† Corresponding author. E-mail:
In this paper we propose a mathematical model of bone remodeling with time delays of both osteoclast-derived paracrine signaling of tumor and tumor-derived paracrine signaling of osteoclast. The effects of time delays on the growth of tumor cells and bone system are studied in multiple myeloma-induced bone disease. In the case of small osteoclast-derived paracrine signaling, it is found that the growth of tumor cells slows down, the oscillation period of the ratio of osteoclasts to osteoblasts is extended with increasing time delay, and there is a competition between the delay and osteoclast-derived paracrine signaling. In the case of large tumor-derived paracrine signaling, the tumor-derived paracrine signaling can induce a more significant decline in tumor growth for long time delay, and thus slowing down the progression of bone disease. There is an optimal coupling between the tumor-derived paracrine signaling of osteoclasts and time delay during the progressions of bone diseases, which suppresses the tumor growth and the regression of bone disease.
Time delay is ubiquitous in most physical and biological systems, arising from finite propagation speeds of signals, which has been widely studied in the context of coupled limit cycle oscillator systems,[1–4] for example, the semiconductor lasers,[5] neuronal networks,[6–8] etc.
Multiple myeloma is a hematologic malignancy that interferes in a destructive way with the natural bone renewal process,[9–13] coupled with several diseases which involve abnormal and uncontrolled proliferation of tumor cells. Multiple myeloma-induced bone disease occurs due to dysregulated bone remodeling, where tumor cells interacting with the bone marrow microenvironment disrupt healthy kinetics, which is dependent on the behaviors of osteoclasts and osteoblasts, and ultimately leads to significant bone destruction.[14–16] Currently, the therapy of multiple myeloma-induced bone disease is still a challenge by virtue of its incurability, but it may be controlled to some extent.[17,18] Some mathematical models[19–22] have been proposed to reveal the kinetics of bone cell populations in healthy and pathological bone remodeling, and underlying cellular and signaling mechanisms. It was demonstrated that the skeleton undergoes constant remodeling through bone resorption carried out by active osteoclasts and bone formation by osteoblasts, restoring the bone mass to its initial value, and exhibits regular periodic oscillations in bone cell populations. A limit cycle is observed in the phase plane of osteoclasts–osteoblasts.
Although there are a number of theoretical and experimental studies in various aspects of bone physiology, little work deals with the role of time delay in multiple myeloma-induced bone disease. However, there is increasing evidence for the existence of the delay during the bone remodeling cycles.[23,24] For example, it is observed that the increase in bone cell populations takes place after 10–15 days in mice and 6–8 days in humans through therapy with G-CSF.[24] Moreover, some clinical observations also demonstrated that signals transmission delays are inherent to bone systems due to the finite propagation speeds in the bone remodeling process.[25–27]
Interesting questions now arise: how do the time delays influence the oscillation in a number of osteoclasts and osteoblasts? What are the effects of time delays on bone tumor (i.e., myeloma cells) in response to different treatments? To address these issues, according to the models[19–22] of bone remodeling with multiple myeloma, we propose a mathematical model of bone remodeling kinetics with delay times of both osteoclast-derived paracrine signaling of tumor and tumor-derived paracrine signaling of osteoclast in this paper. Our model can characterize both the kinetics of bone remodeling during the disease progression and the time delays of osteoclasts on tumor cells due to the promotion of bone resorption for the growth of the tumor.
The rest of this paper is organized as follows. In Section
In the presence of multiple myeloma, the bone remodeling process involves the activities of pre-osteoclasts, osteoclasts, pre-osteoblasts, osteoblasts, and myeloma cells. The scheme of interactions between bone cells and myeloma cells is shown by Fig.
Power law approximations are effective tools for describing and analyzing highly nonlinear biochemical systems.[28] Bone remodeling is a nonlinear biological process involving complex mechanical and biochemical signaling pathways. Power law approximations are applied to the descriptions of biochemical autocrine and paracrine factors for bone system by Komarova et al., and are widely used in the subsequent studies of bone remodeling process.[19–21] When the delay times of both osteoclast-derived paracrine signaling of tumor and tumor-derived paracrine signaling of osteoclast are considered, the time evolutions of osteoclasts, osteoblasts, and myeloma cells can be written as
![]() | (1) |
![]() | (2) |
![]() | (3) |
![]() | (4) |
![](cpb_26_3_030503/cpb_26_3_030503_005.gif)
![](cpb_26_3_030503/cpb_26_3_030503_006.gif)
![](cpb_26_3_030503/cpb_26_3_030503_007.gif)
![](cpb_26_3_030503/cpb_26_3_030503_008.gif)
![](cpb_26_3_030503/cpb_26_3_030503_009.gif)
![](cpb_26_3_030503/cpb_26_3_030503_010.gif)
![](cpb_26_3_030503/cpb_26_3_030503_011.gif)
![](cpb_26_3_030503/cpb_26_3_030503_012.gif)
![](cpb_26_3_030503/cpb_26_3_030503_013.gif)
![](cpb_26_3_030503/cpb_26_3_030503_014.gif)
![](cpb_26_3_030503/cpb_26_3_030503_015.gif)
![](cpb_26_3_030503/cpb_26_3_030503_016.gif)
![](cpb_26_3_030503/cpb_26_3_030503_017.gif)
![](cpb_26_3_030503/cpb_26_3_030503_018.gif)
On the other hand, for the multiple myeloma-induced bone disease, there are three possible pharmaceuticals for the treatment of disease: bisphosphonates which can suppress the production of osteoclasts,[29–31] proteasome inhibitors which can stimulate osteoblast production and bone formation,[16,32,33] and the anti-tumor therapy [13,20,29] which inhibits the growth of tumor cells. Thus, in the bone remodeling process involving the activities of osteoclasts, osteoblasts, and myeloma cells, the time evolution of cell populations Eqs. (
![]() | (5) |
![]() | (6) |
![]() | (7) |
![](cpb_26_3_030503/cpb_26_3_030503_022.gif)
![]() | (8) |
![](cpb_26_3_030503/cpb_26_3_030503_024.gif)
![](cpb_26_3_030503/cpb_26_3_030503_025.gif)
The above bone remodeling model can be studied by numerical simulation. Here, we take the parameters in accordance with Refs. [19] and [20]:
In the absence of time delays, the kinetic behaviors of bone remodeling without or with myeloma cells are shown in Fig.
![](cpb_26_3_030503/cpb_26_3_030503_042.gif)
The osteoclast-derived paracrine signaling of tumor can be characterized by the parameter
![](cpb_26_3_030503/cpb_26_3_030503_053.gif)
When
When
![](cpb_26_3_030503/cpb_26_3_030503_072.gif)
The tumor-derived paracrine signaling of osteoclast can be characterized by the parameter
![](cpb_26_3_030503/cpb_26_3_030503_081.gif)
When
The kinetic behaviors of myeloma cells and bone system in the presence of time delays are simulated for anti-tumor therapy, then together with either bisphosphonates or proteasome inhibitors, and numerical results are presented in Fig.
Figures
In this paper, we proposed a mathematical model of bone remodeling with time delays for multiple myeloma-induced bone disease. The effects of time delays on myeloma cells and bone system are analyzed from two representative systemic parameters: osteoclast-derived paracrine signaling of tumor
Our model with time delays for multiple myeloma-induced bone disease not only can be used to analyze many qualitative properties of time delays and systemic parameters, but also might provide guidelines for probing the mechanism of bone destruction and exploring more effective therapeutic approaches for multiple myeloma-induced bone disease.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] | |
[37] | |
[38] | |
[39] | |
[40] | |
[41] | |
[42] | |
[43] |