The Model of Sugar Metabolism and Exercise Energy Expenditure Based on Fractional Linear Regression Equation

We use random sampling to select 16 students from a particular university (voluntarily participate in this experiment). Among them, 8 were young men of average weight, and 8 were obese young men [3]. The inclusion criteria are as follows: 1) The consequence has been stable in the past three months. According to the Chinese adult BMI standard: average body weight BMI 18.5–23.9, obesity BMI ≥ 28. 2) No history of motion sickness and no organic disease. Through incremental exercise load test and physical examination, and eliminate risk factors such as cardiovascular dysfunction. 3) I have not taken hormone drugs recently. 4) No bad habits such as smoking and drinking.

All subjects performed 4 experiments. First, we obtain the respective maximum oxygen uptake based on the incremental load results. Second, we determine the treadmill speed of 50% VO₂max, 75% VO₂max, and 100% VO₂max exercise intensity. Third, three exercise tests were carried out after 5 minutes of preparation for the experiment: (1) Volunteers continued to run for 30 minutes and recover for 30 minutes at an exercise intensity of 50% VO₂max. (2) Volunteers run at 75% VO₂max exercise intensity for 4 minutes and intermittently recover for 2 minutes as a group of intermittent exercises, perform 5 consecutive groups, and recover for 30 minutes after 5 groups exercises. (3) Volunteers run for 3 minutes at an exercise intensity of 100% VO₂max and intermittently recover for 3 minutes as a group of intermittent exercises. Finally, the volunteers recovered for 30 minutes after performing 5 consecutive exercises [4]. Throughout the experiment, the MAX-II test system was used to measure the body's gas metabolism indicators.

This experiment uses complete rest during the intermittent period. We performed the statistics in the above table to prove that the body function did not return to a normal state after taking complete rest. Table 2 shows that after 75% VO₂max and 100% VO₂max are intermittent for 2 min and 3 min, the final heart rate (HR) is still 135 beats/min. This shows that the subjects have not fully recovered before the next exercise and ensure the primary conditions for intermittent exercise.

We collect data through a gas analyzer and create a data table. We use Excle2010 and SPSS20.0 statistical software for data processing. The paper uses ±s to describe the statistical indicators, uses two-way analysis of variance and independent sample T-test to analyze the data and draw conclusions [6]. The difference was statistically significant when P<0.05.

During the development of the fractional calculus theory, many definitions of fractional calculus of middle function appeared, such as the definition of Grünwald-Letnikov fractional calculus, the definition of Riemann-Liouville fractional calculus, and the definition of Caputo. This section will introduce the fractional Caputo to define the differential operator and introduce the equivalent transformation relations and related functions used in this article: (1)

Definition of Caputo fractional differential (1) Dtaf(t)=1Γ(1−a)∫0tfm+1(τ)(t−τ)adτ \begin{equation} D_{t}^{a} f(t)=\frac{1}{\Gamma(1-a)} \int_{0}^{t} \frac{f^{m+1}(\tau)}{(t-\tau)^{a}} d \tau \end{equation} Where: is an integerα = m + γ, 0 < γ < 1.

Time fractional lossy transmission line equation analysis. Take the Laplace to transform defined by the Caputo fractional order of time t on both sides of equation (5), then: (5) uxx(x,s)=RC[Sau(x,s)−∑k=00u(k)(x.0)Sa−1−k]=RC[Sau(x,s)−u(x,0)Sa−1] \begin{equation} u_{x x}(x, s)=R C\left[S^{a} u(x, s)-\sum_{k=0}^{0} u^{(k)}(x .0)_{S^{a-1-k}}\right]=R C\left[S^{a} u(x, s)-u(x, 0) S^{a-1}\right] \end{equation}

Organize to get: (6) u(x,s)=1RCSauxxuxx(x,s)+u(x,0)s \begin{equation} u(x, s)=\frac{1}{R C S^{a}} u_{x x} u_{x x}(x, s)+\frac{u(x, 0)}{s} \end{equation}

Assuming u(x,s)=∑0∞un(x,s) u(x,s) = \sum\nolimits_{0}^{\infty}{u_{n}(x,s)} then: (7) ∑0∞un(x,s)=u(x,0)s+1RCSauxx(x,s) \begin{equation} \sum_{0}^{\infty} u_{n}(x, s)=\frac{u(x, 0)}{s}+\frac{1}{R C S^{a}} u_{x x}(x, s) \end{equation}

The paper uses the Adomian decomposition method to decompose u(x, s), and the initial condition is considered to be u(x, s) = V₀cos, x, and the recurrence formula is obtained: (8) {u0(x,s)=u(x,0)s=V0cosxsun+1(x,t)=1RCSauxx(x,s) \begin{equation} \left\{\begin{array}{l} u_{0}(x, s)=\frac{u(x, 0)}{s}=\frac{V_{0} \cos x}{s} \\ u_{n+1}(x, t)=\frac{1}{R C S^{a}} u_{x x}(x, s) \end{array}\right. \end{equation}

We can use MATLAB software to get: (9) u(x,s)=∑0∞un(x,s)=V0cosxs(1−1RCSa+1(RCSa)2+⋯+(−1)n1(RCSa)n+⋯)=sa−1sa+1/RCV0cosx \begin{equation} \begin{array}{r} u(x, s)=\sum_{0}^{\infty} u_{n}(x, s)=\frac{V_{0} \cos x}{s}\left(1-\frac{1}{R C S^{a}}+\frac{1}{\left(R C S^{a}\right)^{2}}+\cdots+(-1)^{n} \frac{1}{\left(R C S^{a}\right)^{n}}+\cdots\right)= \\ \frac{s^{a-1}}{s^{a}+1 / R C} V_{0} \cos x \end{array} \end{equation}

Finally, the inverse Laplace transformation of the Mittag-Leffler function can be used to obtain the voltage response: (10) u(x,s)=L−1(u(x,s)=L−1(sa−1sa+1/RCV0cosx)=V0cosxEa,1(−taRC)=V0cosx∑k=00(−ta/RC)kΓ(ak+1) \begin{equation} \begin{array}{r} u(x, s)=L^{-1}\left(u(x, s)=L^{-1}\left(\frac{s^{a-1}}{s^{a}+1 / R C} V_{0} \cos x\right)=V_{0} \cos x E_{a, 1}\left(-\frac{t^{a}}{R C}\right)=\right. \\ V_{0} \cos x \sum_{k=0}^{0} \frac{\left(-t^{a} / R C\right)^{k}}{\Gamma(a k+1)} \end{array} \end{equation}

Table 3 shows that the sugar oxidation amount, sugar oxidation energy supply ratio, fat oxidation amount, and oxidation energy supply ratio per kilogram bodyweight of the obese group are lower than average body weight under the same exercise intensity.

Exercise intensity and body weight are two factors that affect substrate metabolism and energy consumption [7]. Table 4 shows that the main effects of exercise intensity on substrate metabolism and energy consumption reached a significant level during exercise through the two-factor analysis of variance. Still, the main results of bodyweight-only on sugar and total energy consumption reached a considerable level. Therefore, there is only interaction between strength and body weight in the amount of sugar oxidation.

Table 5 shows that under the same exercise intensity, the obesity group only had higher sugar oxidation per kilogram body weight and oxidation energy supply ratio than average body weight. And the total energy consumption, fat oxidation amount, and oxidation energy supply ratio per kilogram weight were lower than the medium bodyweight group.

The results of the two-way analysis of variance are shown in Table 6. In the recovery period after exercise, the main effects of exercise intensity on substrate metabolism and energy consumption reached a significant level (P<0.05). In addition, the main products of body weight on fat oxidation, total energy consumption. And sugar oxidation energy supply ratio reached significant levels (P<0.05), but there was no significant effect on sugar oxidation and fat oxidation energy supply ratio [8]. Thus, there is no interaction between exercise intensity and body weight on substrate metabolism and energy consumption during the recovery period.

Continuous exercise and intermittent exercise on substrate metabolism and energy consumption are all related to exercise intensity. With the increase of exercise intensity, the amount of sugar oxidation and the ratio of oxidative energy supply, and the total energy consumption gradually increase, the amount of fat oxidation and the percentage of oxidation energy supply decrease progressively [9]. These changes are related to output power. Thus, with the increase of exercise intensity, the body puts forward higher requirements for the energy supply rate.

In the primary effect analysis, it was found that body weight has a significant effect on sugar and total energy consumption but has no significant impact on the ratio of fat and substrate metabolism. The reason may be that different body fat levels, and body weight has a particular effect on exercise capacity. Among them, it has the most significant impact on lean body weight. The slender body mass is highly correlated with the aerobic and anaerobic capacity of the human body, and the lean body mass varies greatly between body weights [10]. Therefore, there is a significant difference between body weights when sugars are needed for rapid oxidation for energy. In addition, it may be because the absolute exercise intensity of normal-weight people is greater than that of obese people.

Exercise weight loss mainly considers total energy expenditure. Therefore, for fat loss, the total energy consumption during exercise and recovery period is more important than the effect of energy consumption during exercise alone. This is also why some scholars have strongly advocated high-intensity interval training (HIIT) in recent years [11]. This study found that exercise intensity significantly impacts substrate metabolism and energy consumption during the recovery period after exercise. With the increase of exercise intensity, the amount of sugar oxidation and the ratio of energy supply for sugar metabolism gradually decrease after training, the amount of oxidation of fat, the proportion of energy supply for fat metabolism, and the total energy consumption increase progressively. This is mainly because endocrine hormones, body temperature, pulmonary ventilation, etc., remain at high levels after intensive exercise. Therefore, oxygen reserves need to be restored as soon as possible. The ATP-CP system needs to be quickly replenished, and lactic acid needs to be cleared as soon as possible. Currently, the primary energy supply mode has been changed from anaerobic energy supply to aerobic energy supply, and glycogen in the body has almost been exhausted during exercise. Therefore, the primary substance for aerobic energy is changed from sugar to fat. As a result, the free fatty acid level in the blood increased significantly after the training. Studies have proved that the energy supply ratio of fat after exercise is closely related to exercise intensity and exercise time. Exercise intensity affects the amount of excess oxygen consumption after exercise, and exercise time will prolong excessive oxygen consumption after exercise. In theory, high-intensity training to reduce fat is mainly in the excess oxygen consumption after exercise. The greater the exercise intensity, the higher the excess oxygen consumption after exercise.

There was no significant difference in comparing substrate metabolism and energy consumption during the recovery period between 50% VO₂max and 75% VO₂max. It shows that when the intensity exceeds 50% VO₂max, excessive oxygen consumption and substrate metabolism changes are not apparent after exercise. Or at this time, the excessive oxygen consumption after exercise temporarily enters the plateau period. Exercise intensity as an independent factor can affect substrate metabolism and energy consumption after practice remains to be studied [12]. There is a significant difference in the ratio of substrate metabolism energy supply between interval and continuous exercise. The reasons may be as follows: during high-intensity intermittent exercise, the pituitary-adrenal axis reacts more intensely, which leads to increased hormone secretion and improves fat oxidation energy supply ratio. It is also possible that intermittent exercise consists of the exercise period and the recovery period. The two periods continue to intersect, increasing the number of substrate mobilizations, producing numerous metabolites, increasing the amount of fat and the ratio of the oxidative energy supply after exercise.

Exercise intensity as the main effect significantly impacts the amount of sugar oxidation and total energy consumption. During the whole exercise process, with the increase of exercise intensity, the oxidation amount of sugar, the ratio of oxidation energy supply, and the total energy consumption gradually increase. In contrast, the oxidation amount of fat and the percentage of oxidation energy supply decrease progressives; there are significant differences in substrate metabolism and energy consumption characteristics during and after exercise. Still, the trends of the two are exactly opposite. This is because the body's regulation realizes the sugar oxidation energy supply ratio and the fat oxidation energy supply ratio.

The whole process of comprehensive exercise and recovery period found that moderate-intensity continuous training is better than an intermittent exercise in terms of the fat burning effect. The amount of fat oxidation and the ratio of oxidative energy supply in the constant movement are higher than that in intermittent exercise. Although the total energy consumption of intermittent exercise is higher than that of continuous practice, the energy is mainly provided by the burning of sugar. Keeping the hormone content at a high level after exercise also increases excessive oxygen consumption after exercise. Due to the short rest time during the recovery period after training in this study, the gas of excessive oxygen consumption after training was not fully collected, which resulted in the relative value of substrate metabolism, and the energy consumption is significantly lower than in other studies.