Mathematical modeling and projection of vitamin D levels before and during COVID-19: a cross-sectional study including 86.772 samples

doi:10.21203/rs.3.rs-1986452/v1

Purpose

Vitamin D (Vit-D) is defined as a steroid hormone produced in the body by exposure of the human skin to UV (ultraviolet) B rays and has an important role in calcium and phosphorus metabolism as well as bone mineralization. In this study, it was aimed to examine Vit-D deficiency not only biologically but also mathematically and produce a mathematical model to determine the level of Vit-D if the COVID-19 measures continue.

Methods

This study included 86.772 patient samples. Vit-D levels have a wave-curved form. Mathematical modeling has been done for such a curve, and this model has been defined. For describing the model, the MATLAB Curve Fitting Toolbox has been used. SSE has been optimized by the Trust-Region algorithm where all the other parameters are selected as default.

Results

There was no significant difference between the sexes according to Vit-D levels (p > .05). It was determined that Vit-D levels were significantly higher in the summer months and lower in the winter months (p < .001). When comparing the spring months, Vit-D levels in 2020 were found to be significantly lower than in 2019 (p < .001); on the contrary, when examined based on the summer, autumn, and winter months, it was determined that Vit-D levels increased in 2020 compared to 2019 (p < .001). A mathematical model was produced using Vit-D data and a future prediction was made.

Conclusion

Restrictions, partial or complete closures, and curfews can significantly affect individuals' Vit-D levels during the COVID-19 outbreak. There is a need for multicenter studies with larger populations covering different regions to strengthen and support our results.

Vitamin D

mathematical modeling

mathematical projection

COVID-19

SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2), which caused the COVID-19 pandemic, exerts its effect by infecting pulmonary epithelial cells through the ACE-2 (angiotensin-converting enzyme-2) receptor [1]. While SARS-CoV-2 causes damage to the pulmonary epithelium, it also infects macrophages using ACE-2 receptors and leads to their activation [2]. Macrophages, neutrophils, and T cells can be activated through sustained elevation of important cytokines such as interleukin (IL)-1, IL-6, and tumor necrosis factor (TNF) alpha. Eventually, type 2 pneumocyte apoptosis may be induced and patients may directly face acute respiratory distress syndrome (ARDS) [2]. At this point, host responses can be enhanced in some cases by the overwhelming expression of proinflammatory cytokines. This "cytokine storm" can be blamed for some of the serious events of COVID-19, such as ARDS [3, 4].

Vitamin D (Vit-D) is defined as a steroid hormone produced in the body by exposure of the human skin to UV (ultraviolet) B rays and has important roles in calcium and phosphorus metabolism as well as bone mineralization [5, 6]. Subcutaneous production can be described as the major source of Vit-D synthesis, although foods such as dairy products, fish oil, and liver also contain noticeable amounts of Vit-D [6, 7]. Synthesis and availability of Vit-D depend on many important factors, including exposure to sunlight, latitude, the season of the year, the hour of the day, pigmentation of the skin, age, sex, and body mass index. Vit-D binds to nuclear Vit-D receptors called VDRs during cellular events, forming a VDRE. In this case, the activated VDR can attach to the cathelicidin gene promoter VDRE and thus cause the host to initiate defense against some viral infections. Vit-D can also directly affect the innate immune system through the expression of lysosomal enzymes and the release of nitric oxide. In both cases, Vit-D directly contributes to the fight against infection. Thus, Vit-D plays an immunomodulatory role in SARS-CoV-2 infection by suppressing the adaptive immune system of respiratory epithelial cells [5, 8, 9].

It is a well-known fact that Vit-D deficiency is a risk factor for many diseases, such as autoimmune diseases [10], cardiovascular diseases [11], obesity [12], and cancer [13]. Vit-D deficiency in individuals has been clearly demonstrated in several studies conducted in many countries, including Turkey [14, 15]. Furthermore, with the curfew within the scope of the measures taken during the COVID-19 pandemic process, the amount of daylight that individuals are exposed to is decreasing day by day, and that contributes to Vit-D deficiency.

In recent years, there has been an increasing interest in studies on Vit-D. Most of these studies are focused on both disease pathogenesis and the etiology of Vit-D [18–23]. Degerud et al. (2016) investigated the seasonal Vit-D levels in cardiovascular diseases [40]. In another study conducted by Zgaga et al. (2022), statistical power was evaluated in randomized controlled trials investigating Vit-D [41]. However, there is no study aimed at the mathematical modeling of Vit-D. In this study, it was aimed to examine Vit-D deficiency not only biologically but also mathematically and produce a mathematical model to determine the level of Vit-D if the COVID-19 measures continued.

SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2), which caused the COVID-19 pandemic, exerts its effect by infecting pulmonary epithelial cells through the ACE-2 (angiotensin-converting enzyme-2) receptor [1]. While SARS-CoV-2 causes damage to the pulmonary epithelium, it also infects macrophages using ACE-2 receptors and leads to their activation [2]. Macrophages, neutrophils, and T cells can be activated through sustained elevation of important cytokines such as interleukin (IL)-1, IL-6, and tumor necrosis factor (TNF) alpha. Eventually, type 2 pneumocyte apoptosis may be induced and patients may directly face acute respiratory distress syndrome (ARDS) [2]. At this point, host responses can be enhanced in some cases by the overwhelming expression of proinflammatory cytokines. This "cytokine storm" can be blamed for some of the serious events of COVID-19, such as ARDS [3,4].

Vitamin D (Vit-D) is defined as a steroid hormone produced in the body by exposure of the human skin to UV (ultraviolet) B rays and has important roles in calcium and phosphorus metabolism as well as bone mineralization [5,6]. Subcutaneous production can be described as the major source of Vit-D synthesis, although foods such as dairy products, fish oil, and liver also contain noticeable amounts of Vit-D [6,7]. Synthesis and availability of Vit-D depend on many important factors, including exposure to sunlight, latitude, the season of the year, the hour of the day, pigmentation of the skin, age, sex, and body mass index. Vit-D binds to nuclear Vit-D receptors called VDRs during cellular events, forming a VDRE. In this case, the activated VDR can attach to the cathelicidin gene promoter VDRE and thus cause the host to initiate defense against some viral infections. Vit-D can also directly affect the innate immune system through the expression of lysosomal enzymes and the release of nitric oxide. In both cases, Vit-D directly contributes to the fight against infection. Thus, Vit-D plays an immunomodulatory role in SARS-CoV-2 infection by suppressing the adaptive immune system of respiratory epithelial cells [5,8,9].

It is a well-known fact that Vit-D deficiency is a risk factor for many diseases, such as autoimmune diseases [10], cardiovascular diseases [11], obesity [12], and cancer [13]. Vit-D deficiency in individuals has been clearly demonstrated in several studies conducted in many countries, including Turkey [14,15]. Furthermore, with the curfew within the scope of the measures taken during the COVID-19 pandemic process, the amount of daylight that individuals are exposed to is decreasing day by day, and that contributes to Vit-D deficiency.

In recent years, there has been an increasing interest in studies on Vit-D. Most of these studies are focused on both disease pathogenesis and the etiology of Vit-D [18-23]. Degerud et al. (2016) investigated the seasonal Vit-D levels in cardiovascular diseases [40]. In another study conducted by Zgaga et al. (2022), statistical power was evaluated in randomized controlled trials investigating Vit-D [41]. However, there is no study aimed at the mathematical modeling of Vit-D. In this study, it was aimed to examine Vit-D deficiency not only biologically but also mathematically and produce a mathematical model to determine the level of Vit-D if the COVID-19 measures continued.

Materials and methods

This study suggests two different approaches: one is to create an empirical model, and the other is to generate a statistical analysis of the data. In this retrospective, the cross-sectional and methodological study included 125.643 patients (18–75 years) who were admitted to the Dokuz Eylul University Hospital for various reasons and whose Vit-D levels were measured in the Biochemistry unit between 2019-2020 and 2020-2021 (before and after the COVID-19 outbreak). It has been accepted that 2020 reflects the period of the COVID-19 pandemic, while 2019 reflects the non-pandemic period. The dependent variable of the study was Vit-D level, and the independent variables were defined as age, sex, seasons, and months. This study was approved by the Dokuz Eylul University Non-Interventional Research Ethics Committee (Date: 05.01.2022, Approval No: 2022/01-26). Due to the retrospective observational nature of the study, written informed permission was not required, and all techniques followed the Declaration of Helsinki.

Vitamin D total analytical properties

The Vit-D total test was measured in vitro for the quantitative determination of a total of 25 (OH) Vit-D in human serum and plasma (EDTA, lithium-heparin, sodium-heparin) using ADVIA Centaur XP systems. The ADVIA Centaur Vit-D test is measured with the chemiluminescence immune method. It is a competitive immunoassay using an anti-25(OH) Vit-D monoclonal mouse antibody labeled with acridinium ester (AE) and a fluorescently labeled Vit-D analog. Analysis can be monitored with the REF 10493589 ADVIA Centaur Vit-D Calibrator. Three levels of REF 10632229 ADVIA Centaur Vit-D Control are used as internal quality control.

Performance characteristics of the analysis

Measuring range: The ADVIA Centaur Vit-D test measures 25(OH) Vit-D in the concentration range from 4.2 to 150 ng/ml (10.5 to 375 nmol/l); Sensitivity: The functional sensitivity of the ADVIA Centaur Vit-D test is 3.33 ng/ml (8.33 nmol/l); Repeatability (Precision): 4.93% Within-Run 5.44% Between-Runs; Method Comparison: ADVIA Centaur Vit-D = 1.15 (LC/MS/MS) + 0.70 ng/ml, r = 0.91. In the verification study performed in the laboratory, the total error was found to be 12.97%.

Description of the dataset

As mentioned earlier, the dataset was taken by "The Biochemical Laboratory of Dokuz Eylul University". The observed raw data includes 125.643 analysis results. Moreover, the features of the dataset contain information about age, the date of birth, sex, the date of the analysis, and the score Vit-D of the patient. Among these features, "sex" is a categorical value, the date of analysis and the date of birth are time series, and the rest of them are numerical values.

The raw data has been cleaned elaborately with the consensus of the authors. First, the unnecessary columns are removed. This step is followed by limiting the study to the range of 18-75 years due to a potential bias that may occur from age. Then, both the incidental values and the null values have been removed. Additionally, the outliers are also removed. To investigate the seasonal effects of COVID-19 on the levels of Vit-D, the 'Date of Analysis' time series has been manipulated to convert it into "Year" and "Month". Finally, the years are grouped by considering the mean of the Vit-D Score.

After the cleaning process has been completed successfully, the dataset contains 86.772 samples and 5 features such as "age", "sex", "Vit-D Score", "year" and "month".

Mathematical model

Understanding the existing truth behind a disease to interpret the clinical approaches is not simple. Mathematical models can be instrumental in our understanding of the biological behavior of such a system. A convenient mathematical model relies on many complex patterns, such as the selection of the crucial features of the system, the choice of specific parameters, the use of the correct mathematical tools, physical compatibility, etc. Besides the complexity of the anatomy of a human, all these issues make developing a mathematical model a much more challenging problem.

Various types of mathematical models can be used for different purposes in the applied sciences. Among them, due to the use of experiments rather than physical theory, empirical models are more original. While describing an empirical model, statistical inferences are used for model parameter selections. However, the creation process still has its own difficulties and limitations. One of the greatest challenges in describing an empirical model is the non-availability of the use of categorical values in the dataset. This makes our task more difficult but more attractive.

In our study, the collected dataset contains both numerical and categorical values. Therefore, the first aim has been designed to study the categorical values priorly. For this purpose, the significance levels of the mean score of Vit-D were carried out for both sexes in a month. Moreover, for a continuous mathematical model, the age column was also excluded. Thereafter, for the purpose of seasonal study, the mean levels of Vit-D are grouped by years.

Vitamin-D levels have a wave-curved form. It is worth noting that the results can vary from data to data. The following functions can be utilized to describe a model for such a curve:

where ${a}_{i},{b}_{i},{c}_{i}$ and $\omega$ are constants. Moreover, ${\widehat{L}}_{Vit-D}\left(t\right)$ represents the approximated value of Vit-D levels where $t$ stands for time (in months). Furthermore, ${N}_{e}$ denotes the number of terms of the approximation.

The rearranged data were randomly divided into training (70%) and test (30%) sets. The goodness of the models was controlled by${ R}^{2}$and Adjusted-${R}^{2}$ (A_${R}^{2}$) such that:

$${R}^{2}=1-\frac{SSE}{SST}$$

3

$$A\_{R}^{2}=1-\left(\frac{n-1}{n-p}\right)\frac{SSE}{SST}$$

4

Notice that SSE and SST denote the sum of squared error and the sum of the squared total, respectively. Additionally, n is the number of observations, and p is the number of regression coefficients.

Statistical analysis

In the evaluation of the data, descriptive statistics, means, median values (percentiles), and standard deviations (sd) of the patients were calculated. Compliance with the normal distribution was checked with the Kolmogorov-Smirnov test. The Student’s t-test and ANOVA were used to examine the differences between continuous variables. All computational studies were performed using MATLAB software (version 2021b). For describing the model, the MATLAB Curve Fitting Toolbox has been used. In so doing, SSE has been optimized by the Trust-Region algorithm, where all the other parameters are selected as default. The SPSS (version 24.0) and R (version 4.1.3) package programs were also used for the analysis and visualization of statistical data. The statistical significance level was determined as a 2-sided p < .05.

A total of 86.772 samples were included in this study, and 67.5% (n = 58.545) of the study group were women. The median age of the research group was 54.0 (40.0–65.0) and the mean age was determined to be 51.8 ± 15.6 years. The median and mean Vit-D levels at the time of measurement were also calculated at 19.8 (13.8–26.7) ng/ml and 21.6 ± 11.5 ng/ml, respectively. Descriptive statistics of the research group are available in Table 1.

Table 1

Descriptive features of the research group
Variables	Total (n = 86.772)
Sex, n (%) Women Men	58.545 (67.5) 28.227 (32.5)
Age (median, interquartile range), year	54.00 (40.00–65.00)
Vitamin D level (median, interquartile range), ng/ml	19.81 (13.81–26.76)
Age group, n (%) ≤ 18–30≤ < 30–45≤ > 45	11.189 (12.9) 17.383 (20.0) 58.200 (67.1)
Season, n (%) Spring Summer Autumn Winter	17.224 (19.8) 20.466 (23.6) 23.106 (26.6) 25.976 (29.9)

The relationship between Vit-D levels and sex was investigated. There is no significant difference between the sexes according to Vit-D levels (p > .05) (Fig. 1). The association between vitamin D levels and age group was also investigated. According to the analysis results, it has been found that Vit-D levels increase significantly as age increases (p < .001) (Fig. 2). When Vit-D levels were examined according to the seasons, as shown in Fig. 2 and Fig. 3, it was determined that Vit-D levels were significantly higher in the summer months and lower in the winter months (p < .001). Vit-D levels were found to be higher in autumn than in spring, which can be described as the transition season (p.001) (Fig. 3). In addition to these, seasonal Vit-D changes according to years were also evaluated. According to the test results, when comparing the spring months, Vit-D levels in 2020 were found to be significantly lower than in 2019 (p < .001); on the contrary, when examined based on the summer, autumn, and winter months, it was determined that Vit-D levels increased in 2020 compared to 2019 (p < .001) (Fig. 4).

From a mathematical point of view, Fig. 1 emphasizes that the mean Vit-D levels in both men and women have a similar trend line. That is, the sex column can be considered negligible. Supplementary Fig. S1 is evidence that the Vit-D levels have a wave-curved form. Once all the preprocessing has been fulfilled, the trend line of the Vit-D levels in years is presented in Supplementary Fig. S1. In this study, two mathematical models have been suggested to approximate the cleaned dataset. The results are listed in Table 2.

Table 2

Comparisons of PM1 and PM2 in terms of${\text{N}}_{\text{e}}$
	PM1 (sine)
	${N}_{e}=3$	${N}_{e}=4$	${N}_{e}=5$	${N}_{e}=6$	${N}_{e}=3$	${N}_{e}=4$	${N}_{e}=5$	${N}_{e}=6$
R²	0.8206	0.8506	0.9230	0.9682	0.8097	0.8451	0.9404	0.9165
A R²	0.7309	0.7241	0.8153	0.8910	0.7313	0.7521	0.8900	0.8178
$\#$ of coefficients	9	12	15	18	8	10	12	14

Table 3

Future predictions of the proposed model
Date	Predictions
Jan 2022	18.0859
Feb 2022	18.7034
Mar 2022	18.2834
Apr 2022	16.8195
May 2022	16.8326
Jun 2022	20.2541
Jul 2022	25.4126
Aug 2022	28.3784
Sep 2022	27.3337
Oct 2022	24.4880
Nov 2022	23.1418
Dec 2022	23.7753

As seen in Table 2, as ${N}_{e}$ increases the accuracies of the models are getting well. However, the goodness of the PM2 interrupted its consistency when ${N}_{e}=6$. This probably happened because of the over-estimation. Considering the accuracies and the number of coefficients altogether, the PM2 for ${N}_{e}=5$ seems much preferable than those PM1. Thus, the mathematical model of the estimations of levels of Vit-D is

Supplementary Fig. S2 indicates how fits the proposed model to the raw data set.

After the estimated values are computed the proposed model has been tested on the test set. For a reliable test, the root means square error (RMSE) and the relative error (RE) have been considered.

$$RMSE={\left(\frac{1}{11}\sum _{i=1}^{11}{\left|{L}_{Vit-D}\left({t}_{i}\right)-{\widehat{L}}_{Vit-D}\left({t}_{i}\right)\right|}^{2}\right)}^{1/2}$$

7

$$RE=\frac{\sum _{i=1}^{11}\left|{L}_{Vit-D}\left({t}_{i}\right)-{\widehat{L}}_{Vit-D}\left({t}_{i}\right)\right|}{\sum _{i=1}^{11}\left|{L}_{Vit-D}\left({t}_{i}\right)\right|}$$

8

where ${L}_{Vit-D}\left({t}_{i}\right)$ is the value of raw data at $t={t}_{i}$, and ${\widehat{L}}_{Vit-D}\left({t}_{i}\right)$ is the predicted data values,$i=\text{1,2},..11.$

For the proposed model the errors are as follows: RE = 0.1110 and RMSE = 2.8839. This guarantees that the proposed model is a good agreement with the test data with an 11% error margin. A further investigation has been done for predicting the Vit-D levels in the upcoming months. Before ending the section, the graph of the future predictions has been illustrated in Supplementary Supplementary Fig. S3.

Mathematical models are used as a tool not only to understand the truth behind real-life problems but also to develop measures to be taken to solve these problems. In this study, even though two mathematical models have been suggested, a more convenient one has been selected to analyze the dataset. Since there is no such study in the literature, this model can not be discussed with any other model. Therefore, considering all the presented tables and figures together, the described mathematical model fits well with the real data.

Previous studies indicated no specific data about gender preference for vitamin D deficiency. Some studies demonstrated that vitamin D deficiency was more common in females, whereas others stated that it was more common in males [24, 25]. Our data show that there is no statistically significant difference between males and females.

Many reports indicate that older age is a risk factor for vitamin D deficiency [24, 26, 27]. Even healthy elderly people are advised to take higher doses of vitamin D [28]. Our analysis, on the contrary, points out that vitamin D levels increase in older age groups. This may be related to higher vitamin D supplementation in older age groups in this pandemic period.

When the spring months were examined, we discovered that Vit-D levels were significantly lower in 2020 than in 2019 (p < .001). The rapid increase in COVID-19 cases and the emergence of restriction measures can be shown as the reasons for the decrease in Vit-D levels. Because the COVID-19 outbreak was declared a pandemic by the World Health Organization (WHO) on March 11, 2020 [29, 30]. On the same date (11 March 2020), the first case of COVID-19 was detected in Turkey [31]. On April 3, entrances and exits to the city were banned in 30 metropolitan cities in Turkey for 15 days, and later, the capacity of restrictions was expanded across the country day after day [32]. Therefore, due to restrictions, people were not exposed to getting enough sunlight, so Vit-D levels may have dropped.

We compared the years 2020 and 2021 and discovered that Vit-D levels increased significantly in 2021 compared to 2020 (p < .001). In various randomized controlled and observational studies, it has been emphasized that the use of various types and amounts of Vit-D supplements may have a protective effect against COVID-19 [33–37]. Therefore, this rise in Vit-D levels suggests that it may be due to the increase in the use of Vit-D supplements, as they may be protective against COVID-19 or reduce its side effects.

When 2019 and 2021 are compared, we found that although Vit-D levels in 2021 were higher than in 2020, they were lower than in 2019. There was a second wave during the COVID-19 period [38, 39]. This decrease in Vit-D may have been caused by restrictions in the second wave of COVID-19. However, this decrease in Vit-D levels is relatively less than the decrease in the onset of COVID-19.

There were some limitations to this research. The first is that all the included data was obtained from the hospital database. While it is an advantage to have data from a large hospital database in the region, data from a single center may not reflect the population's Vit-D levels. Likewise, hospital access has decreased because of restrictions. This may involve a certain amount of bias in determining actual Vit-D levels in the population. Another limitation is that the study was carried out in a relatively sunny region of Turkey, and this may be misleading as to actual Vit-D levels in the country. In addition to all this, from the mathematical point of view, the obtained coefficient can vary due to the random division of the training and test sets. This means that the returned values of levels of Vit-D can change according to the coefficient.

This study was undertaken to propose a mathematical model for the level of Vit-D and predict an average level for the upcoming months. Restrictions, partial or complete closures, and curfews can significantly affect individuals' Vit-D levels during the COVID-19 outbreak. Therefore, possible decreases in Vit-D in individuals may cause an increase in the incidence of Vit-D-related diseases. Despite the limitations, the empirical findings of this study provide a new understanding of insights for describing such models in medicine. Vit-D is an essential and necessary vitamin for individuals of all ages. In this respect, it is very important to make future forecasts for the development of measures that can be taken in terms of public health during pandemic periods. Thus, policies can be created to raise awareness of the community about Vit-D in cases of partial or complete closure. There is a need for multicenter studies with larger populations covering different regions to strengthen and support our results.

Data availability statement: The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Ethical approval: This study was approved by Dokuz Eylul University Non-Interventional Research Ethics Committee with decision number 2022/01-26 and date 05.01.2022.

Funding statement: This research has received no specific support from any funding agency, commercial or non-profit sectors.

Conflict of interest statement: Mehmet Emin Arayici, Sila Ovgu Korkut Uysal, Asim Leblebici, Zeynep Akcali, Deniz Tuna Edizer, Seher Kabul, Dilek Cimrin and Ender Berat Ellidokuz declare that they have no conflict of interest.

Authors’ Contributions: Mehmet Emin Arayici, Sila Ovgu Korkut Uysal, Asim Leblebici, Zeynep Akcali, Deniz Tuna Edizer, Seher Kabul, Dilek Cimrin and Ender Berat Ellidokuz contributed significantly to the work's idea, design, and data collection. Mehmet Emin Arayici, Sila Ovgu Korkut Uysal, Asim Leblebici contributed to data analysis. Mehmet Emin Arayici, Sila Ovgu Korkut Uysal, Asim Leblebici, Zeynep Akcali, Deniz Tuna Edizer, Seher Kabul, Dilek Cimrin and Ender Berat Ellidokuz all contributed to the work's data interpretation. All authors contributed to the preparation and critical revision of the paper. The final manuscript was reviewed and approved by all authors.

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No competing interests reported.

Supplementalmaterials.docx

Mathematical modeling and projection of vitamin D levels before and during COVID-19: a cross-sectional study including 86.772 samples

Status:

Version 1

Abstract

Purpose

Methods

Results

Conclusion

Figures

Introduction

Materials And Methods

Statistical analysis

Results

Discussion

Limitations

Conclusion

Declarations

References

Additional Declarations

Supplementary Files

Status:

Version 1

	PM1 (sine)
	\({N}_{e}=3\)	\({N}_{e}=4\)	\({N}_{e}=5\)	\({N}_{e}=6\)	\({N}_{e}=3\)	\({N}_{e}=4\)	\({N}_{e}=5\)	\({N}_{e}=6\)
R²	0.8206	0.8506	0.9230	0.9682	0.8097	0.8451	0.9404	0.9165
A R²	0.7309	0.7241	0.8153	0.8910	0.7313	0.7521	0.8900	0.8178
\(\#\) of coefficients	9	12	15	18	8	10	12	14