# 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 .single_shell import eps_cell_single_shell
from .suspensionmodels import bhcubic_eps_model, eps_sus_MW
[docs]
def eps_cell_single_shell_wall(omega, km, em, kcp, ecp, kw, ew, dm, Rc, dw):
r"""Single 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:`R_\mathrm{c}`
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 = (1.0 - dw / (Rc + dw)) ** 3
epsi_w = ew - 1j * kw / (e0 * omega)
epsi_p = eps_cell_single_shell(omega, km, em, kcp, ecp, dm, Rc)
# model
epsi_cell = epsi_w * (
(2.0 * epsi_w + epsi_p - 2.0 * w * (epsi_w - epsi_p))
/ (2.0 * epsi_w + epsi_p + w * (epsi_w - epsi_p))
)
return epsi_cell
[docs]
def single_shell_wall_model(
omega, km, em, kcp, ecp, kw, ew, kmed, emed, p, c0, dm, Rc, dw
):
r"""Impedance of single 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, 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}`
ew: double
cell wall permittivity, value for :math:`\varepsilon_\mathrm{w}`
kw: double
cell wall conductivity, value for :math:`\sigma_\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}`
dw: double
cell wall thickness, value for :math:`d_\mathrm{w}`
Returns
-------
:class:`numpy.ndarray`, complex
Impedance array
Notes
-----
.. warning::
The unit capacitance is in pF!
The membrane conductivity is in uS/m!
See Also
--------
:meth:`impedancefitter.single_shell.single_shell_wall_model`
"""
c0 *= 1e-12 # use pF as unit
km *= 1e-6
# cell model
epsi_cell = eps_cell_single_shell_wall(omega, km, em, kcp, ecp, kw, ew, dm, Rc, dw)
epsi_med = emed - 1j * kmed / (e0 * omega)
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 single_shell_wall_bh_model(
omega, km, em, kcp, ecp, kw, ew, kmed, emed, p, c0, dm, Rc, dw
):
r"""Impedance of single shell model 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, 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}`
ew: double
cell wall permittivity, value for :math:`\varepsilon_\mathrm{w}`
kw: double
cell wall conductivity, value for :math:`\sigma_\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}`
dw: double
cell wall thickness, value for :math:`d_\mathrm{w}`
Returns
-------
:class:`numpy.ndarray`, complex
Impedance array
Notes
-----
.. warning::
The unit capacitance is in pF!
The membrane conductivity is in uS/m!
See Also
--------
:meth:`impedancefitter.single_shell.single_shell_wall_model`
"""
c0 *= 1e-12 # use pF as unit
km *= 1e-6
# cell model
epsi_cell = eps_cell_single_shell_wall(omega, km, em, kcp, ecp, kw, ew, dm, Rc, dw)
epsi_med = emed - 1j * kmed / (e0 * omega)
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