**5. Experimental evaluation**

In this section the experimental results obtained by the execution of both versions of our simulator of the innate system, sequential and parallel, are presented. The experiments were performed on a 2.8 GHz Intel Core i7-860 processor, with 8 GB RAM, 32 KB L1 data cache, 8 MB L2 cache with a NVIDIA GeForce 285 GTX. The system runs a 64-bits version of Linux kernel 2.6.31 and version 3.0 of CUDA toolkit. The *gcc* version 4.4.2 was used to compile all versions of our code. The NVIDIA GeForce 285 GTX has 240 stream processors, 30 multiprocessors, each one with 16KB of shared memory, and 1GB of global memory. The number of threads per block are equal to 879, and each block has 512 threads. The codes were executed 3 times to all versions of our simulator, and the average execution time for each version of the code is presented in Table 1. The standard deviation obtained was negligible. The execution times were used to calculate the speedup factor. The speedup were obtained by dividing the sequential execution time of the simulator by its parallel version.


Table 1. Serial and parallel execution times. All times are in seconds.

The results reveal that our CUDA version was responsible for a significant improvement in performance: a speedup of 276 was obtained. This expressive gain was due to the embarrassingly parallel nature of computation that must be performed. In particular, the same computation must be performed for a huge amount of data, and there are no dependency and/or communication between parallel tasks.

#### **6. Simulation**

To study the importance of some cells, molecules and processes in the dynamics of the innate immune response, a set of simulations were performed for distinct values of parameters and initial conditions. Table 2 presents the initial conditions and the values of the parameters used in the simulations of all cases. Exceptions to the values presented in Table 2 are highlighted in the text.

6 Will-be-set-by-IN-TECH

of 2%) of their initial values, except for some parameters, that were also adjusted from -100% to + 100%, but in steps of 20%. The combination of all different set of parameters and initial conditions give us a total of 450,000 system of ODEs that must be evaluated in this work.

The sequential code that implements the sensitivity analysis was first implemented in C. Then the code was parallelized using CUDA. The parallel code is based on the idea that each combination of distinct values of parameters and initial conditions can be computed independently by a distinct CUDA thread. The number of threads that will be used during computation depends on the GPU characteristics. In particular, the number of blocks and threads per block are chosen taking into account two distinct values defined by the hardware:

The forward Euler method was used for the numerical solution of the systems of ODEs with a time-step of 0.0001 days. The models were simulated to represent a total period equivalent

In this section the experimental results obtained by the execution of both versions of our simulator of the innate system, sequential and parallel, are presented. The experiments were performed on a 2.8 GHz Intel Core i7-860 processor, with 8 GB RAM, 32 KB L1 data cache, 8 MB L2 cache with a NVIDIA GeForce 285 GTX. The system runs a 64-bits version of Linux kernel 2.6.31 and version 3.0 of CUDA toolkit. The *gcc* version 4.4.2 was used to compile all versions of our code. The NVIDIA GeForce 285 GTX has 240 stream processors, 30 multiprocessors, each one with 16KB of shared memory, and 1GB of global memory. The number of threads per block are equal to 879, and each block has 512 threads. The codes were executed 3 times to all versions of our simulator, and the average execution time for each version of the code is presented in Table 1. The standard deviation obtained was negligible. The execution times were used to calculate the speedup factor. The speedup were obtained

by dividing the sequential execution time of the simulator by its parallel version.

Table 1. Serial and parallel execution times. All times are in seconds.

and/or communication between parallel tasks.

**6. Simulation**

the text.

Sequential 285 GTX Speedup Factor 4,315.47s 15.63s 276.12

The results reveal that our CUDA version was responsible for a significant improvement in performance: a speedup of 276 was obtained. This expressive gain was due to the embarrassingly parallel nature of computation that must be performed. In particular, the same computation must be performed for a huge amount of data, and there are no dependency

To study the importance of some cells, molecules and processes in the dynamics of the innate immune response, a set of simulations were performed for distinct values of parameters and initial conditions. Table 2 presents the initial conditions and the values of the parameters used in the simulations of all cases. Exceptions to the values presented in Table 2 are highlighted in

a) the warp size and b) the maximum number of threads per block.

to 5 days after the initial infection.

**5. Experimental evaluation**

The complete set of equations that has been simulated, including the initial values used, are presented by Equation 6:

$$\begin{cases} \frac{dM}{dt} = -\mu\_A A - (\lambda\_{NA|A} N + \lambda\_{AM|A} A M + \lambda\_{RM|A} R M).A\\ A(0) = 20 \text{[} 40 \\\\varphi\_{P} = (P\_{\text{max}}^{\text{max}} - P\_{\text{min}}^{\text{min}}) \frac{CH}{CH + \log R} + P\_{\text{N}}^{\text{min}} \\quad s\_{\text{surr}S\_{N}} = perm \text{adj}\_{\mathcal{N}} (N^{\text{max}} - N) \\\frac{dN}{dt} = -\mu\_N N - \lambda\_{A|N} A.N + \text{source}\_N \\ N(0) = 0 \\\\varphi\_{P} = (P\_{RM}^{\text{max}} - P\_{RM}^{\text{min}}) \frac{CH}{CH + \log R} + P\_{RM}^{\text{min}} \\\frac{dRM}{dt} = -\mu\_{RM} R M - \lambda\_{RM|A} R M.A + \text{source}\_{RM} \\\ R M(0) = 1 \\\\ \frac{dM}{dt} = -\mu\_{AM} AM + \lambda\_{RM|A} R M.A \\ \lambda M(0) = 0 \\\\ \frac{dM}{dt} = -\mu\_{CH} C H + (\beta\_{CH|N} N + \beta\_{CH|AM} A M).A \cdot (1 - \frac{C H}{\log R}) \\\ C M(0) = 0 \end{cases} (6)$$

It should be noticed that in this case two distinct initial values for *A*(0) will be used: *A*(0) = 20 and *A*(0) = 40.

The sensitivity analysis has shown that two parameters are relevant to the model: the capacity of the tissue to support the entrance of new neutrophils (*Nmax*) and the phagocytosis rate of LPS by neutrophils (*λN*|*A*).

*Nmax* is the most sensitive parameter in the model. The capacity of the tissue to support the entrance of new neutrophils is directed related to the permeability of the endothelial cells, which form the linings of the blood vessels. If a positive adjustment is made in the parameter related to the permeability, then there are more neutrophils entering into the tissue. This larger amount of neutrophils into the tissue has many consequences: first, more cells are phagocyting, so the amount of LPS reduces faster. Second, a smaller amount of resting macrophages becomes active, because there is less LPS into the tissue. Third, a larger amount of cytokines are produced, since neutrophils are the main responsible for this production. If a negative adjustment is made, the inverse effect can be observed: with a smaller amount of neutrophils in the tissue, more resting macrophages become active. Also, a smaller amount of cytokines are produced.

Figures 2 to 6 illustrate this situation. It can be observed that the LPS decays faster when *Nmax* achieves its maximum value.

Fig. 3. Temporal evolution of neutrophils with *A*(0) = 20 and for distinct values of *Nmax*.

Modelling the Innate Immune System 359

Fig. 4. Temporal evolution of LPS with *A*(0) = 20 and for distinct values of *Nmax*.

Figures 7 to 11 present the complete scenario.

In the second scenario, with the double of LPS and starting with just one resting macrophage, it can be observed that bringing more neutrophils into the tissue do not reduce the number of resting macrophages that become active. This happens due to the larger amount of LPS in this scenario when compared to the previous one. The larger amount of activated macrophages also explains why the amount of cytokines in this scenario is larger than in the previous one.

The third scenario presents the results obtained when the initial amount of LPS is again equal to 20. This scenario revels that the second most sensitive parameter is *<sup>λ</sup>N*|*A*. *<sup>λ</sup>N*|*<sup>A</sup>* is responsible for determining how effective is the phagocitosis of the neutrophils in tissue. It can be observed in Figures 12 to 16 that a negative adjustment in this tax makes the neutrophil response to be less effective against LPS, while a positive adjustment in the tax makes the neutrophil response to be more effective. Resting macrophages and activated macrophages are also affected by distinct values of *<sup>λ</sup>N*|*A*. Increasing the value of *<sup>λ</sup>N*|*<sup>A</sup>* causes the neutrophils


Table 2. Initial conditions, parameters and units.

Fig. 2. Temporal evolution of cytokines with *A*(0) = 20 and for distinct values of *Nmax*.

8 Will-be-set-by-IN-TECH

*cell*.*day* Su et al. (2009)

*cell*.*day* Su et al. (2009)

*cell*.*day* Su et al. (2009)

*cell*.*day* estimated

*cell*.*day* estimated

*day* estimated

*day* estimated

*day* estimated *chInf* 3.6 *cell* based on de Waal Malefyt et al. (1991)

*cell*.*day* estimated

*day* based on Price et al. (1994)

*Nmax* 8 *cell* estimated *MRmax* 6 *cell* estimated

*keqch* 1 *cell* estimated

Fig. 2. Temporal evolution of cytokines with *A*(0) = 20 and for distinct values of *Nmax*.

**Parameter Value Unit Reference** *N*<sup>0</sup> 0 cell estimated *CH*<sup>0</sup> 0 cell estimated *A*<sup>0</sup> 20 cell estimated *RM*<sup>0</sup> 1 cell estimated *AM*<sup>0</sup> 0 cell estimated *μCH* 7 1/*day* estimated *μ<sup>N</sup>* 3.43 1/*day* estimated *μ<sup>A</sup>* 0 1/*day* Su et al. (2009) *μRM* 0.033 1/*day* Su et al. (2009) *μAM* 0.07 1/*day* Su et al. (2009)

*<sup>λ</sup>N*|*<sup>A</sup>* 0.55 <sup>1</sup>

*<sup>λ</sup>A*|*<sup>N</sup>* 0.55 <sup>1</sup>

*<sup>λ</sup>AM*|*<sup>A</sup>* 0.8 <sup>1</sup>

*<sup>β</sup>CH*|*<sup>N</sup>* <sup>1</sup> <sup>1</sup>

*<sup>β</sup>CH*|*AM* 0.8 <sup>1</sup>

*<sup>N</sup>* 11.4 <sup>1</sup>

*<sup>N</sup>* 0.0001 <sup>1</sup>

*RM* 0.1 <sup>1</sup>

*RM* 0.01 <sup>1</sup>

*<sup>λ</sup>RM*|*<sup>A</sup>* 0.1 <sup>1</sup>

Table 2. Initial conditions, parameters and units.

*Pmax*

*Pmin*

*Pmax*

*Pmin*

Fig. 3. Temporal evolution of neutrophils with *A*(0) = 20 and for distinct values of *Nmax*.

Fig. 4. Temporal evolution of LPS with *A*(0) = 20 and for distinct values of *Nmax*.

In the second scenario, with the double of LPS and starting with just one resting macrophage, it can be observed that bringing more neutrophils into the tissue do not reduce the number of resting macrophages that become active. This happens due to the larger amount of LPS in this scenario when compared to the previous one. The larger amount of activated macrophages also explains why the amount of cytokines in this scenario is larger than in the previous one. Figures 7 to 11 present the complete scenario.

The third scenario presents the results obtained when the initial amount of LPS is again equal to 20. This scenario revels that the second most sensitive parameter is *<sup>λ</sup>N*|*A*. *<sup>λ</sup>N*|*<sup>A</sup>* is responsible for determining how effective is the phagocitosis of the neutrophils in tissue. It can be observed in Figures 12 to 16 that a negative adjustment in this tax makes the neutrophil response to be less effective against LPS, while a positive adjustment in the tax makes the neutrophil response to be more effective. Resting macrophages and activated macrophages are also affected by distinct values of *<sup>λ</sup>N*|*A*. Increasing the value of *<sup>λ</sup>N*|*<sup>A</sup>* causes the neutrophils

Fig. 7. Temporal evolution of cytokines with *A*(0) = 40 and for distinct values of *Nmax*.

Modelling the Innate Immune System 361

Fig. 8. Temporal evolution of neutrophils with *A*(0) = 40 and for distinct values of *Nmax*.

the active phagocytes; c) tissue damage; and d) anti-inflammatory mediators.

A model of inflammation composed by ODEs in a three-dimensional domain considering three types of cells/molecules has been proposed by Kumar et al. (2004): the pathogen and two inflammatory mediators. The model was able to reproduce some experimental results depending on the values used for initial conditions and parameters. The authors described the results of the sensitivity analysis and some therapeutic strategies were suggested from this analysis. The work was then extended (Reynolds et al. (2006)) to investigate the advantages of an anti-inflammatory response dependent on time. In this extension, the mathematical model was built from simpler models, called reduced models. The mathematical model (Reynolds et al. (2006)) consists of a system of ODEs with four equations to model: a) the pathogen; b)

A new adaptation of the first model (Kumar et al. (2004)) was proposed to simulate many scenarios involving repeated doses of endotoxin (Day et al. (2006)). In this work the results

**7.1 ODEs models**

Fig. 5. Temporal evolution of resting macrophages with *A*(0) = 20 and for distinct values of *Nmax*.

to produced more cytokines, so more macrophages can migrate into the tissue through blood vessel, and also there are more cells into the tissue that can phagocyte LPS.

The last scenario is presented by Figures 17 to 21. In this scenario, the amount of LPS is doubled when compared to the previous one. It can be observed that distinct values used as initial conditions for LPS only changes how long it takes to the complete elimination of LPS. It can also be observed that both macrophages populations are affected by the larger amount of LPS. In particular, the amount of macrophages is slightly higher in this scenario due to the larger amount of LPS.

#### **7. Related works**

This section presents some models and simulators of the HIS found in the literature. Basically two distinct approaches are used: ODEs and PDEs.

10 Will-be-set-by-IN-TECH

Fig. 5. Temporal evolution of resting macrophages with *A*(0) = 20 and for distinct values of

Fig. 6. Temporal evolution of activate macrophages with *A*(0) = 20 and for distinct values of

to produced more cytokines, so more macrophages can migrate into the tissue through blood

The last scenario is presented by Figures 17 to 21. In this scenario, the amount of LPS is doubled when compared to the previous one. It can be observed that distinct values used as initial conditions for LPS only changes how long it takes to the complete elimination of LPS. It can also be observed that both macrophages populations are affected by the larger amount of LPS. In particular, the amount of macrophages is slightly higher in this scenario due to the

This section presents some models and simulators of the HIS found in the literature. Basically

vessel, and also there are more cells into the tissue that can phagocyte LPS.

*Nmax*.

*Nmax*.

larger amount of LPS.

two distinct approaches are used: ODEs and PDEs.

**7. Related works**

Fig. 7. Temporal evolution of cytokines with *A*(0) = 40 and for distinct values of *Nmax*.

Fig. 8. Temporal evolution of neutrophils with *A*(0) = 40 and for distinct values of *Nmax*.

#### **7.1 ODEs models**

A model of inflammation composed by ODEs in a three-dimensional domain considering three types of cells/molecules has been proposed by Kumar et al. (2004): the pathogen and two inflammatory mediators. The model was able to reproduce some experimental results depending on the values used for initial conditions and parameters. The authors described the results of the sensitivity analysis and some therapeutic strategies were suggested from this analysis. The work was then extended (Reynolds et al. (2006)) to investigate the advantages of an anti-inflammatory response dependent on time. In this extension, the mathematical model was built from simpler models, called reduced models. The mathematical model (Reynolds et al. (2006)) consists of a system of ODEs with four equations to model: a) the pathogen; b) the active phagocytes; c) tissue damage; and d) anti-inflammatory mediators.

A new adaptation of the first model (Kumar et al. (2004)) was proposed to simulate many scenarios involving repeated doses of endotoxin (Day et al. (2006)). In this work the results

Fig. 11. Temporal evolution of activate macrophages with *A*(0) = 40 and for distinct values

Modelling the Innate Immune System 363

Fig. 12. Temporal evolution of cytokines with *<sup>A</sup>*(0) = 20 and for distinct values of *<sup>λ</sup>N*|*A*.

The model proposed by Su et al. (2009) uses a system of partial differential equations (PDEs) to model not only the functioning of the innate immune system, as well as the adaptive immune system. The model considers the simplest form of antigen, the molecular constituents of pathogens patterns, taking into account all the basic factors of an immune response: antigen, cells of the immune system, cytokines and chemokines. This model captures the following stages of the immune response: recognition, initiation, effector response and resolution of infection or change to a new equilibrium state (*steady state*). The model can reproduce important phenomena of the HIS such as a) temporal order of arrival of cells at the site of infection, b) antigen presentation by dendritic cells, macrophages to regulatory T cells d) production of pro-inflammatory and anti-inflammatory cytokines and e) the phenomenon of

of *Nmax*.

**7.2 PDEs models**

chemotaxis.

Fig. 9. Temporal evolution of LPS with *A*(0) = 40 and for distinct values of *Nmax*.

Fig. 10. Temporal evolution of resting macrophages with *A*(0) = 40 and for distinct values of *Nmax*.

obtained through experiments with mouse are used to guide the *in silico* experiments seeking to recreate these results qualitatively.

A one-dimensional model to show if and when leukocytes successfully defend the body against a bacterial infection is presented in Keener & Sneyd (1998). A phase-plane method is then used to study the influence of two parameters, the enhanced leukocyte emigration from bloodstream and the chemotactic response of the leukocytes to the attractant.

Finally, one last work (Vodovotz et al. (2006)) developed a more complete system of ODEs of acute inflammation, including macrophages, neutrophils, dendritic cells, Th1 cells, the blood pressure, tissue trauma, effector elements such as iNOS, *NO*− <sup>2</sup> and NO<sup>−</sup> <sup>3</sup> , pro-inflammatory and anti-inflammatory cytokines, and coagulation factors. The model has proven to be useful in simulating the inflammatory response induced in mice by endotoxin, trauma and surgery or surgical bleeding, being able to predict to some extent the levels of TNF, IL-10, IL-6 and reactive products of NO (NO− <sup>2</sup> and NO<sup>−</sup> 3 ).

12 Will-be-set-by-IN-TECH

Fig. 9. Temporal evolution of LPS with *A*(0) = 40 and for distinct values of *Nmax*.

Fig. 10. Temporal evolution of resting macrophages with *A*(0) = 40 and for distinct values of

obtained through experiments with mouse are used to guide the *in silico* experiments seeking

A one-dimensional model to show if and when leukocytes successfully defend the body against a bacterial infection is presented in Keener & Sneyd (1998). A phase-plane method is then used to study the influence of two parameters, the enhanced leukocyte emigration

Finally, one last work (Vodovotz et al. (2006)) developed a more complete system of ODEs of acute inflammation, including macrophages, neutrophils, dendritic cells, Th1 cells, the blood

and anti-inflammatory cytokines, and coagulation factors. The model has proven to be useful in simulating the inflammatory response induced in mice by endotoxin, trauma and surgery or surgical bleeding, being able to predict to some extent the levels of TNF, IL-10, IL-6 and

<sup>2</sup> and NO<sup>−</sup>

<sup>3</sup> , pro-inflammatory

from bloodstream and the chemotactic response of the leukocytes to the attractant.

3 ).

pressure, tissue trauma, effector elements such as iNOS, *NO*−

<sup>2</sup> and NO<sup>−</sup>

*Nmax*.

to recreate these results qualitatively.

reactive products of NO (NO−

Fig. 11. Temporal evolution of activate macrophages with *A*(0) = 40 and for distinct values of *Nmax*.

Fig. 12. Temporal evolution of cytokines with *<sup>A</sup>*(0) = 20 and for distinct values of *<sup>λ</sup>N*|*A*.

#### **7.2 PDEs models**

The model proposed by Su et al. (2009) uses a system of partial differential equations (PDEs) to model not only the functioning of the innate immune system, as well as the adaptive immune system. The model considers the simplest form of antigen, the molecular constituents of pathogens patterns, taking into account all the basic factors of an immune response: antigen, cells of the immune system, cytokines and chemokines. This model captures the following stages of the immune response: recognition, initiation, effector response and resolution of infection or change to a new equilibrium state (*steady state*). The model can reproduce important phenomena of the HIS such as a) temporal order of arrival of cells at the site of infection, b) antigen presentation by dendritic cells, macrophages to regulatory T cells d) production of pro-inflammatory and anti-inflammatory cytokines and e) the phenomenon of chemotaxis.

Fig. 15. Temporal evolution of resting macrophages with *A*(0) = 20 and for distinct values of

Modelling the Innate Immune System 365

Fig. 16. Temporal evolution of activate macrophages with *A*(0) = 20 and for distinct values

(Muraille et al. (1996)). Klein (1980) presents and compares three mathematical models of B

ImmSim (Bezzi et al. (1997); Celada & Seiden (1992)) is a simulator of the HIS that implements the following mechanisms: immunological memory, affinity maturation, effects of hypermutation, autoimmune response, among others. CAFISS (a Complex Adaptive Framework for Immune System Simulation) (Tay & Jhavar (2005)) is a framework used for modelling the immune system, particularly HIV attack. SIMMUNE (Meier-Schellersheim & Mack (1999)) allows users to model cell biological systems based on data that describes cellular behaviour on distinct scales. Although it was developed to simulate immunological phenomena, it can be used in distinct domains. A similar tool is CyCells (Warrender (2004)),

*<sup>λ</sup>N*|*A*.

of *<sup>λ</sup>N*|*A*.

cell differentiation and proliferation.

designed to study intercellular relationships.

Fig. 13. Temporal evolution of neutrophils with *<sup>A</sup>*(0) = 20 and for distinct values of *<sup>λ</sup>N*|*A*.

Fig. 14. Temporal evolution of LPS with *<sup>A</sup>*(0) = 20 and for distinct values of *<sup>λ</sup>N*|*A*.

Pigozzo et al. (2011) present a PDE model to simulate the immune response to lipopolysaccharide (LPS) in a microscopic section of a tissue, reproducing, for this purpose, the initiation, maintenance and resolution of immune response.

#### **7.3 Other works**

Several proposals which attempt to model both the innate and the adaptive HIS can be found in the literature. An ODE model is used to describe the interaction of HIV and tuberculosis with the immune system (Denise & Kirschner (1999)). Other work focus on models of HIV and T-lymphocyte dynamics, and includes more limited discussions of hepatitis C virus (HCV), hepatitis B virus (HBV), cytomegalovirus (CMV) and lymphocytic choriomeningitis virus (LCMV) dynamics and interactions with the immune system (Perelson (2002)). An ODE model of cell-free viral spread of HIV in a compartment was proposed by Perelson et al. (1993). Another interesting work tries to integrate the immune system in the general physiology of the host and considers the interaction between the immune and neuroendocrine system 14 Will-be-set-by-IN-TECH

Fig. 13. Temporal evolution of neutrophils with *<sup>A</sup>*(0) = 20 and for distinct values of *<sup>λ</sup>N*|*A*.

Fig. 14. Temporal evolution of LPS with *<sup>A</sup>*(0) = 20 and for distinct values of *<sup>λ</sup>N*|*A*.

the initiation, maintenance and resolution of immune response.

**7.3 Other works**

Pigozzo et al. (2011) present a PDE model to simulate the immune response to lipopolysaccharide (LPS) in a microscopic section of a tissue, reproducing, for this purpose,

Several proposals which attempt to model both the innate and the adaptive HIS can be found in the literature. An ODE model is used to describe the interaction of HIV and tuberculosis with the immune system (Denise & Kirschner (1999)). Other work focus on models of HIV and T-lymphocyte dynamics, and includes more limited discussions of hepatitis C virus (HCV), hepatitis B virus (HBV), cytomegalovirus (CMV) and lymphocytic choriomeningitis virus (LCMV) dynamics and interactions with the immune system (Perelson (2002)). An ODE model of cell-free viral spread of HIV in a compartment was proposed by Perelson et al. (1993). Another interesting work tries to integrate the immune system in the general physiology of the host and considers the interaction between the immune and neuroendocrine system

Fig. 15. Temporal evolution of resting macrophages with *A*(0) = 20 and for distinct values of *<sup>λ</sup>N*|*A*.

Fig. 16. Temporal evolution of activate macrophages with *A*(0) = 20 and for distinct values of *<sup>λ</sup>N*|*A*.

(Muraille et al. (1996)). Klein (1980) presents and compares three mathematical models of B cell differentiation and proliferation.

ImmSim (Bezzi et al. (1997); Celada & Seiden (1992)) is a simulator of the HIS that implements the following mechanisms: immunological memory, affinity maturation, effects of hypermutation, autoimmune response, among others. CAFISS (a Complex Adaptive Framework for Immune System Simulation) (Tay & Jhavar (2005)) is a framework used for modelling the immune system, particularly HIV attack. SIMMUNE (Meier-Schellersheim & Mack (1999)) allows users to model cell biological systems based on data that describes cellular behaviour on distinct scales. Although it was developed to simulate immunological phenomena, it can be used in distinct domains. A similar tool is CyCells (Warrender (2004)), designed to study intercellular relationships.

Fig. 20. Temporal evolution of resting macrophages with *A*(0) = 40 for distinct values of

Modelling the Innate Immune System 367

Fig. 21. Temporal evolution of activate macrophages with *A*(0) = 40 for distinct values of

In this chapter we presented the sensitivity analysis of a mathematical model that simulates the immune response to LPS in a microscopic section of a tissue. The results have shown that the two most relevant parameters of the model are: the capacity of the tissue to support the

The sensitivity analysis can be a time consuming task due to the large number of scenarios that must be evaluated. This prohibitive computational cost leads us to develop a parallel version of the sensitivity analysis code using GPGPUs. Our experimental results showed that the parallelization was very effective in improving the sensitivity analysis performance, yielding

entrance of more neutrophils and the phagocytosis rate of LPS by neutrophils.

*<sup>λ</sup>N*|*A*.

*<sup>λ</sup>N*|*A*.

**8. Conclusion and future works**

speedups up to 276.

Fig. 17. Temporal evolution of cytokines with *<sup>A</sup>*(0) = 40 for distinct values of *<sup>λ</sup>N*|*A*.

Fig. 18. Temporal evolution of neutrophils with *<sup>A</sup>*(0) = 40 for distinct values of *<sup>λ</sup>N*|*A*.

Fig. 19. Temporal evolution of LPS with *<sup>A</sup>*(0) = 40 for distinct values of *<sup>λ</sup>N*|*A*.

16 Will-be-set-by-IN-TECH

Fig. 17. Temporal evolution of cytokines with *<sup>A</sup>*(0) = 40 for distinct values of *<sup>λ</sup>N*|*A*.

Fig. 18. Temporal evolution of neutrophils with *<sup>A</sup>*(0) = 40 for distinct values of *<sup>λ</sup>N*|*A*.

Fig. 19. Temporal evolution of LPS with *<sup>A</sup>*(0) = 40 for distinct values of *<sup>λ</sup>N*|*A*.

Fig. 20. Temporal evolution of resting macrophages with *A*(0) = 40 for distinct values of *<sup>λ</sup>N*|*A*.

Fig. 21. Temporal evolution of activate macrophages with *A*(0) = 40 for distinct values of *<sup>λ</sup>N*|*A*.

#### **8. Conclusion and future works**

In this chapter we presented the sensitivity analysis of a mathematical model that simulates the immune response to LPS in a microscopic section of a tissue. The results have shown that the two most relevant parameters of the model are: the capacity of the tissue to support the entrance of more neutrophils and the phagocytosis rate of LPS by neutrophils.

The sensitivity analysis can be a time consuming task due to the large number of scenarios that must be evaluated. This prohibitive computational cost leads us to develop a parallel version of the sensitivity analysis code using GPGPUs. Our experimental results showed that the parallelization was very effective in improving the sensitivity analysis performance, yielding speedups up to 276.

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#### **9. Acknowledgement**

The authors would like to thank FAPEMIG, CNPq (479201/2010-2), CAPES and UFJF for supporting this study.

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**19** 

**A Stochastically Perturbed** 

Mehmet Sevkli1 and Aise Zulal Sevkli2

*Industrial Engineering, Riyadh* 

*Information Technology, Riyadh Kingdom of Saudi Arabia* 

**Particle Swarm Optimization for Identical** 

*2King Saud University, College of Computer and Information Sciences, Department of* 

Identical parallel machine scheduling (PMS) problems with the objective of minimizing makespan (Cmax) is one of the well known NP-hard [1] combinatorial optimization problems. It is unlikely to obtain optimal schedule through polynomial time-bounded algorithms. Small size instances of PMS problem can be solved with reasonable computational time by exact algorithms such as branch-and-bound [2, 3], and the cutting plane algorithm [4]. However, as the problem size increases, the computation time of exact methods increases exponentially. On the other hand, heuristic algorithms generally have acceptable time and memory requirements, but do not guarantee optimal solution. That is, a feasible solution is obtained which is likely to be either optimal or near optimal. The well-known longest processing time (LPT) rule of Graham [5] is a sort of so called list scheduling algorithm. It is known that the rule works very well when makespan is taken as the single criterion [6]. Later, Coffman et al. [7] proposed MULTIFIT algorithm that considers the relation between bin-packing and maximum completion time problems. Yue [8] showed that the MULTIFIT heuristic is not guaranteed to perform better than LPT for every problem. Gupta and Ruiz-Torres [9] developed a LISTFIT algorithm that combines the bin packing method of the MULTIFIT heuristic with multiple lists of jobs. Min and Cheng [10] introduced a genetic algorithm (GA) that outperformed simulated annealing (SA) algorithm. Lee et al. [11] proposed a SA algorithm for the PMS problems and compared their results with the LISTFIT algorithm. Tang and Luo [12] developed a new iterated local search (ILS) algorithm that is based on varying number of cyclic exchanges.

Particle swarm optimization (PSO) is based on the metaphor of social interaction and communication among different spaces in nature, such as bird flocking and fish schooling. It is different from other evolutionary methods in a way that it does not use the genetic operators (such as crossover and mutation), and the members of the entire population are maintained through out the search procedure. Thus, information is socially shared among

**1. Introduction** 

**Parallel Machine Scheduling Problems** 

*1King Saud University, Faculty of Engineering, Department of* 

Zhang, B., Hirahashi, J., Cullere, X. & Mayadas, T. N. (2003). Elucidation of molecular events leading to neutrophil apoptosis following phagocytosis, *The Journal of biological chemistry* 278: 28443–28454.
