^{1}

^{2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

_{2}, Colgate Pro-relief, and Sensodyne repair (_{2} and Colgate Pro-relief. It was, however, found that the trajectory for the Sensodyne repair was a bit complex, thus making the prediction difficult.

Over the last decade, dentin hypersensitivity (DH) has been extensively researched owing to its widespread prevalence and noticeable painful oral health problem affecting many individuals [

Although bioactive glass occludes patent dentinal tubules by supplying calcium (Ca^{2+}) and phosphate (^{2+} and

In an attempt to address the aforementioned concerns of limited salivary flow, proargin technology was developed by Kleinberg in 2002 based on the role saliva plays in naturally occluding dentinal tubules [

Given the differences in the occluding characteristics for the aforementioned biomaterials in saliva and without saliva, a new biomaterial from eggshell waste and titanium dioxide (EB@TiO_{2}) is proposed as an alternative occluding material for DH management. Whilst a recent study has demonstrated the occluding characteristics of EB@TiO_{2} [

Importantly, mathematical modeling offers a different research perspective by overcoming some of the problems frequently encountered in an experimental study [

The logistic equation was first proposed by the seminal work of Pierre-Francois Verhulst (1844-1845). Verhulst derived the logistic equation to describe the self-limiting growth of biological population [

Ever since the discovery, the logistic equation has been extensively used in many scientific fields such as ecology, chemistry, population dynamism, mathematical psychology, political science, geoscience, statistics, economics, and sociology [

Another typical application of the logistic equation is in medicine, where the logistic differential equation is used to model the growth of tumors or to study the reaction of pharmacokinetics [_{2}, proargin, and bioactive glass (NovaMin), in saliva and without saliva.

Food grade anatase titanium dioxide (CAS No: 13463677) was purchased from Sigma-Aldrich (Germany). Citric acid monohydrate was supplied by Merck (South Africa). Two different brands of toothpaste, namely, Sensodyne repair (GlaxoSmithKline, UK) and Colgate Pro-relief (Colgate-Palmolive, Poland), were bought from a popular shopping mall located at Durban (South Africa).

Eggshell and titanium dioxide composite was prepared in accordance with the method reported in the literature [_{2}).

The experimental parameter was obtained from the remineralization test conducted in our laboratory. Twenty-one dentin specimens measuring 5 mm × 5 mm × 1 mm were prepared by sectioning perpendicular to the long axis of the teeth below the enamel-dentinal junction using a low-speed diamond saw under water cooling conditions. A sensitive model was simulated by soaking the specimens in 4% wt citric acid solution for 2 min. The specimens were then randomly assigned into three groups, namely, EB@TiO_{2}, Colgate Pro-relief, and Sensodyne repair (

Each specimen from the respective groups was brushed twice daily (morning and evening) for seven days with a toothbrush powered with 1.5 v alkaline battery (Oralwise, China) for 1 min and allowed to dry for 30 s before rinsing with deionized water. Brushing was performed at room temperature using 100 mg of each respective toothpaste. The slurry of EB@TiO_{2} was prepared by mixing 100 mg of the powder with 200

The mathematical model considers that the size of dentin tubules (

The following logistic equation model was proposed:

By using the method of solving the first-order ordinary differential equation (separation of variable method), we obtain the analytical solution of the model equation (3):

A model fitting and parameter estimation were conducted using our model to fit real data for the three samples (EB@TiO_{2}, Colgate Pro-relief, and Sensodyne repair) for the two cases: with saliva and without saliva. The aim of these analyses is to show that the model we considered can be used to study as well as make future predictions on these samples. For the model fitting, we take the carrying capacity (_{x}) as 100% while the growth rate is estimated from the model fittings for all the samples. The model fittings were carried out using the built-in MATLAB least-squares fitting routine fmincon in the optimization toolbox.

Table _{2}-treated group, the % occluded area ratios observed for the specimens with saliva were significantly higher than those without saliva for day 2, 3, 4, 6, and 7 (

Mean and standard deviation for the area ratios of the occluded tubules brushed in seven days with and without saliva.

Day 1 | Day 2 | Day 3 | Day 4 | Day 5 | Day 6 | Day 7 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

WS | S | WS | S | WS | S | WS | S | WS | S | WS | S | WS | S | |

EB@TiO_{2} |
19 ± 2.6^{a} |
15.6 ± 2.2^{b} |
35.1 ± 4.4^{a} |
43.6 ± 4.7^{b} |
50.4 ± 4.3^{a} |
71.4 ± 2.8^{b} |
62.0 ± 3.7^{a} |
82.0 ± 3.1^{b} |
93.3 ± 2.3^{a} |
93.9 ± 3.0^{a} |
95.1 ± 1.6^{a} |
98.4 ± 1.9^{b} |
97.9 ± 1.3^{a} |
99.3 ± 1.0^{b} |

Colgate Pro-relief | 32.3 ± 3.0^{a} |
16 ± 3.6^{b} |
45.7 ± 6.8^{a} |
21.4 ± 2.2^{b} |
62.3 ± 6.7^{a} |
20.3 ± 2.7^{b} |
62.0 ± 6.7^{a} |
20.9 ± 2.9^{b} |
68.6 ± 6.1^{a} |
19.9 ± 3.4^{b} |
74.3 ± 6.8^{a} |
61.9 ± 4.8^{b} |
88.9 ± 3.2^{a} |
80.1 ± 8.6^{b} |

Sensodyne repair | 12.9 ± 3.6^{a} |
19.3 ± 3.5^{b} |
13.6 ± 2.5^{a} |
29.0 ± 2.2^{b} |
13.4 ± 2.8^{a} |
44.1 ± 4.9^{b} |
15.0 ± 3.3^{a} |
60.9 ± 4.6^{b} |
16.0 ± 3.1^{a} |
55.3 ± 5.3^{b} |
17.0 ± 3.7^{a} |
53.3 ± 6.2^{b} |
70.1 ± 4.2^{a} |
90.4 ± 3.0^{b} |

Values are mean ± standard deviations (

For the Sensodyne repair group, the % occluded area ratios observed in days 1–7 for the group with saliva were statistically significantly higher than those observed without saliva (

The results of the model fitting for the three samples are presented in Figure _{2} and Colgate Pro-relief (for the two cases: with and without saliva), but did not produce a very good fitting for the Sensodyne repair.

Model fitting of the desensitizing paste in saliva and without saliva.

The estimated parameters for the tubule occlusion rate (_{x}) is given in Table _{2}, it was observed that the _{x} was higher (0.9500) with saliva when compared with the rate measured without saliva (0.7762). Similarly, the _{x} measured for Sensodyne repair was higher in the specimens treated with saliva (0.4635) when compared against those treated without saliva (0.2646). In contrast, it was found that the _{x} for specimens treated with Colgate Pro-relief without saliva was higher than those measured with the specimens treated with saliva.

Estimation of model parameters.

Parameters (_{x}) |
Estimated by model fitting | |
---|---|---|

Without saliva | With saliva | |

EB@TiO_{2} |
0.7762 | 0.9500 |

Colgate Pro-relief (proargin technology) | 0.4230 | 0.3558 |

Sensodyne repair (NovaMin) | 0.2646 | 0.4635 |

The purpose of this study was to investigate the use of the logistic equation model to predict the remineralization characteristics of desensitizing paste. Ilie [

Overall, the remineralization characteristics measured for EB@TiO_{2} were significantly better than those observed for Colgate Pro-relief and Sensodyne repair (_{2} in saliva and without saliva appear to follow the same pattern. More importantly, the model produces a good fitting for the dentin specimens treated with or without saliva (Figure _{2} would achieve complete occlusion of the dentin tubules at the end of seven days brushing. The remineralization ability observed for samples treated with EB@TiO_{2} may be attributed to the modification of the carbonate structure content in eggshell with titanium dioxide. This is in agreement with the suggestion of Cutler [_{2} sample group could be attributed to the nanosized particle distribution in EB@TiO_{2} [

With regard to the Colgate Pro-Argin paste, the model fitting suggests that the behavior with and without saliva differs. It was found that the model produced a better fitting for the samples treated without saliva (Figure

In terms of the Sensodyne repair, we could not achieve a good model fit to make an accurate prediction of the treatment. This may, however, be related to the quality of the input parameters [^{2+}) and phosphate (

In this paper, we have demonstrated that our model can be used to study and make a future prediction for the two samples: EB@TiO_{2} and Colgate Pro-relief (for both cases: with and without saliva). For the Sensodyne repair, we discover from the figure that the trajectory for its real data is a bit complex. This makes it difficult to obtain a good fitting using our model. We hope to consider a more complex model that can fit the real data for Sensodyne repair in our future work.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

The authors acknowledged the financial support from the National Research Foundation of South Africa (no. 104824).