def eps_cell_double_shell(omega, km, em, kcp, ecp, kne, ene, knp, enp, dm, Rc, dn, Rn):
r"""Complex permittivity of double shell model
Parameters
----------
omega: :class:`numpy.ndarray`, double
list of frequencies
value for :math:`c_0`, unit capacitance in pF
em: double
membrane permittivity,membrane permittivity, value for :math:`\varepsilon_\mathrm{m}`
km: double
membrane conductivity, value for :math:`\sigma_\mathrm{m}` in :math:`\mu`\ S/m
ecp: double
cytoplasm permittivity, value for :math:`\varepsilon_\mathrm{cp}`
kcp: double
cytoplasm conductivity, value for :math:`\sigma_\mathrm{cp}`
ene: double
nuclear envelope permittivity, value for :math:`\varepsilon_\mathrm{ne}`
kne: double
nuclear envelope conductivity, value for :math:`\sigma_\mathrm{ne}` in mS/m
enp: double
nucleoplasm permittivity, value for :math:`\varepsilon_\mathrm{np}`
knp: double
nucleoplasm conductivity, value for :math:`\sigma_\mathrm{np}`
dm: double
membrane thickness, value for :math:`d_\mathrm{m}`
Rc: double
cell radius, value for :math:`R_\mathrm{c}`
dn: double
nuclear envelope thickness, value for :math:`d_\mathrm{n}`
Rn: double
nucleus radius, value for :math:`R_\mathrm{n}`
Returns
-------
:class:`numpy.ndarray`, complex
Permittivity array
"""
v1 = (1. - dm / Rc)**3
v2 = (Rn / (Rc - dm))**3
v3 = (1. - dn / Rn)**3
epsi_m = em + km / (1j * omega * e0)
epsi_cp = ecp + kcp / (1j * omega * e0)
epsi_ne = ene + kne / (1j * omega * e0)
epsi_np = enp + knp / (1j * omega * e0)
E3 = epsi_np / epsi_ne
E2 = ((epsi_ne / epsi_cp) * (2. * (1. - v3) + (1. + 2. * v3) * E3)
/ ((2. + v3) + (1. - v3) * E3))
E1 = ((epsi_cp / epsi_m) * (2. * (1. - v2) + (1. + 2. * v2) * E2)
/ ((2. + v2) + (1. - v2) * E2))
epsi_cell = (epsi_m * (2. * (1. - v1) + (1. + 2. * v1) * E1)
/ ((2. + v1) + (1. - v1) * E1))
return epsi_cell
[docs]def double_shell_model(omega, km, em, kcp, ecp, ene, kne, knp, enp, kmed, emed, p, c0, dm, Rc, dn, Rn):
r"""Impedance of double shell model
Parameters
----------
omega: :class:`numpy.ndarray`, double
list of frequencies
c0: double
value for :math:`c_0`, unit capacitance in pF
em: double
membrane permittivity,membrane permittivity, value for :math:`\varepsilon_\mathrm{m}`
km: double
membrane conductivity, value for :math:`\sigma_\mathrm{m}` in :math:`\mu`\ S/m
ecp: double
cytoplasm permittivity, value for :math:`\varepsilon_\mathrm{cp}`
kcp: double
cytoplasm conductivity, value for :math:`\sigma_\mathrm{cp}`
ene: double
nuclear envelope permittivity, value for :math:`\varepsilon_\mathrm{ne}`
kne: double
nuclear envelope conductivity, value for :math:`\sigma_\mathrm{ne}` in mS/m
enp: double
nucleoplasm permittivity, value for :math:`\varepsilon_\mathrm{np}`
knp: double
nucleoplasm conductivity, value for :math:`\sigma_\mathrm{np}`
emed: double
medium permittivity, value for :math:`\varepsilon_\mathrm{med}`
kmed: double
medium conductivity, value for :math:`\sigma_\mathrm{med}`
p: double
volume fraction
dm: double
membrane thickness, value for :math:`d_\mathrm{m}`
Rc: double
cell radius, value for :math:`R_\mathrm{c}`
dn: double
nuclear envelope thickness, value for :math:`d_\mathrm{n}`
Rn: double
nucleus radius, value for :math:`R_\mathrm{n}`
Returns
-------
:class:`numpy.ndarray`, complex
Impedance array
Notes
-----
.. warning::
The unit capacitance is in pF!
Equations for the single-shell-model [Feldman2003]_:
.. math::
\nu_1 = \left(1-\frac{d_\mathrm{m}}{R_\mathrm{c}}\right)^3
.. math::
\nu_\mathrm{2} = \left(\frac{R_\mathrm{n}}{R-d}\right)^3
.. math::
\nu_\mathrm{3} = \left(1-\frac{d_\mathrm{n}}{R_\mathrm{n}}\right)^3
.. math::
\varepsilon_\mathrm{c}^\ast = \varepsilon_\mathrm{m}^\ast\frac{2(1-\nu_\mathrm{1})+(1+2\nu_\mathrm{1})E_\mathrm{1}}{(2+\nu_\mathrm{1})+(1-\nu_\mathrm{1})E_\mathrm{1}}
.. math::
E_\mathrm{1} = \frac{\varepsilon_\mathrm{cp}^\ast}{\varepsilon_\mathrm{m}^\ast} \frac{2(1-\nu_\mathrm{2})+(1+2\nu_\mathrm{2})E_\mathrm{2}}{(2+\nu_\mathrm{2})+(1-\nu_\mathrm{2})E_\mathrm{2}}
.. math::
E_\mathrm{2} = \frac{\varepsilon_\mathrm{ne}^\ast}{\varepsilon_\mathrm{cp}^\ast}\frac{2(1-\nu_\mathrm{3})+(1+2\nu_\mathrm{3})E_\mathrm{3}}{(2+\nu_\mathrm{3})+(1-\nu_\mathrm{3})E_\mathrm{3}}
.. math::
E_\mathrm{3} = \frac{\varepsilon_\mathrm{np}^\ast}{\varepsilon_\mathrm{ne}^\ast}
.. math::
\varepsilon_\mathrm{m} = \varepsilon_\mathrm{m} - j \frac{\sigma_\mathrm{m}}{\varepsilon_0 \omega}
.. math::
\varepsilon_\mathrm{cp} = \varepsilon_\mathrm{cp} - j \frac{\sigma_\mathrm{cp}}{\varepsilon_0 \omega}
.. math::
\varepsilon_\mathrm{ne} = \varepsilon_\mathrm{ne} - j \frac{\sigma_\mathrm{ne}}{\varepsilon_0 \omega}
.. math::
\varepsilon_\mathrm{np} = \varepsilon_\mathrm{np} - j \frac{\sigma_\mathrm{np}}{\varepsilon_0 \omega}
.. math::
\varepsilon_\mathrm{cell}^\ast = \varepsilon_\mathrm{m}^\ast \frac{2 (1 - \nu_1) + (1 + 2 \nu_1) E_1}{(2 + \nu_1) + (1 - \nu_1) E_1}
.. math::
\varepsilon_\mathrm{sus}^\ast = \varepsilon_\mathrm{med}^\ast
\frac{(2 \varepsilon_\mathrm{med}^\ast + \varepsilon_\mathrm{cell}^\ast) - 2 p
(\varepsilon_\mathrm{med}^\ast - \varepsilon_\mathrm{cell}^\ast)}
{(2 \varepsilon_\mathrm{med}^\ast + \varepsilon_\mathrm{cell}^\ast) + p
(\varepsilon_\mathrm{med}^\ast - \varepsilon_\mathrm{cell}^\ast)}
.. math::
Z = \frac{1}{j \varepsilon_\mathrm{sus}^\ast \omega c_0}
In [Ermolina2000]_ , there have been reported upper/lower limits for certain parameters.
They could act as a first guess for the bounds of the optimization method.
+-----------------------------------+---------------+---------------+
| Parameter | lower limit | upper limit |
+===================================+===============+===============+
| :math:`\varepsilon_\mathrm{m}` | 1.4 | 16.8 |
+-----------------------------------+---------------+---------------+
| :math:`\sigma_\mathrm{m}` | 8e-8 | 5.6e-5 |
+-----------------------------------+---------------+---------------+
| :math:`\varepsilon_\mathrm{cp}` | 60 | 77 |
+-----------------------------------+---------------+---------------+
| :math:`\sigma_\mathrm{cp}` | 0.033 | 1.1 |
+-----------------------------------+---------------+---------------+
| :math:`\varepsilon_\mathrm{ne}` | 6.8 | 100 |
+-----------------------------------+---------------+---------------+
| :math:`\sigma_\mathrm{ne}` | 8.3e-5 | 7e-3 |
+-----------------------------------+---------------+---------------+
| :math:`\varepsilon_\mathrm{np}` | 32 | 300 |
+-----------------------------------+---------------+---------------+
| :math:`\sigma_\mathrm{np}` | 0.25 | 2.2 |
+-----------------------------------+---------------+---------------+
| R | 3.5e-6 | 10.5e-6 |
+-----------------------------------+---------------+---------------+
| :math:`R_\mathrm{n}` | 2.95e-6 | 8.85e-6 |
+-----------------------------------+---------------+---------------+
| d | 3.5e-9 | 10.5e-9 |
+-----------------------------------+---------------+---------------+
| :math:`d_\mathrm{n}` | 2e-8 | 6e-8 |
+-----------------------------------+---------------+---------------+
.. [Ermolina2000] Ermolina, I., Polevaya, Y., & Feldman, Y. (2000). Analysis of dielectric spectra of eukaryotic cells by computer modeling. European Biophysics Journal, 29(2), 141–145. https://doi.org/10.1007/s002490050259
See Also
--------
:meth:`impedancefitter.single_shell.single_shell_model`
"""
c0 *= 1e-12
km *= 1e-6
kne *= 1e-3
epsi_med = emed + kmed / (1j * omega * e0)
epsi_cell = eps_cell_double_shell(omega, km, em, kcp, ecp, kne, ene, knp, enp, dm, Rc, dn, Rn)
esus = eps_sus_MW(epsi_med, epsi_cell, p)
Ys = 1j * esus * omega * c0 # cell suspension admittance spectrum
Z_fit = 1 / Ys
return Z_fit
[docs]def double_shell_bh_model(omega, km, em, kcp, ecp, ene, kne, knp, enp, kmed, emed, p, c0, dm, Rc, dn, Rn):
r"""Impedance of double shell model
Parameters
----------
omega: :class:`numpy.ndarray`, double
list of frequencies
c0: double
value for :math:`c_0`, unit capacitance in pF
em: double
membrane permittivity,membrane permittivity, value for :math:`\varepsilon_\mathrm{m}`
km: double
membrane conductivity, value for :math:`\sigma_\mathrm{m}` in :math:`\mu`\ S/m
ecp: double
cytoplasm permittivity, value for :math:`\varepsilon_\mathrm{cp}`
kcp: double
cytoplasm conductivity, value for :math:`\sigma_\mathrm{cp}`
ene: double
nuclear envelope permittivity, value for :math:`\varepsilon_\mathrm{ne}`
kne: double
nuclear envelope conductivity, value for :math:`\sigma_\mathrm{ne}` in mS/m
enp: double
nucleoplasm permittivity, value for :math:`\varepsilon_\mathrm{np}`
knp: double
nucleoplasm conductivity, value for :math:`\sigma_\mathrm{np}`
emed: double
medium permittivity, value for :math:`\varepsilon_\mathrm{med}`
kmed: double
medium conductivity, value for :math:`\sigma_\mathrm{med}`
p: double
volume fraction
dm: double
membrane thickness, value for :math:`d_\mathrm{m}`
Rc: double
cell radius, value for :math:`R_\mathrm{c}`
dn: double
nuclear envelope thickness, value for :math:`d_\mathrm{n}`
Rn: double
nucleus radius, value for :math:`R_\mathrm{n}`
Returns
-------
:class:`numpy.ndarray`, complex
Impedance array
Notes
-----
.. warning::
The unit capacitance is in pF!
Note that here the Bruggeman-Hanai formula is used.
See Also
--------
:meth:`impedancefitter.double_shell.double_shell_model`
"""
c0 *= 1e-12
km *= 1e-6
kne *= 1e-3
epsi_med = emed + kmed / (1j * omega * e0)
epsi_cell = eps_cell_double_shell(omega, km, em, kcp, ecp, kne, ene, knp, enp, dm, Rc, dn, Rn)
esus = bhcubic_eps_model(epsi_med, epsi_cell, p)
Ys = 1j * esus * omega * c0 # cell suspension admittance spectrum
Z_fit = 1 / Ys
return Z_fit