**6. The significance of mathematical modeling as an instrument of scientific discovery in mental disorders**

Here, let me demonstrate the potential value of modeling for knowledge discovery in the aging brain. Brain diseases create a worldwide health problem. For instance, the total cost for brain disorders in the European Union amounted almost 800 billion*€* in 2010. Thus, the average cost per capita was about *€*5500. Nature medicine quoted a similar expensive picture for the United States [20]. Alzheimer disease is the most common cause of dementia. Mental disorders are at the top of the list of most costly health conditions in the United States. According to the Information Technology and Innovation Foundation (ITIF), annually \$1.5 trillion are spent on mental disorders, a cost to the economy of about 8.8% of the GDP [21].

**33**

**Figure 12.**

**Figure 11.**

*Mathematical Modeling: The Art of Translating between Minds and Machines…*

to encourage pharmaceutical companies to invest in more research.

These diseases also have a personal face. Parkinson's disease and Alzheimer's disease affect popular personalities as well as loved ones. Due to the severity of the crisis, the ITIF policy recommendations include expansion funding for NIH/NINDS and

But how can progress be made to address diseases of the brain especially in an aging population? Here, I point out preliminary data about research in mathematical model to better understand metabolic changes that affect the aging brain. **Figure 11** shows microcirculatory changes that affect cerebral perfusion. In this preliminary research, we can see that the capillary blockage caused by lymphocytes or the increase of tortuosity is capable of causing subtle changes in blood microperfusion and also enhances intracranial resistance. Even though these results are still in preliminary stages, mathematical models at the microcirculatory level offer a unique perspective capable of answering questions at the macroscopic scale which is very difficult to access experimentally. There are numerous additional questions concerning brain pathologies that mathematical models can effectively address. The example in **Figure 12** depicts the computation of hemodynamic risk factors in the human arterial tree. We envision a system in which *virtual vascular intervention planning* can be conducted on the computer. Physicians will have an interface for loading patientspecific images onto a computer system, enabling immersed visualization of the diseased vascular territory potentially improving diagnosis. For example, location and extent of arterial or venous stenoses can be detected and rigorously analyzed. In order to quantify the risk to the patient stemming from such disorder of the cerebral vasculature, the endovascular interventionist will have access to computer

*Preliminary simulation results for the aging brain. In a simulated blockage of the capillaries (5, 20, and 20%* 

*occlusion), color codes show a drastic reduction in blood flow.*

*Computer model of blood follow in the entire arterial tree in a human subject.*

*DOI: http://dx.doi.org/10.5772/intechopen.83691*

*Mathematical Modeling: The Art of Translating between Minds and Machines… DOI: http://dx.doi.org/10.5772/intechopen.83691*

These diseases also have a personal face. Parkinson's disease and Alzheimer's disease affect popular personalities as well as loved ones. Due to the severity of the crisis, the ITIF policy recommendations include expansion funding for NIH/NINDS and to encourage pharmaceutical companies to invest in more research.

But how can progress be made to address diseases of the brain especially in an aging population? Here, I point out preliminary data about research in mathematical model to better understand metabolic changes that affect the aging brain. **Figure 11** shows microcirculatory changes that affect cerebral perfusion. In this preliminary research, we can see that the capillary blockage caused by lymphocytes or the increase of tortuosity is capable of causing subtle changes in blood microperfusion and also enhances intracranial resistance. Even though these results are still in preliminary stages, mathematical models at the microcirculatory level offer a unique perspective capable of answering questions at the macroscopic scale which is very difficult to access experimentally. There are numerous additional questions concerning brain pathologies that mathematical models can effectively address. The example in **Figure 12** depicts the computation of hemodynamic risk factors in the human arterial tree. We envision a system in which *virtual vascular intervention planning* can be conducted on the computer. Physicians will have an interface for loading patientspecific images onto a computer system, enabling immersed visualization of the diseased vascular territory potentially improving diagnosis. For example, location and extent of arterial or venous stenoses can be detected and rigorously analyzed. In order to quantify the risk to the patient stemming from such disorder of the cerebral vasculature, the endovascular interventionist will have access to computer

#### **Figure 11.**

*Technology, Science and Culture - A Global Vision*

experimentally. These results allowed us to predict blood flow and oxygen exchange in a large section of the somatosensory cortex for a 3 × 3 × 3 millimeter section [18]. These two examples show how model generation can create mathematical representations of complex biological domains to make them amenable to mathematical analysis. Specifically, these models allow nonintuitive inferences about cerebral circulation. The first conclusion concerned the uneven distribution of hemodynamic states in the microcirculation and the role that the network plays in ensuring even oxygenation. The second example of vascular synthesis enabled predictions of the oxygen change in humans where currently there is no imaging modality capable of penetrating into the human brain at the level of individual capillaries. Having demonstrated the practical role of model generation and automatic formulation of process models for the normal brain, I now ask the question, is model generation significant?

*Depiction of synthetic and in vivo cortical architectures. A synthetic network of a 3 × 3 × 3 mm section of the microcirculatory network in humans (left). Rendering of a blood flow simulation performed on a murine somatosensory cortical section (1 × 1 × 1 mm, see [17]), which was acquired by photon microscopy image acquired in the Kleinfeld laboratory [22]. The morphology of the synthetic and the real trees have a striking* 

**6. The significance of mathematical modeling as an instrument of** 

Here, let me demonstrate the potential value of modeling for knowledge discovery in the aging brain. Brain diseases create a worldwide health problem. For instance, the total cost for brain disorders in the European Union amounted almost 800 billion*€* in 2010. Thus, the average cost per capita was about *€*5500. Nature medicine quoted a similar expensive picture for the United States [20]. Alzheimer disease is the most common cause of dementia. Mental disorders are at the top of the list of most costly health conditions in the United States. According to the Information Technology and Innovation Foundation (ITIF), annually \$1.5 trillion are spent on mental disorders, a cost to the economy of about 8.8% of the GDP [21].

**scientific discovery in mental disorders**

**32**

**Figure 10.**

*similarity.*

*Preliminary simulation results for the aging brain. In a simulated blockage of the capillaries (5, 20, and 20% occlusion), color codes show a drastic reduction in blood flow.*

models with subject-specific vascular tree representations that are automatically generated from medical images. Based on subject-specific data, computational network representations are generated, and the equations of momentum and mass transfer are automatically built and solved. This computer-assisted analysis will give physicians access to detailed 3D CFD simulation results including velocity profiles and streamlines as well as hemodynamic risk factor such as the average wall-shear stress or the RRT values. Currently, only engineers and scientists can perform these expensive simulations, limiting benefit to patients. But improving access to these rigorous computational results would inform physicians about the possible risk that the endovascular pathology stenosis poses to downstream blood vessels or possible redistributions of cerebral blood flow. Same day, or even better, real-time CFD results may also influence the physician's decision for intervention planning such as the need to place a flow-diverting stent. Before placing the stent in the real patient, the physician may choose to assess the effect on the stenosed vessel by performing a virtual stent procedure using the subject-specific virtual representation on the computer to predict expected blood flow changes. **Figure 13** shows the post treatment simulation, the virtual angioplasty would be able to remove the problematic wallshear stress and RRT values. This result would encourage the physician to proceed with this intervention. Post-treatment is also possible to compare the actual flow measurements with the computational predictions in order to inform and refine the computational model for future interventions. By integrating computational

#### **Figure 13.**

*Virtual intervention planning. Physicians will be able to plan interventions using subject-specific blood flow simulations.*

**35**

drugs to market.

**Figure 14.**

*Mathematical Modeling: The Art of Translating between Minds and Machines…*

methods into virtual endovascular planning, we expect to advance the clinical

A second area of significance for future intelligent modeling environments concerns the rational design of intrathecal drug delivery methods into the central nervous system. A prototype interface is depicted in **Figure 14**. Currently, the fate of drugs inserted into the central nervous system is difficult to predict, so new drugs need to undergo trial and error testing in animals. We are working on a three-dimensional virtual reality tool that will enable physicians to perform virtual infusion experiments with drug pumps to decide on continuous infusion or bolus injection for the purpose of achieving desired biodistribution of drugs in the central nervous system. Innovative treatments with gene vectors or antisense oligonucleotide (ASO) therapies that are designed to treat patients with brain diseases have never been used clinically before. For these situations, the use of mathematical models can help optimize drug dosing and anticipate risks *in silico* before animal or human experimentation occurs. Together with experimental assessments, these virtual methods may improve the capability to introduce new drugs with lower risk to the patient, as well as shorten development times for

*Virtual drug injection. Physicians can optimally plan intrathecal drug injection procedures on a virtual patient.*

It is therefore possible to conclude that modeling can dramatically accelerate the discovery about complex systems as we have shown in the aging brain and the rational design of drug delivery methods. Formulation of a process model is rather expensive, meaning machine generation is both lucrative in terms of cost savings, and effective because it allows a faster feedback/inference cycle. There is an intel-

lectual demand for system engineering research to generate these models.

**7. Implication for teaching students the art of mathematical modeling**

In the last section, I would like to apply conclusions from the discussion of mathematical modeling in sciences to engineering education. Throughout this essay, mathematical modeling has been characterized as an effort of replacing a theory-less domain of facts by another in which all theories are known. This definition explains why pure mathematical property exploration—the solution of equation—does not necessarily lead to insights about the prototype. The

practice and improve outcomes for patients in the future.

*DOI: http://dx.doi.org/10.5772/intechopen.83691*

*Mathematical Modeling: The Art of Translating between Minds and Machines… DOI: http://dx.doi.org/10.5772/intechopen.83691*

**Figure 14.** *Virtual drug injection. Physicians can optimally plan intrathecal drug injection procedures on a virtual patient.*

methods into virtual endovascular planning, we expect to advance the clinical practice and improve outcomes for patients in the future.

A second area of significance for future intelligent modeling environments concerns the rational design of intrathecal drug delivery methods into the central nervous system. A prototype interface is depicted in **Figure 14**. Currently, the fate of drugs inserted into the central nervous system is difficult to predict, so new drugs need to undergo trial and error testing in animals. We are working on a three-dimensional virtual reality tool that will enable physicians to perform virtual infusion experiments with drug pumps to decide on continuous infusion or bolus injection for the purpose of achieving desired biodistribution of drugs in the central nervous system. Innovative treatments with gene vectors or antisense oligonucleotide (ASO) therapies that are designed to treat patients with brain diseases have never been used clinically before. For these situations, the use of mathematical models can help optimize drug dosing and anticipate risks *in silico* before animal or human experimentation occurs. Together with experimental assessments, these virtual methods may improve the capability to introduce new drugs with lower risk to the patient, as well as shorten development times for drugs to market.

It is therefore possible to conclude that modeling can dramatically accelerate the discovery about complex systems as we have shown in the aging brain and the rational design of drug delivery methods. Formulation of a process model is rather expensive, meaning machine generation is both lucrative in terms of cost savings, and effective because it allows a faster feedback/inference cycle. There is an intellectual demand for system engineering research to generate these models.

## **7. Implication for teaching students the art of mathematical modeling**

In the last section, I would like to apply conclusions from the discussion of mathematical modeling in sciences to engineering education. Throughout this essay, mathematical modeling has been characterized as an effort of replacing a theory-less domain of facts by another in which all theories are known. This definition explains why pure mathematical property exploration—the solution of equation—does not necessarily lead to insights about the prototype. The

*Technology, Science and Culture - A Global Vision*

models with subject-specific vascular tree representations that are automatically generated from medical images. Based on subject-specific data, computational network representations are generated, and the equations of momentum and mass transfer are automatically built and solved. This computer-assisted analysis will give physicians access to detailed 3D CFD simulation results including velocity profiles and streamlines as well as hemodynamic risk factor such as the average wall-shear stress or the RRT values. Currently, only engineers and scientists can perform these expensive simulations, limiting benefit to patients. But improving access to these rigorous computational results would inform physicians about the possible risk that the endovascular pathology stenosis poses to downstream blood vessels or possible redistributions of cerebral blood flow. Same day, or even better, real-time CFD results may also influence the physician's decision for intervention planning such as the need to place a flow-diverting stent. Before placing the stent in the real patient, the physician may choose to assess the effect on the stenosed vessel by performing a virtual stent procedure using the subject-specific virtual representation on the computer to predict expected blood flow changes. **Figure 13** shows the post treatment simulation, the virtual angioplasty would be able to remove the problematic wallshear stress and RRT values. This result would encourage the physician to proceed with this intervention. Post-treatment is also possible to compare the actual flow measurements with the computational predictions in order to inform and refine the computational model for future interventions. By integrating computational

*Virtual intervention planning. Physicians will be able to plan interventions using subject-specific blood flow* 

**34**

**Figure 13.**

*simulations.*

proposed model-based learning model delineated a continuing feedback cycle of sharpening the problem formulation, solution, and interpretation of results. Accordingly, the rigorous solution of mathematical properties is only a subtask, but not the essence of mathematical modeling which requires translation between physical prototype and mathematical relations and between computational predictions and actual process system states. It is a key that the interpretation of mathematical results (predictions) informs knowledge of the behavior of the original study system. The repeated translations pose a linguistic, more than a mere logical challenge. We therefore suggest that problem formulation of process models is similar to a communication and composition task. The realization about the linguistic nature of mathematical modeling has implications on how it ought to be taught.

Mathematical modeling involved frequent translation between the physical and mathematical languages. The view that mathematical modeling is a form of translation and composition between languages gives indications on how modeling can effectively be learned and taught. First, let us appreciate that languages requires a grammar and syntax. In the world of mathematics, these are the mathematical properties that need to be studied before any serious composition can commence. In this aspect, students are often at a loss, not because they fail to comprehend the logic of mathematics, but because they fail to parse its terminology. Even if the logic is clear, we do not comprehend wisdom written in a foreign tongue. It requires good reading practice before students are able to compose in this language.

I have implemented the "math-as-a-foreign language" pedagogy in several course offerings in the past 10 years, for instance, a course in biological system analysis. Accordingly, we have reading exercises to make sure that students are familiar with the words of the mathematical syntax. Grammatical rules are introduced as the properties of linear and nonlinear systems. All assignments are given as a natural language memo, which forces students to translate instruction in natural language into mathematical expressions. There are translation exercises in which students practice physical prototypes conversion into quantitative expressions. This involves the choice of mathematical entities (scalars, vectors, and matrices) and a suitable mapping into biological properties. Temperatures are an example of scalar fields, velocities form vector fields. For the description of the state of stress, a tensor field is needed. The characterization of an anisotropic porous medium requires a diffusion tensor field. We also have composition exercises in which multi-physical phenomena are transcribed from the physical world of reactors with connecting processing streams into networks of mathematical relationships using property vectors and connectivity matrices. Finally, students are tasked in the validation steps in which mathematical predictions are interpreted in terms of physical model behavior.

This course has been running the last 10 years with a high success in terms of student retention rates. Typically, our students are more confident about mathematical modeling than they were before the course or the sequence of math courses before. The biggest obstacle to mathematic learning was removed by recognizing that mathematics is a language.

## **Acknowledgements**

An earlier version of this text was delivered as a presentation of the 2040 Visions of Process Systems Engineering—A Symposium on the Occasion of George Stephanopoulos's 70th Birthday and Retirement from MIT, June 2, 2017.

**37**

**Author details**

Andreas A. Linninger

\*Address all correspondence to:

provided the original work is properly cited.

University of Illinois, Chicago, United States

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*Mathematical Modeling: The Art of Translating between Minds and Machines…*

*DOI: http://dx.doi.org/10.5772/intechopen.83691*

*Mathematical Modeling: The Art of Translating between Minds and Machines… DOI: http://dx.doi.org/10.5772/intechopen.83691*
