The Nutritional Density Ratio Dilemma: Developing a Scale for Nutritional Value Paul D. Q. Campbell

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1 The Nutritioal Desity Ratio Dilemma: Developig a Scale for Nutritioal Value Paul D. Q. Campbell Shared by Paul D. Q. Campbell The author(s) would appreciate your feedback o this article. Click the yellow butto below. Thak you! Server Geerated Time Stamp This file was shared o peerevaluatio.org o Fri, 4 May :56:

2 The Nutritioal Desity Ratio Dilemma: Developig a Scale for Nutritioal Value The Nutritioal Desity Ratio Dilemma Oe of the importat priciples Uiversity utritio classes attempt to teach is the Nutritio Desity Ratio. It's oe of those thigs that has the appearace of a real mathematical expressio, but is't. The Nutritio Desity Ratio is the ratio of a food's Nutritioal Value (N V ) i relatio to its caloric cotet (K, measured i kilocalories {Kcal}). (Groder, Roth, & Walkigshaw, 2011) So the formula is explaied as a simple ratio: : The explaatio is so eloquetly simple that it takes oly a ratioal huma beig to uderstad it ituitively. A glass of orage juice has far more utriet value (N V ) tha a commesurate glass of soda pop i proportio to its Kcal (K) value. Agai, it is axiomatically ituitive. The problem is that oe ca defie ad quatify exactly what a Calorie, ad therefore a Kilocalorie, are, but what is the uit of utritioal value so it is possible to complete the formula of the ratio? Without a value for a uit of utritio the Nutritio Desity Ratio is about as useful as Drake's Equatio! Drake's Equatio was developed to determie the probability of the umber of civilizatios withi our galaxy with a society techologically similar eough to our ow that we could commuicate with them. However the values that ca be plugged ito the equatio are so broad ad speculative that the equatio is largely meaigless. (Crichto, 2003) Oe could plug virtually ay positive umber ito the formula ad call the result a aswer. Whether the aswer was true or ot is completely u-provable. The problem becomes eve more troublig o closer examiatio. While the orage juice ad soda pop example is perfectly ituitive, the problem becomes much harder to coceptualize i a compariso betwee soda pop ad wheat bra, or a egg ad a bea. The reaso is because there is o perfectly liear relatioship betwee N V ad Kcal. I fact, the relatioship, while techically o-liear, turs out to be a very recogizable relatioship, at least graphically. Graphically it is the Normal Distributio Fuctio: Page 1 of 8

3 The Nutritioal Desity Ratio Dilemma: Developig a Scale for Nutritioal Value Mathematically it looks pretty frighteig: = τ But this is o more tha the mathematical expressio of the above graphic. τ (lower case Tau) is used i place of 27 at the author's discretio, as τ is a more logical expressio of circular fuctios. (Hartl,2010) The optimal N V exists at the poit where total N V is maximized relative to the Kcal cotet of the food source. While techically this fuctio does ot meet Roles Theorem, it is visually obvious that such a maximum (maxima) exists without havig to go through the rigors of calculatig the first, secod ad third derivatives of the iitial fuctio. The maxima exists at the crest of the ormal distributio fuctio. There is a relatively simple explaatio why the N V fuctio has to be a ormal distributio fuctio. Cosider a theoretical situatio of a food with very low Kcal cotet. It seems ratioal that as Kcal approaches zero that the N V of the food would have to likewise either approach zero, remai costat or potetially icrease. We have already observed i the orage juice ad soda pop example that ituitively N V does ot ecessarily always decrease as Kcal decreases. However, to extrapolate this cocept to its ultimate logical coclusio where a decrease i Kcal is directly related to a icrease i N V is mathematically impossible. lim >0 This meas that ay food with o caloric cotet, ad ay utriet value at all, is mathematically udefied. A simple way to visualize such a asymptotic theoretical relatioship mathematically is: = Page 2 of 8

4 The Nutritioal Desity Ratio Dilemma: Developig a Scale for Nutritioal Value This does ot implicitly suggest that the iverse mathematically ca be true. While the author does coted that both N V ad Kcal ca simultaeously equal zero, either ca idepedetly be zero. While it seems axiomatic, for purpose of this paper it is assumed that there exists o food with either a egative N V or a egative Kcal cotet. Commercial claims that there are foods oe ca eat which cosume more Kcal tha they cotai are mathematically ufouded. If: The: lim >0 lim 0 >0 To further visualize that the Nutritio Desity Ratio must be a ormal distributio curve fuctio cosider some examples of foods ad where they would fall alog the Kcal cotiuum. Sugar is geerally cosidered to have a high Kcal value with a relatively low N V. Yet ot all sugar is created equal. Ideed ot all sugars are eve recogizable as sugar outside of their biochemical defiitios. The followig discussio of sugar will restrict itself to moosaccharides to simplify the mathematics. It is uderstood that sucrose (dextrose) is the most commoly familiar form of sugar, but also that it is a disaccharide cosistig of varyig ratios of glucose ad fructose. I the ame of simplicity we will restrict our discussio of sugars to two: glucose ad fructose. (Reece, Urry, Cai, Wasserma, Miorsky, & Jackso, 2011) Glucose ca be directly delivered to the cells ad used i metabolism. Coversely fructose must first travel to the liver, be coverted to glucose, ad the delivered to the cells for utilizatio. (Heiz, Walther, & Kirsch, 1968) It is therefore a reasoable deductio that glucose may have a higher N V tha fructose. However, this is oe of those situatios where the poit represetig the itersectio betwee N V ad Kcal may fall uder the curve of the fuctio. Glucose cotais approximately 4 Kcal per gram (U.S. Food ad Drug Admiistratio, 2004), whereas fructose also cotais about 4 Kcal per gram (Saulo, 2005), but because of the extra processig required i the liver has a lower N V. This is perfectly reasoable sice everythig uder the curve is a cotiuous radom variable, ad othig exists above the curve. Remember, the ideal food lies directly o the curve at the curve's maxima. There are foods, that although cosidered "healthy" are ot ecessarily utritive, which would fall o the opposite side of the distributio. High fiber foods would fall i this area. At this poit i history to imply that high fiber foods are ot utritive is tatamout to cultural apostasy, but mathematically it is true. The huma body ca o more digest the cellulose from a piece of celery tha it ca from a piece of oak. High fiber foods, such as celery, do of course have some N V, ad therefore some Kcal value. Page 3 of 8

5 The Nutritioal Desity Ratio Dilemma: Developig a Scale for Nutritioal Value I the illustratio above we ca see that high fiber has relatively low N V ad relatively low Kcal cotet. Neither fiber or sugar are itrisically bad for you, ad from a purely eergy based poit of view oe ca live loger o a diet of pure sugar tha pure fiber, but either diet will be ultimately fatal. This leads us to the coclusio that for a food to be healthy, or ideally healthy, the Nutriet Desity Ratio, N V :K, must be equal to Perhaps more properly put, the umber of uits of N V must be exactly equal to its Kcal cotet to be perfectly utriet dese. So if a Kcal is a uit of eergy, what is a uit of N V? Well, for some theoretical food to be ideally utritious it must be 100% utritious, ad i the world of math, 100% equals 1. The perfectly utritious food therefore has oe uit of Nutritioal Value for every Kcal. Seems simple eough, but what is the uit? Previous attempts at defiig a Uit of Nutritio do ot seem to have come to a irrefutable coclusio. (Drewowski, 2005) If you have't already guessed, this is where the mathematics are goig to start gettig a little complex. First we have to cosider the problem from two differet perspectives: food versus diet. I the aforemetioed pages it has bee demostrated that the N V of a food is some value that allow the Nutritio Desity Ratio to be oe. People, i geeral, do ot live o a specific food, they live o a variety of foods, or a diet. Sice to date o oe has attempted to calculate or ivet a uit for N V, dietary sciece has used the shotgu approach, meaig that with a wide eough variety of foods eate i some moderate amout based solely o their Kcal cotet that the average, (mea), N V will geerally be reached. I other words, they hope that: Where: N V = Average utriet value m = the th meal N V = m 1 =1 Page 4 of 8

6 The Nutritioal Desity Ratio Dilemma: Developig a Scale for Nutritioal Value... the average of all meals over oe's lifetime is diverse eough that the Nutritio Desity Ratio is 1. More formally this could be expressed as: N V m = m 1 ( xp wp ) + ( xl wl ) + ( xc wc ) + ( xv wv ) + ( xm vm ) + xw = K 1 I the aforemetioed formula m is the th meal i the series of all meals eate over time, with each meal cosistig of food(s) of varyig compositio ad Nutriet Value. But o a idividual basis, with oly oe food cosidered, the perfectly utritious food that has oe uit of N V for every Kcal, must meet the followig formula: ( xp wp ) ( xl wl ) + ( xc wc ) + ( xv wv ) + ( xm vm ) + + xw = K 1 Where: x = Quatity w = Weight (Value) P = Protei L = Lipid C = Carbohydrate V = Vitami M = Mieral W = Water K = Kilocalories So for example: xp would be the Amout of the th Protei, ad wp would be the particular weight of the value that the th Protei has for the body. This aloe ca become mathematically eormously complex, sice eve with scales like the Glycemic Idex (Groder, Roth, & Walkigshaw, 2011) or the Protei Efficiecy Ratio (Shils, Shike, Olso, & Ross, 2005), comparig apples ad orages is o loger just a fuy metaphor. Theoretically, if the N V of a food is: 1 ( xp wp ) ( xl wl ) + ( xc wc ) + ( xv wv ) + ( xm vm ) + xw The the perfectly utritious food with a N V :K ratio of 1 is where: Meaig Page 5 of 8

7 dy dx The Nutritioal Desity Ratio Dilemma: Developig a Scale for Nutritioal Value ( xp wp ) ( xl wl ) + ( xc wc ) + ( xv wv ) + ( xm vm ) + + xw = the maxima of the poit o the fuctio curve should be the maximum Nutritioal Value for its relative Kcal cotet. K 0 The Nutriet Value Cojecture Developig a Scale for Nutritioal Value: For all of the foregoig to be true, we arrive at what I shall call the Nutriet Value Cojecture: A food that has a collective sum total of oe (1) Nutri (the ame we shall assig for the uit of utritioal value) will satisfy: 1 ( xp wp ) ( xl wl ) + ( xc wc ) + ( xv wv ) + ( xm vm ) + xw = Kurtosis Agai this ituitively implies that over the course of a lifetime, with a diverse eough diet, the Theory of Very Large Numbers dictates that everythig teds towards the mea, ad so overall diet should be o average highly utritious. However, malourishmet ad obesity statistics suggest the kurtosis of this curve overall appears more platykurtic. So the effectiveess of relyig o extremely large data samples for diet, is at best subjective as suggested earlier. Diet however is metioed here oly tagetially ad is beyod the purview of this paper. Page 6 of 8

8 Skewess The Nutritioal Desity Ratio Dilemma: Developig a Scale for Nutritioal Value The cocept of skewess, like that of kurtosis, is tagetial to the object of this paper, but I will metio it briefly for the beefit of those who may choose to extrapolate the utritioal desity ratio o a broader scale of applicatio. It is mathematically possible to maitai a optimal overall utritio thatt is ot perfectly ormally distributed, but this applies oly to the cosumptio of may foods, ad over the course of time. Ideed it is largely a part of popular diet culture. Foods which have a low Nutritioal Desity Ratio (NDR) ca remai mathematically utritious by reducig the quatity of cosumptio, resultig i a egatively skewed utritioal distributio. Coversely, diets with high NDR, which are equally udesirable, but ofte mistake for beig "healthier", have very low Kcal cotet, but also very low N V, ad result i a positively skewed utritioal distributio, ad ca be mathematically utritious by icreasig the volume of cosumptio. Bibliography Crichto, M. (2003). Alies Cause Global Warmig. Caltech Micheli Lecture (pp. 1-12). Staford, Califoria : Stephe H. Scheider, Staford Uiversity. Groder, M., Roth, S. L., & Walkigshaw, B. C. (2011). Nutritioal Foudatios ad Cliical Applicatios (5th Editio ed.). St. Louis, MO: Mosby. Hartl, M. (2010). The Tau Maifesto. Tau Day, 2010 (pp. 1-35). Pasadea, Califoria: Michael Hartl. Heiz, F., Walther, L., & Kirsch, J. (1968). Ezymes of fructose metabolism i huma liver. The Houral of Cliical Ivestigatio, 47 (8), Reece, J. B., Urry, L., Cai, M., Wasserma, S., Miorsky, P., & Jackso, R. (2011). Campbell biology (9th editio; Iteratioal editio.). Harlow: Pearso Educatio. Saulo, A. A. (2005). Sugars ad sweeteers i foods. Hoolulu, HI: Uiversity of Hawaii. Shils, M., Shike, M., Olso, J., & Ross, C. (2005). Moder Nutritio i Health ad Disease (Teth editio ed.). Philadelphia, PA.: Lippicott Williams & Wilkis. Page 7 of 8

9 The Nutritioal Desity Ratio Dilemma: Developig a Scale for Nutritioal Value U.S. Food ad Drug Admiistratio. (2004). Calories Cout: Report of the Workig Group o Obesity. U.S. Food ad Drug Admiistratio. Washigto D.C.: U.S. Food ad Drug Admiistratio. Page 8 of 8