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An increasing number of studies are beginning to show that both low-density lipoprotein and high-density lipoprotein cholesterol can constitute risk factors for myocardial infarction. Such a behaviour has been called by experts in the field the “chameleonic effect” of cholesterol. In the present paper, a fractal/multifractal model for low-density lipoprotein and high-density lipoprotein cholesterol dynamics is proposed. In such a context, a fractal/multifractal tunneling effect for systems with spontaneous symmetry breaking is analyzed so that if the spontaneous symmetry breaking is assimilated to an inflammation (in the form of a specific scalar potential), then a coupling between two fractal/multifractal states can be observed. These two states, which have been associated to biological structures such as low-density lipoprotein and high-density lipoprotein, transfer their states through a fractal/multifractal tunneling effect. Moreover, in our opinion, the widely used notions of “good” and “bad” cholesterol must be redefined as two different states (low-density lipoprotein and high-density lipoprotein) of the same biological structure named “cholesterol.” In our work, for the first time in the specialized literature, low-density lipoprotein and high-density lipoprotein have been regarded as two different states of the same biological structure (named “cholesterol”), such as in nuclear physics, the neutron and proton are two different states of the same particle named nucleon.

Cholesterol fractions, especially low-density lipoprotein (LDL) and high-density lipoprotein (HDL) cholesterol, are frequently analyzed biomarkers in clinical laboratories [

Observational studies have shown that LDL and HDL have opposing associations with risk of myocardial infarction, with LDL cholesterol being a positive factor and HDL cholesterol being a negative (protective) factor [

However, in the following paragraphs we want to show, by doing a synopsis of the literature in this field that, in recent years, more and more evidence points to the fact that a chameleonic behavior can be attributed to LDL and HDL cholesterol.

Observational studies cannot separate the causal role in the pathological process from the role of a marker of the underlying pathophysiology. The results of both randomized trials of LDL cholesterol-lowering treatments [

If particular plasma biomarkers are directly involved in a pathological process, then inherited variation changing plasma concentrations of these biomarkers should affect the risk of disease in the direction and magnitude predicted by the plasma concentrations. This approach has been used in the past to analyze plasma HDL cholesterol; although with restricted sample sizes, a small number of single nucleotide polymorphisms (SNPs) at a few genes and with SNPs that affect multiple lipid fractions [

This is a main reason for which current studies have not been able to completely resolve the possible causal relevance of HDL cholesterol concentrations for risk of myocardial infarction.

It is our strong opinion that we cannot simply divide cholesterol fractions into “good” or “bad ones”. In the following, we will present arguments for this statement, based on recent published articles [

Zewinger et al. [

If an acute phase response or systemic inflammation are absent, the HDL proteome constitutes anti-inflammatory particles, but if an acute phase response or systemic inflammation are present, then the HDL proteome is remodeled to constitute particles that increase the inflammatory response. This type of system possibly evolved to provide protection against viral and bacterial infections in the past, when humans did not have long enough lifespans to suffer from chronic inflammatory diseases such as atherosclerosis or rheumatoid arthritis. The study of Zewinger et al. [

The main protein in LDL is apoB (apolipoprotein B), a protein that does not exchange between particles. The main protein in HDL is apoA-I (apolipoprotein A-I), which can be exchanged between particles. It is well known that all proteins associated with HDL are continuously moving on and off the HDL particles. It was hypothesized that HDL evolved as part of the innate immune system and is a chameleon-like lipoprotein [

Moreover, evidence has been found that atherosclerosis is not the only inflammatory disease with abnormal HDL. Watanabe et al. [

We must also mention that new clinical studies support all laboratory findings listed above. Let us note some of them:

Ravnskov et al. [

During 2018, at the European Society Congress in Munich, Dr. Marc Allard-Ratick, from Emory University School of Medicine in Atlanta, presented a study carried out as part of the Emory Cardiovascular Biobank. The participants were 63 years old on average and about one-third were women. The conclusions were the following: (a) patients with HDL levels in the middle range of the spectrum, between 41 and 60 mg/dL of blood, had the lowest risk for heart attack or death from heart diseases. Opposite to this, (b) patients with HDL readings below 41 or above 60 faced a significantly increased risk for both health outcomes, demonstrating what the researchers called a “U-shaped” risk pattern. Scientists showed that patients with HDL levels exceeding 60 were found to have a 50 percent greater risk of heart disease death or heart attack, compared with those in the middle range. Race and gender did not appear to affect the findings. “The mechanism behind this finding remains unclear,” Dr. Marc Allard-Ratick said. During the same Congress, Dr. Gregg Fonarow, director of the Ahmanson-UCLA Cardiomyopathy Center and co-director of the UCLA Preventative Cardiology Program in Los Angeles, said that “research from UCLA established more than two decades ago that HDL cholesterol could—in certain individuals (including those with very high levels of HDL) and in certain circumstances—be dysfunctional and proinflammatory,” and contribute to narrowing of the arteries. “In others words, the so-called ‘good’ cholesterol in terms of cardiovascular risk could go ‘bad’ and be associated with excess risk,” added Fonarow, who was not involved in this work [

Previously, Madsen et al. [

El Khoudary et al. [

Dr. Christopher Cannon, professor of medicine at Harvard Medical School and a cardiologist at Brigham and Women's Hospital, in a 2017 Harvard Heart Letter [

All these controversies regarding HDL are generated by expanding and deepening studies which were lately focused on these two lipoproteins (LDL and HDL). Nowadays we cannot divide the cholesterol between “good” and “bad” anymore, in a simplistic approach, because slowly the research involves more and more specialists outside medicine, as biophysicists, biologists, geneticists, etc., who focus the area of interest at a molecular and submolecular level, searching for the primary processes, before any clinical and paraclinical evidence (atherosclerotic plaques in our case); HDL cholesterol appears to be one of the most important risk factors in plaque formation, but this theory has not been proved yet, although there are multiple studies that pointed towards this possibility [

Also, in a previous work [

Chaoticity and nonlinearity are both structural and functional for any biostructure assimilated to a complex system [

Classically, commonly used models are usually founded on the otherwise unjustified supposition that variables describing the dynamics of any biostructure as a complex system are differentiable (see, for example, the kinetic models for blood dynamics [

This method of describing the dynamics of any biostructure as a complex system clearly implies the development of new geometrical structures and also of new mathematical models for which the motion laws, invariant to spatial and temporal transformations, are integrated with scale laws, invariant to spatial and temporal scales transformations. In our opinion, such a geometrical structure can be based on the concept of a fractal and the corresponding mathematical model can be based on the Fractal Theory of Motion [

The fundamental assumption of this model is the one that the dynamics of any biostructure as a complex system will be described by continuous but nondifferentiable motion curves (fractal motion curves). These fractal motion curves exhibit the property of self-similarity in every point, which can be translated into a property of holography (every part reflects the hole). Basically, we are discussing about “holographic” implementations of dynamics of any biostructure as a complex system through Schrödinger type fractal “regimes” (i.e., describing the dynamics of any biostructure as a complex system by using Schrödinger type equations at various scale resolutions).

Therefore, the fundamental assumption of our mathematical model (in accordance with the Fractal Theory of Motion) is the one that the motions of blood’s structural units (elements, HDL, LDL, colloids, etc.) [

In this context, let us analyze the dynamical behaviour of specific structural units of the blood in the form of HDL and LDL cholesterol particles. The functionality of the above stated hypothesis implies the following [

Any continuous but nondifferentiable curve of cholesterol particles (cholesterol fractal/multifractal curve) is explicitly scale resolution

The dynamics of cholesterol particles are related to the behaviour of a set of functions during the zoom operation of the scale resolution

The dynamics of cholesterol particles are described through fractal/multifractal variables, i.e., mathematical functions depending on both the space and time coordinates and the scale resolution since the differential time reflection invariance of any dynamical variable is broken. Then, in any point of the cholesterol fractal/multifractal curve, two derivatives of the variable field

The “+” sign corresponds to forward processes of cholesterol particles, while the “−” sign corresponds to the backwards ones. We must mention that in the differentiable case relations (

are equivalent (one passes from one definition to the other by the

The differential of the spatial coordinate field

The nondifferentiable part of the spatial coordinate field, by means of which we can describe the fluctuations of cholesterol dynamics, satisfies the fractal/multifractal equation [

where _{F} defines the fractal dimension of the cholesterol nondifferentiable curve.

Any definition can be chosen for _{F} (fractal dimension in Kolmogorov sense, fractal dimension in Hausdorff–Besikovici sense, etc. [_{F} = 2, coherent-type processes are generated in cholesterol dynamics. For _{F} < 2, correlative-type processes are induced, while for _{F} > 2 noncorrelative-type ones can be found [

Let us note that equation (

The differential time reflection invariance of any cholesterol dynamical variable is recovered by combining the derivatives

This is a natural result of the prolongation procedure on the complex space of any dynamics and, particularly, cholesterol dynamics [

with

The real part

In the absence of any external constraint, an infinite number of fractal/multifractal curves (geodesics) can be found relating any pair of points, and this is true on all scales of cholesterol dynamics. Then, in the fractal/multifractal space of cholesterol, all cholesterol particles are substituted with the geodesics themselves so that any external constraint can be interpreted as a selection of geodesics. The infinity of geodesics in the bundle, their nondifferentiability, and the two values of the derivative imply a generalized statistical fluid-like description [

through (

The previous relation (

Cholesterol dynamics can be described through a covariant derivative in the following form. For this, let us consider that the cholesterol fractal/multifractal curves are immersed in a 3-dimensional space and that ^{i} is the spatial coordinate field of a point on this fractal/multifractal curve. In these conditions, any variable field

where the summation over repeated indices is understood. We will keep this convention in the following.

We want to remind the fact that the

These relations are valid in any point and more for the points ^{i} on the cholesterol fractal/multifractal curve which we have selected in (

In the following, we suppose that the average values of the all variable field

Even the average value of

In this condition, (

If we divide by d

These relations also allow us to define the operators:

Under these circumstances, taking into account (

Now, from (

In the following, this operator will be identified with the derivative of the scale covariant based on the scale covariance principle.

Let us now consider the principle of scale covariance (the physics laws applied to cholesterol specific dynamics are invariant with respect to scale resolution transformations) and postulate that the passage from the classical (differentiable) biophysics to the fractal/multifractal (nondifferentiable) biophysics can be implemented by replacing the standard time derivative

This means that the local fractal/multifractal acceleration

If the fractalisation/multifractalisation is achieved by Markov-type stochastic processes, which involve Lévy type movements [

In these conditions, the cholesterol geodesics take the simple form

For irrotational motions of the fractal/multifractal cholesterol particles, the complex velocity field (

Then, substituting this relation in (

The fractal/multifractal cholesterol variable

Let us make explicit such a situation for

Multiplying (

In (

According to the aforementioned statements, let us analyze the state transfer between LDL and HDL cholesterol. Thus, when cholesterol particles are subjected to an external constraint, i.e., an inflammation which can be assimilated to a scalar potential

In the one-dimensional case, the above equation becomes

If the external scalar potential

If, in such a context, we suppose that the state transfer between LDL and HDL cholesterol implies spontaneous symmetry breaking [

The effective potential for the case of a fractal/multifractal tunneling effect for biological systems with spontaneous symmetry breaking.

In these conditions, the stationary fractal/multifractal equation becomes

For each of the three regions, the solutions of the equations are

Due to the infinite potential in the two extreme regions,

Since the states density

These, along with

Due to the algebraic form of the two equation pairs, in order to establish the concrete expression of the “secular equation” (for eigenvalues ^{th} order determinant,

We find

For

For

Because both eigenvalues equations are strongly transcendent, a direct estimation of solutions

It results, for now, at least qualitatively, that the presence of the barrier (of finite height _{0}) between −

In the following, the previous results will be calibrated to cholesterol dynamics. Indeed, the identification of LDL and HDL states can be performed by admitting that

Taking the above into account, we can thus state that LDL and HDL are two different states of the same biological structure, like in the case of neutron and proton which are two different states of the same particle, named nucleon. The state transfer between LDL and HDL occurs by means of a fractal/multifractal tunneling effect.

The main conclusions of the present paper are the following:

We build a mathematical model for describing cholesterol dynamics, based on recent scientific evidence that cholesterol (especially HDL and LDL) has a chameleonic behaviour.

Our mathematical model is founded on the hypothesis that both HDL and LDL dynamics are described by means of continuous and nondifferentiable motion curves (fractal/multifractal curves). In such a context, the dynamics equations in the form of fractal/multifractal-type geodesics are obtained, and from here, in the stationary case, a fractal/multifractal tunneling effect for systems with spontaneous symmetry breaking is analyzed. If the spontaneous symmetry breaking is assimilated to an inflammation (in the form of a scalar potential), then two fractal/multifractal states through a fractal/multifractal tunneling effect can be observed. These two states can be associated to biological structures such as LDL and HDL. In minimal terms, we can observe here a coupling between fractal/multifractal states of cholesterol, generated by a fractal/multifractal tunneling effect. In this case, we have two potential local parabolic wells with minimal points at

The fact presented above are in accordance with the latest studies results. Thus, we can unequivocally state that the role of cholesterol fractions must be clearly redefined. Our model could offer an explanation of why high values of HDL cholesterol can be “toxic” or why, in certain conditions, LDL cholesterol can be a protective factor. We can practically discuss about different states of the same entity, HDL and LDL being expressions of a unique entity—cholesterol—with a pro or antiatherogenic effect modelled by the instant state and the alternation between the two possible sides. As a consequence, as long as cholesterol fractions maintain a continuous “fluidity,” the maximum benefit will be attained if the total cholesterol, in absolute value, is decreased. Our mathematical model only enforces the recent medical findings in the field, which are more and more frequent. At the same time, in our opinion, the present mathematical model confirms and explains the apparent paradoxes from clinical studies. Furthermore, our model can be used to analyze biological dynamics at nanoscale [

There are no data sets used in this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

All authors have contributed equally to this paper.

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