import numpy as np
import dcmri.pk as pk
[docs]
def aif_parker(t, BAT: float = 0.0) -> np.ndarray:
"""Population AIF model as defined by `Parker et al (2006) <https://onlinelibrary.wiley.com/doi/full/10.1002/mrm.21066>`_
Args:
t (array_like): time points in units of sec.
BAT (float, optional): Time in seconds before the bolus arrives. Defaults to 0 sec (no delay).
Returns:
np.ndarray: Concentrations in M for each time point in t. If t is a scalar, the return value is a scalar too.
References:
Adapted from a contribution by the QBI lab of the University of Manchester to the `OSIPI code repository <https://github.com/OSIPI/DCE-DSC-MRI_CodeCollection>`_.
Example:
>>> import numpy as np
>>> import dcmri as dc
Create an array of time points covering 20sec in steps of 1.5sec, which rougly corresponds to the first pass of the Paeker AIF:
>>> t = np.arange(0, 20, 1.5)
Calculate the Parker AIF at these time points, and output the result in units of mM:
>>> 1000*dc.aif_parker(t)
array([0.08038467, 0.23977987, 0.63896354, 1.45093969,
2.75255937, 4.32881325, 5.6309778 , 6.06793854, 5.45203828,
4.1540079 , 2.79568217, 1.81335784, 1.29063036, 1.08751679])
"""
# Check input types
if not np.isscalar(BAT):
raise ValueError('BAT must be a scalar')
# Convert from secs to units used internally (mins)
t_offset = np.array(t) / 60 - BAT / 60
# A1/(SD1*sqrt(2*PI)) * exp(-(t_offset-m1)^2/(2*var1))
# A1 = 0.833, SD1 = 0.055, m1 = 0.171
gaussian1 = 5.73258 * np.exp(
-1.0 * (t_offset - 0.17046) * (t_offset - 0.17046) / (2.0 * 0.0563 * 0.0563))
# A2/(SD2*sqrt(2*PI)) * exp(-(t_offset-m2)^2/(2*var2))
# A2 = 0.336, SD2 = 0.134, m2 = 0.364
gaussian2 = 0.997356 * np.exp(
-1.0 * (t_offset - 0.365) * (t_offset - 0.365) / (2.0 * 0.132 * 0.132))
# alpha*exp(-beta*t_offset) / (1+exp(-s(t_offset-tau)))
# alpha = 1.064, beta = 0.166, s = 37.772, tau = 0.482
sigmoid = 1.050 * np.exp(-0.1685 * t_offset) / \
(1.0 + np.exp(-38.078 * (t_offset - 0.483)))
pop_aif = gaussian1 + gaussian2 + sigmoid
return pop_aif / 1000 # convert to M
[docs]
def aif_tristan_rat(t, BAT=4.6 * 60, duration=30) -> np.ndarray:
"""Population AIF model for rats measured with a standard dose of
gadoxetate.
Args:
t (array_like): time points in units of sec.
BAT (float, optional): Time in seconds before the bolus arrives.
Defaults to 4.6 min.
duration (float, optional): Duration of the injection. Defaults to 30s.
Returns:
np.ndarray: Blood concentrations in M for each time point in t. If
t is a scalar, the return value is a scalar too.
References:
- Melillo N, Scotcher D, Kenna JG, Green C, Hines CDG, Laitinen I,
et al. Use of In Vivo Imaging and Physiologically-Based Kinetic
Modelling to Predict Hepatic Transporter Mediated Drug-Drug
Interactions in Rats.
`Pharmaceutics 2023;15(3):896 <https://doi.org/10.3390/pharmaceutics15030896>`_.
- Gunwhy, E. R., & Sourbron, S. (2023). TRISTAN-RAT (v3.0.0).
`Zenodo <https://doi.org/10.5281/zenodo.8372595>`_
Example:
.. plot::
:include-source:
>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> import dcmri as dc
Create an array of time points over 30 minutes
>>> t = np.arange(0, 30*60, 0.1)
Generate the rat input function for these time points:
>>> cb = dc.aif_tristan_rat(t)
Plot the result:
>>> plt.plot(t/60, 1000*cb, 'r-')
>>> plt.title('TRISTAN rat AIF')
>>> plt.xlabel('Time (min)')
>>> plt.ylabel('Blood concentration (mM)')
>>> plt.show()
"""
# Constants
dose = 0.0075 # mmol
Fb = 2.27 / 60 # mL/sec
# https://doi.org/10.1021/acs.molpharmaceut.1c00206
# (Changed from 3.61/60 on 07/03/2022)
# From Brown the cardiac output of rats is
# 110.4 mL/min (table 3-1) ~ 6.62L/h
# From table 3-4, sum of hepatic artery and portal vein
# blood flow is 17.4% of total cardiac output ~ 1.152 L/h
# Mass of liver is 9.15g, with density of 1.08 kg/L,
# therefore ~8.47mL
# 9.18g refers to the whole liver, i.e. intracellular tissue
# + extracellular space + blood
# Dividing 1.152L/h for 8.47mL we obtain ~2.27 mL/min/mL liver
# Calculation done with values in Table S2 of our article
# lead to the same results
Hct = 0.418 # Cremer et al, J Cereb Blood Flow Metab 3, 254-256 (1983)
VL = 8.47 # mL
# Scotcher et al 2021, DOI: 10.1021/acs.molpharmaceut.1c00206
# Supplementary material, Table S2
GFR = 1.38 / 60 # mL/sec (=1.38mL/min)
# https://doi.org/10.1152/ajprenal.1985.248.5.F734
P = 0.172 # mL/sec
# Estimated from rat repro study data using PBPK model
# 0.62 L/h
# Table 3 in Scotcher et al 2021
# DOI: 10.1021/acs.molpharmaceut.1c00206
VB = 15.8 # mL
# 0.06 X BW + 0.77, Assuming body weight (BW) = 250 g
# Lee and Blaufox. Blood volume in the rat.
# J Nucl Med. 1985 Jan;26(1):72-6.
VE = 30 # mL
# All tissues, including liver.
# Derived from Supplementary material, Table S2
# Scotcher et al 2021
# DOI: 10.1021/acs.molpharmaceut.1c00206
E = 0.4 # Liver extraction fraction, estimated from TRISTAN data
# Using a model with free E, then median over population of controls
# Derived constants
VP = (1 - Hct) * VB
Fp = (1 - Hct) * Fb
K = GFR + E * Fp * VL # mL/sec
# Influx in mmol/sec
J = np.zeros(np.size(t))
Jmax = dose / duration # mmol/sec
J[(t > BAT) & (t < BAT + duration)] = Jmax
# Model parameters
TP = VP / (K + P)
TE = VE / P
E = P / (K + P)
Jp = pk.flux_2cxm(J, [TP, TE], E, t)
cp = Jp / K
return cp * (1 - Hct)
