Given the complexity of biological systems, modelling the immune response to the spread of infectious disease is non-trivial and has been the subject of considerable computational efforts. HIV infection, if untreated, leads to a slow depletion of the immune system over many years, resulting in terminal AIDS disease and death. Despite decades of research, the biological mechanisms underpinning the immune system response and implicit in its impairment remain a subject of discussion. Recent biological findings indicate the importance of functional lymphatic tissue in maintaining homoeostasis of the immune system, which is vital for stability. HIV infection-related immune activation leads to chronic inflammation of this tissue and formation of non-functional scar tissue with detrimental effects on homeostasis. We propose a computational model based on the Cellular Automata formalism to quantify these effects in a simulated fraction of a lymphatic tissue. We also include effects of antiretroviral treatment and highlight implications for system representation through larger-scale simulations, as well as optimizations of treatment schedules.
1.1
Acquired Human Immunodeficiency Syndrome (AIDS) remains a global health concern even three decades or more after the discovery of the Human Immunodeficiency Virus as its root cause, [1]. The virus preferentially infects white blood cells of a certain type (CD4+), which are a crucial part of the immune response, leading to failure of the immune system approximately 10 years after initial infection. Potent drugs, termed Antiretroviral Therapy (ART), have been developed, which aim at the suppression of viral activities. However, these medications do not provide a cure for the disease, due to the persistence of impermeable viral reservoirs in the body; instead they have to be administered life-long to maintain sufficient viral suppression, [2]. Daily administration of multiple drugs can become a problem and interruptions are common, especially in resource-limited settings, where drug availability is not guaranteed or social factors interfere with adherence, [3].
Moreover, Structured Treatment Interruptions (STIs) have been suggested to enable recovery from drug toxicities, to promote resilience of the immune system and to reduce drug costs, [4]. Though initial clinical studies on STIs provided encouraging evidence regarding safety of interrupted treatment, [5, 6], a subsequent large-scale study (SMART) indicated potential harmful effects, [7]. Despite these effects have not been characterized in detail and may have been related to study design, [8], the results led to a general loss of interest in assessing treatment interruptions in a clinical setting. Questions regarding safe margins of interruption regimen and the nature of detrimental effects thus remain open. 1.2
It is well-known that viral replication occurs predominantly in lymphatic tissue, where most of the susceptible cells are located, [9]. The lymphatic system is a widespread network of nodes and ducts which is crucial for immune response but also for the spread of infected cells throughout the body.
Recent advances in in-vivo microscopy have enabled determination of the detailed structure of a certain kind of lymphatic tissue, the Fibroblastic Reticular Cell Network (FRCn). Its mesh-like structure has been reported to mimic a small world topology with lattice-like properties, [10]. This view is complemented by histological imaging showing that the Fibroblastic Reticular Cells (FRCs) are fixed to a scaffolding of collagen fibers. During inflammation, this formation of collagen fibers is greatly increased, [11], leading to collagen deposits (see Fig. 1) which block off FRCs from the chemical signals (cytokines) which maintain their viability, leading to depletion of these cells and formation of non-functional collagen tissue. Antiretroviral treatment has been shown to stop the formation of the scar tissue but not to enable its reversal, [12]. The formation of FRCn has not yet been observed in vivo due to the long timescales involved (weeks to years), however. Various models have been proposed in the HIV context to help understanding the spread of the virus between, [13], and within hosts, [14]. The majority of these models have found to be based on systems of Ordinary Differential Equations (ODEs), [15], representing a variant of the SIR model originally used to model spread of communicable diseases throughout populations. In context of within-host models, each DE denotes the change of a quantitative property related to HIV like viral load, healthy and infected cell concentrations over time. Starting from basic forms with two or three equations, models have been extended to cover additional features, like effects of antiretroviral treatment or immune response (resulting in systems of 20 or more equations). Simpler forms allow further mathematical analysis, [16], whereas more complex forms may be used for numerical simulation studies, [17]. For a more detailed review on ODE based HIV-modelling see [14] and references therein.
In general, ODE-based approaches assume virus und susceptible cells being in an environment satisfying the conditions of well-mixed. That assumption fits loosely best to the bloodstream, which initially has been considered a main source of viral replication, [14]. However, following the increasing availability of data on tissue involvement, the suitability of the well-mixed assumption has been questioned recently, [18].
Treatment interruptions and optimizations have also been subject to modelling studies using ODEs, [15]. A popular approach, besides pure simulation studies, [19], considers an optimal control problem, [20–22], to obtain treatment regimen which keep biological properties, like viral load or healthy cell count within predefined bounds which are considered to be non-critical. This view of the biological system as a kind of thermostat was especially useful with only limited data on anatomical compartments other than the bloodstream available. However, as clinical studies have indicated the complexity of HIV infection progression, consideration of alternative modelling approaches might be worthwhile.
We therefore propose a computational model, which considers both spatial structure of lymphatic tissue and the effects of inflammation, to examine long-term development and to allow predictions to be made on effects of treatment interruptions, where clinical data is not available for reasons mentioned in section 1. Our model presents an extension of the approach reported in [15] with improved biological realism obtained from drawing upon recent studies.
For our model we assume two effects to be responsible for the depletion of CD4+ cells, as reported by clinical studies, [11]. • HIV-induced direct killing of susceptible CD4+ cells • Impairment of survival through depletion of FRCs Both factors influence the concentration of CD4+ cells in the body, with the time-scale of the former relatively short (days to weeks) while the latter is effective at longer timescales (months to years). However due to regenerative capabilities of CD4+ cells, the first effect is reversible under antiretroviral treatment, while only minimal regeneration has been observed, [12], for the second.
We apply a bottom-up modelling approach, by using the Stochastic Cellular Automata (SCA) formalism, i.e. a stochastic extension of Cellular Automata (CA), [23]. The CA formalism has successfully been applied for modelling environments involving fixed matrices, including HIV, [24]. It utilizes a regular grid or lattice structure, where each site represents a single cell and its neighborhood the cell-to-cell contacts. We further assume that each cell has a connection to its 4 immediate neighbors as suggested from analysis of in-vivo imaging data, [10]. In terms of Cellular Automata computation, this corresponds to a von Neumann neighborhood.
An average lymph node measures about 1 cm in diameter and is of approximately spherical shape. From murine experiments, it is known that in FRCn, mean separation of the centers of mass of single cells is ~ 23 µm, [10]. A human lymph node, would therefore consist of around 40 Million FRCs. For our simulations, however, we consider a smaller fraction, namely a 2-dimensional slice through a lymph node, consisting approximately of 250,000 single cells. This slice represents less than 1% of the total volume, however this simplified view is acceptable in this case since we do not aim at reproducing an anatomically correct representation of the organ (see e.g. [25]).
From clinical data, we know that collagenation mainly starts from connections in the blood circulation (High Endothelial Venules). These connections appear to be scattered randomly throughout the tissue, [12]. We assume that collagenation increase occurs predominantly through elongation of existing collagen fibers as suggested by those data available to date on this aspect. 2.2
Before simulated infection occurs, the lattice is populated with both healthy FRCs and co-located CD4+ cells. The CD4+ cells are assumed to maintain equilibrium (homeostasis) and their concentrations in lymphatic tissue to be stable, [26]. However, HIV infection has a two-fold effect in our model: destroying CD4+ cells through viral infection and mechanisms of cellular apoptosis and pyroptosis, and through upregulation of the immune system, [26]. The upregulated immune response in turn causes increased collagenation of lymphatic tissue.
The fraction of CD4+ cells which is co-located with collagenated tissue, tends to exhibit poorer survival than usual (in healthy tissue), in addition to the depletion due to HIV infection. Treatment, however reduces the latter, as well as inflammation stimulation, since viral load is reduced. CD4+ cells are therefore regenerated and the collagenation processes in lymphatic tissue are slowed down. Upon reinstatement of upregulation of the immune system due to a treatment interruption, however, homeostasis disruption occurs again.
In terms of model rules, no direct death and/or inflammation takes place other than by viral infection. This context gives rise to the state transition model shown in Fig. 1, which contains the four possible combinations of CD4+ cells and collagen status as well as permitted changes. As mentioned in the previous section, collagenation originates from the High Endothelial Venules, where CD4+ cells enter the lymph nodes from the blood circulatory system. The assumption, that the blood circulation acts as a means to spread infection throughout the body, allows us to calibrate the model using available clinical data on CD4+ cell long-term degradation under HIV infection, obtained from blood samples, [27]. Table 1 lists the model parameters together with values obtained by calibration. Using the model parametrization in this table, we aim to test the effect of different patterns of unstructured interrupted treatment in the long term on collagenation (and thus immune system degradation). To simulate interrupted treatment, we assume ‘average’ adherence (the probability that the simulated patient takes the medication each week). As the treatment initiation timing has also been deemed to be important in clinical studies, [28], we allow for this in the simulations as well. Table 2. summarizes the parameter values chosen for the simulation runs. After the treatment period (corresponding to ~ 5 years), the increase in collagenation compared to that evident at the start of treatment is determined. In a second experiment, we aim to investigate the effects of structured treatment interruptions. Thus, we adopt the approach of CD4+ guidance, which has been pursued in recent clinical trials, [4].This approach uses a threshold value of CD4+ cell count in a patient as a surrogate marker to decide on interruption or (re-)initiation of antiretroviral treatment. For our simulations, we apply the limits used in two major clinical trials, [7, 29] and again observe the outcomes after 500 time steps.
An example model run is depicted in Fig. 3, exhibiting unstructured treatment interruptions due to incomplete adherence. In general, the model runs demonstrate good correspondence to clinical results, [6]. For an infection simulated to occur at time step 0, the count of CD4+ cells drops rapidly, (characteristic of the acute phase). The cell count continues to drop at a slower rate until initiation of simulated treatment, which leads to a significant recovery in number of CD4+ cells. Incomplete adherence to the treatment regime causes temporary drops in cell numbers (signified by the irregular saw tooth pattern), as well as an overall decline in the long term. This decline accelerates once the treatment period concludes. Collagenation, however, exhibits a constant increase, albeit with less-pronounced slope during treatment.
In Fig. 4 we show the loss of CD4+ cells and the increase in collagenation during treatment periods. The results illustrate the influence of treatment initiation timing and of adherence to the regimen. We observe that the latter has a fundamental impact on long-term progression of collagenation and, consequently, on immune system fitness. 200 300 time step [weeks]
Treatment initiation timing has a less-pronounced effect. Results show that, if treatment is introduced during the early phases of infection (within the first 2 years), initiation time has greater impact than for later stages. However, with high adherence (80% and above), collagenation can be kept within acceptable bounds even for late treatment initiation. As an acceptable outcome, we take a total collagenated area of less than 75% at the end of the simulated 10 years. This level has been determined as a threshold from clinical data, [12], above which AIDS disease is imminent. 3.2
Example results of our second experiment concerning structured treatment interruptions are shown in Fig. 5. Using the same thresholds for CD4+ guidance as two largescale clinical studies (SMART, LOTTI, see Table 3.), simulated data is in good correspondence with clinical data. The two sets of clinical data indicate that counts of CD4+ cells are constantly decreasing during the course of these studies, reflected by the simulation data. Since the duration of the clinical trials was limited due to practical reasons, we extended simulation times to allow for an estimation of long-term effects of the interruption thresholds used in the studies.
LOTTI
SMART 1200 l]1000 µ / s l l ce 800 [ t n uo 600 c + 4 D 400 C study duration [months] Fig. 5. Comparison of clinical data from two CD4+ guided STI studies (LOTTI, SMART; dots) with model output (lines) from corresponding simulation experiments.
On this behalf, absolute values of CD4+ count limits from clinical literature were normalized for use with the model by assuming an initial cell count of 1000 cells/µl.and results were again obtained after 500 time steps (~ 10 years). The results (see Table 3.) show a relatively small difference both in terms of collagenation and CD4+ counts. These results are remarkable in a way since the SMART study strongly indicated adverse effects of interrupted treatment, whereas the LOTTI study did not find significant differences in this regard compared to continuous treatment. The different outcomes of our simulations in terms of collagenation and CD4+ count may therefore account for some threshold between non-critical progression and adverse effects. We have proposed a simple Stochastic Cellular Automata model, which describes the effects of HIV infection and antiretroviral therapy on both CD4+ cell counts and collagen deposition in lymphatic tissue. Additionally, the model can predict possible outcomes of incomplete antiretroviral treatment, both unstructured and structured (difficult to obtain from controlled clinical studies). Simulated adherence patterns indicated the importance for long-term immunological fitness, of overall adherence to medication, and of treatment initiation timing during the early phases of infection, with regard to improved management. These experiments on antiretroviral treatment timing and adherence explore the implications for outcomes of Structured Treatment Interruptions, formerly discounted due to lack of long term predictions, despite some evident patient benefits.
Further steps might include an extension of our approach to evaluate strategies for an improved monitoring of antiretroviral therapy by helping to identify patterns, like certain concentration thresholds, which may indicate a future loss of viral control given poor adherence by employing AI techniques. Applicability in real-life settings may furtherly be improved by incorporation of realistic sampling intervals to obtain biological properties from blood or tissue with least patient stress.
Regarding structured interruptions, numerical simulation in conjunction with appropriate optimization techniques could be utilized for identifying optimal treatment regimen with improved biological realism compared to ODE-based approaches. A conceivable way would be the use of evolutionary algorithms to optimize treatment on/off patterns.
In addition, the proposed model offers further development potential, not least as it represents to date only a small part of human lymphatic tissue. A realistic simulation size even for one lymph node requires around 100 times more lattice sites than the current model. This problem can be alleviated using state-of-the-art parallelization approaches, [30], to simulate different regions and this forms part of ongoing investigation.
Unfortunately, clinical data for validation of model performance are sparse at present due to ethical considerations as well as measurement limitations, but recent advances in biological imaging and in-vivo microscopy may well provide additional insights and enable refinement of key parameters.
In summary, computational modelling of lymphatic tissue features involving cellto-cell connections offers considerable potential for further investigation and optimization of long-term HIV treatment. Promising foundations using bottom-up modelling have already been laid for investigating infectious diseases. We think it worthwhile to pursue further, given growing availability of appropriate experimental and clinical data. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.