PyMC3 PK modelling. Model cant resolve to parameters used to create the data set - bayesian

Question

I am new to PK modelling and pymc3, but I have been playing around with pymc3 and trying to implement a simple PK model as part of my own learning. Specifically a model that captures this relationship...
Where C(t)(Cpred) is concentration at time t, Dose is the dose given, V is Volume of distribution, CL is clearance.
I have generated some test data (30 subjects) with values of CL =2 , V=10, for 3 doses 100,200,300, and generated data at timepoints 0,1,2,4,8,12, and also included some random error on CL (normal distribution, 0 mean, omega =0.6) and on the residual unexplained error DV = Cpred + sigma, where sigma is normally distributed the SD =0.33. In addition I have included a transformation on C and V with respect to the weight (uniform distribution 50-90) CLi = CL * WT/70; Vi = V * WT/70.
# Create Data for modelling
np.random.seed(0)
# Subject ID's
data = pd.DataFrame(np.arange(1,31), columns=['subject'])
# Dose
Data['dose'] = np.array([100,100,100,100,100,100,100,100,100,100,
200,200,200,200,200,200,200,200,200,200,
300,300,300,300,300,300,300,300,300,300])
# Random Body Weight
data['WT'] = np.random.randint(50,100, size =30)
# Fixed Clearance and Volume for the population
data['CLpop'] =2
data['Vpop']=10
# Error rate for individual clearance rate
OMEGA = 0.66
# Individual clearance rate as a function of weight and omega
data['CLi'] = data['CLpop']*(data['WT']/70)+ np.random.normal(0, OMEGA )
# Individual Volume as a function of weight
data['Vi'] = data['Vpop']*(data['WT']/70)
# Expand dataframe to account for time points
data = pd.concat([data]*6,ignore_index=True )
data = data.sort('subject')
# Add in time points
data['time'] = np.tile(np.array([0,1,2,4,8,12]), 30)
# Create concentration values using equation
data['Cpred'] = data['dose']/data['Vi'] *np.exp(-1*data['CLi']/data['Vi']*data['time'])
# Error rate for DV
SIGMA = 0.33
# Create Dependenet Variable from Cpred + error
data['DV']= data['Cpred'] + np.random.normal(0, SIGMA )
# Create new df with only data for modelling...
df = data[['subject','dose','WT', 'time', 'DV']]
Create arrays ready for model...
# Prepare data from df to model specific arrays
time = np.array(df['time'])
dose = np.array(df['dose'])
DV = np.array(df['DV'])
WT = np.array(df['WT'])
n_patients = len(data['subject'].unique())
subject = data['subject'].values-1
I have built a simple model in pymc3 ....
pk_model = Model()
with pk_model:
# Hyperparameter Priors
sigma = Lognormal('sigma', mu =0, tau=0.01)
V = Lognormal('V', mu =2, tau=0.01)
CL = Lognormal('CL', mu =1, tau=0.01)
# Transformation wrt to weight
CLi = CL*(WT)/70
Vi = V*(WT)/70
# Expected value of outcome
pred = dose/Vi*np.exp(-1*(CLi/Vi)*time)
# Likelihood (sampling distribution) of observations
conc = Normal('conc', mu =pred, tau=sigma, observed = DV)
My expectation was that I should have been able to resolve from the data the constants and error rates that were originally used to generate the data, although I have not been able to do this, although I can get close. In this example...
data['CLi'].mean()
> 2.322473543135788
data['Vi'].mean()
> 10.147619047619049
And the trace shows....
So my questions are..
Is my code structured correctly and are there any glaring mistakes that I have overlooked that might account for this difference?
Can I structure the pymc3 model to better reflect the relationship from which I have generated the data?
What would be your suggestions to improve the model?
Thanks in advance!

Answer 1

I'm going to answer my own question!
But I implemented a hierarchal model following the example found here...
GLM -hierarchical
and it works a treat. Also I noticed errors in the way I was applying the errors in the dataframe - should use
data['CLer'] = np.random.normal(scale=OMEGA, size=30)
To ensure each subject has a different value for the error