Optimization of a stem cell culture system from the perspective of minimizing medium consumption

Transcription

1 Optimization of a stem cell culture system from the perspective of minimizing medium consumption Jianming Sang April 25 th, 2014 Abstract Stem cells hold valuable potential applications in both basic developmental science research as well as in clinic areas like regenerative medicine. One of the biggest challenges for stem cells to be widely used is the high cost of large-scale culture of stem cells as many expensive biological factors must be applied to maintain the survival and self-renewal of stem cells. In this project, I try to design a system composed of a medium flowing channel and a cell growth well for culturing stem cells and optimize the medium flowing rate, in other word to minimize the consumption of the medium and keeping the stem cells healthy at the same time. Multiple factors will influence the fat e of stem cells and here only the oxygen consumed and carbon dioxide produced by the stem cells will be taken as the constraints in this system for simplicity. Thus the medium functions by providing oxygen to the cells and removing the extra carbon dioxide. To optimize the system, the influence of system geometry on the velocity distribution, nutrition supply rate, medium utilization efficiency and waste deletion rate was first analyzed by using the COMSOL simulation. After setting the geometry, the minimum flow rate for supply adequate oxygen and the minimum flow rate for removing the extra carbon dioxide have been analytically solved based on some assumptions. And the optimal solution is identified as the larger one of the two flow rates.

2 Introduction Stem cells hold great potential applications in regenerative medicine [1] as well as in developmental science [2], because of its inherent abilities of indefinitely self-renewal and potentials to differentiate into a wide variety of somatic cells [3]. The fate of stem cells is determined in vivo by their surrounding microenvironment called niche composed of biological, chemical and mechanical cues [4]. Many important factors in this niche have been identified, which lays the foundations to precisely control stem cell fate in vitro for clinical applications. The production of large masses of stem cells and their derivatives now represents another major hurdle to be overcome in developing stem cell-based therapies as the amount of stem cells needed in clinic is far more than that in lab. For example, it has been estimated that at least 10 9 (the equivalent of cmtissue culture dishes of cells) cells would be needed for heart repair after myocardial infarction [5]. To address this problem, many large scale-up stem cell culture platforms have been developed like bioreactors [6]. While these bioreactor culture systems have been proved to be successful in producing large amount of stem cells, or even directed differentiated cells, another issue, namely culturing cost, which is usually neglected in developing these prototype culture systems, has to be taken into consideration. The culture of stem cells always involves many biological factors like bfgf [7], either naturally derived proteins, human recombinant proteins or synthetic peptides, which are pretty expensive and makes it impossible for stem cells-based therapies to be available for everyone. And the high cost of large-scale culture of stem cells may be the last hinder for stem cells to be widely used. From this prospective, this project will be dedicated to developing a system for culture stem cells and optimizing the medium utilization efficiency of the culture system. Both simulation and analytical analysis will be carried out. And consider the complexity of stem cell culture, some important assumptions will be made for simplicity. 2. Design problem 2.1 Mathematical model

3 2.1.1 Problem statement Figure 1: 3D and 2D schematic illustration of the stem cell culture system, which contains a medium flow channel and a cell culture well. The stem cells grow in the well with the flowing medium in the channel providing nutrition and removing the metabolic waste. As showed in figure 1, the stem cell culture system contains a medium flowing channel and a stem cell growth well. The flowing medium in the channel can provide the nutrition to the stem cells and remove the extra metabolic waste at the same time. Both the high nutrition concentration and low metabolic waste concentration around stem cells are vital to maintain the survival and self-renewal of stem cells. And both of them will be determined by the flowing rate of medium if the geometry of the system is set. Moreover, the number of stem cells in the growth well will increase as the time goes by, so as the amount of oxygen needed and amount of carbon dioxide produced by the cells, which means this is a dynamic culture system and the flow rate of the medium needs to be adjusted continually in order to optimize this culture system, namely to minimize the consumption of the medium Important assumptions First as showed below in figure 2, the classical growth curve of cells contain four distinct phases. Here I assume that the stem cells always belong to the log-phase during the entire culture time so as the number of stem cells can be precisely calculated by using the equation N(t) = N 0 e kt where

4 N 0 is the initial cell number, N(t) is the number of cells at time t and k is the cell growth rate. Actually if culture time is limited in a short period, this assumption can be true. Figure 2. Typical cell growth curve[8], which can be divided into four distinct phases and by assuming that the cells always belong to the log phase, can the number of cells can be precisely calculated. Second, the stem cells will not differentiate into other cell types during the culture period of time. Stem cells can easily differentiate into other somatic cells, which poses challenges for characterizing the cells and calculating the number of cells. However, as long as enough nutrition and biological factors are provided, few stem cells will differentiate. Figure 3. Schematic illustration of the assumptions that cell only need the nutrition oxygen and produce the waste carbon dioxide. The medium functions by supplying the oxygen and removing the extra carbon dioxide.

5 Third, hundreds of different factors are needed for culturing stem cells in reality. Here for simplicity, only oxygen supply and carbon dioxide deletion will be taken into account as showed in figure Mathematical model Object function minimize ( ) g1 g2 g3 g4 = 0 g5 where g1 is the exponential growth pattern of stem cells, g2 describes that the amount of oxygen diffused into the culture well needs to be larger than that cells need at any time, g3 describes that the concentration of carbon dioxide in the culture well should be lower than the dangerous level at any time, g4 and g5 denotes the diffusion of oxygen and carbon dioxide between the channel and the well. The meaning and values of the parameters used are listed in table 1. Table 1. List of parameters used in the model building. Parameter Symbol Value Initial Time t 0 (s) 0 End time t(s) 10 6 Flow rate of the medium v/(µm/s) Variable Initial cell number N o 10 5 Cell number at time t N t Variable Cell growth rate k(1/s) x10-7 [9]

6 O 2 concentration in medium C 1 (mol/m 3 ) 2.2[9] CO 2 concentration in medium C 2 (mmhg) 38 CO 2 tolerance of cells C limit (mmhg) 150 O 2 diffusivity in medium (cm 2 /s) 3.29 x 10-5 [11] C0 2 diffusivity in medium (cm 2 /s) 1.92 x 10-5[ 12] O 2 diffusion density J O2 Variable CO 2 dioxide diffusion density J CO2 Variable O 2 consumption rate R O2 (mol/(cell*s)) 5.55x10-17 [10] CO 2 produce rate R C02 (mol/(cell*s)) 5.36x10-17 [10] Medium channel side length a(cm) 1 Size of the cell culture well a x b x h (cm 3 ) 1x1x10 Model analysis The monotonicity analysis table is showed below: f + v g1 g2 g3 g4 g5 N/A N/A N/A N/A N/A

7 From the monotonicity table, it is hard to tell whether the constraints are active or inactive as the variable of flow rate v is implicit in all the constraints. However, from another perspective we can easily show the existence of the feasible solution. The medium has two functions: providing adequate oxygen for the stem cells and remove the extra carbon dioxide. The larger the flow rate is, the more oxygen the medium can provide to the stem cells. So at any time t there always exits a minimum flow rate V O2, at which the amount of oxygen provided by the flow medium is what the cells just need. And similarly, as the flow rate increases, the medium can take away more carbon dioxide. So there exits another flow rate V CO2, at which the medium can remove the amount of carbon dioxide produced by the cells at time t. Since here we only consider these two factors, so the optimal flow rate V would be the larger one of V O2 and V CO2. Design optimization Optimization results Geometry analysis: the geometry of the medium flow channel and the cell culture well will influence the velocity distribution, oxygen supply rate, oxygen utilization efficiency and carbon dioxide deletion rate. To visually investigate the effect of geometry on these factors, simulation has been carried out using the COMSOL and the results are showed below in figure 4. Here I define R as the ratio between the height of the channel and the height of the culture well. As expected, when the R is small, the oxygen supply rate is low, so as the carbon dioxide deletion rate (not shown here). The advantage when R is small is that most of the oxygen will be consumed by the cells, which means that the medium utilization efficiency is high. As the R increases, the oxygen supply rate increases, which can ensure that the stem cells would get enough oxygen at any time. But the problem when R is large is that much of oxygen won t be consumed by the cells, which means that a lot of medium will be wasted. And what s more, when R is large, the velocity distribution (Figure 4C) shows that the magnitude of velocity near the cells are very large, and the large velocity means large shear stress around stem cells, which will have significantly influence the fate of stem cells[9].

8 Figure 4. The COMSOL simulation results: the effect of the ratio between medium flchannel height to cell culture well height on velocity distribution and oxygen supply. Similarly, simulation of effect of the cell culture well length has also been carried out and the results are showed in figure 6. As we can see from the figure, when the length is small, the oxygen supply rate is large but the medium utilization efficiency is small. As the length increases, the efficiency increases, but at the same time the cells may not get enough oxygen and carbon dioxide may accumulate at the end of the culture well. And also shear stress around cells becomes large.

9 Figure 5. The COMSOL simulation results: the influence of stem cell culture well on velocity distribution and oxygen supply. The difficulty for optimizing the geometry is that the optimal geometry will vary with the medium flow rate as the flow rate itself will greatly influence the oxygen supply and carbon dioxide deletion. That is to say the optimal geometry needs to vary as the time goes by, which is impossible in reality as we cannot adjust the geometry of the system once it has been fabricated. So here to continue the analysis, I just roughly determine the geometry based on the simulation results. Based on the discussion in the monotonicity analysis, the optimal solution of flow rate would be the larger one of the minimum velocity for providing the enough oxygen denoted as V O2 and the

10 minimum velocity for removing the carbon dioxide denoted as V CO2. And the analytical calculation results are showed below: (1) (2) ( ) Substitute the variables with the values listed in the Table 1, we can get the value of the velocity showed below: (3) (4) The corresponding relationship between velocity and time is showed in figure 6. As we can see, the minimum velocity for supplying adequate oxygen for stem cells is always larger than the minimum one for removing the carbon dioxide. Since the purpose of the project is to minimum the medium consumption on the premise that stem cells can grow well, and here we only consider oxygen supply and carbon dioxide deletion these two constraints, so obviously the optimal solution is. Figure 6. The minimum velocity for providing adequate oxygen and the minimum velocity for removing the extra carbon dioxide varies with time and the latter is always smaller.

11 Discussion We can rearrange the velocity equations as following: ( ) ( ) (5) ( ) ( ) (6) Here again we define (the ratio of medium flow channel height to the cell culture well height). Keeping a as 1 cm, the relationship between and R is showed in figure 7A. As we can see, increases as the R increases, which means more medium is needed as R increases and the medium utilization efficiency decreases. And this result is compatible with the simulation results. Figure 7. A) has a positive linear relationship with R while keeping other factors unchanged; B) Setting R = 1, is negatively related with the size of the medium flow channel. Figure 7B shows that if we confine R = 1, decreases as a increases, which means that the medium flow rate would decrease as the geometry of the culture system increase. This is understandable as the number of cells is equal for all geometries, which means that the total amount of oxygen needed is constant for all geometries. And at the same flow rate, more medium will flow through the channel for the larger channel. So to transport the same amount of oxygen, meaning the same amount of medium, smaller velocity is needed for the larger channel. So decreases as a increases. The same analysis method can be applied to. What

12 unexpected is that the equation shows that is unrelated with the ratio R, which is not compatible with the simulation results. And this may result from the simplification for calculation of the velocity. Another important issue is that we only consider oxygen and carbon dioxide these two constraints here, which makes it easier to calculate the optimal velocity. But in reality, as explained above, hundreds of different factors are needed for maintaining stem cell survive and self-renewal. And true result would be more like that showed in figure 8 and even more complex. As it is difficulty to take every factor into consideration, actually even not all factors influencing stem cell fate have been identified up to now, here I just optimize the culture system considering oxygen and carbon dioxide transportation, which will provide some useful information in designing stem cell culture system. Figure 8. Multiple factors in reality will influence the fate of stem cells and the optimal velocity would be much more complicated. Reference 1. Gerecht-Nir, Sharon, and Joseph Itskovitz-Eldor. "Cell therapy using human embryonic stem cells." Transplant immunology 12.3 (2004): Clark, Amander T., and Renee A. Reijo Pera. "Modeling human germ cell development with embryonic stem cells." (2006):

13 3. Metallo, Christian M., et al. "Engineering tissue from human embryonic stem cells." Journal of cellular and molecular medicine 12.3 (2008): Discher, Dennis E., David J. Mooney, and Peter W. Zandstra. "Growth factors, matrices, and forces combine and control stem cells." Science (2009): Zweigerdt, Robert. "Large scale production of stem cells and their derivatives."engineering of Stem Cells. Springer Berlin Heidelberg, Tandon, Nina, et al. "Bioreactor Engineering of Stem Cell Environments."Biotechnology advances (2013) Zhao, Feng, et al. "Effects of oxygen transport on 3 d human mesenchymal stem cell metabolic activity in perfusion and static cultures: Experiments and mathematical model." Biotechnology progress 21.4 (2005): Gray, David R., et al. "CO2 in large-scale and high-density CHO cell perfusion culture." Cytotechnology (1996): Chow, Dominic C., et al. "Modeling po< sub> 2</sub> Distributions in the Bone Marrow Hematopoietic Compartment. II. Modified Kroghian Models."Biophysical journal 81.2 (2001):