Mathematical modeling of disease dynamics in SDHB- and SDHD-related paraganglioma: Further step in understanding hereditary tumor differences and future therapeutic strategies

Succinate dehydrogenase subunit B and D (SDHB and SDHD) mutations represent the most frequent cause of hereditary pheochromocytoma and paraganglioma (PPGL). Although truncation of the succinate dehydrogenase complex is thought to be the disease causing mechanism in both disorders, SDHB and SDHD patients exihibit different phenotypes. These phenotypic differences are currently unexplained by molecular genetics. The aim of this study is to compare disease dynamics in these two conditions via a Markov chain model based on 4 clinically-defined steady states. Our model corroborates at the population level phenotypic observations in SDHB and SDHD carriers and suggests potential explanations associated with the probabilities of disease maintenance and regression. In SDHB-related syndrome, PPGL maintenance seems to be reduced compared to SDHD (p = 0.04 vs 0.95) due to higher probability of tumor cell regression in SDHB vs SDHD (p = 0.87 vs 0.00). However, when SDHB-tumors give rise to metastases, metastatic cells are able to thrive with decreased probability of regression compared with SDHD counterparts (p = 0.17 vs 0.89). By constrast, almost all SDHD patients develop PGL (mainly head and neck) that persist throughout their lifetime. However, compared to SDHB, maintenance of metastatic lesions seems to be less effective for SDHD (p = 0.83 vs 0.11). These findings align with data suggesting that SDHD-related PPGL require less genetic events for tumor initiation and maintenance compared to those related to SDHB, but fail to initiate biology that promotes metastatic spread and metastatic cell survival in host tissues. By contrast, the higher number of genetic abnormalities required for tumor initiation and maintenance in SDHB PPGL result in a lower penetrance of PGL, but when cells give rise to metastases they are assumed to be better adapted to sustain survival. These proposed differences in disease progression dynamics between SDHB and SDHD diseases provide new cues for future exploration of SDHx PPGL behavior, offering considerations for future specific therapeutic and prevention strategies.


Introduction
Pheochromocytomas and paragangliomas (PPGLs) are rare neuroendocrine tumors that arise either in head and neck parasympathetic paraganglia or paraaortic chromaffin tissue, which comprise sympathetic adrenal medulla and extra-adrenal paraganglia. Tumors that derive from either parasympathetic or sympathetic paraganglia are collectively named paragangliomas (PGLs) with the term pheochromocytoma (P) being restricted to adrenal PGL [1].
In up to 70% of cases, PPGL are associated with germline and somatic mutations in 15 wellcharacterized PPGL driver or fusion genes. The contribution to tumor initiation or progression of these disease driving genes is still not fully understood [2,3]. This is well illustrated by the example of hereditary PGL syndromes. In 2000, Baysal et al. described the first PGL syndrome related to deficiency in succinate dehydrogenase (SDH) enzyme activity due to mutations in SDH subunit D (SDHD), part of mitochondrial complex II and the tricarboxylic acid (TCA) cycle [4]. This major discovery represents the first unequivocal genetic link between a mitochondrial defect and PPGL development. Association between the TCA cycle and PPGL was later confirmed by the identification of mutations in other genes encoding subunits B [5], C [6], and A [7] of the SDH complex or its flavination factor (SDHAF2) [8] and more recently, mutations in fumarate hydratase [9] and malate dehydrogenase type 2 [10].
These genes are related to the TCA cycle and they are considered tumor supressors with biallelic inactivation of the healthy allele through a somatic event in paraganglial cells. This results in the accumulation of succinate which has pro-oncogenic effects via intracellular and extracellular (« hormone » like) actions and tumorigenesis [11,12]. Activation of the hypoxiainducible factor (HIF) signaling pathway despite normal oxygen supply (also called pseudohypoxia), as well as DNA and histone demethylases inhibition resulting in a hypermethylated genome, are two processes that were uniquely identified to contribute to transformation of a paraganglial cell into PPGL [12,13].
Currently, SDHB and SDHD mutations represent the most frequent cause of hereditary PPGLs associated with TCA defects. In SDHD-patients, PGLs in the head and neck region and anterior/medium mediastinum can be found in 85% of cases wheras less than 5% have abdominal PGL. HNPGL and abdominal PGL coexist in 10% of cases. SDHB-linked PGL syndrome is characterized by a high rate of abdominal PGL (70-80%). HNPGLs occur in 20-30% of cases. The coexistence of HNPGL and abdominal PGL is rare (<3%). It is also notable that multifocality mainly occurs in SDHD cases (at least 60% vs 20% for SDHB) [14]. PGLs with underlying SDHB mutations are associated with a higher risk of aggressive behavior, development of metastatic disease, and ultimately, death [15]. Overall, the risk of metastatic disease in SDHB mutation-associated tumors has been estimated to be 30% vs <5% for SDHD. The transmission of disease is also different. Although SDHD and SDHB are both autosomal dominant diseases, the penetrance of SDHD-related PPGL is modulated by maternal imprinting. Overall disease penetrance of SDHB and SDHD diseases is dependent on the use of high sensitive imaging investigations in the work-up of non probands but also the molecular severity of the variants. SDHD-related mutations (paternally inherited) have very high penetrance (90-100%), in contrast to SDHB ones that have an estimated penetrance of only 20-40% [16][17][18][19]. A lower SDHD disease penetrance may be observed in studies that included low severity mutations [20].
As there is no biological experimental system that successfully replicates the human phenotype, we sought to understand the phenotypic heterogenity of TCA cycle-related PGL syndromes by computational modeling. Our results provide novel insights on potential causes of differential dymanics of PPGL tumorigenesis in SDH carriers.

Results and discussion
We have chosen to use a Markov chain model because it is a well-accepted probabilistic approach for modeling a change between a fixed number of disease states over time. For more than twenty years, Markov models have been used in various areas of medical research, such as cost-effectiveness studies [21], epidemiologic analysis [22,23] or genome research [24]. Our Markov model simulates transitions between various clinically defined states.

In SDHB-related PPGL syndrome
For modeling disease dynamics in SDHB PPGL, the following parameters, derived from clinical studies were used: P b obs = [0.70, 0.06, 0.14, 0.10] (where the superscript "b" stands for SDHB); i.e: 70% will remain without disease corresponding to 30% penetrance, and among patients with active disease: 6% HNPGL and 14% sympathetic PPGL, 10% metastatic disease). For the particular value p 11 = 0.75, a transition matrix T sdhb was estimated and is shown in Fig  1 (in this matrix, for i = 1..4, j = 1..4, the number in row i and column j is transition probability p ij ). This choice is p 11 is discussed in the Sensitivity Analysis section.
The model allowed us to simulate PPGL initiation and spontaneous regression of tumors and demonstrate how these two processes can contribute to the disease phenotype. In this case, the asymptotic probability distribution P Ã sdhb (at steady-state, which gives the probabilities to be in one of the four states in the course of life) was: P Ã sdhb = [0.7, 0.06, 0.14, 0.10], describing exactly the clinical picture of the disease P b obs. Our principal findings show that for SDHB sub-group, the stabilization into a clinical state with the development of PPGL is a process driven by high rate of regression to a state without a (clinically detectable) tumor (low p 21 and p 31 ). However, when PPGL develops, it easily gives rise to metastasis and metastatic PPGL sustains survival (high p 44 ).

In SDHD-related PGL syndrome
A first model was parameterized to simulate SDHD disease dynamics with the following set of P d obsv = [0.00, 0.75, 0.20, 0.05], where the superscript "d" stands for SDHD (i.e, 100% penetrance if the mutation is inherited from the father, 75% HNPGL and 20% sympathetic PPGL, 5% metastatic disease). The T sdhd matrix is shown in Fig 2 with the following probabilities at steady state: P Ã sdhd = [0.00, 0.75, 0.20, 0.05]. These results accurately describe the observed probabilities.
The observed SDHD PPGL steady-state values were characterized by high values of p 12 , p 11 , p 22 and p 43 and low values of p 21 and p 31 . Therefore, it seems that SDHD PPGL (mainly those from parasympathetic paraganglia) develop more easily than SDHB ones and persist throughout the life (high p 12 ) of a patient. Furthermore, although SDHD primary PPGL can transition into metastatic disease, it mostly fails to develop a stable advanced disease state and are therefore not diagnosed as metastatic (low p 44 ).

Differences between SDHB and SDHD
Our model corroborates clinical observations by showing that SDHB and SDHD carriers have different dynamics in PPGL occurrence, regression, and progression thus, providing new insights into the understanding of these tumors. Tables 1 and 2 summarizes the main differences between SDHB and SDHD regarding the probabilities for tumor maintenance or regression. First, tumor maintenance is reduced in SDHB patients (p 22 and p 33 = 0.04) compared to SDHD patients (p 22 and p 33 = 0.95). Second, the probability of staying in metastatic state is, however, higher for SDHB than SDHD patients (p 44 = 0.83 vs 0.11).
The sensitivity analysis shows robustness of the model with respect to reasonable variations in the 4 percentages of disease, which supports the validity of the modeling method. One parameter (p 11 ) had to be fixed prior to the estimation procedure, which was unavoidable to ensure that the number of free-parameters to estimate was equal to 4, the number of observations, as described in the Sensitivity Analysis section.
Our discrete-time Markov chains model simulates transition into clinically-defined steady states and describes qualitative differences in disease dynamics between the 2 syndromes. However, unlike continuous-time Markov chains, it does not provide informations of disease dynamics over time.
Our model does not define the mechanisms underlying transition probabilities that could in theory reflect any biological mechanism. Nevertheless, through an integration of current  Table 1 and Table 2. For example, p 12 is the probability of moving from state 1 to state 2 and p 11 is the probability to stay in this state 1.
Here, genetics stipulates that biallelic inactivation of SDHB or SDHD loci is not enough to cause PPGL. It is only through deregulation of additional genes, cellular, epigenetic, microenvironmental, and other events that PPGL can be formed. Due to location in the genome, SDHD could require fewer genetic hits than SDHB to form PPGL. SDHD as well as SDHAF2related PPGL are characterized by a specific loss of maternal chromosome 11 [27,28], a finding which is consistent with paternal transmission of the diseases (both genes being located in  Table 1 and Table 2 chromosome 11). This is also observed in VHL-related pheochromocytoma [29], suggesting the potential role of several maternally expressed genes in tumorigenesis. By contrast, SDHB PPGL are characterized by a lower frequency of chromosome 11 loss (31% of cases) with a more complex pattern and a greater genomic instability compared to SDH with gains and losses confined to other chromosomes. It is, therefore, possible that chromosome 11p loss is necessary and sufficient to trigger SDHD and SDHAF2 tumorigenesis, whereas SDHB tumors require more complex changes with amplification or deletion of multiple driver genes located on different chromosomes, especially 1p. Assuming a constant rate of genetic instability, this could be in agreement with differences in penetrance of the SDHD and SDHB disease. A second assumption is that additional genetic events contribute to the transformation of PPGL cells into a biology that favours metastatic spread [2,29]. For SDHB-related PPGL, these would have accumulated more genetic events at the time of primary tumor development, therefore having a higher probability to aquire such metastasis promoting genetic events. This speculation potentially explains the very low probabilities of SDHB PPGL compared to SDHD to return from state 4 (metastatic disease) to state 2 (p 42 ) and 3 (p 43 ) presenting with no metastasis. This is reflected by the probability of staying in state 4 as being very high for SDHB and very low for SDHD (p 44 ). Interestingly, tumor maintenance is reduced in SDHB patients (p 22 and p 33 = 0.04) due to a high probability for tumor regression (p 21 and p 31 = 0.87). This dynamic is inverted in SDHD-related PPGL, which are characterized by a high probability for tumor maintenance (p 22 and p 33 = 0.95). These opposite pathways (tumor formation/regression) suggest that genetic abnormalities present in SDHB tumors could involve genes that play a role in execution of gene programming and signaling that control G1-S and G2-M cell cycle checkpoints and death receptor/apoptosis events. A precise threshold of these proteins could be required for maintaining a specific tumor state, where there is a switch from proliferation to a state of proliferative arrest and apoptosis. This property has been illustrated for MYC [30]. We acknowledge that this theory is still only supported by indirect evidence from genetic data that also fails to explain the absence of somatic biallelic SDH inactivation. The metastatic capability of SDHB-deficient tumor cells could also be related to intrinsic capacity of sympathetic nervous system cells to develop metastasis. By contrast in SDHD mutated patients, embryological development could result in the abnormal foundation of a parasympathetic nervous system that lacks an intrinsic capacity to form metastasis. This model remains theoretically plausible, but is not currently supported by experimental evidence.
In light of what is currently known of PPGL tumorigenesis, we suggest that our results could be interpreted as follows: in SDHD, a limited number of genetic abnormalities seem effective for tumor initiation and maintenance (high p 12 ), but fail to initiate a biology that promotes metastatic spread and cell survival in host tissues (low p 44 ). By contrast, the higher number of genetic abnormalities required for tumor initiation and maintenance in SDHB-related PPGL result in a lower penetrance of PGL (low p 21 and p 31 ), but when cells give rise to a tumor followed by metastases, they seem to be more adapted to sustain survival (high p 44 ).
These findings suggest that therapeutic strategies against SDHB should be prioritized for killing cells at early stages of metastatic spread, either with no detected tumors by imaging (e.g. adjuvant systemic therapies) or detectable tumors (e.g. radio-or immunotherapies). By constrast, for SDHD, a major goal would be to prevent mechanisms involved in tumor development and maintenance (prevention of second somatic hit via antioxydants or drugs that reduce endogenous mutations, limiting of exposure to ionizing radiations) since metastatic cells are more instable. The model could also be used to test the disease dynamics following a particular treatment. These findings would ultimately have to be supported by various clinical trials and interventions in patients or experimental models.

Model design
TCA-related hereditary PPGL occurrence, progression, or regression are assumed to be a population-level random processes in which it is currently only known that metastasis occurs more frequently from large and sympathetic primary PPGLs. Such assumptions and knowledge are well suited to be simulated with a Markov chain model that links together a series of stochastically generated events that, over time, result in a clinically significant steady state.
Only 4 PPGL-related disease states were considered: The transition probabilities are denoted by p ij and p ii , where p ij is the probability of moving from state i to state j and p ii is the probability to stay in this state i.
Usually, for a Markov chain with 4 states, the transition probabilities are represented in a 4-4 matrix T (i.e with 4 rows and 4 columns, Fig 3).
For i,j = 1,2,3,4, the coefficients p ij of the 4-4 transition matrix T can be defined as follows for any step n: 1. p 11 is the probability of an SDHB or SDHD mutation carrier not having any tumor (state 1) to stay in this state 2. p 12 is the probability of a patient moving from state 1 to state 2 and developing HNPGL, which could be microscopic in size 3. p 21 is the probability of a patient with HNPGL to return to state 1 from state 2 4. p 13 is the probability of moving from state 1 to state 3 and developing sympathetic PPGL, which could be microscopic in size 5. p 31 is the probability of a patient with sympathetic PPGL to return to state 1 from state 3 6. p 24 , p 34 are the probabilities of a PPGL (state 2 or 3, respectively in HNPGL or sympathetic PPGL) to develop metastasis 7. p 42 , p 43 are the probabilities of returning to state 2 and 3 due to death of metastatic cell(s) from the state 4 8. p 23 , p 32 are the probabilities of moving from HNPGL to sympathetic PPGL and vice versa These four states reflect potential clinical scenarios in a given population of patients. Transition between these states will occur until stabilization of the population into each of the four states. Thus, the steady ("final") state represents the health status of a group of patients observed/diagnosed by a physician. However, the chain of intermediary states is not  Table 1. According to international nomenclature of stochastic matrix, the row vector of the Markov chains has been written in order to meet the following criteria: non negative coefficients and the sum of each row equal to 1.
https://doi.org/10.1371/journal.pone.0201303.g003 observable, because it is a chain of hidden events, driven by randomness that eventually results in the observable health condition of a group. The probability for a group of patients to be stable in any of these four states is based on clinical evaluations and here, by studies related to PPGLs. Changes within one state are called transitions and the model was designed to allow for a change of state in both directions i.e., both tumor development or its regression. Underlying this design is the assumption that PPGL could regress or metastasize through an intermediate step (PPGL).
Furthemore, the existence of transitions back from the metastatic state to one of the two non-metastatic states (HNPGL or sympathetic PPGL) should not be misunderstood as a spontaneous regression from a diagnosed state of metastasis to a benign tumor state. Instead, the model assumes a scenario where not all metastases survive and only a small proportion of them establish into actual metastatic disease. Hence, it is assumed that the chain of intermediary states will always contain a (random) number of transient metastatic steps before the population of patients will stabilize into the four states at time of clinical intervention.

Model estimation
Transition probabilities, which are "invisible" for clinicians, were estimated in order to fit with the limits P Ã corresponding to the phenotype observed in SDHB and SDHD PPGL syndromes: P b obvs or P d obvs . Hence, the transition probabilities p ij selected will be those that produce the theorical value P Ã [i], to be in state i throughout life, which is closest to observed value P obvs [i]. In others words, mathematically, these transition probabilities were estimated by using the method of the mean least squares, which select the p ij, which minimizes the quantity: (P Ã [1]-P obvs [1] The following assuptions were made in calculating transition probabilities: 1. the transition probability p 11 was fixed; this choice being discussed below; 2. to match the number of free parameters with the number of steady-states, it was assumed that transition probabilities do not depend on tumor location (i.e. head and neck PGL vs. sympathetic PGL): p 21 = p 31 and p 22 = p 33 ; 3. since the coexistence of head and neck PGL and sympathetic PPGL is very rare condition, the transition probabilities p 23 and p 32 were fixed at zero; These assumptions have allowed reduction of free-parameters to 4: (p 21 , p 22 , p 24 , p 44 ), the other parameters being either fixed (p 11 = 0.75 for SDHB and p 11 = 0.10 for SDHD), or set at zero (p 23 = 0, p 32 = 0), or constrained by the two assumed relationships (p 21 = p 31 ), (p 22 = p 33 ), and, obviously, by the fact that the transition probabilities from one state must sum to one. Then the 4 free-parameters were estimated to fit the 4 clinically-defined steady states.
The estimation procedure was done with lsqnonlin of the MATLAB software, which implements a constrained non-linear least-square minimization routine.

Sensitivity analysis
Since PPGL penetrance and the proportion of patients with either HNPGL, sympathetic PGL, or metastatic disease are uncertain numbers, it is important to assess the stability of transition probabilities if one moves observed probabilities within a range of realistic values. A model that would produce large swings of transition probabilities for small changes in observed probabilities of disease should be rejected for lack of stability (or robustness). Therefore, the stability of the model has been assessed by running many computations against different probabilities of stable states drawn at random: the 95% confidence interval for SDHB at state 1 was set to 60-80% (corresponding to a 20-40% penetrance, which corresponds to the established penetrance range) and to 0-10% for SDHD (corresponding to a 90-100% penetrance already cited). The uncertainty on the proportion of patients in states (2, 3) has been described with two independant Gaussian laws of distribution with a relative standard deviation of 20% (the choice of 20% is arbitrary, but it is conservative as it is relatively large and challenges the stabilitiy of the model). The proportion of patients in a metastatic state was logically chosen so that the sum of the four probabilities would be one. For each random observation generated by this procedure, a check was done to verify that the four probabilities were in [0,1], then the model was solved with the least-square method already described; the quality of the fit was verified by checking residuals and the output of the solving algorithm and the probabilities of transition were stored. Eventually (after 500 runs), the 95% confidence intervals on the transition probabilities were extracted.
The robustness of the estimation procedure has been evaluated for various values of p 11 , since this parameter has been fixed in the model. Simulations showed a good fit between clinical observations and model predictions in a wide range of p 11 values: [70%, 99%] for SDHB and [1%, 99%] for SDHD. In SDHB-related PPGL, the quality of fit decreased rapidly for values of p 11 below 0.70. However, the results remain qualitatively unchanged and lead to the same biological interpretations regarding transition probabilities. For simplicity, it was decided to provide the estimated results of transition probabilities for p 11 = 0.75 for SDHB and p 11 = 0.10 for SDHD, since these values are linked to the overall disease penetrance of both SDHB and SDHD diseases.