Robust Extraction of Quantitative Information from Histology Images

Quentin Caudron

Romain Garnier

with Bryan Grenfell and Andrea Graham

Outline

  • Image processing
  • Extracted measures
  • Preliminary analysis
  • Future directions
  1. Age as random effect <---

["interface_hepatitis", "confluent_necrosis", "portal_inflammation", "ln_ap_ri"]

In [3]: 
def normalise(df, skip = []) :
	for i in df.columns :
		if i not in skip :
			df[i] -= df[i].mean()
			df[i] /= df[i].std()
	return df






def rescale(df, skip = []) :
    for i in df.columns :
        if i not in skip :
            df[i] -= df[i].min()
            df[i] /= df[i].max()
    return df



# Remove a layer from a list
def delayer(m) :
	out = []
	for i in m :
		if isinstance(i, list) :
			for j in i :
				out.append(j)
		else :
			out.append(i)
	return out







# Remove all layers from a list
def flatten(m) :
	out = m[:]

	while out != delayer(out) :
		out = delayer(out)

	return out








# Generate all combinations of objects in a list
def combinatorial(l) :
	out = []

	for numel in range(len(l)) :
		for i in itertools.combinations(l, numel+1) :
			out.append(list(i))

	return out










def pcaplot(df) :

	# PCA
	pca = decomposition.PCA(whiten = True)
	pca.fit(df)
	p1 = pca.components_[0] / np.abs(pca.components_[0]).max() * np.sqrt(2)/2
	p2 = pca.components_[1] / np.abs(pca.components_[1]).max() * np.sqrt(2)/2

	# Normalise
	norms = np.max([np.sqrt((np.array(zip(p1, p2)[i])**2).sum()) for i in range(len(p1))])
	c = plt.Circle( (0, 0), radius = 1, alpha = 0.2)
	plt.axes(aspect = 1)
	plt.gca().add_artist(c)

	plt.scatter(p1 / norms, p2 / norms)
	plt.xlim([-1, 1])
	plt.ylim([-1, 1])

	for i, text in enumerate(df.columns) :
		plt.annotate(text, xy = [p1[i], p2[i]])

	plt.tight_layout()











def test_all_linear(df, y, x, return_significant = False, group = None) :

    # All possible combinations of independent variables
	independent = combinatorial(x)

	fits = {}
	pval = {}
	linmodels = {}
	qsum = {}
	aic = {}

	# For all dependent variables, one at a time
	for dependent in y :

		print "Fitting for %s." % dependent

		# For all combinations of independent variables
		for covariate in independent :

			# Standard mixed model
			if group is None :

				# Fit a linear model
				subset = delayer([covariate, dependent])
				df2 = df[delayer(subset)].dropna()
				df2["Intercept"] = np.ones(len(df2))
                
				ols = sm.GLS(endog = df2[dependent], exog = df2[delayer([covariate, "Intercept"])]).fit()

				# Save the results
				if (return_significant and ols.f_pvalue < 0.05) or (not return_significant) :
					linmodels.setdefault(dependent, []).append(ols)
					fits.setdefault(dependent, []).append(ols.rsquared)
					pval.setdefault(dependent, []).append(ols.f_pvalue)
					aic.setdefault(dependent, []).append(ols.aic)


			# Mixed effects model
			else :
				subset = delayer([covariate, dependent, group])
				df2 = df[delayer(subset)].dropna()

				# Fit a mixed effects model
				ols = MixedLM(endog = df2[dependent], exog = df2[covariate], groups = df2[group]).fit()

				# Calculate AIC
				linmodels.setdefault(dependent, []).append(ols)
				fits.setdefault(dependent, []).append(2 * (ols.k_fe + 1) - 2 * ols.llf)
				pval.setdefault(dependent, []).append(ols.pvalues)

	if group is not None :
		for i in y :
			f = np.array(fits[i])
			models = np.array(linmodels[i])
			idx = np.where(f - f.min() <= 2)[0]
			bestmodelDoF = [j.k_fe for j in np.array(linmodels[i])[idx]]
			bestmodels = [idx[j] for j in np.where(bestmodelDoF == np.min(bestmodelDoF))[0]]
			qsum[i] = models[idx[np.where(f[bestmodels] == np.min(f[bestmodels]))]]


		return linmodels, fits, pval, qsum

	return linmodels, fits, pval, aic

	
		















def summary(models) :

	# Generate list of everything
	r2 = np.array([m.r2 for dependent in models.keys() for m in models[dependent]])
	p = np.array([m.f_stat["p-value"] for dependent in models.keys() for m in models[dependent]])
	mod = np.array([m for dependent in models.keys() for m in models[dependent]])
	dependent = np.array([dependent for dependent in models.keys() for m in models[dependent]])

	# Sort by R2
	idx = np.argsort(r2)[::-1]

	# Output string
	s = "%d significant regressions.\n\n" % len(r2)
	s += "Ten most correlated :\n\n"

	# Print a summary of the top ten correlations
	for i in idx[:10] :
		s += ("%s ~ %s\n" % (dependent[i], " + ".join(mod[i].x.columns[:-1])))
		s += ("R^2 = %f\tp = %f\n\n" % (r2[i], p[i]))

	print s
    
    
    
    
    
    
    
def rstr(y, x) :
    formatstr = "%s ~ " % y
    for i in x[:-1] :
        formatstr += str(i)
        formatstr += " + "
    formatstr += str(x[-1])
    return formatstr
In [14]: 
import numpy as np
from sklearn.neighbors import KernelDensity
from matplotlib import rcParams
import matplotlib.pyplot as plt
import seaborn
import pandas as pd
import itertools
from sklearn import linear_model, ensemble, decomposition, cross_validation, preprocessing
from statsmodels.regression.mixed_linear_model import MixedLM
import statsmodels.api as sm
from statsmodels.regression.linear_model import OLSResults
from statsmodels.tools.tools import add_constant


%matplotlib inline
rcParams["figure.figsize"] = (14, 8)


# RAW DATA

raw_physical = pd.read_csv("../data/physical.csv")
raw_histo = pd.read_csv("../data/tawfik.csv")
ent = pd.read_csv("../4x/results/entropy.csv").drop(["Unnamed: 0"], 1)
foci = pd.read_csv("../4x/results/foci.csv").drop(["Unnamed: 0"], 1)
lac = pd.read_csv("../4x/results/normalised_lacunarity.csv").drop(["Unnamed: 0"], 1)
gabor = pd.read_csv("../4x/results/gabor_filters.csv").drop(["Unnamed: 0"], 1)
ts = pd.read_csv("../4x/results/tissue_sinusoid_ratio.csv").drop(["Unnamed: 0"], 1)

raw_image = pd.merge(lac, ent,
	on=["Sheep", "Image"]).merge(foci, 
	on=["Sheep", "Image"]).merge(gabor,
	on=["Sheep", "Image"]).merge(ts, 
    on=["Sheep", "Image"])
raw_image.rename(columns = {	"meanSize" : "FociSize", 
								"TSRatio" : "TissueToSinusoid",
								"Count" : "FociCount" }, inplace=True)



# CLEAN DATA

physcols = ["Weight", "Sex", "AgeAtDeath", "Foreleg", "Hindleg"]
imagecols = ["Entropy", "Lacunarity", "Inflammation", "Scale", "Directionality", "FociCount", "FociSize", "TissueToSinusoid"]
histcols = ["Lobular_collapse", "Interface_hepatitis", "Confluent_necrosis", "Ln_ap_ri", "Portal_inflammation", "BD_hyperplasia", "Fibrosis", "TawfikTotal", "Mean_hep_size", "Min_hep_size", "Max_hep_size"]





# IMAGE

# Set FociSize to zero if FociCount is zero
# Drop stdSize
image = raw_image
image = image.drop("stdSize", 1)
image.FociSize[raw_image.FociCount == 0] = 0



# HISTO

histo = raw_histo
histo = histo.drop(["Vessels", "Vacuol", "Pigment", "Std_hep_size"], 1)



# PHYSICAL

physical = raw_physical
physical = physical.drop(["CurrTag", "DeathDate", "Category"], 1)
physical




# COMPLETE DATASET

raw_data = pd.merge(pd.merge(image, histo, on="Sheep", how="outer"), physical, on="Sheep", how="outer")
raw_data.to_csv("../data/tentative_complete.csv")




# AVERAGED BY SHEEP
data = raw_data
data["Inflammation"] = data.FociCount * data.FociSize

sheep = rescale(data.groupby("Sheep").mean())
age = rescale(data.groupby("AgeAtDeath").mean())







# REGRESSIONS : fixed effects, grouped by sheep

df = sheep[["Portal_inflammation", "FociSize"]].dropna()
df["Intercept"] = np.ones(len(df))
portal_inflammation = sm.GLS(endog = df.Portal_inflammation, exog = df[["FociSize", "Intercept"]]).fit().summary()
#portal_inflammation = portal_inflammation.summary()
del portal_inflammation.tables[2]



df = sheep[["BD_hyperplasia", "Scale", "Directionality", "FociSize"]].dropna()
df["Intercept"] = np.ones(len(df))
hyperplasia = sm.GLS(endog = df.BD_hyperplasia, exog = df[["FociSize", "Scale", "Directionality", "Intercept"]]).fit().summary()
#hyperplasia.summary()
del hyperplasia.tables[2]






# REGRESSIONS : fixed effects, grouped by age

df = age[["Max_hep_size", "Entropy", "Directionality"]].dropna()
df["Intercept"] = np.ones(len(df))
maxhepsize = sm.GLS(endog = df.Max_hep_size, exog = df[["Entropy", "Directionality", "Intercept"]]).fit().summary()
del maxhepsize.tables[2]




df = age[["Lobular_collapse", "FociSize"]].dropna()
df["Intercept"] = np.ones(len(df))
lobular_collapse = sm.GLS(endog = df.Lobular_collapse, exog = df[["FociSize", "Intercept"]]).fit().summary()
del lobular_collapse.tables[2]


df = age[["Interface_hepatitis", "Lacunarity"]].dropna()
df["Intercept"] = np.ones(len(df))
interface_hepatitis = sm.GLS(endog = df.Interface_hepatitis, exog = df[["Lacunarity", "Intercept"]]).fit().summary()
del interface_hepatitis.tables[2]


df = age[["Fibrosis", "Inflammation"]].dropna()
df["Intercept"] = np.ones(len(df))
fibrosis = sm.GLS(endog = df.Fibrosis, exog = df[["Inflammation", "Intercept"]]).fit().summary()
del fibrosis.tables[2]




# PCA

s = sheep.dropna(subset=delayer([imagecols, histcols]))
pca = decomposition.PCA(n_components=1)
pcax = pca.fit_transform(s[imagecols])
pcay = pca.fit_transform(s[histcols])
pca = sm.GLS(endog = pcay[:, 0][:, np.newaxis], exog = add_constant(pcax)).fit().summary()
del pca.tables[2]





# REGRESSIONS : mixed effects, intercept on age at death

df = age[["Fibrosis", "Inflammation"]].dropna()
df["Intercept"] = np.ones(len(df))
fibrosis = sm.GLS(endog = df.Fibrosis, exog = df[["Inflammation", "Intercept"]]).fit().summary()
del fibrosis.tables[2]
In [19]: 
a = portal_inflammation.summary()
del a.tables[2]
a
Out[19]:
GLS Regression Results
Dep. Variable: Portal_inflammation R-squared: 0.280
Model: GLS Adj. R-squared: 0.273
Method: Least Squares F-statistic: 37.34
Date: Tue, 28 Oct 2014 Prob (F-statistic): 2.12e-08
Time: 23:35:30 Log-Likelihood: 14.996
No. Observations: 98 AIC: -25.99
Df Residuals: 96 BIC: -20.82
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [95.0% Conf. Int.]
FociSize 0.5627 0.092 6.111 0.000 0.380 0.746
Intercept 0.3368 0.043 7.855 0.000 0.252 0.422

Image Processing

Extraction

  • Automagical
  • Reasonably quick

Robust

  • Invariant to staining, slicing, field-related variation
  • Capture intersample variation

image

image

image

image

Structural and Textural Measures

  • characteristic scale of sinusoid widths
  • directional amplitude of preferred sinusoid alignment
  • tissue to sinusoid ratio
  • count of inflammatory foci per image
  • mean size of inflammatory foci per image
  • information entropy of sinusoid distribution
  • lacunarity ( clustering ) of sinusoids

image

image

Exploratory Analysis

by individual

In [29]: 
portal_inflammation
Out[29]:
GLS Regression Results
Dep. Variable: Portal_inflammation R-squared: 0.280
Model: GLS Adj. R-squared: 0.273
Method: Least Squares F-statistic: 37.34
Date: Tue, 28 Oct 2014 Prob (F-statistic): 2.12e-08
Time: 23:40:10 Log-Likelihood: 14.996
No. Observations: 98 AIC: -25.99
Df Residuals: 96 BIC: -20.82
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [95.0% Conf. Int.]
FociSize 0.5627 0.092 6.111 0.000 0.380 0.746
Intercept 0.3368 0.043 7.855 0.000 0.252 0.422

image

In [31]: 
hyperplasia
Out[31]:
GLS Regression Results
Dep. Variable: BD_hyperplasia R-squared: 0.306
Model: GLS Adj. R-squared: 0.284
Method: Least Squares F-statistic: 13.83
Date: Tue, 28 Oct 2014 Prob (F-statistic): 1.52e-07
Time: 23:40:10 Log-Likelihood: -3.9632
No. Observations: 98 AIC: 15.93
Df Residuals: 94 BIC: 26.27
Df Model: 3
Covariance Type: nonrobust
coef std err t P>|t| [95.0% Conf. Int.]
FociSize 0.6698 0.113 5.902 0.000 0.444 0.895
Scale 0.5811 0.243 2.394 0.019 0.099 1.063
Directionality -0.4419 0.190 -2.330 0.022 -0.819 -0.065
Intercept -0.0504 0.079 -0.642 0.523 -0.206 0.105

image

In [15]: 
pca
Out[15]:
GLS Regression Results
Dep. Variable: y R-squared: 0.075
Model: GLS Adj. R-squared: 0.065
Method: Least Squares F-statistic: 7.723
Date: Wed, 29 Oct 2014 Prob (F-statistic): 0.00657
Time: 14:38:47 Log-Likelihood: -70.082
No. Observations: 97 AIC: 144.2
Df Residuals: 95 BIC: 149.3
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [95.0% Conf. Int.]
const -2.949e-17 0.051 -5.77e-16 1.000 -0.102 0.102
x1 0.3865 0.139 2.779 0.007 0.110 0.663

image

Exploratory Analysis

by age class

In [6]: 
fibrosis
Out[6]:
GLS Regression Results
Dep. Variable: Fibrosis R-squared: 0.800
Model: GLS Adj. R-squared: 0.778
Method: Least Squares F-statistic: 36.07
Date: Wed, 29 Oct 2014 Prob (F-statistic): 0.000201
Time: 11:13:48 Log-Likelihood: 7.8003
No. Observations: 11 AIC: -11.60
Df Residuals: 9 BIC: -10.80
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [95.0% Conf. Int.]
Inflammation 1.0159 0.169 6.006 0.000 0.633 1.399
Intercept -0.0105 0.083 -0.126 0.902 -0.198 0.177

image

In [7]: 
lobular_collapse
Out[7]:
GLS Regression Results
Dep. Variable: Lobular_collapse R-squared: 0.586
Model: GLS Adj. R-squared: 0.540
Method: Least Squares F-statistic: 12.73
Date: Wed, 29 Oct 2014 Prob (F-statistic): 0.00605
Time: 11:13:48 Log-Likelihood: 2.2626
No. Observations: 11 AIC: -0.5252
Df Residuals: 9 BIC: 0.2706
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [95.0% Conf. Int.]
FociSize 1.1379 0.319 3.567 0.006 0.416 1.860
Intercept 0.0460 0.159 0.289 0.779 -0.314 0.406

image

In [8]: 
interface_hepatitis
Out[8]:
GLS Regression Results
Dep. Variable: Interface_hepatitis R-squared: 0.659
Model: GLS Adj. R-squared: 0.621
Method: Least Squares F-statistic: 17.38
Date: Wed, 29 Oct 2014 Prob (F-statistic): 0.00242
Time: 11:13:48 Log-Likelihood: 2.3063
No. Observations: 11 AIC: -0.6126
Df Residuals: 9 BIC: 0.1832
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [95.0% Conf. Int.]
Lacunarity -1.0224 0.245 -4.168 0.002 -1.577 -0.468
Intercept 0.9504 0.143 6.669 0.000 0.628 1.273

image

Exploratory analysis

with a random effect on age at death

Dependent variable Models
AIC < 2 + AICmin		 Primary explanatory variables
Ishak score 7 entropy, tissue-to-sinusoid, focus count, focus size
Lobular collapse 5 entropy, lacunarity, tissue-to-sinusoid, focus count
Confluent necrosis 1 entropy
Interface hepatitis 2 entropy, tissue-to-sinusoid
Portal inflammation 4 entropy, focus size, lacunarity, focus count, scale, directionality
Fibrosis 2 entropy, lacunarity, tissue-to-sinusoid
Biliary hyperplasia 1 focus size
Necrosis, apoptosis, random inflammation This_is_bla2This_is_bla entropy, lacunarity
  • entropy consistently explains histological measures when controlled for age
  • also important : tissue to sinusoid ratio, focus count and size, lacunarity
  • biological / historical reasoning for this potential cohort effect
  • interpretation of these models
  • quality of fit

Conclusions

  • our semi-educated guess measures may capture relevant information
  • underlying structure in the data needs thought
  • still no map from image or histological measures to condition of individual

Future directions

Further exploration of the dataset

  • 145 sheep ( 89 females )
  • 11 age classes
  • potential redundancy in various measures
  • 4460 entries across 27 variables
  • 3330 with full image and histological information
  • 1196 for which complete information is available

More data

  • nutritional information
  • immunity data

Narrow-field images

  • 12536 images
  • spatial distribution of nuclei

image

image

image