fedbatch-Bi的GEKKO不可行常微分方程组

Question

我对GEKKO和生物反应器建模都是新手，所以我可能遗漏了一些明显的东西。你知道吗

我有一个由10个代码组成的系统来描述一个补料分批生物反应器。给出了所有常数。下图显示了此模型的预期行为（摘自一篇论文）。然而，我发现的唯一可行的解决方案是当活细胞密度（XV）=0，并且在所有时间t内保持0，或者如果时间t非常小（<；20）。如果下边界>；=0或初始值设置为XV且t>；20，则系统不可行。你知道吗

对方程和常数进行了多次检查。我试着给变量赋值，但也没用。我只能想到两个问题：我没有正确地初始化变量，或者我没有正确地使用GEKKO。有什么想法吗？谢谢！！你知道吗

import numpy as np
from gekko import GEKKO
import matplotlib.pyplot as plt

m = GEKKO(remote=False)    # create GEKKO model

#constants 3L continuous fed-batch
KdQ = 0.001        #degree of degradation of glutamine (1/h)
mG = 1.1*10**-10   #glucose maintenance coefficient (mmol/cell/hour)
YAQ = 0.90         #yield of ammonia from glutamine
YLG = 2            #yield of lactate from glucose
YXG = 2.2*10**8    #yield of cells from glucose (cells/mmol)
YXQ = 1.5*10**9    #yield of cells from glutamine (cells/mmol)
KL = 150           #lactate saturation constant (mM)
KA = 40            #ammonia saturation constant (mM)
Kdmax = 0.01       #maximum death rate (1/h)
mumax = 0.044      #maximum growth rate (1/h)
KG = 1             #glucose saturation constant (mM)
KQ = 0.22          #glutamine saturation constant (mM)
mQ = 0             #glutamine maintenance coefficient (mmol/cell/hour)
kmu = 0.01         #intrinsic death rate (1/h)
Klysis = 2*10**-2  #rate of cell lysis (1/h)
Ci_star = 100      #inhibitor saturation concentration (mM)
qi = 2.5*10**-10   #specific inhibitor production rate (1/h)

#Flow, volume and concentration
Fo = 0.001         #feed-rate (L/h)
Fi = 0.001         #feed-rate (L/h)
V = 3              #volume (L)
SG = 653           #glucose concentration in the feed (mM)
SQ = 58.8          #glutamine concentration in the feed (mM)

# create GEKKO parameter
t = np.linspace(0,120,121)
m.time = t

XT = m.Var(name='XT')            #total cell density (cells/L)
XV = m.Var(lb=0, name='XV')      #viable cell density (cells/L)
XD = m.Var(name='XD')            #dead cell density (cells/L)
G = m.Var(value = 30, name='G')  #glucose concentration (mM)
Q = m.Var(value = 4.5, name='Q') #glutamine concentration (mM)
L = m.Var(name='L')              #lactate concentration (mM)
A = m.Var(name='A')              #ammonia concentration (mM)
Ci = m.Var(name='Ci')            #inhibitor concentration (mM)
mu = m.Var(name='mu')            #growth rate (1/h)
Kd = m.Var(name='Kd')            #death rate(1/h)

# create GEEKO equations
m.Equation(XT.dt() == mu*XV - Klysis*XD - XT*Fo/V)
m.Equation(XV.dt() == (mu - Kd)*XV - XV*Fo/V)
m.Equation(XD.dt() == Kd*XV - Klysis*XD - XV*Fo/V)
m.Equation(G.dt() == (Fi/V)*SG - (Fo/V)*G + (-mu/YXG - mG)*XV)
m.Equation(Q.dt() == (Fi/V)*SQ - (Fo/V)*Q + (-mu/YXQ - mQ)*XV - KdQ*Q)
m.Equation(L.dt() == -YLG*(-mu/YXG -mG)*XV-(Fo/V)*L)
m.Equation(A.dt() == -YAQ*(-mu/YXQ - mQ)*XV +KdQ*Q-(Fo/V)*A)
m.Equation(Ci.dt() == qi*XV - (Fo/V)*Ci)
m.Equation(mu.dt() == (mumax*G*Q*(Ci_star-Ci)) / (Ci_star*(KG+G)*(KQ+Q)*(L/KL + 1)*(A/KA + 1)))
m.Equation(Kd.dt() == Kdmax*(kmu/(mu+kmu)))

# solve ODE
m.options.IMODE = 4
m.open_folder()
m.solve(display = False)

plt.plot(m.time, XV.value)

使用完全相同模型的文章：

1）利用GEKKO的硕士论文“哺乳动物细胞培养的建模” 链接：

https://search.proquest.com/openview/e4df2d115cbc48ec63235a64b352249c/1.pdf?pq-origsite=gscholar&cbl=18750&diss=y

2）描述方程式的原始论文：“过程模型比较和跨生物反应器规模和 哺乳动物细胞生物过程的操作模式“

链接：https://sci-hub.tw/10.1002/btpr.1664

3）使用该模型的控制系统的论文：“使用非线性模型预测控制器的补料哺乳动物细胞生物过程的葡萄糖浓度控制”

链接：https://sci-hub.tw/https://doi.org/10.1016/j.jprocont.2014.02.007

Answer 1

最后，这不是一个编程问题，而是一个阅读公式并正确翻译它们的问题。你知道吗

mu和Kd不是动态变量，它们是状态向量的普通函数（此时只有维数8）。对于这类中间变量，Gekko具有构造函数m.Intermediate

mu = m.Intermediate((mumax*G*Q*(Ci_star-Ci)) / (Ci_star*(KG+G)*(KQ+Q)*(L/KL + 1)*(A/KA + 1)), name='mu') #growth rate (1/h)
Kd = m.Intermediate(Kdmax*(kmu/(mu+kmu)))    #death rate(1/h)

有了这个变化和XT和XV的初始值5e8，脚本给出了下面的曲线图，看看你引用的论文中能找到什么。你知道吗

Answer 2

有几个问题：

• 最后两个方程是代数方程，不是微分方程。它应该是mu==...而不是mu.dt()==...
• 一些方程式有被零除的可能性。像x.dt() = z/y这样的方程可以用y * x.dt()==z代替，这样当y接近零时，方程就变成0 * x.dt() == z。你知道吗
• 一些初始条件没有设置，因此它们使用默认值0。这可能会产生一个零的解决方案。你知道吗

我输入了一些不同的值，并使用m.options.COLDSTART=2帮助它找到一个初始解决方案。我还使用中间词来帮助可视化任何正在变大的术语。我把细胞浓度以每公升数百万个细胞为单位，以帮助进行缩放。你知道吗

import numpy as np
from gekko import GEKKO
import matplotlib.pyplot as plt

m = GEKKO(remote=False)    # create GEKKO model

#constants 3L continuous fed-batch
KdQ = 0.001        #degree of degradation of glutamine (1/h)
mG = 1.1e-10   #glucose maintenance coefficient (mmol/cell/hour)
YAQ = 0.90         #yield of ammonia from glutamine
YLG = 2            #yield of lactate from glucose
YXG = 2.2e8    #yield of cells from glucose (cells/mmol)
YXQ = 1.5e9    #yield of cells from glutamine (cells/mmol)
KL = 150           #lactate saturation constant (mM)
KA = 40            #ammonia saturation constant (mM)
Kdmax = 0.01       #maximum death rate (1/h)
mumax = 0.044      #maximum growth rate (1/h)
KG = 1             #glucose saturation constant (mM)
KQ = 0.22          #glutamine saturation constant (mM)
mQ = 0             #glutamine maintenance coefficient (mmol/cell/hour)
kmu = 0.01         #intrinsic death rate (1/h)
Klysis = 2e-2  #rate of cell lysis (1/h)
Ci_star = 100      #inhibitor saturation concentration (mM)
qi = 2.5e-10   #specific inhibitor production rate (1/h)

#Flow, volume and concentration
Fo = 0.001         #feed-rate (L/h)
Fi = 0.001         #feed-rate (L/h)
V = 3              #volume (L)
SG = 653           #glucose concentration in the feed (mM)
SQ = 58.8          #glutamine concentration in the feed (mM)

# create GEKKO parameter
t = np.linspace(0,50,121)
m.time = t

XTMM = m.Var(value=1,name='XT')            #total cell density (MMcells/L)
XVMM = m.Var(value=1,lb=0, name='XV')      #viable cell density (MMcells/L)
XDMM = m.Var(value=1.0,name='XD')          #dead cell density (MMcells/L)
G = m.Var(value = 20, name='G')            #glucose concentration (mM)
Q = m.Var(value = 4.5, name='Q')           #glutamine concentration (mM)
L = m.Var(value=1,name='L')                #lactate concentration (mM)
A = m.Var(value=1.6,name='A')              #ammonia concentration (mM)
Ci = m.Var(value=0.1,name='Ci')            #inhibitor concentration (mM)
mu = m.Var(value=0.1,name='mu')            #growth rate (1/h)
Kd = m.Var(value=0.5,name='Kd')            #death rate(1/h)

# scale back to cells/L from million cells/L
XT = m.Intermediate(XTMM*1e7)
XV = m.Intermediate(XVMM*1e7)
XD = m.Intermediate(XDMM*1e7)

e1 = m.Intermediate((mu*XV - Klysis*XD - XT*Fo/V)/1e7)
e2 = m.Intermediate(((mu - Kd)*XV - XV*Fo/V)/1e7)
e3 = m.Intermediate((Kd*XV - Klysis*XD - XV*Fo/V)/1e7)
e4 = m.Intermediate((Fi/V)*SG - (Fo/V)*G + (-mu/YXG - mG)*XV)
e5 = m.Intermediate((Fi/V)*SQ - (Fo/V)*Q + (-mu/YXQ - mQ)*XV - KdQ*Q)
e6 = m.Intermediate(-YLG*(-mu/YXG -mG)*XV-(Fo/V)*L)
e7 = m.Intermediate(-YAQ*(-mu/YXQ - mQ)*XV +KdQ*Q-(Fo/V)*A)
e8 = m.Intermediate(qi*XV - (Fo/V)*Ci)
e9a = m.Intermediate((Ci_star*(KG+G)*(KQ+Q)*(L/KL + 1)*(A/KA + 1)))
e9b = m.Intermediate((mumax*G*Q*(Ci_star-Ci)))
e10a = m.Intermediate((mu+kmu))
e10b = m.Intermediate(Kdmax*kmu)

# create GEEKO equations
m.Equation(XTMM.dt() == e1)
m.Equation(XVMM.dt() == e2)
m.Equation(XDMM.dt() == e3)
m.Equation(G.dt() == e4)
m.Equation(Q.dt() == e5)
m.Equation(L.dt() == e6)
m.Equation(A.dt() == e7)
m.Equation(Ci.dt() == e8)
m.Equation(e9a * mu == e9b)
m.Equation(e10a*Kd == e10b)

# solve ODE
m.options.IMODE = 4
m.options.SOLVER = 1
m.options.NODES = 3
m.options.COLDSTART = 2
#m.open_folder()
m.solve(display=False)

plt.figure()
plt.subplot(3,1,1)
plt.plot(m.time, XV.value,label='XV')
plt.plot(m.time, XT.value,label='XT')
plt.plot(m.time, XD.value,label='XD')
plt.legend()
plt.subplot(3,1,2)
plt.plot(m.time, G.value,label='G')
plt.plot(m.time, Q.value,label='Q')
plt.plot(m.time, L.value,label='L')
plt.plot(m.time, A.value,label='A')
plt.legend()
plt.subplot(3,1,3)
plt.plot(m.time, mu.value,label='mu')
plt.plot(m.time, Kd.value,label='Kd')
plt.legend()
plt.xlabel('Time (hr)')

plt.figure()
plt.plot(m.time, e1.value,'r-.',label='eqn1')
plt.plot(m.time, e2.value,'g:',label='eqn2')
plt.plot(m.time, e3.value,'b:',label='eqn3')
plt.plot(m.time, e4.value,'b ',label='eqn4')
plt.plot(m.time, e5.value,'y:',label='eqn5')
plt.plot(m.time, e6.value,'m ',label='eqn6')
plt.plot(m.time, e7.value,'b-.',label='eqn7')
plt.plot(m.time, e8.value,'g ',label='eqn8')
plt.plot(m.time, e9a.value,'r:',label='eqn9a')
plt.plot(m.time, e9b.value,'r ',label='eqn9b')
plt.plot(m.time, e10a.value,'k:',label='eqn10a')
plt.plot(m.time, e10b.value,'k ',label='eqn10b')
plt.legend()

plt.show()