A Mathematical Model for Determining the Body’s Fluctuating Need for and Synthesis of Active Vitamin D

The process by which 1,25(OH)2D3 is synthesized and degraded and how it is transported out of the cell and body is described. The changing demand for the synthesis of 1-25(OH)2D3 during different conditions experienced by the body is reviewed. A method of determining 1,25(OH)2D3 synthesis and demand, and the percent utilization of 25(OH)D3 to make 1,25(OH)2D3 is presented based on the measurement of the end metabolites of 1,25(OH)2D3 and of its immediate precursor, 25(OH)D3. A mathematical model has been developed to allow the calculation of 1,25(OH)2 D synthesis, and demand, and the percent utilization of 25(OH)D3. Simple algebraic equations have been derived which allow the calculation of these new parameters using the concentrations of the end metabolites of 1,25(OH)2D3 and its immediate precursor, 25(OH)D3 in the serum and urine. Vitamin D plays an important role in combating invading bacteria and viruses and in subduing the body’s associated inflammatory response. This new approach to evaluating vitamin D status may help clinicians determine 25(OH)D3 and 1,25(OH)2D3 levels needed to suppress bacterial infections, viral replication during new viral infections and the reactivation of latent viruses, and to downregulate the inflammatory responses caused by bacteria and viruses.


Introduction
Vitamin D is a fat-soluble vitamin and a hormone that has receptors in many tissues of the body. It plays an important role in many different normal and abnormal conditions that routinely affect the human body. One of the first roles that was discovered for vitamin D was the maintenance of normal calcium and phosphorous levels in bone in order to prevent rickets in children and osteoporosis in adults. It is necessary for maintaining normal muscle strength. It has a very important role in immune system function in order to fight off infection and modulate the inflammatory response to infection. It is important for a developing embryo and fetus during pregnancy. Finally, there is growing evidence that it is important to reduce the risk of heart disease, type 2 diabetes, cancer, and the risk of premature death and cognitive decline [1].
There have been numerus studies recently which have linked increased COVID-19 morbidity and mortality to low vitamin D states [2][3][4][5][6]. Likewise, there have been recent review articles which discuss reasons why low vitamin D states may play a role in a patient's ability to fight off or mitigate the morbidity and mortality associated with a COVID-19 infection [7][8][9]. Central to this new focus on vitamin D status is the important role that vitamin D plays in the modulation of the immune system [10][11][12]. Immune system cells with vitamin D receptors exist in large numbers in many tissues of the body including primary lymphoid organs (bone marrow and thymus) and secondary lymphoid organs (lymph nodes, the spleen, the tonsils, skin, and various mucous membrane layers in the body including those of the nose, throat, and bowel) [13]. An important recent article reviews the relationship between co-morbidities in COVID-19 patients known to be associated with increased morbidity and mortality and those same co-morbidities in low vitamin D states [14].
Persistent low vitamin D states may chronically impair the immune system in individuals in two ways. First, in low vitamin D states, the immune system may be unable to maximally suppress viruses such as COVID-19 in part due to inadequate production of cathelicidin and defensin β2 [15][16][17]. Second, vitamin D is a hormone which acts directly on the immune system including B and T lymphocyte cells to downregulate the inflammatory reaction triggered by viral antigens, or other microorganisms [18].
Plasma 25(OH)D 3 vitamin D precursor levels represent vitamin D stores in the body and have been traditionally measured to assess vitamin D status. This paper will build on and review evidence supporting the following two premises: (1) plasma 25(OH)D 3 levels do not consistently represent 1,25(OH) 2 D 3 synthesis (or total body levels), demand for 1,25(OH) 2 D 3 , or percent utilization of 25(OH)D 3 to form 1,25(OH) 2 D 3 by the body, and (2) the need for and utilization of active vitamin D by tissues of the body can vary between individuals and dramatically in the same individual under different circumstances. Many factors can reduce a person's ability to produce active vitamin D including low vitamin D precursor stores, increased body mass index, BMI (i.e., obesity) which dilutes the concentration of vitamin D precursor in the serum, decreased exposure to ultra-violate B, UVB, light, increasing age, exogenous medications, and genetic factors.
The measurement of active vitamin D, 1,25(OH) 2 D 3 , plasma levels is possible and in certain situations is indicated. This paper will present information to support the premise that serum active vitamin D levels often do not correlate with vitamin D precursor levels and may not correlate well with active vitamin D synthesis in the body.

Vitamin D Molar Balance Model
A molar balance approach to analyze active vitamin D synthesis and degradation is presented in this paper. Instead of focusing on the front end (input) of the vitamin D synthesis pathway, this paper will focus only on the final metabolism of active vitamin D and its immediate precursor. This molar balance approach may permit (1) an estimation, at one point in time (t), of the amount of active vitamin D that has been recently produced in the body, (2) the determination (based on measurable quantities) of the synthesis of active vitamin D, 1,25(OH) 2 D 3 , over 24 h, and (3) the evaluation of the body's overall demand for 1,25(OH) 2 D 3 at a given point in time (t). Overall demand will be determined by comparing the amount of active vitamin D precursor, 25(OH)D 3 , used to synthesize active vitamin D to the amount of active vitamin D precursor, that is diverted away from the synthesis of active vitamin D and metabolized (wasted). Knowing the demand for 1,25(OH) 2 D 3 , and the percent utilization of active vitamin D precursor to produce active vitamin D, may help to determine whether inadequate, adequate, or excess vitamin D precursor is present. Measurement of the above vitamin D characteristics may enable clinicians to better identify individuals who may have impaired immune system function associated with inadequate active vitamin D synthesis and to better treat these individuals when they have COVID-19 viral or other acute infection infections. Similarly, measurement of these characteristics may allow clinicians to better treat those individuals with chronic inflammation associated with partially reactivated, latent viruses.
The 25(OH)D 3 molar balance model is depicted qualitatively in Figure 1. This figure provides a diagrammatic picture of the movement of vitamin D through the body and between the intracellular and extracellular compartments of the body. The blue arrow in Figure 1 depicts an intracellular synthesis pathway for 1,25(OH) 2 D 3 without its immediate precursor, 25(OH)D 3 , ever having to pass through or to be initially stored in the extracellular plasma space.  The molar balance model is described quantitatively in several equations included in this text as well as in a more rigorous manner in the Appendix A. The figures and equations included in this text along with the Appendix A describe the active vitamin D precursor (25(OH)D 3 ) transport, conversion to active vitamin D, metabolism, and excretion from the body. In this molar balance model, the body is divided into two main compartments for simplicity. The extracellular compartment is made up of all those spaces in the body which are not made up of cells and which contain fluids such as plasma, lymphatic fluid, bile, interstitial fluid, etc. The intracellular compartment is the space made up all those cells in the tissues of the body. It is important to remember that the synthesis of active vitamin D and its breakdown to its end metabolites is an intracellular process. See Figure 2a,b below. the tissues of the body. It is important to remember that the synthesis of active vitamin D and its breakdown to its end metabolites is an intracellular process. See Figure 2a,b below.
(a)  For the purpose of this model, the spaces which are made up of urine in the bladder and stool in the distal small bowel and large bowel are considered outside the body and the end metabolites of vitamin D are assumed lost to the intracellular and extracellular spaces of the body once they enter urine and stool in these spaces. The cells of bladder and bowel tissues including the cells of their mucosal linings are in the intracellular space/compartment of the body (see Figure 2c).
This model describes the synthesis and transport of active vitamin D, 1,25(OH) 2 D 3 , and its inactive metabolites into and out of the intracellular and extracellular compartments of the body at any arbitrary time (t) and over an arbitrary, 24 h, interval of time. The relationship between the end inactive metabolites of 1,25(OH) 2 D 3 and the end inactive metabolites of 25(OH)D 3 (that are diverted away from 1,25(OH) 2 D 3 synthesis, i.e., wasted) are used to establish changing 1,25(OH) 2 D 3 demand and changing vitamin D precursor, 25(OH)D 3 , utilization by the body to synthesize active vitamin D.
This model may also provide a better understanding of why: (1) the same store or concentration of 25(OH)D 3 in the plasma may result in different overall (increased or decreased) rates of 1,25(OH) 2 D 3 synthesis in cells of the tissues of the body, (2) the same store or concentration of 25(OH)D 3 in the plasma may result in an inadequate rate of 1,25(OH) 2 D 3 synthesis for one individual but not for another or for one individual experiencing increased 1,25(OH) 2 D 3 demand over their usual vitamin D demand state [19][20][21]. and (3) a significant portion of D 3 can be converted from vitamin D 3 to 25(OH)D 3 and then to 1,25(OH) 2 D 3 in the same cell without this portion of the body's vitamin D precursor, 25(OH)D 3 , ever passing through the plasma (sub-extracellular) space before being converted to 1,25(OH) 2 D 3 [22]. The active form of vitamin D is a hormone and is structurally different from its precursor molecules and its inactive metabolites. This difference allows active vitamin D to attach to receptor sites in or on target cells in order to activate different genes and chemical reactions. As a result of a body's changing demand or need for active vitamin D, cells/tissues in the body can produce increasing or decreasing amounts of the active form of vitamin D, 1,25(OH) 2 D 3 , from its precursor, 25(OH)D 3 . The source of vitamin D precursor needed for this synthesis can come from vitamin D precursor already present in the body's cells/tissues, or from vitamin D precursor entering the body's cells/tissues from the plasma space.
The changing need for active vitamin D can occur in cells/tissues which cannot completely synthesize their own active form of vitamin D but rely on plasma vitamin D precursor, 25(OH)D 3 , or rely on active vitamin D, 1,25(OH) 2 D 3 from other cells/tissues of the body (endocrine source). There are also the cells/tissues that can synthesize their own active vitamin D (autocrine or paracrine source for active vitamin D). Thus, some cells/tissues that store vitamin D precursors, can produce their own active vitamin D independently of the extracellular plasma source of the vitamin D precursor, 25(OH)D 3 .
Changing demand and synthesis of active vitamin D has been documented in several studies. One recent study which compared active vitamin D serum levels in non-pregnant women with those in pregnant women found significantly increased active vitamin D levels in those who were pregnant. Mean levels of active vitamin D, 1,25(OH) 2 D 3 , at 15 weeks were as follows: non-pregnant controls, 85.6 picomoles per liter (pmols/L), non-preeclamptic pregnant women, 336.3 pmols/L, and preeclamptic pregnant women 388.8 pmols/L [23]. Mean serum 25(OH)D 3 levels in the same groups, respectively, were 46.8 nmols/L, 44.7 nmols/L, and 33.1 nmols/L. In the first two groups of women, mean serum 25(OH)D 3 levels were similar, but mean serum 1.25(OH) 2 D 3 levels varied by a factor of 4.
In comparing the mean serum 25(OH)D 3 levels of the last two groups of pregnant women, the pre-eclamptic women had a lower mean serum 25(OH)D 3 level but a higher mean serum 1,25(OH) 2 D 3 level. These findings represent a dramatic example of uncoupling of mean serum 25(OH)D 3 and 1,25(OH) 2 D 3 levels when comparing non-pregnant and pregnant women's levels and to a lesser extent when comparing pregnant and pre-eclamptic pregnant women's levels.
An increase in vitamin D utilization/demand has been documented by serial measurements of dropping serum vitamin D precursor, 25(OH)D 3 levels during military training of short duration [24]. During a 1993 study of nutritional status in young healthy US Army soldiers undergoing an arduous 21-day Special Forces Evaluation and Selection Program, serum vitamin D precursor, 25(OH)D 3 levels dropped from an initial mean level of 61 ng/mL (range 34-100 ng/mL) on the first day to a mean level of 55 ng/mL (range 38-97 ng/mL) on day 10, to a mean level of 51 ng/mL (range 42-60) by day 20. The statistical significance of this drop was not tested at the time perhaps in part because vitamin D was only one of many nutritional status biomarkers that were measured and because mean vitamin D levels remained above the upper limits of normal (50 ng/mL at the time). To approximately convert ng/mL to nmols/L, multiply ng/mL by 2.5.
Vitamin D binding protein and total and free vitamin D metabolites have been shown to have a changing or diurnal rhythm (i.e., repeated daily pattern of change in vitamin D metabolite levels during the day) [25]. A recent case study of a middle-aged woman, who was taking 5000 IUs of vitamin D 3 orally each afternoon for over one year, demonstrated not only a daily serum 25(OH)D 3 diurnal rhythm but also a temporary significant drop in serum 25(OH)D 3 level associated with an acute but short duration respiratory illness [26]. The study included four separate days of evaluation, 11 October, 18 October, 15 November, and 28 November 2017) and the midday serum 24(OH)D 3 levels on each of these days was: 67 ng/mL, 65 ng/mL, 51 ng/mL (at the time of a brief cold), and 67 ng/mL, respectively. The approximate drop of 25% on day three was significant, p < 0.013.
The level of vitamin D precursor, 25(OH)D 3 , in the plasma is often not proportional to the level of active vitamin D, 1,25(OH)2D 3 in the plasma and does not directly indicate the rate of active vitamin D synthesis in all the tissues of the body. This occurs in part because paracrine/autocrine 1,25(OH) 2 D 3 can be synthesized and metabolized in cells/tissues of the body without active vitamin D precursor, 25(OH)D 3 , passing from the plasma into the affected cell and without paracrine/autocrine produced active 1,25(OH) 2 D 3 passing from the tissue/cells where it is produced and consumed into the plasma space. This observation is supported by the concurrent presence of the enzymes that allow the conversion of D 3 to 25(OH)D 3 [27,28].
The RNA expression of these enzymes has been examined recently in 27 different tissues of 95 human subjects as part of a larger study of gene expression [29]. The units of gene transcript expression are RPKM (i.e., Reads per Kilobase of transcript, per Million mapped reads) [30]. In comparing/interpreting the RPKM reads, (i.e., comparing the ability of cells/tissues to produce 25(OH)D 3 and its metabolites), it is important to note that RPKM reads represent the density of RNA in the tissue sample and not the overall size of the tissue or organ. Thus, the ability of tissues and organs to produce enzymes or process substrate will also depend on the total relative size of the tissue or organ and other factors such as the rate of diffusion or perfusion of the substrates through the tissue. The following four paragraphs outline the presence of the enzymes of vitamin D metabolism in many of the cells/tissues of the body.
The RNA expression study from which the above data are taken did not include skeletal muscle tissue, and it is not known if enzyme RNA, for both steps in the conversion of D 3 to 1,25(OH) 2 D 3 , exists in skeletal muscle. However, CYP27B1 has been found to be present in skeletal muscle [36][37][38]. In this molar balance analysis, D2 and its metabolites, and C-3-epi-25(OH)D 3 and 1,25(OH) 2 -3-Epi-D 3 are ignored to simplify the model [39].
The concurrent existence of DNA genes and their resulting RNA that code for enzymes capable of converting D 3 to 25(OH)D 3 (3 possible enzymes) and 25(OH)D 3 to 1,25(OH) 2 D 3 (1 enzyme) in many different tissue locations suggests the existence of a synthesis pathway from D 3 to 1,25(OH) 2 D 3 in cell/tissue types that are independent of the need for transported extracellular (plasma) compartment 25(OH)D 3 generated from the liver. The presence of RNA for a particular enzyme does not necessarily equate to a functional protein being translated from the RNA and the functionality of the proteins would need to be verified. The large presence of immune system cells in many tissues of the body makes the immune system potentially one of the largest producers and consumers of active vitamin D during certain conditions such as severe infections especially involving multiple tissues such as with COVID-19 virus infections.
An alternate, strictly intracellular, 1,25(OH) 2 D 3 synthesis and degradation pathway supports the idea that extracellular (plasma) compartment 25(OH)D 3 and 1,25(OH) 2 D 3 levels are partially uncoupled from intracellular compartment active vitamin D synthesis. An intracellular active vitamin D synthesis pathway independent of plasma, 25(OH)D 3 , may explain why the rate of active vitamin D, 1,25(OH) 2 D 3 , synthesis is partially independent from the aqueous (plasma) vitamin D precursor, 25(OH)D 3 concentration.
Serum 25(OH)D 3 levels and 1,25(OH) 2 D 3 levels do not appear to have a proportional relationship, and serum 25(OH)D 3 concentration in the plasma space may not necessarily reflect the body's ability to meet the need for active vitamin D synthesis. The following mathematical equations for the molar balance analysis are based on and depend on the accurate measurement of changing concentrations of vitamin D end metabolites in plasma and urine samples. This model can be used to estimate active vitamin D synthesis and to assess whether the body's demand for vitamin D is being met. This molar balance method, of assessing vitamin D status, analyzes and depends on primarily the backend of the vitamin D metabolic pathway rather than the frontend.
Some data concerning where vitamin D 3 and 25(OH)D 3 are stored in the body exist based on intravenous radiolabeled vitamin D studies. D 3 has been shown to accumulate the most in adipose tissue but D 3 also accumulates in muscle, skin, plasma, and other organs. Total body storage of vitamin precursors has been estimated to be approximately 1 micro-mole (10 −6 moles) with two-thirds stored as D 3 (primarily in adipose tissue) and the rest primarily stored as 25(OH)D 3 in various tissues and fluids including adipose tissue 35%, muscle 20%, and plasma/extracellular fluid 30% [40,41]. The molar balance model divides the storage of vitamin D into two compartments: intracellular and extracellular. Plasma is considered a sub-space of the extracellular compartment. The concentrations of 25(OH)D 3 , 1,25(OH) 2 D 3 and their end metabolites are assumed to be approximately uniform or homogeneous in both the plasma and total extracellular spaces. In contrast, levels of vitamin D precursors and active vitamin D are not assumed to be homogeneous within the cells/tissues of the intracellular compartment or spaces.
Using radioactive H 3 tagging, attempts have been made in several studies to determine the portion of vitamin D and its metabolites excreted in the urine verses in the stool (through bile) [42][43][44]. In three studies approximate differences between excretion in the urine verses in the stool have been as follows: 32% versus 68% (n = 7), 25% versus 75% (n = 7), and 22% versus 78% (n = 16). The molar balance model uses the values of 25% excretion in the urine and 75% in the stool.

Constant Compartment Volumes
The volume of the body's extracellular compartment or space, V e , (or plasma subcompartment, V p ) can change in certain circumstances such as pregnancy, severe dehydration, or congestive heart failure. However, for our model, it is assumed that the extracellular (or plasma) volumes at some initial time, t 0 , V e (t 0 ), equals the extracellular compartment volume at some later time, t 1 , V e (t 1 ), and equals a constant V e . Similarly, the body's intracellular volume, V i is assumed to be constant [45,46] 2.
Intracellular volume = 2 × Extracellular Volume For a complete list of different volume assumptions for men and women, see Table 1, "Intracellular and Extracellular Compartment Volume Analysis", and Table 2, "Total Male and Female Intracellular, Extracellular, and Plasma Volumes".

Water Makes up Approximately 60% of Body Weight in the Male and 55% Body Weight in the Female
The C n Intracellular Concentration of End Metabolites = β Times the C n Extracellular Concentration The mean intracellular concentrations of the end metabolites, m n , will be assumed to be proportional to and a factor β times the same concentrations in the extracellular compartment. The β factor is greater than 1 in this model because the end metabolites are assumed to be transported from the intracellular compartment (where they are made) to the extracellular compartment by a concentration gradient in order to be excreted from the body (see Figure 2d).

Extracellular Sub-Compartment C n Concentrations Are Equal
Vitamin D metabolite, m n , concentrations may differ somewhat between extracellular sub-compartments/spaces and their concentration gradients around cells may vary. However, in our model, the concentrations of 25(OH)D 3 and its many metabolites (including 1,25(OH) 2 D 3 ) are assumed to be uniform for the same metabolites across different extracellular sub-compartments at any given point in time.

5.
Nanomoles, 10 −9 Moles (nmols) of Active Vitamin D Synthesized over 24 h Equals the nmols of Active Vitamin D End Metabolites Synthesized over the Same 24 h The Molar Balance Model for active vitamin D synthesis assumes that the number of nmols of active vitamin D synthesized over a 24 h period equals the sum of the number of nmols of end metabolites of active vitamin D synthesized over the same 24 h. This steady state assumption further depends on the following: (1) any change in the total body's number of nmols of active vitamin D or in the number of nmols of active vitamin D intermediate metabolites in the intracellular plus extracellular compartments is small compared to the synthesis of active vitamin D end metabolites over 24 h, and (2) the number of nmols of active vitamin D and active vitamin D intermediate metabolites excreted is relatively small compared to the nmols of end metabolites excreted over the same 24 h period. All the measurements made using this and the following assumptions are in units of nmols. The fact that this measurement is made over 24 h allows this measurement also to represent active vitamin D synthesis per 24 h (a synthesis rate).
The equations derived to determine the nmols of active vitamin D synthesized over 24 h depend on the above assumptions. The validity of these assumptions could be checked by doing second serum and urine assays which measures total end metabolites of active vitamin D precursor, active vitamin D, and all their intermediate metabolites after all active vitamin D precursor, active vitamin D, and its intermediate metabolites are driven to their end metabolites by adding significant excess CYP24A1 hydroxylase enzyme. If, after adding measured nmols of serum active vitamin D and its precursor, 25(OH)D 3 , to the measurement of their end metabolites in the first sample, a discrepancy between the sum of these metabolites in the first sample and the total end metabolites of 25(OH)D 3 and 1,25(OH) 2 D 3 in the second sample exists, then a correction factor can be added to the calculation (see further discussions below and in the Appendix A, Section 2, Assumptions).

6.
The nmols of End Metabolites, m 8:10 , in the Intracellular plus Extracellular Spaces Can Be Used to Estimate the nmols of Active Vitamin D Synthesis (Levels) at Time (t).
The total nmols of active vitamin D, 1,25(OH) 2 D 3 end metabolites (m 8 , m 9 , and m 10 ) in the extracellular plus intracellular spaces, can be used to estimate the body's total active vitamin D synthesis at any point in time, (t). The assumption in the sentence above allows for a quicker method for assessing active vitamin D synthesis at any given time (t) from a single plasma sample and to track changes in active vitamin D synthesis due to exogenous vitamin D supplementation. The use of only the end metabolites of active vitamin D as a biomarker for active vitamin D synthesis only approximates active vitamin D synthesis when the levels (synthesis) of active vitamin D and its intermediate metabolites are changing significantly (going up or down) as the demand and synthesis of active vitamin D changes.
As with the 24 h assessment above, a second plasma serum assay could be performed which measures the total end metabolites of active vitamin D after all active vitamin D and its intermediate metabolites are driven to their end metabolites by the CYP24A1 hydroxylase enzyme. This assay could be used to check for a significant buildup of active vitamin D intermediate metabolites relative to active vitamin D end metabolites. If the discrepancy is large then a factor to correct for this discrepancy must be added to the molar balance model calculation. This same analysis could be performed on a spot urine although the dilution variability of spot urines would have to be accounted for (see Figure 2a

Model Derivation through Figures and Equations
Based on the above assumptions, Figure 2a-d, and Tables 1 and 2 in this paper and the following equations in this paper (including in the Appendix A), the synthesis of and demand for 1,25(OH) 2 D 3 by the many cells/tissues of the body is estimated, and the percent utilization of vitamin D 3 precursor, 25(OH)D for the synthesis of active vitamin D by the body is determined. Finding a way to estimate the nmols of intracellular 1,25(OH) 2 D 3 synthesized in the body at any point in time (t) or over a period (t 1 − t 0 ) such as 24 h (in nmols) could be used to identify changing demand for 1,25(OH) 2  The small letter m and its subscripts n (0-10) are used to identify specific metabolites in the vitamin D synthesis pathway. Figure 2a,b describe the enzyme pathways that metabolize 25(OH)D 3 , m 0 , into the production of active vitamin D hormone, 1,25(OH) 2 D 3 , m 1 , and its metabolites. The enzyme pathways that metabolize m 0 away from the synthesis of active vitamin D into inactive metabolites are part of the body's regulation of the synthesis of active vitamin D and are also described in Figure 2a (1) and (2) from the Appendix A and repeated below, represents the amount of 1,25(OH) 2 D 3 produced by cells/tissues of the body over a time interval (t 1 − t 0 ). This integral of S 0,1 can also be used to estimate the amount of active vitamin D that has been synthesized in the body at any time (t). See Appendix A, Equations (9a), (10a) and (13a). One formula uses the integral of the rate of active vitamin D synthesis, (S 0,1 ) over an arbitrary 24 h time interval to estimate the total synthesis of 1,25(OH) 2 D 3 over 24 h. This synthesis estimate is accomplished by summing the change in the total amount of the end metabolites of 1,25(OH) 2 D 3 in the extracellular and intracellular compartments and the accumulation of end metabolites of 1,25(OH) 2r D 3 in urine and stool over 24 h. So far, this method has focused on the use of the end metabolites of active vitamin D and the end metabolites of its precursor, 25(OH)D 3 (which are formed from vitamin D precursor which is diverted away from the synthesis of active vitamin D) in order to evaluate the actual amount of and the need for active vitamin D synthesis. There is one other consideration concerning the synthesis of active vitamin D and the serum concentration of active vitamin D precursor. Gene CYP27B1 of enzyme C1 Hydroxylase is responsible for the synthesis of active vitamin D from its precursor (see Figure 2a and the yellow oval, Figure 2b) and the gene CYP2R1 is responsible for the synthesis of active vitamin D precursor, 25(OH)D 3 , from D 3 . When these genes become mutated and are no longer able to produce functioning 1-α Hydroxylase or 25-hydroxylase enzymes, respectively, severe active vitamin D deficiency can occur [47].
There may be some genetically variable forms of these deficiencies which may impair the production of their hydroxylase enzymes and effect the production of active vitamin D or its precursor without totally stopping the production of these forms of vitamin D. Measurement of the end metabolites of active vitamin D, of its precursor, 25(OH)D 3 , and of the end metabolites of diverted active vitamin D precursor, may help to determine whether either of these two genes and the production rate of their enzymes may be altered.
A second formula has been derived to determine M 1 (t) (Equation (3) below) or the amount of end metabolites of active vitamin D, m 1 , that are present in the body at any point in time (t). The amount of end metabolites of active vitamin D in the body at time (t) is then assumed to represent or to be proportional to the amount of active vitamin D synthesized at that moment. The following numbered equations are taken from derivation of equations in Appendix A that quantify active vitamin D synthesis and demand as well as the percent utilization of the precursor to active vitamin D for active vitamin D synthesis.
for men where V p = 3.0 L, V u (t 0 ) = 0, and β is initially set at 1.1 (Appendix A) where V e = 4.66V p for men and 4.4V p for women and β is set at 1.1 In the Appendix A, formulas for a 1,25(OH) 2 D 3 demand ratio, Drm 1 , i+e (t), Equation (4) below and a 1,25(OH) 2 D 3 24 h Urine demand ratio, Drm 1 , u (t 0 ,t 1 ), Equation (5) below are also derived. The rationale for using the ratio of the end inactive metabolites of 1,25(OH) 2 D 3 , m 1 , as the numerator and the end inactive metabolites of 25(OH)D 3 , m 0 , as the denominator, is that this ratio will decrease if excess 25(OH)D 3 exists and is shunted away from the synthesis of active vitamin D when the body's active vitamin D supply is adequate and vice versa. Knowing this ratio may be helpful clinically to judge whether there is sufficient vitamin D precursor to meet the needs of the body for active vitamin D synthesis. This concept has been introduced previously based on calculating a serum calcitriol/calcifediol ratio using this ratio as an indicator of vitamin D hydroxylation efficiency or as an indicator of relative vitamin D CYP enzyme activities rather than the actual indirect measurement of actual molar substrate 1,25(OH) 2 D 3 /25(OH)D 3 ratios in the body using the end metabolites of active vitamin D and its immediate precursor [40,48] (see p. 641, Figure 37.3 in Ref. [46]). An increase in m 1 demand should result in an increase in m 1 synthesis from m 0 accompanied by a decrease in diversion of m 0 away from m 1 synthesis. (i.e., an increase in the m 1 demand ratio) This pattern is suggested in the results of a recent study comparing serum vitamin D metabolites between non pregnant women (controls) and pregnant women (at 15 weeks). One group of pregnant women were without the complication of preeclampsia and one group of pregnant women (at 15 weeks) who went on to develop preeclampsia [23]. As stated above, demand for active vitamin D increases during pregnancy. Mean serum active vitamin D levels, m 1 , in these three groups above were, respectively: 85.6, 336.3, and 388.8 pmoles/L. The levels of initial inactive 24,25(OH) 2 D 3 , m 2 , metabolites of m 0 , changed in the opposite direction from active vitamin D synthesis and were, respectively: 9.7, 6.5, and 3.2 nmoles/L These metabolites decreased in parallel with higher m 1 levels. The differences were not statistically significant but the numbers of subjects were small, respectively: 9, 25, and 25.
A somewhat different pattern was seen in urine samples of one of the initial metabolites of active vitamin D precursor, m 0 which was diverted away from the synthesis of active vitamin D. Mean metabolite m 2 , 24,25(OH) 2 D 3 , levels in the urine of these three groups were, respectively: 55.4, 84.1, and 35.6 nmols/liter [23]. The mean m 2 concentrations in the urine samples decreased in the preeclampsia pregnant group compared to the nonpreeclampsia pregnant group suggesting higher utilization % in the pre-eclampsia group. The difference between the m 2 levels between the two pregnant groups was statistically significant. Of note, the other initial intermediate metabolite of m 0 , 23,25(OH)D 3 or m 3 which also diverts m 0 away from the synthesis of active vitamin D was not measurable in the urine samples of this study.
A second method for estimating active vitamin D demand would be determination of the % of vitamin D precursor, 25(OH)D 3 , m 0 , that is used to synthesize active vitamin D, 1,25(OH) 2 D 3 , m 1 , at one point in time (t) or over 24 h. The first % utilization measurement, Ut % m 0 , i+e (t), is presented in Equation (6)   This molar balance approach depends on the accurate and standardized measurement of the serum end metabolites of active vitamin D and of the diverted end metabolites of its precursor, 25(OH)D 3 . These measurements are the critical foundation of this method. Serum, which is plasma with platelets and clotting factors removed, is used to make these measurements. The serum samples will be formed by spinning down a whole blood sample after clotting has occurred in order to separate the serum from the blood sample cells, platelets, and clotting factors.
Liquid-chromatography-tandem-mass spectrometry (LC-MS-MS) is considered the measurement of choice for vitamin D metabolites [49,50]. With LC-MS-MS, up to twelve vitamin D metabolites have been measured simultaneously [51,52]. Analytical challenges exist with this technology. Sample type, protein precipitation, analyte extraction, derivatization, chromatographic separation ionization, and capabilities of the mass spectrometer must be addressed [50]. Calibration, standardization, and use of internal standards are also important requirements to achieve consistent, accurate results.
Purified standard compounds for only two of the three end metabolites of active vitamin D, m 8 , calcitroic acid, and m 10 , calcioic acid are available commercially [53][54][55]. A purified standard compound for the remaining end metabolite of active vitamin D, m 9 , or calcitriol-26,23-lactone, and standard compounds for the diverted, m 0 precursor, end metabolites, m 6 or cholacalcioic acid and m 7 or 25(OH)D 3 -26-23 lactones will have to be synthesized since these standard compounds are required for the liquid-chromatographytandem-mass spectrometry measurements of these serum end metabolite measurements.
About 90% of circulating 25(OH)D 3 is protein bound to vitamin D binding protein (VDBP), albumin, and lipoproteins [49]. The percent binding of active vitamin D end metabolites, and the diverted vitamin D precursor end metabolite, to serum VDBP may not be known. The binding of end metabolites and their relative fat and water solubilities may have to be determined. Measurement of the three end metabolites of active vitamin D and the two end metabolites of the precursor, 25(OH)D 3 which are diverted away from the synthesis of active vitamin D, may also depend on their actual concentrations in plasma and urine and the range of measurability. The measurements of these end metabolites would be made using the same instruments that are currently measuring other forms of vitamin D.
A source of human CYP24A1 hydroxylase enzyme will be needed to further evaluate and test the proposed assays used by this model. This enzyme will be used to drive the intermediate metabolites to their end metabolites. This reagent (enzyme) will be required to verify that there is not a significant change in total intermediate metabolites during a transitory increase or decrease in active vitamin D synthesis or in the rate of vitamin D precursor diversion. This enzyme is currently available from several commercial sources Biocompare, "https:www.biocompare.com/Search-Biomolecules/?search=CYP2 4A1 (accessed on 17 July 2020)" [53]. CYP24A1 has been used in previous vitamin D metabolic studies [54].
Vitamin D metabolite measurements have also been made from human spot urine specimens including 25(OH)D 3 and 24,25(OH) 2 D 3 [23]. Measurement of human spot urine 1,25(OH) 2 D 3 and 23,25(OH) 2 D 3 concentrations were below the limits of detection. The normal volume range of an adult 24 h urine is 800-2000 17mL with a normal intake of 2 L of fluid per day. Since the 24 h urine can be concentrated, concentrating the urine will make the detection of end metabolites in urine easier.
Finally, the technologies used in measuring or monitoring vitamin D metabolites, although challenging, are well established. The only new aspect is adding five different end metabolites of vitamin D to the two currently dominant forms of vitamin D measured, (i.e., active vitamin D 3 , 1,25(OH) 2 D 3 and its immediate precursor, 25(OH)D 3 ).

How Can and Why Should These Vitamin D Parameters Be Used?
Using a molar balance model approach, the following vitamin D parameters can be calculated: (1) an estimate of active vitamin D synthesis in the body at any time (t) and over a 24 h time interval, (2) an active vitamin D demand ratio, and (3) an active vitamin D % utilization. Clinical lab software will perform the calculations listed in Equation (9a) through 18 using the measured concentrations of the end metabolites m 8:10 and m 6:7 as well as different constants for men and women. Each of these new vitamin D parameters can then be determined and monitored in many different conditions where the demand by the body for active vitamin D changes dramatically. These parameters cannot be currently determined by simply measuring serum 25(OH)D 3 and 1,25(OH) 2 D 3 levels in the serum.
During severe infections caused by many different agents including the SARS-CoV-2 virus, the immune system may require significantly greater synthesis of active vitamin D to suppress the offending agent (i.e., virus) and the inflammatory response caused by the agent [56]. These clinical vitamin D parameters along with other measurements of an individual's clinical response to their COVID-19 infection will assist clinicians to establish when significant additional vitamin D supplementation is helpful and to determine the required type and dose of vitamin D supplementation.

Molar Balance Model Limitations
The molar balance model depends on several assumptions some of which may need modification as the end metabolites are initially measured and some of the assumptions will not be accurate in all situations. If the extracellular volume or plasma volumes change significantly due to swelling, the assumed volume values will have to be modified. Kidney failure will also prevent the measurement of end metabolites in the urine and may lead to build up of some end metabolites in the body including the extracellular compartment and plasma, and lead to greater excretion of these metabolites in the stool.

Summary
A new method using a molar balance model is proposed to determine total body active vitamin D, m 1 , levels including synthesis over a 24 h period, the level of demand in an individual for active vitamin D, and the % utilization of vitamin D precursor, m 0 , to synthesize active vitamin D. Determination of these parameters associated with vitamin D levels and synthesis will require the measurement of the end vitamin D metabolites m 6:10 in both serum and urine using the same equipment and techniques used to measure active vitamin D and its immediate precursor. Knowing this new information may enable clinicians to improve the body's immune system response to COVID-19 viral infections as well as to other viruses going forward by maximizing the positive effects of active vitamin D on immune system function. where the first subscript, n, indexes metabolite type (see list above for metabolite definitions) and the second subscript, k, designates three dimensional spaces in which metabolites can be located. There are four primary spaces or compartments: intracellular, extracellular, plasma, and collected urine. The plasma space is a sub-compartment of the extracellular space. Two secondary spaces used in the mathematical derivation include the body's stool space and the space or volume of the collected plasma samples used to make metabolite measurements. The spaces are denoted as follows:

Conclusions
(1) k = i for intracellular space (2) k = e for extracellular space (3) k = p or pla for plasma sub-extracellular space (4) k = u collected urine space (5) k = s collected stool space (6) k = cp collected plasma sample Note: number subscripts always refer to metabolites or time and letter subscripts to spaces.
The total body nmols of metabolite m n at any time (t) in the intracellular and extracellular compartments is denoted: The total nmols of metabolite m n over any time period (t 1 − t 0 ) in a urine collected specimen is denoted: M n,u (t 1 ) − M n,u (t 0 ) = M n,u (t 1 ) (D. 3) since the nmols at the start of a urine collection will always be = to 0. Metabolite concentrations in each space are defined as: where Vk is the volume in compartment k.

Metabolite Transport, Synthesis, and Loss
The transport rate of any metabolite n from the intracellular to extracellular compartment (space) at time (t) is R n (t) (D.5) We consider several pathways of vitamin D metabolite synthesis and use S to denote rates of synthesis/conversion. All synthesis/conversion occurs in the intracellular compartment. The rate of synthesis/conversion between metabolites i and j, where i = j, is denoted S i,j (t) (D.6) Some specific synthesis rates of interest are:  One goal of this model is to develop a method for measuring the synthesis of the active vitamin D metabolite, m 1 . Using our notation, the total quantity of m 1 synthesized from m 0 between two arbitrary time periods t 0 and t 1 is

Metabolite Loss and Urine Volume Creation
The total loss of metabolite m n from the extracellular space is divided between the urine volume space, 25% and the stool volume space 75%. Total rate of metabolite, m n , loss: L* n (t) = L n,u (t) + L n,s (t) = 4L n,u (t) (D. 12) and we denote the rate of urine volume creation at time t as The collected urine volume consists of a urine volume created between the time at which collection starts, t s or t 0 , and the time at which collection ends, t e or t 1 . Clinically, the urine volume is always 0 at the start of a urine collection; therefore, terms involving t s or t 0 always drop out.
t 0 U(t) dt since V u at the start of collection is 0 (D.14) The quantity of metabolite m n in the urine space at the end of the collection period consists of 25% of metabolites m n excreted from the body between the time at which collection starts, t s or t 0 , and the time at which collection ends, t e or t 1 . (Note: the remaining 75% of metabolite excretion occurs through the stool). Therefore, we can write The concentration of metabolite n in the urine space at the end of the collection period is also a function of the end collection time only since the nmoles of m n and urine volume equal 0 at the start of the urine collection (t 0 ):

Assumptions A
Plasma volumes are somewhat different between men and woman based on different physical sizes (average woman 55 kg and average male 70 kg) and they have slightly different mean hematocrit % (Hct.%) levels (mean for women 40% and mean for men 45.5%). Thus, the average plasma volume for a 55 kg woman is 2.5 L and for an average 70 kg male 3.0 L. An average male's plasma volume is thus 20% greater than a women's plasma volume. See Figure 2c for more detailed explanation of volume determinations.
For the purpose of this model, the plasma, extracellular, and intracellular volumes are assumed to be known constants for the average female 155 kg and the average male 170 kg, denoted as V pla , V e , and V i . The intracellular and extracellular volumes can be expressed as multiples of the plasma volume [29,30]: For this model, the total nmols of vitamin D metabolites, m n , in the intracellular and extracellular compartments or spaces will be determined from measurement of metabolite m n concentrations in a collected plasma sample.
The next assumption is that metabolite concentrations of m n in the extracellular, plasma, and collected plasma sample are equal at the time (t) of specimen collection, i.e., C n,e (t) = C n,p (t) = C n,cp (t) (A.4) Mean metabolite concentrations in the intracellular and extracellular compartments are assumed to be proportional with a constant of proportionality of β. β equilibrates the mean concentrations of m n metabolites between the non-homogeneous intracellular and assumed approximately homogeneous extracellular compartments. End metabolites of m 0 and m 1 are assumed to transfer passively from the intracellular to the extracellular compartment and visa versa passively depending on concentration gradients. β is assumed to be a constant over physiological concentrations of end metabolites, m n , and to be somewhat greater than 1 since a mean higher to lower concentration gradient overall between the intracellular and extracellular spaces must exist in order for the end metabolites to be excreted from the body. Initially β will be assumed to be 1.1-see Figure 2d. Assumption A.8 below assumes that the rate of synthesis of m 1 from m 0 is the same as the sum of the rates of synthesis of its three end metabolites. That is, every time 12 molecules of m 1 are synthesized a total of 12 molecules of end metabolites of m 1 , (m 8 + m 9 + m 10 ), will also be synthesized. Similarly, A.9 assumes that the sum of the rates of synthesis of m 2 from m 0 and m 3 from m 0 is the same as the sum of the rates of synthesis of m 6 from m 2 and m 7 from m 3 S 0,1 (t) = S 4,8 (t) + S 5,9 (t) + S 5,10 (t) (A.8) S 0,2:3 (t) = S 2,6 (t) + S 3,7 (t) (A.9) Assumptions A.8 and A.9 do not reflect what is truly going on in the two intracellular vitamin D m 0 pathways because these assumptions neglect possible changing concentration levels and the presence of changing nmols of m n intermediate metabolites. Assumption A.8 assumes that the rate of active vitamin D synthesis equals the sum of the rate of end metabolite synthesis (a presumed steady state) without a buildup or decrease in intermediate metabolites. Assumption A.9 likewise assumes that the rate of m 0 diversion away from active vitamin D synthesis equals the sum of the rate of its end metabolite synthesis (a presumed steady state) without a buildup or decrease in its intermediate metabolites. Even under non steady state conditions, these two assumptions might introduce a tolerable error.
The validity of A.8 and A.9 assumptions can be checked by doing second serum and urine assays which measures total end metabolites of active vitamin D precursor, active vitamin D, and all of their intermediate metabolites after all active vitamin D precursor, active vitamin D, and its intermediate metabolites are driven to their end metabolites by excess CYP24A1 hydroxylase enzyme added to a second plasma sample drawn at time t 0 and t 1 as well as by a second determination from the urine specimen collected over 24 h. If, after subtracting out the end metabolites which would have been generated by the remaining active vitamin D and its precursor in the first sample, the measured nmoles of end metabolite in the second sample are significantly larger than those in the first serum sample, then a correction factor can be added to the calculation to account for a buildup of intermediate metabolites in either the 25(OH)D 3 or 1,25(OH) 2 D 3 degradation pathways when the synthesis of active vitamin D increases.
The collected serum plasma sample can be divided into two equal parts. In the first part, concentrations of Then, measured concentrations of C 0,cp , C 1,cp , C 6-7,cp , and C 8-10,cp in the first sample must be subtracted from the concentration of C 6:7,cp (t) plus C 8:10,cp (t) in the second sample. Any discrepancy between the two values will represent the concentration of intermediate metabolites not measured in the first collected plasma sample. If the discrepancy is significant a correction factor can be introduced to correct for an increase in intermediate metabolites in the 25(OH)D 3 and 25(OH) 2 D 3 degradation pathways when the synthesis of either of these vitamin D forms increases significantly. The same assumptions and processes described in this and the above paragraphs can be used in the evaluation of the end metabolites of active vitamin D and its precursor in urine collections. This two-measurement comparison technique along with the following equations could be used to see whether the assumption that the buildup of intermediates in these two degradation pathways is not significant or needs addressing.

3.
Equations From Assumptions (A.1) to A.10, total m 1 synthesis and the total diversion of m 0 away from the synthesis of active vitamin D, m 1 , over an arbitrary time interval (t 1 − t 0 ) can be estimated in men or women using the (1) Measurement of C 8:10,cp at times t 0 and t 1 (2) Measurement of C 6:7,cp at times t 0 and t 1 (3) Measurement of C 8:10,u at time t 1 (since there is no urine when the collection starts) (4) Measurement of C 6:7,u at time t 1 (since there is no urine when the collection starts) (5) Measurement of V u (t 1 ) (6) Measurement of C 0,cp at times t 0 and t 1 and (7) Standard values for V i , V e , and V pla, for men and women.
An expression for the synthesis of m 1 over an arbitrary time interval (t 1 − t 0 ) is developed in three steps. In the first step, an expression for the change in quantity of m 1 in the body's intracellular and extracellular compartments over the time interval, (t 1 − t 0 ) is derived using m 1 s end metabolites, m 8:10 . In the second step, an expression for the quantity of m 1 excreted or lost to the body in urine and stool over the same time interval is derived using m 1 's end metabolites, m 8:10 . In the third step, the expressions derived in parts 1 and 2 are summed to solve for the total synthesis of active vitamin D, m 1 , based on the assumption that the change in intracellular plus extracellular nmols of m 8:10 over the time interval, plus the loss of m 8:10 in the urine and stool over the same time interval will represent a good estimate of the synthesis of m 1 over this time interval. Using the same approach, an expression for the amount of m 0 diverted away from the synthesis of m 1 will be developed using the end metabolites, m 6:7 .
Using Assumption A.8, the following equations can be written: The terms on the righthand side of Equation 5a will be replaced with known quantities (either from quantities that have been measured before such as volumes or from measurements that we can make based on our definitions or assumptions). The first substitution will be for the metabolite loss term involving urine. Note time subscripts s for start and e for end have been replaced by 0 for the initial start of the collection where the urine volume and end metabolites of m 1 would be 0 and e would be replaced by 1 at the end time of the collection. (i.e., t s = t 0 and t e = t 1 ). Using definitions (D15) and (D16), the total loss of metabolites m 8 , m 9 , and m 10 from the body between t 0 and t 1 (using a factor four times the loss from urine to make up for the loss of metabolites from both urine and stool equals: whereC 8:10,p (t) = C 8,p (t) + C 9,p (t) + C 10,p (t) · · · · · · nmoles/L (Eq. 8a) substituting the right side of Equation (8a) into the right side of Equation (7a) (for men) results in: where V p for an average size woman is 2.5 L and β = 1.1.

The m 1 Demand Ratio
At any point in time (t), the 1,25(OH)D 3 , m 1 , demand ratio (D r,i+e ), expresses the ratio of the sum of end inactive metabolites of active vitamin D, 1,25(OH) 2 D 3 , m 1 (i.e., m 8 + m 9 + m 10 ) to the sum of non m 1 , end inactive metabolites of 25(OH)D 3 (i.e., m 6 + m 7 ) in the intracellular plus extracellular compartments. The end inactive metabolites of 1,25(OH) 2 D 3 track the portion of 25(OH)D 3 that goes on to make active vitamin D, m 1 . The non m 1 end inactive metabolites of 25(OH)D 3 track the portion of 25(OH)D 3 that is diverted away from making active vitamin D especially when the supply of active vitamin D is sufficient or in excess. See Figure 2a,b and expressions below from Appendix A. This ratio may reflect the body's demand for active vitamin D.
The total body nmols of metabolite m n at any time (t) in the intracellular and extracellular compartments is denoted: M n,i+e (t) (D. 2) The quantity (nmols) of active vitamin D, (m 1 ), or M 1 in the extracellular plus intracellular compartments at time (t) that has been synthesized can then be estimated using the assumed (A.8) steady state synthesis pathways, [i.e., S 0,1 (t) = S 4,8 (t) + S 5,9 (t) + S 5,10 (t)] and by using the assumptions A.4 and A.5 1 (i.e., the relationship between the mean intracellular compartment m 8 The nmols of m 1 in the intracellular plus extracellular compartments at time (t) can then be estimated to equal the nmols of m 8:10 in the intracellular plus the extracellular compartments: Similarly, by using (A.9) and substituting C 6,7,cp (t) into Equation (Eq.13a), the nmols of m 0 in the intracellular plus extracellular compartments diverted away from active vitamin D synthesis in nmoles can be estimated. The 24 h m 0 utilization percent is defined as the ratio of the change in the nmols of m 8:10 (t 1 ,t 0 ) in the intracellular and extracellular compartments plus the accumulation of m 8:10 (t 0 ,t 1 ) in the urine and stool over 24 h divided by the sum of the total m 8:10 (t 0 ,t 1 ) metabolites described above plus the change in the nmols of m 6:7 (t 1 ,t 0 ) in the intracellular and extracellular compartments plus the accumulation of m 6:7 (t 0 ,t 1 ) nmols in the urine and stool over 24 h.

×100%
(Eq. 18a) The integral in the numerator position of Eq.18a represents the sum of the end metabolites of m 1 (1,25(OH) 2 D 3) . For males it is determined by Eq.9a and for females, Eq.10a. The denominator is made up of the integral in the numerator position plus the integral of the end metabolites of 25(OH)D 3 that are diverted away from the synthesis of 1,25(OH) 2 D 3 and are in fact wasted. This second term in the denominator is determined by Eq. 11a for males or Eq. 12a for females. This integral ratio is then multiplied by 100% to determine the percent utilization of the precursor, 25(OH)D 3 that is used to make active vitamin D, 1,25(OH) 2 D 3 at any given point in time.

Percent Increase or Decrease of the above Parameters over Time
Percent increase or decrease of the parameters above including estimates of synthesis of active vitamin D over 24 h, active vitamin D demand ratios, 25(OH)D 3 percent utilization, and biomarkers for total active vitamin D synthesis at any time t, over the time interval t n+1 and t n can be determined by taking the ratio of a parameter at time t n+1 divided by the same parameter at time t n , minus 1, and then times 100%. For the demand ratio of m 1 as an example: {[Drm 1 , i+e (t n+1 ) ÷ (Drm 1 , i+e (t n )] − 1} × 100% (Eq. 19a) If the same technology to determine the above parameters is used by different labs, and if the measurements which are made at different labs have consistent measurement techniques, variations in parameter values between sequential runs and between labs should be controlled and not significant.