Table of Contents:

This section first describes the results of bivariate, MLP analysis, and reduced retention model (RRM) estimated by the data set of case studies 1, 2, and 3 described in Section 11.2. Afterwards, it illustrates the results of LR model developed using the data set of case study 1.


The pathophysiological process involved in fluorosis has been described by Gupta et al. [2] which states that increased blood fluoride is followed by lowering of serum calcium which is further followed by increase in immature osteoblasts. This leads to increased serum and bone GAG which further cause decrease in calcification of bone tissues. Considering the logical sequence of manifestation of fluorosis proposed here along with the fact that SAP can be considered as an indicator of the osteoblastic bone activity [2], a bivariate hypothesis was postulated “Subjects with higher level of blood fluoride have lower level of serum calcium which is associated with increased level of SAP.” The mean and standard deviation of the biochemical parameters (blood fluoride, serum calcium and SAP) were taken from case study 2: Shivdaspura and Vanasthali Jaipur. More data points were generated assuming a normal distribution within the population for each of the parameters with sufficiently large sample size (100 subjects). The data points were then studied using the scatter plots shown in Figures 11.2 and 11.3.

Scatter plot between data of serum calcium and blood fluoride generated by assuming a normal distribution of population and taking a sample size of 100

FIGURE 11.2 Scatter plot between data of serum calcium and blood fluoride generated by assuming a normal distribution of population and taking a sample size of 100.

Scatter plot between data of serum alkaline phosphate (SAP) and serum calcium generated by assuming a normal distribution of population and taking a sample size of 100

FIGURE 11.3 Scatter plot between data of serum alkaline phosphate (SAP) and serum calcium generated by assuming a normal distribution of population and taking a sample size of 100.

Figure 11.2 which shows the scatter plot of serum calcium and blood fluoride, reveals that serum calcium and blood fluoride are found to have a negative association (inverse relation) and the points getting clustered along a straight line depict the presence of a linear relationship. Figure 11.3 shows the scatter plot between serum calcium and blood fluoride and displays a negative association between the two variables. Although the points cluster along a line, they are not clustered quite as closely as they were for the scatter plot in Figure 11.2. The strength of these linear relationships was further explored using Pearson correlation (p). Following the null hypothesis was taken:

Using SPSS, the correlation matrix between serum calcium and blood fluoride and SAP was developed. Table 11.2 shows Pearson correlation coefficients between the three biochemical parameters maintaining the necessary pairs. Since the significance value was less than 0.05 in both the cases null hypothesis H0 was rejected. Hence, p = -0.673 in the first case indicate blood fluoride has a strong negative correlation with serum calcium. While for other cases p = -0.520 denotes moderately strong negative correlation of SAP with serum calcium. Hence, the bivariate hypothesis postulated from the pathophysiology proves to be correct.

TABLE 11.2 Correlation Matrix of Pearson Correlation Coefficients between Serum Calcium and Blood Fluoride; and SAP and Serum Calcium

Serum Calcium (serum_Ca)

Blood Fluoride(blood_f)


Pearson Correlation (p)



Sig. (2-tailed)



N (sample size)



After a critical examination of the pre-treatment data of biochemical parameters of children from the villages of Jaipur, illustrated in case study 1, it was observed that the data could be used to interpret the fluctuations in biochemical parameters on the basis of the level of fluoride intake through drinking water. Biochemical parameters were classified into four categories by relating them to an ordinal variable 4vater_fluoride_grade’ with values

I, 2, 3, and 4 representing the fluoride levels in drinking water for villages of Ramsagarki Dhani, Rampura, Shivdaspura, and Rapuria respectively. Water_fluoride_grade =1 was used for drinking fluoride levels of 2.4 ppm, 2 for 4.6 ppm, 3 for 5.6, and 4 for 13.6 ppm. These data were then used as an input to MLP neural network analysis in SPSS v20. Tables 11.2-11.5 show the output of the MLP analysis.

Table 11.3 shows that 60.5% of the valid sample data was randomly picked and used for training the network while the rest 39.5% was utilized for testing. Table 11.4 lists all the biochemical parameters that were used as covariates in the input layer and describe the hidden and output layer (Table

II. 6).

TABLE 11.3 Case Processing Summary of Neural Network Training and Testing Data













TABLE 11.4 Network Information of Input, Hidden, and Output Layers of Neural Network

Input Layer






















Number of units’


Rescaling method for covariates


Hidden Layer(s)

Number of hidden layers


Number of units in hidden layer Г'


Activation function

Hyperbolic tangent

Output Layer

Dependent variables



Number of units


Activation function


Error function


’Excluding the bias unit.

TABLE 11.5 Model Summary (Cross Entropy Error and Percent Incorrect Predictions) of Training and Testing Data from the MLP Analysis


Cross entropy error


Percent incorrect predictions


Stopping rule used

1 consecutive step(s) with no decrease in error

Training time



Cross entropy error


Percent incorrect predictions


Dependent variable: Water fluoride grade

Error computations are based on the testing sample.

TABLE 11.6 Importance and Normalized Importance (Sensitivity Analysis) of Independent Variables from the MLP Analysis

Biochemical Parameter


Normalized Importance






















urinary _fluoride









Figure 11.4 displays the structure of a neural network with one hidden layer and four hidden units defined in the MLP neural network. Table 11.5 shows the model summary. The cross-entropy error, which the network tried to minimize during training, was 2.511. Moreover, the percentage of incorrect prediction was 18.8% in the testing samples. So, the percentage of correct prediction was 81.2%, which can be considered as good for physiological systems.

Table 11.6 and Figure 11.5 display normalized importance of various biochemical parameters generated from Independent variable importance analysis. This is a sensitivity analysis, based on the combined training and testing of samples. It computes and creates a table and a chart displaying the importance and normalized importance of each predictor in defining the neural network. It was observed that ‘GAG’ contributed the most in the neural network model construction, followed by ‘blood fluoride,’ ‘serum calcium,’ ‘serum fluoride,’ ‘SSA,’ etc.

Jha et al. [14] suggested that elevated content of GAG in bone and its reflection in serum can be an index to assess fluoride toxicity and detect fluorosis at very early stages, while the above MLP model has reinforced the fact that serum GAG is an important predictor of the level of drinking water fluoride, and stating vice versa the serum GAG values will be affected more significantly in comparison to other biochemical parameters with varying fluoride levels in drinking water.

FIGURE 11.4 Depiction of structure of artificial neural network having various biochemical parameters in the input layer correlating with the drinking water fluoride level of the four target areas as depicted in case study 1 [2]; There is one hidden layer with four units and its activation function is hyperbolic tangent. The structure was produced by MLP analysis in SPSS.

FIGURE 11.5 Bar chart of various biochemical parameters showing their normalized importance in correlating with drinking fluoride level and further in prediction of skeletal fluorosis; serum GAG was found to be significant. The chart was produced by MLP analysis in SPSS.

The observed importance of GAG can be further backed by the reasoning that bone and teeth, which are adversely affected by ingestion of high fluoride for prolonged duration, are made up of calcium and bone matrix [14, 15]. About 80-85% of bone matrix is made of collagen and the remaining 15-20% by glycosaminoglycans (GAG) and glycoproteins [16]. All three components of bone and teeth, i.e., collagen, GAG, and calcium are adversely affected by the ingestion of high fluoride through various food and water sources for a prolonged duration [14,15]. Hence, serum GAG must be assessed as a key indicator of fluorosis.

The significant effect of serum GAG has been therefore used to develop RRM for calcium dose prediction at different water fluoride levels. Serum GAG was found to have the following strong linear relationship (R2 = 0.9698) with fluoride concentration in drinking water.

The above model was resultant of linear curve fitting of data in Figure 11.6 which plots the data of averages of serum GAG and drinking fluoride values of the four villages from case study 1.

Next, according to the proposed pathophysiology, serum GAG is inversely proportional to the retention of calcium in the bones, i.e., bone calcification. However, more calcium will be retained in the bones if there is an increase in calcium intake, which in the fluoride-free condition is governed by the equation [17]: Plot showing the relationship between serum GAG and drinking water fluoride level fit to linear co-relation; the data used was derived from case study 1 [2]

FIGURE 11.6 Plot showing the relationship between serum GAG and drinking water fluoride level fit to linear co-relation; the data used was derived from case study 1 [2].

Further, it was found that the serum GAG level in the human body in the normal conditions is in the range 9-11 mg/dl, and for calculations, it was assumed to be 10 mg/dl. The changes in the level of serum GAG because of the increase in the drinking fluoride level (Eqn. (11.8)), along with the varying calcium intake (Eqn. (11.9)) were used to calculate the reduced calcium retention in bones and the model (Eqn. (11.10)) was derived as follows:

Calcium intake by the sample population of the four villages with varying drinking fluoride level (2.4 to 13.6 mg/1) ranged from 250 to 1010 mg/day [2]. When the calcium dietary intake is not enough to maintain the calcium homeostasis the value of retention was negative, which represents the process of leaching of bone, i.e., instead of bone calcification, calcium was leaching out from the bone. However, for the same level of water fluoride, if there is an increase in calcium intake, then calcium depletion would stop and retention would start. This critical calcium dose was plotted against the fluoride level (Figure 11.7) and the model (Eqn. (11.11)) was obtained by linear fitting:

Plot of critical calcium and drinking water fluoride level

FIGURE 11.7 Plot of critical calcium and drinking water fluoride level.

Model Eqn. (11.11) predicts the supplemental dose of calcium to be taken at different water fluoride levels. The value of Ca dose at which retention starts is not the same as the required adequate Ca dose for a human being. The dosage predicted by RRM shows the amount of calcium required for maintaining calcium homeostasis within the body without the expense of calcium being leached out from the bones. However, for regular body growth and development Ca is a basic element and should be adequate in the diet to avoid other problems.

The above-developed model was extended for predicting the requirement of calcium to combat fluoride ingestion using data from the case study of Delhi by Susheela and Bhatnagar [13]. In this case, study two interventions were adopted to reverse fluorosis, first the fluorosis affected patients were advised to consume water with less fluorine, and second they were given calcium supplements simultaneously. The reported clinical pre-treatment data of blood and urinaiy fluoride were studied for various drinking fluoride levels and following correlations were fit from the reported data (as shown in Figures 11.8a and 11.8b):

(A) Plot of blood fluoride and drinking water fluoride fit to linear co-relation; the data was derived from case study 3 [13]

FIGURE 11.8 (A) Plot of blood fluoride and drinking water fluoride fit to linear co-relation; the data was derived from case study 3 [13].

(B) Plot between urinary fluoride and fluoride through drinking water fit to linear co-relation; the data was derived from case study 3 [13]

FIGURE 11.8(B) Plot between urinary fluoride and fluoride through drinking water fit to linear co-relation; the data was derived from case study 3 [13].

Thereafter, the values of blood and urinary fluoride were estimated using co-relation 1 and 2 after the first intervention of each patient, i.e., after reducing the level of drinking fluoride. These values were found to be greater than the reported values by Susheela and Bhatnagar [13]. because the reported values were taken after the second intervention, i.e., calcium supplementation. This difference in the values was hence used to calculate the effect of calcium in the treatment procedure by assuming that, apart from improving serum calcium levels the presence of extra calcium in daily intake directly inhibits the absorption of fluoride ions into the body, by forming insoluble complex of calcium fluoride [2]. Hence, the following two equations were used from the possible reactions occurring in serum based on its solubility product to find the extra calcium dose required due to fluoride toxicity.

Along with the Eqn. (11.14) and (11.15) following three boundary conditions have been used: (1) According to the 7th edition of World Book Rush-Presbyterian-St. Luke’s Medical Center Medical Encyclopedia (1995), World Book: 120-121. Volume of blood in body is 4.7 liters, (2) According to MedlinePlus Medical Encyclopedia (2013). A person urinates 2 liters in a day (3) 33% of the calcium intake dose gets absorbed from the stomach to serum [18]. The resulting “Delta Ca dose” for each patient with different water fluoride level was estimated and plotted (Figure 11.9) and Eqn. (11.16) was fitted linearly to the data to predict for the extra dose of calcium for different fluoride levels:

Plot between delta calcium and drinking water fluoride level fir to linear co-relation

FIGURE 11.9 Plot between delta calcium and drinking water fluoride level fir to linear co-relation.

However, apart from the above two reactions, calcium is responsible for many other reactions within the body so actual overall calcium dose requirement would be different from delta Ca dose. Gupta et al. [2] had proposed a thumb rale for treatment by Ca dosage stating “25 mg calcium dose is required per mg of fluoride intake.” So, Figure 11.10 was plotted in order to draw a comparison among the calcium doses predicted by different models that have been developed so far. These include calcium dosage predicted by Eqn. (11.11) and Eqn. (11.16) for drinking fluoride level ranging from 1 to 15 ppm, and that recommended by Gupta et al. [2].

Plot between calcium doses predicted by delta calcium model, reduced retention model and recommended calcium dose by Gupta et al. [2] on у-axis and drinking water fluoride level on x-axis

FIGURE 11.10 Plot between calcium doses predicted by delta calcium model, reduced retention model and recommended calcium dose by Gupta et al. [2] on у-axis and drinking water fluoride level on x-axis.

It was observed that RRM predicts values of calcium are almost comparable to the recommended values and the two curves almost overlap for drinking water fluoride level greater than 8 ppm. The deviations of delta Ca model can be accounted to the fact that Ca dose predicted by this model is in an addition to the Calcium requirement for normal bone growth. This is the value of Ca dose at which retention starts and hence should not be misinterpreted as the required adequate Ca dose for a human growth.


Bar chart shown in Figure 11.11 describes that even though the villages differ in the amount of fluoride consumed by its population through drinking water, the distribution of skeletal fluorosis looks similar, with less than 10% population in all the villages suffering due to skeletal fluorosis grade 2, around 30% from grade 1 and rest from grade 0. Grade 2, 1 and 0 presents the severity of fluorosis standing for severe, moderate, and mild respectively.

Bar chart depicting the severity of skeletal fluorosis

FIGURE 11.11 Bar chart depicting the severity of skeletal fluorosis (0,1, and 2 indicate the grade of skeletal fluorosis showing mild, moderate, and severe level) among the population of four target villages of Jaipur depicted in case study 1 [2].

This result was further explored by developing a classification model to predict grade of skeletal fluorosis from a child’s age, weight, and fluoride intake through drinking water resources. Backward logistic regression has been applied on the physical parameters’ data obtained from case study 1. The SPSS v20 output is shown in Table 11.7.

From Table 11.7, it is turned out that none of the independent variables including ‘age,’ weight,’ and ‘fluoridewater’ were significant enough (p<0.5) to develop a statistically significant model. Hence, all the variables got eliminated in step 4.

The model strengthens the result shown by bar graph in Figure 11.11 that grade of skeletal fluorosis is independent of fluoride intake through drinking water and also rules out any influence of age and weight of the subject. The reason behind this nature of result is due to limited data available. The values of daily dose of calcium, vitamin C, D, wider range of age and weight, levels of biochemical parameters within the body, etc., might be decisive in determining the severity of skeletal fluorosis.

TABLE 11.7 Variables Included in the Equation at Various Steps of Backward Logistic Regression*







Step la






























Step 2a






















Step 3a















Step 4a








s Variable(s) entered on step 1: age, weight, fluoride_water.

* The inapt variable gets removed in every subsequent step on the basis of Sig. value in backward logistic regression analysis.


RRM mathematically represented the biological changes as observed in the various biochemical parameters within the human body due to the prolonged intake of high fluoride which caused fluorosis. Delta Calcium model predicted the daily calcium dose requirement in order to prevent the harmful effects due to high fluoride intake as well as to reverse the fluoride deposition in individuals suffering from fluorosis.

The Pearson correlation coefficient between blood fluoride and serum calcium was -0.673 and between SAP and serum calcium was -0.520, both of which being negative demonstrate inverse con-elation between the respective pairs. This lends support to the initial steps of proposed pathophysiology of fluoride intake in human body reported in this paper. A nonnalized importance chart was developed by MLP analysis of the biochemical parameters, which revealed significant importance of serum GAG in detection of onset of fluorosis and prediction of calcium dose. This novel finding was applied to develop RRM, which successfully predicted daily dose of calcium required to compete with fluoride intake through drinking water. These dosages were found to be in accordance with the medically recommended thumb rule of 25 mg calcium requirement per mg of fluoride intake. Further, the logistic regression analysis was applied for case study 1, which deduced that reported data of level of fluoride intake, age, and weight of the children were not sufficient enough to classify the perceived grade of skeletal fluorosis among the population of Ramsagar, Rampura, Shivdaspura, and Raipuria. This is due to the role of other parameters like serum calcium, serum fluoride, GAG, vitamin C, D, etc., against the development of skeletal fluorosis. In sum, this study gives an insight to contemporary mathematical modeling approach used to analyze the reported data of fluoride affected regions. Since, serum GAG was found to be a strong indicator of fluorosis; it may be considered as an early predictor for the onset of fluorosis and thus help in the medical management of this menace.



multilayer perceptron parathyroid hormone Pearson product-moment coefficient reduced retention model serum GAG


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