Journal of Conservative Dentistry

: 2021  |  Volume : 24  |  Issue : 1  |  Page : 10--14

Nonclinical research areas of future importance for clinical therapies: Exploring the concepts of nonlinearity in dentistry

Poorya Jalali1, Gunnar Hasselgren2,  
1 Department of Endodontics, Texas A&M College of Dentistry, Dallas, TX, USA
2 Division of Endodontics, Columbia University College of Dental Medicine, New York, NY, USA

Correspondence Address:
Dr. Poorya Jalali
Department of Endodontics, Texas A&M College of Dentistry, 3302 Gaston Ave, Dallas, TX 75246


Linear system analysis has been dominating medical and dental research, and most of the research achievements in these fields have come from applying a reductionist view of nature. However, biologic systems are fundamentally nonlinear with highly composite dynamics made up of numerous interacting elements and feedback loops, therefore studying them as linear models may not result in an accurate representation of their true features. The authors reviewed and utilized some of the principles of chaos and nonlinearity and extended them to clinical dentistry, from cracked tooth and flare-up after root canal procedures to the outcome of clinical treatments. Utilization of the concepts of chaos and sensitive dependence on initial conditions, and the concepts of self-organization, stigmergy, and fractals may help us to understand some of the puzzles that have not been solved by conventional linear models. The goal of this paper is to present some areas within nonclinical research that we believe will have important roles in the development of future clinical examination methods and therapies.

How to cite this article:
Jalali P, Hasselgren G. Nonclinical research areas of future importance for clinical therapies: Exploring the concepts of nonlinearity in dentistry.J Conserv Dent 2021;24:10-14

How to cite this URL:
Jalali P, Hasselgren G. Nonclinical research areas of future importance for clinical therapies: Exploring the concepts of nonlinearity in dentistry. J Conserv Dent [serial online] 2021 [cited 2021 Aug 4 ];24:10-14
Available from:

Full Text


It is always hard to predict and when you look back with some years of experience into the clinical sciences, you can see that at an earlier stage you would not have been able to fully predict the present states of clinical therapies. Hence, to aim for a look into the future of the clinical sciences today, we are in a fog and can at best rely on educated guesses.

Still, there are some trends and one of them is that knowledge from the true basic sciences has afforded major leaps in clinical approaches. Engineering, also known as applied science, has also markedly contributed to the evolution of clinical diagnostics and therapies. A recent example is the development of three-dimensional radiography which has had a marked improvement on clinic activities. Engineering is and has for a long time been a major part of our clinical world. Just think of the instruments, we are using and depend on in our daily work – the mirrors, probes, tweezers, lights, chairs, etc. Research in mining, metallurgy, design, and many other areas is behind these tools that we depend on and they are often an extension of our hands during patient treatment. We as clinicians know little about all the research, development, and innovation that are part of these everyday blessings in our clinics. To add to our “general ignorance” of the world around us that we are dependent on, there are fields in research that are rather unknown to us today, but most likely will bring markedly new approaches to clinical activities.

The aim for this paper is to look into some areas within nonclinical research that we believe will have important roles in the development of future clinical examination methods and therapies.

 Chaos and Nonlinearity

Linear system analysis has been dominating medical and dental research, and most of the achievements in these fields have come from applying a reductionist view of nature – you simplify events to make it easier to study them. In linear systems, the output is directly proportional to the input, and they express characteristics that are simpler and more predictable than complex nonlinear systems. However, biologic systems are fundamentally nonlinear with highly composite dynamics made up of numerous interacting elements and feedback loops,[1] therefore studying them as linear models may not always result in an accurate representation of their true features.

Craniofacial structures, also, largely due to dynamical complexity at not only the macroscopic level but also at the microscopic level, give nonlinear responses to external stimuli such as infection, trauma, or treatment. For instance, it has been shown that, due to nonlinear dynamics of temporomandibular joint, the stress level in the disc is not proportional to the masticatory muscle activation level.[2] Therefore, a 20% increase in muscle activation level will not result in a 20% increase of the shear stress in the articular disc. Incorporating other elements such as duration of the time spent clenching, the direction of the load, and viscoelasticity of the other orofacial structures (e.g., bone, periodontal ligament [PDL], and dentin) would make it a complex nonlinear system. Not to mention the fact that PDL, which is just one of the many elements involved in this already complex system, independently has a complex nonlinear viscoelastic behavior.[3]

The complex nonlinearity also exists at the cellular and molecular level. The immune system is an example of a system with complex behaviors that emerge as a result of nonlinear interactions between a large number of elements at various levels of organization.[4] Therefore, interaction of the immune system with any stimuli, for example, a target population of bacteria, must be considered as a complex dynamic process.[5] One of the features of complex systems is sensitive dependence on initial conditions. This means that if two states in a nonlinear system differ by only a small amount they will follow very different trajectories over time, making the outcomes markedly different, hence making the fate of each state unpredictable.

The principles of chaos and nonlinearity are also applicable to endodontics; from cracked teeth and flare-up after root canal procedures to the outcome of clinical procedures. For example, in recent years, there has been a growing interest in studying fracture mechanics not only because of its importance in the clinical sciences and structural integrity of various forms but also because of the challenges to experimental techniques and mathematical analysis. There is theoretical evidence in favor of chaotic behavior of fracture development.[6]

Flare-up after endodontic treatment has been proposed as an example of a chaotic nonlinear response to instrumentation.[7] The incidence of severe pain and swelling after instrumentation of a root canal system is reported between 4% and 12%.[8] Although some factors such as over-instrumentation and apical extrusion of bacteria have been suggested as contributing, the exact mechanism of flare-ups is unknown, and anecdotal evidence shows that it can happen even with a careful instrumentation technique.[9],[10],[11] The following hypothetical scenario can be used to better grasp the unpredictability of endodontic flare-ups: root canal treatment is performed on two contralateral mandibular bicuspids on the same patient. The initial pulpal and periapical diagnosis for both premolars is pulp necrosis with asymptomatic apical periodontitis, and the endodontic treatment is performed by the same clinician with the clinician using identical techniques. Two days later, the patient returns to the office with severe pain and abscess involving buccal space on the left side, but having no swelling, pain, or discomfort on the right. Considering that, from the clinical standpoint, both teeth were exposed to very similar conditions how can this large difference in fate be explained? Utilizing the concepts of chaos and sensitive dependence on initial conditions, one can describe the unpredictability and sudden emergence of endodontic flare-ups. Flare-up is an intense inflammatory response to bacterial and/or physical stimulation that ignites severe symptoms and swelling.[9] The host inflammatory response has a complex nonlinear dynamic,[12] therefore incorporating more variables (e.g., different bacterial floras in the two teeth, number of extruded bacteria, or degree of mechanical irritation, etc.) into this already complex system will make it more chaotic. As a chaotic system, it has an infinite number of trajectories in a finite region of phase space. In this system, a periapical trajectory can be defined as the sequence of inflammatory expressions/responses of periapical tissues of an individual tooth over time. It could be conceived as a moving picture of a periapical status that covered several days or months after initiating an endodontic treatment. The moving picture would show continuous reconstructions or destructions that a dynamic periapical status undergoes such as developing an acute abscess or healing of an endodontic lesion. Because of sensitive dependence on initial conditions and due to even a very small, but rapid change in the state of the periapical tissues, the periapical status can skip from one trajectory into another resulting a completely different outcome. Thus, returning to the above scenario, theoretically the extrusion of only a few more bacteria during instrumentation of the left premolar could move the periapical status onto a new trajectory ending as an acute abscess.

There are other clinical situations, in which nonlinearity is applicable. There is a nonlinear dose-response relationship between the number of pathogenic bacteria and apical periodontitis. Therefore, a twofold increase in the number of bacteria in a root canal system would not cause a twofold increase in the signs and symptoms of apical periodontitis. This nonlinearity is also accountable for making the infective dose for causing apical periodontitis and enigma. Not only does the infective dose vary across bacterial species[13] but also it differs according to the patient's age and systemic health.[14] Considering the polymicrobial etiology of endodontic disease,[15] variation in immune response, and gene polymorphism,[16] a nonlinear behavior for endodontic disease and outcome is expected. This would explain the mysterious success of a poorly done root canal treatment in one case, and an unfortunate failure of a seemingly flawless endodontic treatment in another. In other words, the real problem is the impossibility to define normal values.[17] What for one person is subinfective dose or “normal,” for another person may be a very large dose of bacteria causing apical periodontitis, cellulitis with fascial space involvement, or even death. Another factor that makes the exact prediction of endodontic outcome impossible is the nature of adaptive immunity. By definition, adaptive immunity is adaptive, which means its behavior may change through time. Therefore, determining its status at one point in time would not necessarily predict the future immune response.[18]

For the above reasons, the application of nonlinear methods and chaos theory are more suitable to evaluate the outcome of endodontic treatment. Based on the chaos theory's framework provided by Lorenz,[19] due to sensitive dependence, the reliability of forecasts in complex systems declines exponentially over time. Therefore, when there are a large number of variables and feedback loops influencing the endodontic outcome, clinicians and researchers are forced to evaluate treatment outcome based on probability rather than accurate prediction, and their capacity to establish probability will deteriorate as they try to foresee further into the future. However, outcome prediction can be improved by limiting the number of variables in a dynamical system, which can only be achieved by a personalized approach to treatment customized to each patient's immunological and biological profile.


Self-organization, which is one of the key concepts in nonlinear systems, can be appreciated in dental-related biology and pathophysiology of diseases. Self-organization is a spontaneous process, in which a structured form arises from interactions between elements of a seemingly anarchic system.[20] In complex dynamic systems, this self-organization could result in emergence, in which larger existence and new properties come out from smaller entities, and a new emergent pattern becomes greater than the sum of its individual parts.[12] Multiple organ dysfunction syndrome, endodontic interappointment flare-up, and microbial biofilm have been proposed as developing emergence in a complex system,[7],[12] in which novel synergistic characteristics arise from interactions between the lower-level components of the system.

The human microbiome (including oral and gut microbiome) is another example of self-organization, in which an emergent framework arises from the collective behavior of microorganisms. Recent studies have shown that gut microbiome can collectively regulate human cognition and social behavior (e.g., anxiety and depression) through secretion of neurotransmitters. Consequently, it has been proposed that the gut microbiome is an unconscious system shaping human behavior, and its imbalance may result in negative effects on mental health.[21] In a similar fashion, disruptions in the multitude of oral microorganisms (i.e., oral microbiome) may contribute to oral diseases such as caries, periodontitis, peri-implantitis, and various forms of mucosal diseases.[22] Although the different habitats in the oral cavity (e.g., tongue, gingiva, and dental plaque) may harbor different genera of bacteria, there are sufficient common traits among them,[22] which verifies the oral microbiome as a continuum of microorganisms connecting these habitats. Therefore, in the management of oral diseases, clinicians may need to target the microbiome as a whole, instead of eliminating just the pathogenic bacteria.

Similar to a social insect colony, microorganisms in oral plaque or biofilm can be labeled to as a “superorganism,” in which the omnium gatherum of multiple individual organisms (i.e., bacteria), manifest some of the organism-like characteristics such as integrity, strive to a common goal, and task specialization. Such a colony of microorganisms becomes such as a single living organism, preserving its structure in space, and resisting foreign influence. In certain situations, some microorganisms may become disconnected or removed from the initial colony and start forming an auxiliary colony in a new environment. For example, bacteria can breakout from the oral microbiome and reproduce and differentiate independently inside a root canal system and hence forming a biofilm significantly different from that of in the oral cavity.


Stigmergy is another concept that can help to better understand the emergence in complex systems. The principle of stigmergy is that “footmarks and impressions” left in a field by one organism (or an agent) can affect the behavior and fate of other organisms. The result of this process is the development of shared external memory and collective behavior which can be seen in, for example, social insects. The concept of stigmergy has been proposed as one of the main drivers in self-organization and emergence of bacterial biofilms.[23] Stigmergy can also be appreciated in morphogenesis and tissue self-organization.[24] The growth and architectural development of biological tissues rely heavily on memory and history. In other words, the actions of the system are guided not only by the current state but also by past events.[24] In endodontics, clinicians may take advantage of stigmergy by recovering the tissue memory through the release of fossilized proteins in the dentin matrix (e.g., transforming growth factor beta), in order to induce root regeneration.[25] This dormant capacity explains the ability of an immature root with pulp necrosis to grow back into its normal form years after its destined maturation even in instances when the tissues that form the new root are different from dentin.


Fractals are shapes and their pattern and fragmentation remain the same when one magnifies the object. In other words, fractals are patterns that show self-similarity at different scales.[26] Fractal geometries are very common in nature such as lightening, Mississippi River Delta, or the tightly clustered florets of a Romanesco cauliflower. Fractal Dimension (FD) is a ratio that measures the complexity of a fractal pattern,[27] and there are various ways to calculate FD of an object.[28],[29]

Fractal geometry has been applied to a wide range of topics in medicine and physiology. It has been shown that cardiopulmonary tissues have fractal patterns such as the branching patterns of arteries, veins, cardiac muscle bundles, and the bronchial tree.[30],[31] Using advanced imaging technique to study the changes in their fractal pattern may shed light on the progression of a disease or their response to treatment.[31],[32]

Some examples of fractal patterns in dentistry are patterns of alveolar bone and the vascular network in the gingival and vestibular mucosa.[33] Fractal analysis has the potential to be used as a method for evaluation of treatment outcome in dentistry. Few studies, by analyzing the radiographic images, have shown an increase of FD during bone regeneration and healing.[34],[35] Within the root canal system, similar branching patterns can be seen when studying the apical delta, in which a main canal branches off into smaller accessory canals.[36] Zooming further into the dentin, the major dentinal tubules branch into fine branches and eventually into microbranches with 25–200 μm in diameter.[37] Therefore, it shows self-similarity at different scales and may be considered as a fractal model. The potential diagnostic value of the fractals in the pulpal disease should be examined. In the presence of chronic pulpal disease (due to trauma or caries), the root canal system may undergo asymptomatic degeneration which sometimes manifested by only gradual signs of calcification (i.e., decrease in FD). In future, using the advanced imaging technologies and the concept of fractal analysis clinicians may be able to detect these subtle alterations inside the root canal system. The same diagnostic approach can be applied to the pulpal vasculature, which also manifests a fractal pattern, as one can infer from a study by Takahashi et al.[38] Ophthalmology studies made a similar application of fractal patterns to the morphology of the pulpal vasculature.[39] Although using the current technology, the fractal pattern of pulpal vasculature cannot be studied in vivo, in vitro evaluation of its morphology could lay a foundation for future studies. Therefore, the methods to calculate and interpret the FD of the root canal system and pulpal vasculature warrant further investigation.

 Quantum Biology

Quantum Mechanics is known as the science of particular events at the subatomic level. It is hard to grasp quantum theory as it appears weird and illogical. Richard Feynman summarized it well: nobody really understands quantum theory. Still, modern society has benefitted immensely from quantum theory as it has made lasers, CD and DVD players, smartphones, magnetic resonance imaging, and a multitude of other things possible. The quantum field is hard to grasp if you are not fluent in mathematics and most of us are not, but the book “Life on the edge” is an excellent introduction for non-mathematicians to Quantum Biology.[40]

Quantum Biology is an emerging field which has not really been considered in our dental publications yet. Quantum biological events take place in the body below the cellular level and are for example part of the enzymatic processes. The connection between biology and quantum theory was pointed out already in 1943 when Schrödinger stated that life is a quantum-level phenomenon.[41] Quantum Biology holds the explanations for many biological events that we hardly understand today. For example, the exact mechanism of enzyme activity has eluded us for a long time even if enzymes and their activities have been studied intensely. Quantum Biology has an explanation for the ability of enzymes to make chemical reactions possible without the addition of major amounts of energy.[42]

In dentistry, it has been proposed that signals from occlusal loading are transported through clustered water as quanta (protons) and reach the dentin tubules with odontoblastic processes where the signals trigger the formation of secondary dentin.[43]

The mentioned clustered water brings us to another topic: water, a prerequisite for life. It is necessary for all cells and tissues to have a certain water content to function. Water is not always the common H2O that comes out of the tap, the simple molecule consisting of three atoms, two hydrogen and one oxygen, has been found to be a much more complex substance than previously thought. It can form large molecules which are the basis of so-called clustered water which covers biologic surfaces.[44],[45] This clustered water has markedly different properties than regular water and is an excellent conduit for electric currents. The expression water wire has been used to describe the speed of electric current in clustered water. The structure of water is studied extensively, and the knowledge from this will likely change our views on many biologic processes.

The present knowledge of water is gathered at an excellent website:


Biologic systems are complex and often consist of infinite interactions between smaller elements at the cellular level. Nonlinearity thinking helps us understand these systems better even if it may feel counter intuitive compared to the linear models we are familiar with.

Financial support and sponsorship


Conflicts of interest

There are no conflicts of interest.


1Higgins JP. Nonlinear systems in medicine. Yale J Biol Med 2002;75:247-60.
2Commisso MS, Martínez-Reina J, Mayo J. A study of the temporomandibular joint during bruxism. Int J Dent Oral Sci 2014;6:116-23.
3Oskui IZ, Hashemi A. Dynamic tensile properties of bovine periodontal ligament: A nonlinear viscoelastic model. J Biomech 2016;49:756-64.
4Ahmed E, Hashish AH. On modelling the immune system as a complex system. Theory Biosci 2006;124:413-8.
5Rickles D, Hawe P, Shiell A. A simple guide to chaos and complexity. J Epidemiol Community Health 2007;61:933-7.
6Alves LM, Lobo RF. A chaos and fractal dynamic approach to the fracture mechanics. In: Ausloos M, Dirickx M, editor. The Logistic Map and the Route to Chaos. Berlin, Heidelberg: Springer; 2006. p. 295-316.
7Jalali P, Hasselgren G. Endodontic inter-appointment flare-ups: An example of chaos? Dent Hypotheses 2015;6:44-8.
8Tsesis I, Faivishevsky V, Fuss Z, Zukerman O. Flare-ups after endodontic treatment: A meta-analysis of literature. J Endod 2008;34:1177-81.
9Siqueira JF Jr. Microbial causes of endodontic flare-ups. Int Endod J 2003;36:453-63.
10Ghivari SB, Kubasad GC, Chandak MG, Akarte NR. Apical extrusion of debris and irrigant using hand and rotary systems: A comparative study. J Conserv Dent 2011;14:187-90.
11Gyanani H, Chhabra N, Parmar GR. Comparative assessment of efficacy of two different pretreatment single oral doses of betamethasone on inter-appointment and postoperative discomfort: An in vivo clinical evaluation. J Conserv Dent 2016;19:564-8.
12Seely AJ, Christou NV. Multiple organ dysfunction syndrome: Exploring the paradigm of complex nonlinear systems. Crit Care Med 2000;28:2193-200.
13Leggett HC, Cornwallis CK, West SA. Mechanisms of pathogenesis, infective dose and virulence in human parasites. PLoS Pathog 2012;8:e1002512.
14Hara-Kudo Y, Takatori K. Contamination level and ingestion dose of foodborne pathogens associated with infections. Epidemiol Infect 2011;1505-10.
15Fouad AF, Barry J, Caimano M, Clawson M, Zhu Q, Carver R, et al. PCR-based identification of bacteria associated with endodontic infections. J Clin Microbiol 2002;40:3223-31.
16Morsani JM, Aminoshariae A, Han YW, Montagnese TA, Mickel A. Genetic predisposition to persistent apical periodontitis. J Endod 2011;37:455-9.
17Klonowski W. Personalized neurological diagnostics from biomedical physicist's point of view and application of new non-linear dynamics methods in biosignal analysis. Int J Biol Biomed Eng 2011;5:190-200.
18Everson GT, Trotter JF, editors. Liver Transplantation: Challenging Controversies and Topics. Totowa, NJ: Springer Science and Business Media; 2009.
19Lorenz EN. Deterministic nonperiodic flow. J Atmos Sol Terr Phys 1963;20:130-41.
20Dobrescu R, Purcarea VL. Emergence, self-organization and morphogenesis in biological structures. J Med Life 2011;4:82.
21Dinan TG, Stilling RM, Stanton C, Cryan JF. Collective unconscious: How gut microbes shape human behavior. J Psychiatr Res 2015;63:1-9.
22Gao L, Xu T, Huang G, Jiang S, Gu Y, Chen F. Oral microbiomes: More and more importance in oral cavity and whole body. Protein Cell 2018;9:488-500.
23Gloag ES, Turnbull L, Whitchurch CB. Bacterial stigmergy: An organising principle of multicellular collective behaviours of bacteria. Scientifica 2015;2015:387342.
24Sasai Y. Cytosystems dynamics in self-organization of tissue architecture. Nature 2013;493:318.
25Smith AJ, Duncan HF, Diogenes A, Simon S, Cooper PR. Exploiting the bioactive properties of the dentin-pulp complex in regenerative endodontics. J Endod 2016;42:47-56.
26Mandelbrot BB. Fractal geometry: What is it, and what does it do? Proc R Soc London 1989;423:3-16.
27Mandelbrot B. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 1967;156:636-8.
28Smith TG Jr., Lange GD, Marks WB. Fractal methods and results in cellular morphology dimensions, lacunarity and multifractals. J Neurosci Methods 1996;69:123-36.
29Geraets WG, Van Der Stelt PF. Fractal properties of bone. Dentomaxillofac Radiol 2000;29:144-53.
30Iannaccone PM, Khokha M. Fractal Geometry in Biological Systems: An Analytical Approach. Boca Raton, FL: CRC Press; 1996.
31Nagao M, Murase K, Yasuhara Y, Ikezoe J. Quantitative analysis of pulmonary emphysema: Three-dimensional fractal analysis of single-photon emission computed tomography images obtained with a carbon particle radioaerosol. AJR Am J Roentgenol 1998;171:1657-63.
32Moledina S, de Bruyn A, Schievano S, Owens CM, Young C, Haworth SG, et al. Fractal branching quantifies vascular changes and predicts survival in pulmonary hypertension: A proof of principle study. Heart 2011;97:1245-9.
33Sánchez I, Uzcátegui G. Fractals in dentistry. J Dent 2011;39:273-92.
34Wilding R, Slabbert J, Kathree H, Owen C, Crombie K, Delport P. The use of fractal analysis to reveal remodelling in human alveolar bone following the placement of dental implants. Arch Oral Biol 1995;40:61-72.
35Heo M, Park K, Lee S, Choi S, Koak J, Heo S, et al. Fractal analysis of mandibular bony healing after orthognathic surgery. Oral Surg Oral Med Oral Pathol Oral Radiol Endod 2002;94:763-7.
36Gao X, Tay FR, Gutmann JL, Fan W, Xu T, Fan B. Micro-CT evaluation of apical delta morphologies in human teeth. Sci Rep 2016;6:36501.
37Mjör IA, Nordahl I. The density and branching of dentinal tubules in human teeth. Arch Oral Biol 1996;41:401-12.
38Takahashi K, Kishi Y, Kim S. A scanning electron microscope study of the blood vessels of dog pulp using corrosion resin casts. J Endod 1982;8:131-5.
39Masters BR. Fractal analysis of the vascular tree in the human retina. Annu Rev Biomed Eng 2004;6:427-52.
40McFadden J, Al-Khalili J. Life on the Edge. The Coming of Age of Quantum Biology. New York, NY: Broadway Books; 2014.
41Schrödinger E. What is Life? Cambridge: Cambridge University Press; 1967.
42Cha Y, Murray CJ, Klinman JP. Hydrogen tunneling in enzyme reactions. Science 1989;243:1325-30.
43Moss ML, Moss-Salentijn L, Hasselgren G, Ling H. A quantum biological hypothesis of human secondary dentinogenesis. Med Hypotheses 2005;64:479-86.
44Chaplin MF. A proposal for the structuring of water. Biophys Chem 2000;83:211-21.
45Kasemo B. Biological surface science. Surface Sci 2002;500:656-77.