$q_c$ = Amount of Oxygen in Myoglobin... The one that we measure using the BSX Sensor. \begin{equation}VO_2 = HR * SpO_2 \end{equation}
Questions:
Why is $E_u = O_h#C_{Hemoglobin}$? $$ $$ Is $C_{Hemoglobin}$ actually the Pulse Oximeter reading? $$ $$ We know that $C=\frac{q_c}{V}$. Why is $C =\frac{q_c}{C_{Hemoglobin}}$?
$M_m$ = Myoglobin quantity $$ $$ $M_o$ = Oxygen quantity bound to Myoglobin$$ $$ Oxygen in Myoglobin = $O_m = \frac{M_o}{M_m}$
Oxygen concentration in Myoglobin is $O_{Myo} = \frac {M_{O2-Myo}}{M_{Myo}}$ $$ $$ Oxygen concentration in hemoglobin is $O_{Hb} = \frac {M_{O2-Hb}}{M_{Hb}}$ $$ $$ Flux of Oxygen from Hemoglobin to Myoglobin is $f_i = D * [O_{Hb} - O_{Myo}]$ $$ $$ Where, D is a conductance equivalent for fluid dynamics and is equal to $ D = \frac {k}{\mu}$ where k = permeability and $\mu = viscocity $ $$ $$ $\frac{dM_{O2-Myo}}{dt} = D * [\frac {M_{O2-Hb}}{M_{Hb}} - \frac {M_{O2-Myo}}{M_{Myo}}] - f_o $ $$ $$
I am trying to use hypothetical values of the parameters such as oxygen concentrations in hemoglobin and myoglobin to see what waveforms I get. However, I think that the model may not be adequately defined. We are not using any parameters other than oxygen concentrations in hemoglobin (measured by the pulse oximeter) and in myoglobin (measured by the BSX sensor).
Is there a way we can use the heart rate data and speed (or power) in our model? I understand that heart rate and speed are both reflected in the SmO2 reading.
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%matplotlib inline
import scipy.integrate as integrate
import matplotlib.pyplot as plt
import numpy as np
from __future__ import division
import random
pi = np.pi
sqrt = np.sqrt
cos = np.cos
sin = np.sin
# Following parameters will be used to create sinusoidal-like signals as hypothetical signals for input parameters.
samp_f = 100
freq = 72
t = np.linspace(0,120,freq*5)
wt = 2*pi*freq*t #np.linspace(0,1,samp_f)
#Following function removes sets the negatives values of any given signal to zero.
def signal_clipper (signal):
for i in range(len(signal)):
if (signal[i]<=0):
signal[i] = 0
else:
signal[i] = signal[i]
return signal
M_Hb= 10 # mg/dl - an arbitrary value
M_O2Hb= signal_clipper(random.random()*sin(wt)) # random signal
M_Myo = 2 # mg/dl - an arbitrary value
M_O2Myo = signal_clipper(random.random()*cos(wt)) # random signal
D = 4 #signal_clipper(random.random()*cos(wt)) # random signal
fo = 1 # A random constant value for out-flux.
# Following is the actual differential equation.
def o2transport (M_O2Myo,t):
dmo2dt = -D*(M_O2Myo/M_Myo) + D*(M_O2Hb/M_Hb) - fo
return dmo2dt
zinit = [0.9]*len(M_O2Hb) # Initial condition.
z = integrate.odeint(o2transport,zinit,t) # Actual ODE Solver
print "Number of cells that had NaNs were", np.isnan(z).sum(),"out of ",z.size,"elements."
z = np.nan_to_num(z)
# Plot containing subplots showing the input signals such as values of M_O2Myo, M_Myo, M_O2Hb, and M_Hb.
plt.figure()
a=10
b=100
plt.subplot (311)
plt.plot (M_O2Hb[a:b],'r' )
plt.subplot (312)
plt.plot (M_O2Myo[a:b],'b')
plt.subplot (313)
plt.plot (z[2,a:b],'k')
# Plot of a small portion of the output signal M_O2-Myo. This should look like the waveform
# we get from the BSX sensor.
# plt.figure()
# for i in range (0,freq,1):
# plt.plot(z[i,a:b])
plt.figure()
plt.plot (t[a:b],z[2,a:b])
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