Transitioning from acute to chronic pain: a simulation study of trajectories of low back pain

In a clinical study of 131 persons with chronic widespread pain, 267 ‘at risk’ and 56 controls [34] with abnormal HPA-axis function were associated with a high rate of fibromyalgia, a syndrome characterized by chronic widespread body pain. These results confirmed the hypothesis that the psychosocial-stress caused by pain, as well as the pain, are linked to altered HPA-axis functioning. Increased cortisol levels due to alterations in the regulation of the HPA axis at young age are also believed to be one mediating mechanism for several adult conditions, including metabolic syndrome, cardiovascular disease, and psychiatric disorders [34]. Infants born after significant exposure to stressful conditions are often small for gestational age (SGA), based on standardized growth norms. A study of 37 participants, including infants of gestational age ranging from 34 to 41 weeks, suggests SGA neonates have blunted HPA axis responses to stressors [37]. These findings are consistent with animal models showing that adverse intrauterine conditions can result in blunted cortisol responses to acute stressors, and may provide a mechanism for adult susceptibility to disease through HPA axis abnormality. The importance of the hormonal response system to stress through the HPA axis is also seen in other clinical studies, particularly stress hormones known as glucocorticoids (and primarily cortisol) that are key factors in patient alcohol dependence [48].

There is growing evidence that relative hypocortisolism, as a marker of stress-induced HPA axis dysfunction, may increase vulnerability to pain and chronic pain disorders [17‐19, 28, 29, 35, 58]. HPA activities, especially cortisol trajectory, are also a useful tool in longitudinal cohort studies of pain. In a study by Paananen et al. [38], for example, their sample group included 805 participants from the Western Australian Pregnancy Cohort (Raine) Study, who participated in the Trier Social Stress Test (TSST) at age 18-years. The number of pain sites, pain duration, pain intensity and pain frequency were assessed at age 22 in order to measure severity of musculoskeletal (MS) pain. An abnormal HPA axis response to psychosocial stress at 18-years of age was shown to be associated with MS pain alone, and MS pain combined with increased pain sensitivity at 22-years of age.

More recent evidence also suggests that in a disordered stress-system pathway, the HPA axis activity may be responsible for the genesis and maintenance of long-term sensory and emotional problems that lead to post-traumatic pain and disability. In a study by [57], the authors used hair-cortisol and hair-normalized salivary cortisol as biomarkers of distress following traumatic injuries of whiplash or distal radius fractures. Small sample results indicated that the cortisol-waking response may become a useful biomarker of current distress as measured using the Pain Catastrophizing Scale [57], especially when it is normalized to three-month hair cortisol. The hair-normalized cortisol waking-response also had predictive capacity, correlating with three-month self-reported disability. Tomas et al. [55] also provided a review of HPA axis dysfunction in chronic fatigue syndrome patients. The study included evidence of enhanced corticosteroid-induced negative feedback, basal hypocortisolism, attenuated diurnal variation, and a reduced responsivity to challenge. A putative causal role for a genetic profile, childhood trauma, and oxidative stress were considered. In addition, gender was also determined as a factor, in addition to an increased frequency of HPA axis dysregulation in females. The mechanisms of the HPA appears to affect LBP by following similar pathways as chronic fatigue syndrome. Abnormal cortisol concentrations reflect differences in the biological mediation of the stress response, or may be consequent on the differential nature/magnitude of the stressor engendered by the LBP. For example, in the chronic fatigue study, an attenuated diurnal variation has also been shown [39, 55], particularly with a loss of the morning-peak of ACTH or cortisol, while challenge studies often, but not invariably, show a diminished HPA axis responsivity. This has been assessed using ACTH, cortisol, and/or 11-deoxycortisol response to pharmacological challenge employing, for example, dexamethasone combined with corticotropin-releasing hormone (CRH), insulin, inflammatory cytokines, metyrapone, a psychosocial-challenge (e.g., using the Trier Social Stress Test), and to a physiological-challenge (such as a wakening; Tomas et al. [55]).

In the proposed model, we focus on the cortisol-level as a biomarker for pain trajectory for the continuation study over the time. There are a few current in vivo techniques that allow us to monitor the cortisol-level continuously for a long duration of time. For example, [56] used an electrochemical impedance sensor to measure cortisol concentration in the interstitial fluid of a human subject. The interstitial fluid is extracted by means of vacuum pressure from micropores created on the stratum corneum layer of the skin. Other measurements of the cortisol such as the saliva sampling protocol [46], and electrochemical immunosensing platforms that are well-known for their sensitive and selective detection of cortisol in biofluids [44], have also developed as tools for continuous in vivo measurements. They will guide predictive modeling development. The recent nanosheet technology [24] also allows monitoring cortisol in low-volume perspired human sweat (cortisol dynamic range from 1 to 500 ng/mL with a limit of detection of 1 ng/mL) using electrochemical impedance spectroscopy.

In this study, we will focus on the longitudinal study of LBP, in particular the transition from acute pain to chronic pain. The HPA activity history over time is of high interest. Our modeling strategy calls for a coupled system of ordinary differential equations to represent the network of brain regions along the HPA axis, and its cortisol and adrenaline production. Specifically relating to a trajectory modeling study in LBP, we focus on outcome measures such as LBP intensity, LBP frequency (number of LBP days per week), but will ignore activity-limitation which will be a subject of a future study. Trajectory patterns in a group of experimental subjects were identified using Hierarchical Cluster Analysis, Latent Class Analysis, or Latent Class Growth Analysis [27].

Following Sriram et al. [47], in developing the mathematical model of the cortisol molecular network for the HPA, we have made two assumptions: (i) The first order dilution-rate due to the transport of hormones and autonomous degradation are considered together. Apart from dilution/autonomous degradation, Michaelis–Menten kinetics are separately considered for the degradation of the hormones, and hormone complexes, within each specific region of the brain (hypothalamus, pituitary and adrenal); and (ii) A sufficient number of molecules is present for the reactions to take place, using continuum kinetics so that stochastic fluctuations (internal noise) are minimal.

The anatomical structure that mediates the stress response is mainly in the central nervous system (CNS) and its peripheral organs. The principle-effectors of the stress-response are localized in the paraventricular nucleus of the hypothalamus, the anterior lobe of the pituitary gland, and the adrenal gland. This collection of brain areas is commonly referred to as the Hypothalamic–pituitary–adrenal (HPA) axis [45]. The HPA axis plays an important role in balancing hormonal levels for the brain, and generate high concentrations of hormones in response to stress, which leads to “downstream changes” [43]. In response to stress over a period of time, the paraventricular nucleus of hypothalamus, which contains neuroendocrine neurons, releases corticotropin-releasing hormone (CRH). The anterior lobe of the pituitary gland is stimulated by CRH to secrete adrenal corticotrophic hormone (ACTH). The adrenal cortex then produces cortisol hormones in response to stimulation by ACTH. Cortisol is a major stress-related hormone, and has effects on many tissues in the body, particularly in the brain. In the brain, cortisol acts on two types of receptors: mineralocorticoid receptors and glucocorticoid receptors [4, 9, 41]. For “down regulating” CRH, it is known that cortisol inhibits the secretion of CRH through the glucocorticoid-receptors complex [11]. With a strong affinity to the glucocorticoid-receptors complex, cortisol, in turn, acts on the hypothalamus and pituitary in a negative feedback cycle to “down regulate” the production of cortisol [47]. Figure 1 illustrates the conceptual connection of the different brain areas in this process. Based on the biophysical relation, we established the underlying nonlinear ordinary differential equations for the cortisol network, as presented in Tables 1, 2, 3. Table 1 provides the mathematical model as in a system of ordinary differential equations. Table 2 provides the biological meaning of the parameters. A set of HPA parameters for LBP-modeling is provided in Table 3 for LBP modeling study. The model described in Table 1 is constructed by the chemical kinetics relations shown in Fig. 1. The four brain areas have excitatory or inhibitory inputs to other areas through mostly hormonal chemical kinetics relations and synaptic couplings [47]. The coefficients [47] are either from: the scientific literature; obtained by other researchers through direct measurement; or are best estimates to match experimental data through global optimization [60].

We simulated the time-series data of cortisol level during a period of 100 days, based on a computational HPA-axis model, which is described in Tables 1 and 3. MATLAB solver ode45 (https://​www.​mathworks.​com/​help/​matlab/​ref/​ode45.​html) was used for its accuracy and relatively fast speed in computing this model. It should be pointed out that, in our preliminary simulation study to demonstrate how mathematical modeling techniques can be used to develop a better understanding of these trajectory results, it was not our intention to explore all possible scenarios.

In this section, we present some examples based on the cortisol-dynamics model reported by [47], with parameters modified to fit into LBP problems. We then conducted a series of modeling pain-trajectories, based on cortisol that revealed patterns that resemble the pain trajectory patterns reported by Kongsted et al. [26]. Because cortisol secretion can be monitored in real-time, its time trajectory provides insight for studying LBP due to the inherent structure of the HPA in the brain hormone system under stress. The cortisol-level values are oscillatory as the time goes on, in response to the pain-process during LBP.

The time trajectory of cortisol-level is presented in Fig. 2, during the first 100 days for the HPA computational model. The cortisol values are given at each hour. Cortisol data can be simulated for any instance, but we only present the data at each hour, and each day contains 24-cortisol data points. Due to the inherent structure of the HPA brain hormone system under stress, the cortisol level values are oscillatory as the time goes forward, in response to the pain-process during LBP. The values are between 0.13440 and 0.13465 μ/DL for the entire times, and the up and down cycle is about 24 h. This simulated cortisol data can be compared with clinical data in the future for validation purpose, whereas cortisol secretion of patients can be monitored in real time and provide further insight for LBP and pain trajectory patterns.

In Fig. 3, a simulated pain trajectory, based on HPA-modeling analysis, is presented.

The cortisol-level values were classified into two different groups, and they present high-state and low-state patterns. The higher-cortisol level was used to model the high-intensity pain, and the lower-level to represent the low-pain or no-pain. We took a daily average of cortisol values for the study period. Then, for the averaged cortisol values, a subjective threshold of cortisol level was taken to separate the daily pain states into two groups (high-pain or low-pain) for the trajectory study. The simulated pain trajectory in two states (high-pain and low-pain) is presented in Fig. 3. We first took a daily average of cortisol level (24 hourly cortisol data points). A subjective threshold (0.1345 in this model; around the average of extreme cortisol values) of cortisol level was used to separate the first 100 days into two groups: high-pain day for these days when cort ≫ 0.1345, or low-pain day when cort < 0.1345, for pain trajectory study. The simulated pain trajectory depicted in Fig. 3 is representative for a LBP episode that has chronic and intermittent pain.

The HPA-system parameters that reflect the synaptic connectivity are crucial in setting pain-states. One can also analyze the dynamics-pattern changes induced by any parameter variation, in order to understand LBP transition from acute to chronic pain. The HPA-system parameters reflect the synaptic coupling strength or neuron degradation, and they are crucial in setting pain-states. One can systematically analyze the dynamic pattern-changes induced by any parameter variation, in order to understand LBP transition from acute to chronic pain.

In Fig. 4a, b, a case of cortisol dynamics of acute pain was simulated over a 100-day period, with acute pain that quickly subsides in 1–2 weeks, whereas a HPA with a reduced adreno-corticotrophin hormone degradation was considered (Vs4 was reduced from 15 µg d to 0.907 µg d, while others remain at normal level). Figure 4a depicts the cortisol level in the 100 days. The cortisol level quickly returns to a flat level in Fig. 4a, after a few initial oscillations. In Fig. 4b, we presented a 3-state pain trajectory calculated from simulated cortisol data in Fig. 4a. Similarly, we first took a daily average of cortisol level (24 hourly data points). Subjective thresholds (0.23986 and 0.2985 in this model) of cortisol level are used to separate the days into three groups: high-pain day (labeled in the high-pain group on the top) for the days when cort ≫ 0. 23986, or healed day when 0.23986 > cort ≫ 0. 23985 (labeled in the healed or no-pain group in the middle), and low-pain for cort < 0. 23985, for trajectory studies. The simulated pain trajectory is representative for an acute LBP episode, and the corresponding pain trajectory converges quickly to the healed or no-pain state, as shown in Fig. 4b.

Then, in Fig. 5, an HPA-system of “manic” pain was simulated, when the stress is elevated (Kstress is increased from 10.1 to 30 µg days), with other parameters identical as in Fig. 4; the cortisol trajectory transforms into a “manic” oscillatory pattern, as shown in Fig. 5a. The nearly periodic patterns have a period around 10, and alternate between high-amplitude oscillations and low-amplitude ones during the 100 day time period. In Fig. 5b, pain trajectory calculated from data in Fig. 5a shows a similar manic pattern. The cortisol level values obtained from HPA model are classified into three different groups, and they present high pain state, low pain state and no pain (healed) state patterns. We first took a daily average of cortisol level (24 hourly data points). Subjective thresholds (0.23990 and 0.29980 in this model) of cortisol level were used to separate the first 100 days into three groups: high-pain day (labeled in the high-pain group in the top) for the days when cort ≫ 0. 23990; or healed day when 0.23990 > cort ≫ 0. 23980 (labeled in the healed or no-pain group in the middle); and low-pain for cort < 0. 23980 (in the bottom) for trajectory studies. The simulated pain trajectory is representative for a manic-style pain LBP.

We now a present a computer-simulated study of pain trajectories in a group of patients. Unlike an individual patient whose hormonal levels are uniquely decided, we expect a distribution of cortisol levels among different patients. We assume the cortisol dynamics values are normalized (as in [57], so the discrepancy is due to random fluctuation or variation among individual patients. We chose a sample of 100 random values of a reduced adreno-corticotrophin hormone degradation rate (Vs4) and the stress coefficient (Kstress). We first evaluated the cortisol dynamics, and took the averaged values of 100 patients. Then, we developed the pain trajectory for this group of patients. We also calculated the variance of the cortisol-level and pain-trajectories. Figure 6a and b depict the average dynamics and their variances in cortisol level during the first 100 days, We kept all the same parameters in the HPA model as in Fig. 4, except Kstress, and we choose 100 randomized values for Kstress. The 100 values were randomly generated by the computer; the average Kstress is 0.9922, and the variance is 0.2907. Then, we simulated the HPA model for each Kstress. The cortisol trajectories vary among patients, from high-amplitude oscillations and low-amplitude ones during the 100-day time period. Each patient’s cortisol data is presented in a, with different color as shown in Fig. 6a. We also calculated the average daily cortisol-levels to be used to calculate the group-pain trajectory. The level variance is also marked in the group. Both daily values are presented in b. For the group, the average and variance become stabilized after 60 days. The average cortisol level values were then used to obtain a 3-state pain trajectory. We took a daily average of cortisol level (24 hourly data points). Subjective thresholds (0.556 and 0.552 in this model) of cortisol level were used to separate the first 100 days into three groups: high-pain day (labeled in the high-pain group in the top) for the days when cort ≫ 0.556; or healed day when 0.556 > cort ≫ 0.552 (labeled in the healed or no-pain group in the middle); and low-pain for cort < 0.552 (in the bottom) for trajectory studies. Shown in Fig. 6c, the simulated pain trajectory is representative for a group of LBP patients who recovered from acute pain in various severities.