Modeling the dynamics of disease states in depression

Abstract

Major depressive disorder (MDD) is a common and costly disorder associated with considerable morbidity, disability, and risk for suicide. The disorder is clinically and etiologically heterogeneous. Despite intense research efforts, the response rates of antidepressant treatments are relatively low and the etiology and progression of MDD remain poorly understood. Here we use computational modeling to advance our understanding of MDD. First, we propose a systematic and comprehensive definition of disease states, which is based on a type of mathematical model called a finite-state machine. Second, we propose a dynamical systems model for the progression, or dynamics, of MDD. The model is abstract and combines several major factors (mechanisms) that influence the dynamics of MDD. We study under what conditions the model can account for the occurrence and recurrence of depressive episodes and how we can model the effects of antidepressant treatments and cognitive behavioral therapy within the same dynamical systems model through changing a small subset of parameters. Our computational modeling suggests several predictions about MDD. Patients who suffer from depression can be divided into two sub-populations: a high-risk sub-population that has a high risk of developing chronic depression and a low-risk sub-population, in which patients develop depression stochastically with low probability. The success of antidepressant treatment is stochastic, leading to widely different times-to-remission in otherwise identical patients. While the specific details of our model might be subjected to criticism and revisions, our approach shows the potential power of computationally modeling depression and the need for different type of quantitative data for understanding depression.

Figure 1. Dynamical systems model for the dynamics of mood.

A) A schematic showing the mood change as a function of the state variable M without external inputs and noise (I = ε = 0). The arrows at 1, 2, 3, 4 indicate the direction of change in those states. The points labeled with b, c, and d are fix points. At these points, the value of the change is zero (dM/dt  = 0). Therefore, when there is no noise, the state will not change once it has reached a fix point. The fix points b and d are stable, meaning that the system will return to these states if slightly perturbed. The fix point c is unstable and has different properties, the system will move further away from point c even if the system is only slightly perturbed. In that case, the system will evolve until it reaches one of the stable fixed points. If , the system will move towards the fix point d. The system will evolve towards the other fix point b, if . Therefore, the fix point c separates the basins of attraction of the two stable fix points. Samples of the evolution of M over time B) without noise, C) with a moderate level of noise and D) with high level of noise. Note, that with high level of noise the system exhibits stochastic transition between positive and negative values.

Figure 2. Finite state machine modeling the transitions between the disease states in depression.

State diagram for the finite state machine. Ellipses represent the disease states in depression. Grey filled ellipses are clinically relevant disease states; unfilled ellipses are auxiliary disease states that are needed to discount short interruptions of clinically relevant disease states. The arrows indicate transitions between disease states. Transitions only occur when the state variable M changes sign, i.e., either from positive to negative, or vice versa. Each arrow is labeled by the criteria that trigger the transition. represents the length (in days) of the period during M<0 before transition to a positive value occurred. In other words, is the duration that a person meets the syndromal criterion for a depressive episode according to DSM-IV-TR . Accordingly, represents the length (in days) of the period during M>0, i.e., the duration in which a person does not meet the syndromal criterion for a depressive episode. The rectangles indicate a change to previously identified states. Short interruptions of disease states are added to the duration of disease states.

Figure 3. Example of the time course of the state variable M and the disease states identified by the finite state machine.

In this example, a symptomatic period lasting 28d is interrupted by an asymptomatic period of 5d and followed by another symptomatic period of 27d. Therefire, our model identifies the three periods together as a single depressive episode of length 60d. and represent the length (in days) of the period when M<0 and M>0, respectively.

Figure 4. Single population model can account for empirical occurrence rate but not for recurrence rates.

A) The occurrence rate (OR) from our simulation (grey bars) was fit to the result from epidemiological studies (black bars). The parameters of the model are: a = 4.65; b = −3; c = 0.175; d = 5; I = 0.02. However, in our simulation, the recurrence rates, RR(i), decrease with the number of prior depressive episodes, which is contrary to epidemiological data. B) The distribution of the number of depressive episodes (DE). The probability of zero DE is 0.8. The bars were cut off to show more clearly the smaller probabilities for the higher numbers of DE. The epidemiological distribution is clearly bimodal (black bars), whereas the simulated distribution is unimodal (grey bars).

Figure 5. Influence of parameters a and b on the occurrence and recurrence rate in the single population model.

A) Occurrence rate, B) first recurrence rate, and C) second recurrence rate, each represented by color scales, for a range of the parameters a and b. The remaining parameters are: c = 0.175; d = 5; I = 0.02. Note, that for all combinations of the parameters a and b, the rate of first recurrence is lower than the occurrence rate, and the rate of second recurrence is lower than the rate of first recurrence.

Figure 6. Two sub-population model can account for empirical occurrence and recurrence rate.

A) The parameters of the two sub-population model are: a = 5; b = −2.85; c = 0.175; d = 5; I = 0.02 for the low-risk sub-population and a = 4.4; b = −3.75; c = 0.175; d = 4.25; I = 0 for the high-risk sub-population. Our simulation data (grey bars) closely matches the empirical (black bars) occurrence and recurrence rates and B) the distribution of the number of depressive episodes.

Figure 7. Modification of parameters a and d cannot account for the effect of antidepressant treatment.

Shown in the color scales are the occurrence rate (A), the median time-to-remission (B) and the contours of the median time-to-remission (C) in simulated data. Consistent with the assumption that monoamine levels correlate with parameter d and the rate of adult neurogenesis with parameter a, the occurrence rate decreases with increasing parameters a and d (A). However, modeling the effect of antidepressant treatment as increases in parameters a and d would make the paradoxical prediction that antidepressant treatment increases the time-to-remission (B). C) To show this conflict more explicitly we plot both the occurrence rate and the time-to-remission in the same panel. The dashed lines represents contours in the occurrence rate at the indicated values, while the color scale represents median time-to-remission. It is highly unlikely to find parameter combinations of a and d which reduces the time-to-remission while keeping the occurrence rate constant or lowering it.

Figure 8. Increases in parameters a and b are consistent with the effect of antidepressant treatment.

The first row of panels shows the results of simulations for the low-risk sub-population where the color scales in A) and B) represent the occurrence rate and median time-to-remission, respectively. Panel C) displays the same data using contour lines (occurrence rate) and color scale (media time-to-remission). The second row of panels shows the results for the high-risk sub-population where the color scale represents D) the median number of depressive episodes and E) median time-to-remission. Panel F) displays the same data using contour lines (median number of depressive episodes) and color scheme (median time-to-remission). The black and white points mark pre- and post-treatment parameters, respectively. For certain parameter combinations an increase in the parameters a and b reduces the median time-to-remission while keeping the occurrence rate (the median number of depressive episodes for the high risk sub-population) constant or lowering it.

Figure 9. Distribution of the duration of depressive episodes. A), B), and C) show data for control group with pre-treatment parameters. D), E), and F) show data for treatment group with post-treatment parameters.

The first row (A, D) of panels shows the duration of depressive episodes for the low-risk subpopulation, the second row (B, E) for the high-risk subpopulation, and the third row (C, F) for the joint distribution. Note that the distributions have long tails, indicating that some patients take much longer to improve than others, even though they all share the same parameters.

Figure 10. Modeling the effect of cognitive behavioral therapy and life style changes on MDD.

Plotting convention as in Figure 8. An increase in the parameter I and/or decrease in c reduces the occurrence rate (A) (the median number of depressive episodes for the high-risk sub-population, D) and the median time-to-remission (B and E). These results suggests that smaller values of parameter c correlates with more positive attitude and larger values of I correlate with more positive environmental influences.