# The ImpedanceFitter is a package to fit impedance spectra to equivalent-circuit models using open-source software.
#
# Copyright (C) 2021 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 .double_shell import eps_cell_double_shell
from .suspensionmodels import eps_sus_MW, bhcubic_eps_model
[docs]def eps_cell_double_shell_wall(omega, km, em, kcp, ecp, kne, ene, knp, enp, kw, ew, dm, Rc, dn, Rn, dw):
r"""Complex permittivity of double shell model with cell wall
Parameters
-----------
omega: :class:`numpy.ndarray`, double
list of frequencies
em: double
membrane permittivity, value for :math:`\varepsilon_\mathrm{m}`
km: double
membrane conductivity, value for :math:`\sigma_\mathrm{m}`
ecp: double
cytoplasm permittivity, value for :math:`\varepsilon_\mathrm{cp}`
kcp: double
cytoplasm conductivity, value for :math:`\sigma_\mathrm{cp}`
ew: double
cell wall permittivity, value for :math:`\varepsilon_\mathrm{w}`
kw: double
cell wall conductivity, value for :math:`\sigma_\mathrm{w}`
dm: double
membrane thickness, value for :math:`d_\mathrm{m}`
Rc: double
cell radius, value for :math:`R_\mathrm{c}`
dw: double
cell wall thickness, value for :math:`d_\mathrm{w}`
Returns
-------
:class:`numpy.ndarray`, complex
Complex permittivity array
Note
----
Asami, K. (2002). Characterization of biological cells by dielectric spectroscopy. Journal of Non-Crystalline Solids, 305(1–3), 268–277. https://doi.org/10.1016/S0022-3093(02)01110-9
"""
w = (1. - dw / (Rc + dw))**3
epsi_w = ew - 1j * kw / (e0 * omega)
epsi_p = eps_cell_double_shell(omega, km, em, kcp, ecp, kne, ene, knp, enp, dm, Rc, dn, Rn)
# model
epsi_cell = epsi_w * ((2. * epsi_w + epsi_p - 2. * w * (epsi_w - epsi_p))
/ (2. * epsi_w + epsi_p + w * (epsi_w - epsi_p)))
return epsi_cell
[docs]def double_shell_wall_model(omega, km, em, kcp, ecp, kne, ene, knp, enp, kw, ew, kmed, emed, p, c0, dm, Rc, dn, Rn, dw):
r"""Impedance of double shell model with cell wall
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}`
kw: double
cell wall conductivity, value for :math:`\sigma_\mathrm{w}` in S/m
ew: double
cell wall permittivity, value for :math:`\varepsilon_\mathrm{w}`
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}`
dw: double
cell wall thickness
Returns
-------
:class:`numpy.ndarray`, complex
Impedance array
Notes
-----
.. warning::
The unit capacitance is in pF!
"""
c0 *= 1e-12
km *= 1e-6
kne *= 1e-3
epsi_med = emed + kmed / (1j * omega * e0)
epsi_cell = eps_cell_double_shell_wall(omega, km, em, kcp, ecp, kne, ene, knp, enp, kw, ew, dm, Rc, dn, Rn, dw)
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_wall_bh_model(omega, km, em, kcp, ecp, kne, ene, knp, enp, kw, ew, kmed, emed, p, c0, dm, Rc, dn, Rn, dw):
r"""Impedance of double shell model with cell wall using Bruggeman-Hanai approach
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
kw: double
cell wall conductivity, value for :math:`\sigma_\mathrm{w}` in S/m
ew: double
cell wall permittivity, value for :math:`\varepsilon_\mathrm{w}`
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}`
dw: double
wall thickness
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_wall(omega, km, em, kcp, ecp, kne, ene, knp, enp, kw, ew, dm, Rc, dn, Rn, dw)
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