**5. Kinetic model of biomarkers in the response to a mild laboratory stressor: A preliminary description**

The difference in the sensitivity of stress biomarkers observed in our experiment can be restate as the difference in the time constant as described above. Here a mathematical model of the response of biomarkers is suggested to describe the experimental result as the difference in the time constant: from on/off binary response to cumulative one.

#### **5.1 Constitution of the kinetic model**

Basic assumptions for the model are introduced according to our experimental fact, and are quite simple as follows:

Salivary Hormones, Immunes and Other Secretory Substances as Possible Stress Biomarker 265

with an increase of individual density. Moreover, it is the simplest model possessing the homeostatic property. With regard to the recovering process, a simple exponential

where *c* is a positive values representing decreasing coefficient. The stress-induced response of biomarkers, increase by a short-term laboratory stressor and decrease by its removal, was simulated with task/break schedule as the same as in the session A and B in our

Figure 11(a) and 11(b) shows the results of the simulation. As expected the model successfully illustrated on/off or cumulative changing profile of biomarkers depending on the parameters. By solely changing the increasing and decreasing parameters, *a*, *b*, and *c*, the degree of such cumulative effects was able to controlled. Therefore this simple non-linear kinetic model proposed here can be assumed as a basis for the stress induced physiological response in our body. By elaborating this model though a series of experiments targeting on the variety of stressors with different schedule on various biomarkers, the dynamics of human stress reaction pathways, HPA and SAM systems, would be better understood.

In this chapter, the salivary biomarker researches as a new metric for human mental stress, its background, methods, experiments, and kinetic model approaches were introduced. Although there are numbers of technical limitation and problems to be solved, biomarkers introduced in this manuscript can be useful and unique measures for human mental states. Stress estimation by salivary biomarkers has a great methodological advantage, because saliva can be collected less-stressfully and in a noninvasive manner unlike blood and urine. Moreover it is the one and the only secretory fluid that can be collected at anytime and by

On the other hand, mathematical model approach lead us an idea of the estimation of optimal work/break schedule in the limited time avoiding excessive secretion of biochemical substances: for an instance, when one has to take a long-distance drive and reach at a destination within a limited period of time, one could estimate the optimal timing of the stop for the rest. It might be useful for a stress management in the working place as well, such as VDT workload and monitoring work. The molecular analysis techniques are advancing day by day, real-time monitoring of such a tiny amount of biomarkers might be available in the near future. Remember the difficulty of self-management of stress and the necessity of introducing objective criteria. Biomarkers introduced in this manuscript can be

This study was supported in part by "Program to Disseminate Tenure Tracking System", Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science

dx/dt = -cx (2)

decreasing function is introduced, as in

**5.2 Result and discussion of the numerical simulation** 

anyone including children and patients in need of nursing care.

a possible solution, even though it still remains in the initial step.

experiment.

**6. Conclusion** 

**7. Acknowledgment** 

and Technology, Japan.


As the most simple and well-consistent with these assumptions, the logistic function is adopted as the basis of this model. The logistic function is a nonlinear ordinary differential equation consisting of the first order of exponential increasing term and the second order of nonlinear decreasing term, as in

$$d\mathbf{x}/dt \equiv (a\text{-}b\text{-}x)\mathbf{x} \tag{1}$$

where *x*, *t*, *a*, and *b* are all positive values representing the concentration of biomarker, time, and increasing and decreasing coefficient respectively. This formula is called a logistic function or growth curve, and it has been applied to describe the exponential growth in the number of bacteria and its decaying, caused by environmental deterioration accompanying

Fig. 11. Simulated biomarker secretion in session A and B.

with an increase of individual density. Moreover, it is the simplest model possessing the homeostatic property. With regard to the recovering process, a simple exponential decreasing function is introduced, as in

$$\text{d}\mathbf{x}/\text{d}\mathbf{t} = \text{-}\infty \tag{2}$$

where *c* is a positive values representing decreasing coefficient. The stress-induced response of biomarkers, increase by a short-term laboratory stressor and decrease by its removal, was simulated with task/break schedule as the same as in the session A and B in our experiment.
